Daily Papers

Face recognition datasets are often collected by crawling Internet and without individuals' consents, raising ethical and privacy concerns. Generating synthetic datasets for training face recognition models has emerged as a promising alternative. However, the generation of synthetic datasets remains challenging as it entails adequate inter-class and intra-class variations. While advances in generative models have made it easier to increase intra-class variations in face datasets (such as pose, illumination, etc.), generating sufficient inter-class variation is still a difficult task. In this paper, we formulate the dataset generation as a packing problem on the embedding space (represented on a hypersphere) of a face recognition model and propose a new synthetic dataset generation approach, called HyperFace. We formalize our packing problem as an optimization problem and solve it with a gradient descent-based approach. Then, we use a conditional face generator model to synthesize face images from the optimized embeddings. We use our generated datasets to train face recognition models and evaluate the trained models on several benchmarking real datasets. Our experimental results show that models trained with HyperFace achieve state-of-the-art performance in training face recognition using synthetic datasets.

Digital entities such as AI agents and humanoid robots increasingly operate alongside real humans, yet their identity infrastructure is based on credentials rather than embodied biometric identity. We introduce Biometric Identity Provisioning (BIP), a new problem and solution framework that addresses: given an enrollment gallery of real human identities, provision virtual identities that are non-colliding with every enrolled identity, maintain sufficient inter-class separability, and are realizable as high-fidelity face images. The key geometric insight is that real face identities occupy a low-dimensional subspace of the embedding hypersphere, leaving no residual subspace for virtual identities. Hence, virtual identities must instead be allocated as unclaimed gaps within the real face manifold itself. BIP is therefore a constrained packing problem: available gaps vastly exceed any foreseeable enrollment scale, and provisioned identities remain non-colliding even as new real identities are subsequently enrolled. Grounded in this geometry, our repulsion-based allocation is not bounded by any fixed provisioning count; we demonstrate 10M non-colliding virtual identity embeddings against a gallery of 360K real identities. Realizing these embeddings as face images requires a generator that operates outside the training distribution of real face images; we introduce GapGen, a gap-aware generator trained with a curriculum that progressively extends synthesis into non-colliding regions, validated at 1M photorealistic virtual face images. We further construct v-LFW, a virtual counterpart to LFW face dataset, with protocols for virtual face verification, cross-reality matching, real-vs-virtual detection, and unified recognition and detection.

Most real-world 3D measurements from depth sensors are incomplete, and to address this issue the point cloud completion task aims to predict the complete shapes of objects from partial observations. Previous works often adapt an encoder-decoder architecture, where the encoder is trained to extract embeddings that are used as inputs to generate predictions from the decoder. However, the learned embeddings have sparse distribution in the feature space, which leads to worse generalization results during testing. To address these problems, this paper proposes a hyperspherical module, which transforms and normalizes embeddings from the encoder to be on a unit hypersphere. With the proposed module, the magnitude and direction of the output hyperspherical embedding are decoupled and only the directional information is optimized. We theoretically analyze the hyperspherical embedding and show that it enables more stable training with a wider range of learning rates and more compact embedding distributions. Experiment results show consistent improvement of point cloud completion in both single-task and multi-task learning, which demonstrates the effectiveness of the proposed method.

We propose a new transformer-based image and video tokenizer with Binary Spherical Quantization (BSQ). BSQ projects the high-dimensional visual embedding to a lower-dimensional hypersphere and then applies binary quantization. BSQ is (1) parameter-efficient without an explicit codebook, (2) scalable to arbitrary token dimensions, and (3) compact: compressing visual data by up to 100times with minimal distortion. Our tokenizer uses a transformer encoder and decoder with simple block-wise causal masking to support variable-length videos as input. The resulting BSQ-ViT achieves state-of-the-art visual reconstruction quality on image and video reconstruction benchmarks with 2.4times throughput compared to the best prior methods. Furthermore, by learning an autoregressive prior for adaptive arithmetic coding, BSQ-ViT achieves comparable results on video compression with state-of-the-art video compression standards. BSQ-ViT also enables masked language models to achieve competitive image synthesis quality to GAN- and diffusion-based methods.

We study the problem of learning generative models for discrete sequences in a continuous embedding space. Whereas prior approaches typically operate in Euclidean space or on the probability simplex, we instead work on the sphere mathbb S^{d-1}. There the von Mises-Fisher (vMF) distribution induces a natural noise process and admits a closed-form conditional score. The conditional velocity is in general intractable. Exploiting the radial symmetry of the vMF density we reduce the continuity equation on mathbb S^{d-1} to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on (mathbb S^{d-1})^L both decompose into posterior-weighted tangent sums that differ only by per-token scalar weights. This gives access to both ODE and predictor-corrector (PC) sampling. The posterior is the only learned object, trained by a cross-entropy loss. Experiments compare the vMF path against geodesic and Euclidean alternatives. The combination of vMF and PC sampling significantly improves results on Sudoku and language modeling.

Contrastive Language--Image Pre-training (CLIP) has manifested remarkable improvements in zero-shot classification and cross-modal vision-language tasks. Yet, from a geometrical point of view, the CLIP embedding space has been found to have a pronounced modality gap. This gap renders the embedding space overly sparse and disconnected, with different modalities being densely distributed in distinct subregions of the hypersphere. In this work, we aim at answering two main questions: 1. Does sharing the parameter space between the multi-modal encoders reduce the modality gap? 2. Can the gap be mitigated by pushing apart the uni-modal embeddings via intra-modality separation? We design AlignCLIP, in order to answer these questions and show that answers to both questions are positive. Through extensive experiments, we show that AlignCLIP achieves noticeable enhancements in the cross-modal alignment of the embeddings, and thereby, reduces the modality gap, while maintaining the performance across several downstream evaluations, such as zero-shot image classification, zero-shot multi-modal retrieval and zero-shot semantic text similarity.

Textual Inversion (TI) is an efficient approach to text-to-image personalization but often fails on complex prompts. We trace these failures to embedding norm inflation: learned tokens drift to out-of-distribution magnitudes, degrading prompt conditioning in pre-norm Transformers. Empirically, we show semantics are primarily encoded by direction in CLIP token space, while inflated norms harm contextualization; theoretically, we analyze how large magnitudes attenuate positional information and hinder residual updates in pre-norm blocks. We propose Directional Textual Inversion (DTI), which fixes the embedding magnitude to an in-distribution scale and optimizes only direction on the unit hypersphere via Riemannian SGD. We cast direction learning as MAP with a von Mises-Fisher prior, yielding a constant-direction prior gradient that is simple and efficient to incorporate. Across personalization tasks, DTI improves text fidelity over TI and TI-variants while maintaining subject similarity. Crucially, DTI's hyperspherical parameterization enables smooth, semantically coherent interpolation between learned concepts (slerp), a capability that is absent in standard TI. Our findings suggest that direction-only optimization is a robust and scalable path for prompt-faithful personalization.

Image restoration under diverse degradations remains challenging for unified all-in-one frameworks due to feature interference and insufficient expert specialization. We propose SLER-IR, a spherical layer-wise expert routing framework that dynamically activates specialized experts across network layers. To ensure reliable routing, we introduce a Spherical Uniform Degradation Embedding with contrastive learning, which maps degradation representations onto a hypersphere to eliminate geometry bias in linear embedding spaces. In addition, a Global-Local Granularity Fusion (GLGF) module integrates global semantics and local degradation cues to address spatially non-uniform degradations and the train-test granularity gap. Experiments on three-task and five-task benchmarks demonstrate that SLER-IR achieves consistent improvements over state-of-the-art methods in both PSNR and SSIM. Code and models will be publicly released.

Deep learning models have become increasingly useful in many different industries. On the domain of image classification, convolutional neural networks proved the ability to learn robust features for the closed set problem, as shown in many different datasets, such as MNIST FASHIONMNIST, CIFAR10, CIFAR100, and IMAGENET. These approaches use deep neural networks with dense layers with softmax activation functions in order to learn features that can separate classes in a latent space. However, this traditional approach is not useful for identifying classes unseen on the training set, known as the open set problem. A similar problem occurs in scenarios involving learning on small data. To tackle both problems, few-shot learning has been proposed. In particular, metric learning learns features that obey constraints of a metric distance in the latent space in order to perform classification. However, while this approach proves to be useful for the open set problem, current implementation requires pair-wise training, where both positive and negative examples of similar images are presented during the training phase, which limits the applicability of these approaches in large data or large class scenarios given the combinatorial nature of the possible inputs.In this paper, we present a constraint-based approach applied to the representations in the latent space under the normalized softmax loss, proposed by[18]. We experimentally validate the proposed approach for the classification of unseen classes on different datasets using both metric learning and the normalized softmax loss, on disjoint and joint scenarios. Our results show that not only our proposed strategy can be efficiently trained on larger set of classes, as it does not require pairwise learning, but also present better classification results than the metric learning strategies surpassing its accuracy by a significant margin.

Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold. Resources are available at: {https://github.com/IAB-IITJ/Harmonizing-Geometry-and-Uncertainty-Diffusion-with-Hyperspheres/}

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural network layers. This makes it hard to use hyperbolic embeddings in downstream tasks. Here, we bridge this gap in a principled manner by combining the formalism of Möbius gyrovector spaces with the Riemannian geometry of the Poincaré model of hyperbolic spaces. As a result, we derive hyperbolic versions of important deep learning tools: multinomial logistic regression, feed-forward and recurrent neural networks such as gated recurrent units. This allows to embed sequential data and perform classification in the hyperbolic space. Empirically, we show that, even if hyperbolic optimization tools are limited, hyperbolic sentence embeddings either outperform or are on par with their Euclidean variants on textual entailment and noisy-prefix recognition tasks.

Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this property. For this purpose, we introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n-dimensional Poincar\'e ball. Due to the underlying hyperbolic geometry, this allows us to learn parsimonious representations of symbolic data by simultaneously capturing hierarchy and similarity. We introduce an efficient algorithm to learn the embeddings based on Riemannian optimization and show experimentally that Poincar\'e embeddings outperform Euclidean embeddings significantly on data with latent hierarchies, both in terms of representation capacity and in terms of generalization ability.

Hyperbolic embeddings offer excellent quality with few dimensions when embedding hierarchical data structures like synonym or type hierarchies. Given a tree, we give a combinatorial construction that embeds the tree in hyperbolic space with arbitrarily low distortion without using optimization. On WordNet, our combinatorial embedding obtains a mean-average-precision of 0.989 with only two dimensions, while Nickel et al.'s recent construction obtains 0.87 using 200 dimensions. We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding. To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS). We show how to perform exact recovery of hyperbolic points from distances, provide a perturbation analysis, and give a recovery result that allows us to reduce dimensionality. The h-MDS approach offers consistently low distortion even with few dimensions across several datasets. Finally, we extract lessons from the algorithms and theory above to design a PyTorch-based implementation that can handle incomplete information and is scalable.

Scaling laws for large language models depend critically on the optimizer and parameterization. Existing hyperparameter transfer laws are mainly developed for first-order optimizers, and they do not structurally prevent training instability at scale. Recent hypersphere optimization methods constrain weight matrices to a fixed-norm hypersphere, offering a promising alternative for more stable scaling. We introduce HyperP (Hypersphere Parameterization), the first framework for transferring optimal learning rates across model width, depth, training tokens, and Mixture-of-Experts (MoE) granularity under the Frobenius-sphere constraint with the Muon optimizer. We prove that weight decay is a first-order no-op on the Frobenius sphere, show that Depth-μP remains necessary, and find that the optimal learning rate follows the same data-scaling power law with the "magic exponent" 0.32 previously observed for AdamW. A single base learning rate tuned at the smallest scale transfers across all compute budgets under HyperP, yielding 1.58times compute efficiency over a strong Muon baseline at 6times10^{21} FLOPs. Moreover, HyperP delivers transferable stability: all monitored instability indicators, including Z-values, output RMS, and activation outliers, remain bounded and non-increasing under training FLOPs scaling. We also propose SqrtGate, an MoE gating mechanism derived from the hypersphere constraint that preserves output RMS across MoE granularities for improved granularity scaling, and show that hypersphere optimization enables substantially larger auxiliary load-balancing weights, yielding both strong performance and good expert balance. We release our training codebase at https://github.com/microsoft/ArchScale.

We propose a novel neural network architecture, the normalized Transformer (nGPT) with representation learning on the hypersphere. In nGPT, all vectors forming the embeddings, MLP, attention matrices and hidden states are unit norm normalized. The input stream of tokens travels on the surface of a hypersphere, with each layer contributing a displacement towards the target output predictions. These displacements are defined by the MLP and attention blocks, whose vector components also reside on the same hypersphere. Experiments show that nGPT learns much faster, reducing the number of training steps required to achieve the same accuracy by a factor of 4 to 20, depending on the sequence length.

Graph-level anomaly detection aims to identify anomalous graphs from a collection of graphs in an unsupervised manner. A common assumption of anomaly detection is that a reasonable decision boundary has a hypersphere shape, but may appear some non-conforming phenomena in high dimensions. Towards this end, we firstly propose a novel deep graph-level anomaly detection model, which learns the graph representation with maximum mutual information between substructure and global structure features while exploring a hypersphere anomaly decision boundary. The idea is to ensure the training data distribution consistent with the decision hypersphere via an orthogonal projection layer. Moreover, we further perform the bi-hypersphere compression to emphasize the discrimination of anomalous graphs from normal graphs. Note that our method is not confined to graph data and is applicable to anomaly detection of other data such as images. The numerical and visualization results on benchmark datasets demonstrate the effectiveness and superiority of our methods in comparison to many baselines and state-of-the-arts.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.

Image analysis in the euclidean space through linear hyperspaces is well studied. However, in the quest for more effective image representations, we turn to hyperbolic manifolds. They provide a compelling alternative to capture complex hierarchical relationships in images with remarkably small dimensionality. To demonstrate hyperbolic embeddings' competence, we introduce a light-weight hyperbolic graph neural network for image segmentation, encompassing patch-level features in a very small embedding size. Our solution, Seg-HGNN, surpasses the current best unsupervised method by 2.5\%, 4\% on VOC-07, VOC-12 for localization, and by 0.8\%, 1.3\% on CUB-200, ECSSD for segmentation, respectively. With less than 7.5k trainable parameters, Seg-HGNN delivers effective and fast (approx 2 images/second) results on very standard GPUs like the GTX1650. This empirical evaluation presents compelling evidence of the efficacy and potential of hyperbolic representations for vision tasks.

Traditional deep learning models rely on methods such as softmax cross-entropy and ArcFace loss for tasks like classification and face recognition. These methods mainly explore angular features in a hyperspherical space, often resulting in entangled inter-class features due to dense angular data across many classes. In this paper, a new field of feature exploration is proposed known as HyperSpaceX which enhances class discrimination by exploring both angular and radial dimensions in multi-hyperspherical spaces, facilitated by a novel DistArc loss. The proposed DistArc loss encompasses three feature arrangement components: two angular and one radial, enforcing intra-class binding and inter-class separation in multi-radial arrangement, improving feature discriminability. Evaluation of HyperSpaceX framework for the novel representation utilizes a proposed predictive measure that accounts for both angular and radial elements, providing a more comprehensive assessment of model accuracy beyond standard metrics. Experiments across seven object classification and six face recognition datasets demonstrate state-of-the-art (SoTA) results obtained from HyperSpaceX, achieving up to a 20% performance improvement on large-scale object datasets in lower dimensions and up to 6% gain in higher dimensions.

Scene graph representations enable structured visual understanding by modeling objects and their relationships, and have been widely used for multiview and 3D scene reasoning. Existing methods such as MSG learn scene graph embeddings in Euclidean space using contrastive learning and attention based association. However, Euclidean geometry does not explicitly capture hierarchical entailment relationships between places and objects, limiting the structural consistency of learned representations. To address this, we propose Hyperbolic Scene Graph (HSG), which learns scene graph embeddings in hyperbolic space where hierarchical relationships are naturally encoded through geometric distance. Our results show that HSG improves hierarchical structure quality while maintaining strong retrieval performance. The largest gains are observed in graph level metrics: HSG achieves a PP IoU of 33.17 and the highest Graph IoU of 33.51, outperforming the best AoMSG variant (25.37) by 8.14, highlighting the effectiveness of hyperbolic representation learning for scene graph modeling. Code: https://github.com/AIGeeksGroup/HSG.

Out-of-distribution (OOD) generalization is critical for machine learning models deployed in the real world. However, achieving this can be fundamentally challenging, as it requires the ability to learn invariant features across different domains or environments. In this paper, we propose a novel framework HYPO (HYPerspherical OOD generalization) that provably learns domain-invariant representations in a hyperspherical space. In particular, our hyperspherical learning algorithm is guided by intra-class variation and inter-class separation principles -- ensuring that features from the same class (across different training domains) are closely aligned with their class prototypes, while different class prototypes are maximally separated. We further provide theoretical justifications on how our prototypical learning objective improves the OOD generalization bound. Through extensive experiments on challenging OOD benchmarks, we demonstrate that our approach outperforms competitive baselines and achieves superior performance. Code is available at https://github.com/deeplearning-wisc/hypo.

Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central approach to manifold learning, and hyperbolic geometry. Specifically, using diffusion geometry, we build multi-scale densities on the data, aimed to reveal their hierarchical structure, and then embed them into a product of hyperbolic spaces. We show theoretically that our embedding and distance recover the underlying hierarchical structure. In addition, we demonstrate the efficacy of the proposed method and its advantages compared to existing methods on graph embedding benchmarks and hierarchical datasets.

In the graph node embedding problem, embedding spaces can vary significantly for different data types, leading to the need for different GNN model types. In this paper, we model the embedding update of a node feature as a Hamiltonian orbit over time. Since the Hamiltonian orbits generalize the exponential maps, this approach allows us to learn the underlying manifold of the graph in training, in contrast to most of the existing literature that assumes a fixed graph embedding manifold with a closed exponential map solution. Our proposed node embedding strategy can automatically learn, without extensive tuning, the underlying geometry of any given graph dataset even if it has diverse geometries. We test Hamiltonian functions of different forms and verify the performance of our approach on two graph node embedding downstream tasks: node classification and link prediction. Numerical experiments demonstrate that our approach adapts better to different types of graph datasets than popular state-of-the-art graph node embedding GNNs. The code is available at https://github.com/zknus/Hamiltonian-GNN.

EarthVision Embed2Scale challenge (CVPR 2025) aims to develop foundational geospatial models to embed SSL4EO-S12 hyperspectral geospatial data cubes into embedding vectors that faciliatetes various downstream tasks, e.g., classification, regression, etc. In this technical report, we introduce our proposed method for the Top-1 winning solution on the Embed2Scale Challenge.

Knowledge graph (KG) embeddings learn low-dimensional representations of entities and relations to predict missing facts. KGs often exhibit hierarchical and logical patterns which must be preserved in the embedding space. For hierarchical data, hyperbolic embedding methods have shown promise for high-fidelity and parsimonious representations. However, existing hyperbolic embedding methods do not account for the rich logical patterns in KGs. In this work, we introduce a class of hyperbolic KG embedding models that simultaneously capture hierarchical and logical patterns. Our approach combines hyperbolic reflections and rotations with attention to model complex relational patterns. Experimental results on standard KG benchmarks show that our method improves over previous Euclidean- and hyperbolic-based efforts by up to 6.1% in mean reciprocal rank (MRR) in low dimensions. Furthermore, we observe that different geometric transformations capture different types of relations while attention-based transformations generalize to multiple relations. In high dimensions, our approach yields new state-of-the-art MRRs of 49.6% on WN18RR and 57.7% on YAGO3-10.

Leaf-lesion segmentation is topology-sensitive: small merges, splits, or false holes can be biologically meaningful descriptors of biochemical pathways, yet they are weakly penalized by standard pixel-wise losses in Euclidean latents. I explore HyperTopo-Adapters, a lightweight, parameter-efficient head trained on top of a frozen vision encoder, which embeds features on a product manifold -- hyperbolic + Euclidean + spherical (H + E + S) -- to encourage hierarchical separation (H), local linear detail (E), and global closure (S). A topology prior complements Dice/BCE in two forms: (i) persistent-homology (PH) distance for evaluation and selection, and (ii) a differentiable surrogate that combines a soft Euler-characteristic match with total variation regularization for stable training. I introduce warm-ups for both the hyperbolic contrastive term and the topology prior, per-sample evaluation of structure-aware metrics (Boundary-F1, Betti errors, PD distance), and a min-PD within top-K Dice rule for checkpoint selection. On a Kaggle leaf-lesion dataset (N=2,940), early results show consistent gains in boundary and topology metrics (reducing Delta beta_1 hole error by 9%) while Dice/IoU remain competitive. The study is diagnostic by design: I report controlled ablations (curvature learning, latent dimensions, contrastive temperature, surrogate settings), and ongoing tests varying encoder strength (ResNet-50, DeepLabV3, DINOv2/v3), input resolution, PH weight, and partial unfreezing of late blocks. The contribution is an open, reproducible train/eval suite (available at https://github.com/ChimdiWalter/HyperTopo-Adapters) that isolates geometric/topological priors and surfaces failure modes to guide stronger, topology-preserving architectures.

The paper presents an O(N log N)-implementation of t-SNE -- an embedding technique that is commonly used for the visualization of high-dimensional data in scatter plots and that normally runs in O(N^2). The new implementation uses vantage-point trees to compute sparse pairwise similarities between the input data objects, and it uses a variant of the Barnes-Hut algorithm - an algorithm used by astronomers to perform N-body simulations - to approximate the forces between the corresponding points in the embedding. Our experiments show that the new algorithm, called Barnes-Hut-SNE, leads to substantial computational advantages over standard t-SNE, and that it makes it possible to learn embeddings of data sets with millions of objects.

Learning feature representations of geographical space is vital for any machine learning model that integrates geolocated data, spanning application domains such as remote sensing, ecology, or epidemiology. Recent work mostly embeds coordinates using sine and cosine projections based on Double Fourier Sphere (DFS) features -- these embeddings assume a rectangular data domain even on global data, which can lead to artifacts, especially at the poles. At the same time, relatively little attention has been paid to the exact design of the neural network architectures these functional embeddings are combined with. This work proposes a novel location encoder for globally distributed geographic data that combines spherical harmonic basis functions, natively defined on spherical surfaces, with sinusoidal representation networks (SirenNets) that can be interpreted as learned Double Fourier Sphere embedding. We systematically evaluate the cross-product of positional embeddings and neural network architectures across various classification and regression benchmarks and synthetic evaluation datasets. In contrast to previous approaches that require the combination of both positional encoding and neural networks to learn meaningful representations, we show that both spherical harmonics and sinusoidal representation networks are competitive on their own but set state-of-the-art performances across tasks when combined. We provide source code at www.github.com/marccoru/locationencoder

Generalized Category Discovery (GCD) is an intriguing open-world problem that has garnered increasing attention. Given a dataset that includes both labelled and unlabelled images, GCD aims to categorize all images in the unlabelled subset, regardless of whether they belong to known or unknown classes. In GCD, the common practice typically involves applying a spherical projection operator at the end of the self-supervised pretrained backbone, operating within Euclidean or spherical space. However, both of these spaces have been shown to be suboptimal for encoding samples that possesses hierarchical structures. In contrast, hyperbolic space exhibits exponential volume growth relative to radius, making it inherently strong at capturing the hierarchical structure of samples from both seen and unseen categories. Therefore, we propose to tackle the category discovery challenge in the hyperbolic space. We introduce HypCD, a simple Hyperbolic framework for learning hierarchy-aware representations and classifiers for generalized Category Discovery. HypCD first transforms the Euclidean embedding space of the backbone network into hyperbolic space, facilitating subsequent representation and classification learning by considering both hyperbolic distance and the angle between samples. This approach is particularly helpful for knowledge transfer from known to unknown categories in GCD. We thoroughly evaluate HypCD on public GCD benchmarks, by applying it to various baseline and state-of-the-art methods, consistently achieving significant improvements.

Text-attributed graphs are widely used across domains, offering rich opportunities for zero-shot learning via graph-text alignment. However, existing methods struggle with tasks requiring fine-grained pattern recognition, particularly on heterophilic graphs. Through empirical and theoretical analysis, we identify an over-abstraction problem: current approaches operate at excessively large hyperbolic radii, compressing multi-scale structural information into uniform high-level abstractions. This abstraction-induced information loss obscures critical local patterns essential for accurate predictions. By analyzing embeddings in hyperbolic space, we demonstrate that optimal graph learning requires faithful preservation of fine-grained structural details, better retained by representations positioned closer to the origin. To address this, we propose H4G, a framework that systematically reduces embedding radii using learnable block-diagonal scaling matrices and M\"obius matrix multiplication. This approach restores access to fine-grained patterns while maintaining global receptive ability with minimal computational overhead. Experiments show H4G achieves state-of-the-art zero-shot performance with 12.8\% improvement on heterophilic graphs and 8.4\% on homophilic graphs, confirming that radius reduction enables faithful multi-scale representation for advancing zero-shot graph learning.

Autoregressive (AR) models are promising for image generation, yet continuous-token AR variants often trail latent diffusion and masked-generation models. The core issue is heterogeneous variance in VAE latents, which is amplified during AR decoding, especially under classifier-free guidance (CFG), and can cause variance collapse. We propose SphereAR to address this issue. Its core design is to constrain all AR inputs and outputs -- including after CFG -- to lie on a fixed-radius hypersphere (constant ell_2 norm), leveraging hyperspherical VAEs. Our theoretical analysis shows that hyperspherical constraint removes the scale component (the primary cause of variance collapse), thereby stabilizing AR decoding. Empirically, on ImageNet generation, SphereAR-H (943M) sets a new state of the art for AR models, achieving FID 1.34. Even at smaller scales, SphereAR-L (479M) reaches FID 1.54 and SphereAR-B (208M) reaches 1.92, matching or surpassing much larger baselines such as MAR-H (943M, 1.55) and VAR-d30 (2B, 1.92). To our knowledge, this is the first time a pure next-token AR image generator with raster order surpasses diffusion and masked-generation models at comparable parameter scales.

Incorporating spatial information, particularly those influenced by climate, weather, and demographic factors, is crucial for improving underwriting precision and enhancing risk management in insurance. However, spatial data are often unstructured, high-dimensional, and difficult to integrate into predictive models. Embedding methods are needed to convert spatial data into meaningful representations for modelling tasks. We propose a novel multi-view contrastive learning framework for generating spatial embeddings that combine information from multiple spatial data sources. To train the model, we construct a spatial dataset that merges satellite imagery and OpenStreetMap features across Europe. The framework aligns these spatial views with coordinate-based encodings, producing low-dimensional embeddings that capture both spatial structure and contextual similarity. Once trained, the model generates embeddings directly from latitude-longitude pairs, enabling any dataset with coordinates to be enriched with meaningful spatial features without requiring access to the original spatial inputs. In a case study on French real estate prices, we compare models trained on raw coordinates against those using our spatial embeddings as inputs. The embeddings consistently improve predictive accuracy across generalised linear, additive, and boosting models, while providing interpretable spatial effects and demonstrating transferability to unseen regions.

Recent works have demonstrated reasonable success of representation learning in hypercomplex space. Specifically, "fully-connected layers with Quaternions" (4D hypercomplex numbers), which replace real-valued matrix multiplications in fully-connected layers with Hamilton products of Quaternions, both enjoy parameter savings with only 1/4 learnable parameters and achieve comparable performance in various applications. However, one key caveat is that hypercomplex space only exists at very few predefined dimensions (4D, 8D, and 16D). This restricts the flexibility of models that leverage hypercomplex multiplications. To this end, we propose parameterizing hypercomplex multiplications, allowing models to learn multiplication rules from data regardless of whether such rules are predefined. As a result, our method not only subsumes the Hamilton product, but also learns to operate on any arbitrary nD hypercomplex space, providing more architectural flexibility using arbitrarily 1/n learnable parameters compared with the fully-connected layer counterpart. Experiments of applications to the LSTM and Transformer models on natural language inference, machine translation, text style transfer, and subject verb agreement demonstrate architectural flexibility and effectiveness of the proposed approach.

Simulations are the best approximation to experimental laboratories in astrophysics and cosmology. However, the complexity, richness, and large size of their outputs severely limit the interpretability of their predictions. We describe a new, unbiased, and machine learning based approach to obtaining useful scientific insights from a broad range of simulations. The method can be used on today's largest simulations and will be essential to solve the extreme data exploration and analysis challenges posed by the Exascale era. Furthermore, this concept is so flexible, that it will also enable explorative access to observed data. Our concept is based on applying nonlinear dimensionality reduction to learn compact representations of the data in a low-dimensional space. The simulation data is projected onto this space for interactive inspection, visual interpretation, sample selection, and local analysis. We present a prototype using a rotational invariant hyperspherical variational convolutional autoencoder, utilizing a power distribution in the latent space, and trained on galaxies from IllustrisTNG simulation. Thereby, we obtain a natural Hubble tuning fork like similarity space that can be visualized interactively on the surface of a sphere by exploiting the power of HiPS tilings in Aladin Lite.

Metric learning aims to learn a highly discriminative model encouraging the embeddings of similar classes to be close in the chosen metrics and pushed apart for dissimilar ones. The common recipe is to use an encoder to extract embeddings and a distance-based loss function to match the representations -- usually, the Euclidean distance is utilized. An emerging interest in learning hyperbolic data embeddings suggests that hyperbolic geometry can be beneficial for natural data. Following this line of work, we propose a new hyperbolic-based model for metric learning. At the core of our method is a vision transformer with output embeddings mapped to hyperbolic space. These embeddings are directly optimized using modified pairwise cross-entropy loss. We evaluate the proposed model with six different formulations on four datasets achieving the new state-of-the-art performance. The source code is available at https://github.com/htdt/hyp_metric.

Large language models (LLMs) have achieved remarkable success and demonstrated superior performance across various tasks, including natural language processing (NLP), weather forecasting, biological protein folding, text generation, and solving mathematical problems. However, many real-world data exhibit highly non-Euclidean latent hierarchical anatomy, such as protein networks, transportation networks, financial networks, brain networks, and linguistic structures or syntactic trees in natural languages. Effectively learning intrinsic semantic entailment and hierarchical relationships from these raw, unstructured input data using LLMs remains an underexplored area. Due to its effectiveness in modeling tree-like hierarchical structures, hyperbolic geometry -- a non-Euclidean space -- has rapidly gained popularity as an expressive latent representation space for complex data modeling across domains such as graphs, images, languages, and multi-modal data. Here, we provide a comprehensive and contextual exposition of recent advancements in LLMs that leverage hyperbolic geometry as a representation space to enhance semantic representation learning and multi-scale reasoning. Specifically, the paper presents a taxonomy of the principal techniques of Hyperbolic LLMs (HypLLMs) in terms of four main categories: (1) hyperbolic LLMs through exp/log maps; (2) hyperbolic fine-tuned models; (3) fully hyperbolic LLMs, and (4) hyperbolic state-space models. We also explore crucial potential applications and outline future research directions. A repository of key papers, models, datasets, and code implementations is available at https://github.com/sarangp2402/Hyperbolic-LLM-Models/tree/main.

Pixel-space diffusion models bypass the reconstruction bottleneck of Variational Autoencoders (VAEs) but face a fundamental "granularity dilemma": capturing global semantics favors large patch scales, while generating high-fidelity details demands fine-grained inputs. To address this issue, we propose HyperDiT, a unified framework establishing Hyper-Connected Cross-Scale Interactions to bridge the semantic and pixel manifold. Diverging from injecting semantics by AdaLN, HyperDiT utilizes Cross-Attention mechanisms, enabling fine-grained tokens to query multi-level semantic anchors globally. To resolve the spatial mismatch during multi-scale interactions, we introduce Scale-Aware Rotary Position Embedding (SA-RoPE) to ensure precise geometric alignment among tokens of varying patch sizes. Furthermore, we incorporate Registers to learn the dense semantics from a pretrained Visual Foundation Model (VFM), effectively reducing generation hallucination and artifacts. Extensive experiments demonstrate that HyperDiT achieves state-of-the-art (SoTA) FID of 1.56 on ImageNet 256times256 directly within the pixel space. By combining the fine-grained stream with semantic guidance, HyperDiT offers a superior paradigm for high-fidelity pixel generation.

Training large language models at 4-bit precision is critical for efficiency. We show that nGPT, an architecture that constrains weights and hidden representations to the unit hypersphere, is inherently more robust to low-precision arithmetic. This removes the need for interventions-such as applying random Hadamard transforms and performing per-tensor scaling calculations-to preserve model quality, and it enables stable end-to-end NVFP4 training. We validate this approach on both a 1.2B dense model and hybrid (Mamba-Transformer) MoE models of up to 3B/30B parameters. We trace this robustness to the dot product: while quantization noise remains largely uncorrelated in both standard and normalized architectures, the signal behaves differently. In nGPT, the hypersphere constraint enhances weak positive correlations among the element-wise products, leading to a constructive accumulation of the signal across the hidden dimension while the noise continues to average out. This yields a higher effective signal-to-noise ratio and a flatter loss landscape, with the effect strengthening as the hidden dimension grows, suggesting increasing advantages at scale. A reference implementation is available at https://github.com/anonymous452026/ngpt-nvfp4

Reconstructing both objects and hands in 3D from a single RGB image is complex. Existing methods rely on manually defined hand-object constraints in Euclidean space, leading to suboptimal feature learning. Compared with Euclidean space, hyperbolic space better preserves the geometric properties of meshes thanks to its exponentially-growing space distance, which amplifies the differences between the features based on similarity. In this work, we propose the first precise hand-object reconstruction method in hyperbolic space, namely Dynamic Hyperbolic Attention Network (DHANet), which leverages intrinsic properties of hyperbolic space to learn representative features. Our method that projects mesh and image features into a unified hyperbolic space includes two modules, ie. dynamic hyperbolic graph convolution and image-attention hyperbolic graph convolution. With these two modules, our method learns mesh features with rich geometry-image multi-modal information and models better hand-object interaction. Our method provides a promising alternative for fine hand-object reconstruction in hyperbolic space. Extensive experiments on three public datasets demonstrate that our method outperforms most state-of-the-art methods.

Text-to-3D generation represents an exciting field that has seen rapid advancements, facilitating the transformation of textual descriptions into detailed 3D models. However, current progress often neglects the intricate high-order correlation of geometry and texture within 3D objects, leading to challenges such as over-smoothness, over-saturation and the Janus problem. In this work, we propose a method named ``3D Gaussian Generation via Hypergraph (Hyper-3DG)'', designed to capture the sophisticated high-order correlations present within 3D objects. Our framework is anchored by a well-established mainflow and an essential module, named ``Geometry and Texture Hypergraph Refiner (HGRefiner)''. This module not only refines the representation of 3D Gaussians but also accelerates the update process of these 3D Gaussians by conducting the Patch-3DGS Hypergraph Learning on both explicit attributes and latent visual features. Our framework allows for the production of finely generated 3D objects within a cohesive optimization, effectively circumventing degradation. Extensive experimentation has shown that our proposed method significantly enhances the quality of 3D generation while incurring no additional computational overhead for the underlying framework. (Project code: https://github.com/yjhboy/Hyper3DG)

In representation learning, uniformity refers to the uniform feature distribution in the latent space (i.e., unit hypersphere). Previous work has shown that improving uniformity contributes to the learning of under-represented classes. However, most of the previous work focused on classification; the representation space of imbalanced regression remains unexplored. Classification-based methods are not suitable for regression tasks because they cluster features into distinct groups without considering the continuous and ordered nature essential for regression. In a geometric aspect, we uniquely focus on ensuring uniformity in the latent space for imbalanced regression through two key losses: enveloping and homogeneity. The enveloping loss encourages the induced trace to uniformly occupy the surface of a hypersphere, while the homogeneity loss ensures smoothness, with representations evenly spaced at consistent intervals. Our method integrates these geometric principles into the data representations via a Surrogate-driven Representation Learning (SRL) framework. Experiments with real-world regression and operator learning tasks highlight the importance of uniformity in imbalanced regression and validate the efficacy of our geometry-based loss functions.

Semantic representations of text, i.e. representations of natural language which capture meaning by geometry, are essential for areas such as information retrieval and document grouping. High-dimensional trained dense vectors have received much attention in recent years as such representations. We investigate the structure of semantic spaces that arise from embeddings made with Sentence-BERT and find that the representations suffer from a well-known problem in high dimensions called hubness. Hubness results in asymmetric neighborhood relations, such that some texts (the hubs) are neighbours of many other texts while most texts (so-called anti-hubs), are neighbours of few or no other texts. We quantify the semantic quality of the embeddings using hubness scores and error rate of a neighbourhood based classifier. We find that when hubness is high, we can reduce error rate and hubness using hubness reduction methods. We identify a combination of two methods as resulting in the best reduction. For example, on one of the tested pretrained models, this combined method can reduce hubness by about 75% and error rate by about 9%. Thus, we argue that mitigating hubness in the embedding space provides better semantic representations of text.

Modeling the energy and forces of atomic systems is a fundamental problem in computational chemistry with the potential to help address many of the world's most pressing problems, including those related to energy scarcity and climate change. These calculations are traditionally performed using Density Functional Theory, which is computationally very expensive. Machine learning has the potential to dramatically improve the efficiency of these calculations from days or hours to seconds. We propose the Spherical Channel Network (SCN) to model atomic energies and forces. The SCN is a graph neural network where nodes represent atoms and edges their neighboring atoms. The atom embeddings are a set of spherical functions, called spherical channels, represented using spherical harmonics. We demonstrate, that by rotating the embeddings based on the 3D edge orientation, more information may be utilized while maintaining the rotational equivariance of the messages. While equivariance is a desirable property, we find that by relaxing this constraint in both message passing and aggregation, improved accuracy may be achieved. We demonstrate state-of-the-art results on the large-scale Open Catalyst dataset in both energy and force prediction for numerous tasks and metrics.

This paper explores the problem of class-agnostic anomaly detection (AD), where the objective is to train one class-agnostic AD model that can generalize to detect anomalies in diverse new classes from different domains without any retraining or fine-tuning on the target data. When applied for new classes, the performance of current single- and multi-class AD methods is still unsatisfactory. One fundamental reason is that representation learning in existing methods is still class-related, namely, feature correlation. To address this issue, we propose residual features and construct a simple but effective framework, termed ResAD. Our core insight is to learn the residual feature distribution rather than the initial feature distribution. Residual features are formed by matching and then subtracting normal reference features. In this way, we can effectively realize feature decorrelation. Even in new classes, the distribution of normal residual features would not remarkably shift from the learned distribution. In addition, we think that residual features still have one issue: scale correlation. To this end, we propose a feature hypersphere constraining approach, which learns to constrain initial normal residual features into a spatial hypersphere for enabling the feature scales of different classes as consistent as possible. Furthermore, we propose a novel logbarrier bidirectional contraction OCC loss and vector quantization based feature distribution matching module to enhance ResAD, leading to the improved version of ResAD (ResAD++). Comprehensive experiments on eight real-world AD datasets demonstrate that our ResAD++ can achieve remarkable AD results when directly used in new classes, outperforming state-of-the-art competing methods and also surpassing ResAD. The code is available at https://github.com/xcyao00/ResAD.

Graph foundation models represent a transformative paradigm for learning transferable representations across diverse graph domains. Recent methods leverage large language models to unify graph and text modalities into a shared representation space using contrastive learning. However, systematic evaluations reveal significant performance degradation at structural boundaries where distinct topological patterns converge, with accuracy losses exceeding 20 percentage points. This issue arises from a key limitation: current methods assume all graph structures can be encoded within a single Euclidean space. In reality, tree structures require hyperbolic geometry to preserve hierarchical branching, while cyclic patterns depend on spherical geometry for closure properties. At structural boundaries, nodes experience conflicting geometric constraints that uniform encoding spaces cannot resolve. This raises a crucial challenge: Can alignment frameworks be designed to respect the intrinsic geometric diversity of graph structures? We introduce GraphShaper, a geometry-aware framework that enhances graph encoding through multi-geometric specialization. Our approach employs expert networks tailored to different geometric spaces, dynamically computing fusion weights to adaptively integrate geometric properties based on local structural characteristics. This adaptive fusion preserves structural integrity before alignment with text embeddings. Extensive experiments demonstrate that GraphShaper achieves 9.47\% accuracy improvements on citation networks and 7.63\% on social networks in zero-shot settings.

With the ever-increasing volumes of the Earth observation data present in the archives of large programmes such as Copernicus, there is a growing need for efficient vector representations of the underlying raw data. The approach of extracting feature representations from pretrained deep neural networks is a powerful approach that can provide semantic abstractions of the input data. However, the way this is done for imagery archives containing geospatial data has not yet been defined. In this work, an extension is proposed to an existing community project, Major TOM, focused on the provision and standardization of open and free AI-ready datasets for Earth observation. Furthermore, four global and dense embedding datasets are released openly and for free along with the publication of this manuscript, resulting in the most comprehensive global open dataset of geospatial visual embeddings in terms of covered Earth's surface.

Hyper-Connections (HC) generalize residual connections into multiple streams, employing residual matrices for cross-stream feature mixing to enrich model expressivity. However, unconstrained mixing disrupts the identity mapping property intrinsic to the residual connection, causing unstable training. To address this, Manifold-Constrained Hyper-Connections (mHC) and its variant restrict these matrices to the Birkhoff polytope (doubly stochastic matrices) via Sinkhorn iterations or permutation-based parameterizations. We reveal three limitations of this polytope constraint: (1) identity degeneration, where learned matrices collapse around the identity and diminish cross-stream interactions, (2) an expressivity bottleneck, as the non-negativity constraint prevents subtractive feature disentanglement, and (3) parameterization inefficiencies, manifesting as unstable Sinkhorn iterations or the factorial-scaling overhead of permutation-based parameterizations. To overcome these flaws, we propose Spectral-Sphere-Constrained Hyper-Connections (sHC). By geometrically shifting the feasible set from a rigid polytope to a spectral norm sphere, sHC allows negative entries, unlocking subtractive interactions for selective feature diversification. This shift eliminates unstable Sinkhorn projections and factorial parameterization, enabling expressive, non-degenerate residual matrices while preserving training stability.

Contrastive representation learning has been outstandingly successful in practice. In this work, we identify two key properties related to the contrastive loss: (1) alignment (closeness) of features from positive pairs, and (2) uniformity of the induced distribution of the (normalized) features on the hypersphere. We prove that, asymptotically, the contrastive loss optimizes these properties, and analyze their positive effects on downstream tasks. Empirically, we introduce an optimizable metric to quantify each property. Extensive experiments on standard vision and language datasets confirm the strong agreement between both metrics and downstream task performance. Remarkably, directly optimizing for these two metrics leads to representations with comparable or better performance at downstream tasks than contrastive learning. Project Page: https://tongzhouwang.info/hypersphere Code: https://github.com/SsnL/align_uniform , https://github.com/SsnL/moco_align_uniform

The increasing availability of large-scale global datasets has generated a demand for scalable spatial prediction methods defined on spherical domains. Classical spatial models that rely on Euclidean distance representations are inappropriate for spherical data because planar projections distort geodesic distances and spatial neighborhood structures, while traditional kriging-based prediction methods are often computationally prohibitive for massive datasets. To address these challenges, we propose a Spherical DeepKriging framework for spatial prediction on S^2. The proposed approach constructs a flexible prediction model by integrating thin-plate spline (TPS) basis functions defined intrinsically on the sphere. Simulation studies and real data analyses are presented to demonstrate the superior predictive performance of the proposed method.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

We introduce the first method for translating text embeddings from one vector space to another without any paired data, encoders, or predefined sets of matches. Our unsupervised approach translates any embedding to and from a universal latent representation (i.e., a universal semantic structure conjectured by the Platonic Representation Hypothesis). Our translations achieve high cosine similarity across model pairs with different architectures, parameter counts, and training datasets. The ability to translate unknown embeddings into a different space while preserving their geometry has serious implications for the security of vector databases. An adversary with access only to embedding vectors can extract sensitive information about the underlying documents, sufficient for classification and attribute inference.

Recently, a new Continual Learning (CL) paradigm was presented to control catastrophic forgetting, called Interval Continual Learning (InterContiNet), which relies on enforcing interval constraints on the neural network parameter space. Unfortunately, InterContiNet training is challenging due to the high dimensionality of the weight space, making intervals difficult to manage. To address this issue, we introduce HyperInterval, a technique that employs interval arithmetic within the embedding space and utilizes a hypernetwork to map these intervals to the target network parameter space. We train interval embeddings for consecutive tasks and train a hypernetwork to transform these embeddings into weights of the target network. An embedding for a given task is trained along with the hypernetwork, preserving the response of the target network for the previous task embeddings. Interval arithmetic works with a more manageable, lower-dimensional embedding space rather than directly preparing intervals in a high-dimensional weight space. Our model allows faster and more efficient training. Furthermore, HyperInterval maintains the guarantee of not forgetting. At the end of training, we can choose one universal embedding to produce a single network dedicated to all tasks. In such a framework, hypernetwork is used only for training and can be seen as a meta-trainer. HyperInterval obtains significantly better results than InterContiNet and gives SOTA results on several benchmarks.

The performance of reinforcement learning (RL) agents depends critically on the quality of the underlying feature representations. Hyperbolic feature spaces are well-suited for this purpose, as they naturally capture hierarchical and relational structure often present in complex RL environments. However, leveraging these spaces commonly faces optimization challenges due to the nonstationarity of RL. In this work, we identify key factors that determine the success and failure of training hyperbolic deep RL agents. By analyzing the gradients of core operations in the Poincaré Ball and Hyperboloid models of hyperbolic geometry, we show that large-norm embeddings destabilize gradient-based training, leading to trust-region violations in proximal policy optimization (PPO). Based on these insights, we introduce Hyper++, a new hyperbolic PPO agent that consists of three components: (i) stable critic training through a categorical value loss instead of regression; (ii) feature regularization guaranteeing bounded norms while avoiding the curse of dimensionality from clipping; and (iii) using a more optimization-friendly formulation of hyperbolic network layers. In experiments on ProcGen, we show that Hyper++ guarantees stable learning, outperforms prior hyperbolic agents, and reduces wall-clock time by approximately 30%. On Atari-5 with Double DQN, Hyper++ strongly outperforms Euclidean and hyperbolic baselines. We release our code at https://github.com/Probabilistic-and-Interactive-ML/hyper-rl .

Alignment and uniformity are fundamental principles within the domain of contrastive learning. In recommender systems, prior work has established that optimizing the Bayesian Personalized Ranking (BPR) loss contributes to the objectives of alignment and uniformity. Specifically, alignment aims to draw together the representations of interacting users and items, while uniformity mandates a uniform distribution of user and item embeddings across a unit hypersphere. This study revisits the alignment and uniformity properties within the context of multimodal recommender systems, revealing a proclivity among extant models to prioritize uniformity to the detriment of alignment. Our hypothesis challenges the conventional assumption of equitable item treatment through a uniformity loss, proposing a more nuanced approach wherein items with similar multimodal attributes converge toward proximal representations within the hyperspheric manifold. Specifically, we leverage the inherent similarity between items' multimodal data to calibrate their uniformity distribution, thereby inducing a more pronounced repulsive force between dissimilar entities within the embedding space. A theoretical analysis elucidates the relationship between this calibrated uniformity loss and the conventional uniformity function. Moreover, to enhance the fusion of multimodal features, we introduce a Spherical B\'ezier method designed to integrate an arbitrary number of modalities while ensuring that the resulting fused features are constrained to the same hyperspherical manifold. Empirical evaluations conducted on five real-world datasets substantiate the superiority of our approach over competing baselines. We also shown that the proposed methods can achieve up to a 5.4% increase in NDCG@20 performance via the integration of MLLM-extracted features. Source code is available at: https://github.com/enoche/CM3.

Detecting the anomaly of human behavior is paramount to timely recognizing endangering situations, such as street fights or elderly falls. However, anomaly detection is complex since anomalous events are rare and because it is an open set recognition task, i.e., what is anomalous at inference has not been observed at training. We propose COSKAD, a novel model that encodes skeletal human motion by a graph convolutional network and learns to COntract SKeletal kinematic embeddings onto a latent hypersphere of minimum volume for Video Anomaly Detection. We propose three latent spaces: the commonly-adopted Euclidean and the novel spherical and hyperbolic. All variants outperform the state-of-the-art on the most recent UBnormal dataset, for which we contribute a human-related version with annotated skeletons. COSKAD sets a new state-of-the-art on the human-related versions of ShanghaiTech Campus and CUHK Avenue, with performance comparable to video-based methods. Source code and dataset will be released upon acceptance.

Partially Relevant Video Retrieval (PRVR) addresses the critical challenge of matching untrimmed videos with text queries describing only partial content. Existing methods suffer from geometric distortion in Euclidean space that sometimes misrepresents the intrinsic hierarchical structure of videos and overlooks certain hierarchical semantics, ultimately leading to suboptimal temporal modeling. To address this issue, we propose the first hyperbolic modeling framework for PRVR, namely HLFormer, which leverages hyperbolic space learning to compensate for the suboptimal hierarchical modeling capabilities of Euclidean space. Specifically, HLFormer integrates the Lorentz Attention Block and Euclidean Attention Block to encode video embeddings in hybrid spaces, using the Mean-Guided Adaptive Interaction Module to dynamically fuse features. Additionally, we introduce a Partial Order Preservation Loss to enforce "text < video" hierarchy through Lorentzian cone constraints. This approach further enhances cross-modal matching by reinforcing partial relevance between video content and text queries. Extensive experiments show that HLFormer outperforms state-of-the-art methods. Code is released at https://github.com/lijun2005/ICCV25-HLFormer.

Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding quality without labels remains an open challenge. In this work, we propose Persistence, a topology-aware metric based on persistent homology that quantifies the geometric structure and topological richness of embedding spaces in a fully unsupervised manner. Unlike metrics that assume linear separability or rely on covariance structure, Persistence captures global and multi-scale organization. Empirical results across diverse domains show that Persistence consistently achieves top-tier correlations with downstream performance, outperforming existing unsupervised metrics and enabling reliable model and hyperparameter selection.

Visual and linguistic concepts naturally organize themselves in a hierarchy, where a textual concept ``dog'' entails all images that contain dogs. Despite being intuitive, current large-scale vision and language models such as CLIP do not explicitly capture such hierarchy. We propose MERU, a contrastive model that yields hyperbolic representations of images and text. Hyperbolic spaces have suitable geometric properties to embed tree-like data, so MERU can better capture the underlying hierarchy in image-text data. Our results show that MERU learns a highly interpretable representation space while being competitive with CLIP's performance on multi-modal tasks like image classification and image-text retrieval.

Large language models (LLMs) have shown great success in text modeling tasks across domains. However, natural language exhibits inherent semantic hierarchies and nuanced geometric structure, which current LLMs do not capture completely owing to their reliance on Euclidean operations. Recent studies have also shown that not respecting the geometry of token embeddings leads to training instabilities and degradation of generative capabilities. These findings suggest that shifting to non-Euclidean geometries can better align language models with the underlying geometry of text. We thus propose to operate fully in Hyperbolic space, known for its expansive, scale-free, and low-distortion properties. We thus introduce HELM, a family of HypErbolic Large Language Models, offering a geometric rethinking of the Transformer-based LLM that addresses the representational inflexibility, missing set of necessary operations, and poor scalability of existing hyperbolic LMs. We additionally introduce a Mixture-of-Curvature Experts model, HELM-MICE, where each expert operates in a distinct curvature space to encode more fine-grained geometric structure from text, as well as a dense model, HELM-D. For HELM-MICE, we further develop hyperbolic Multi-Head Latent Attention (HMLA) for efficient, reduced-KV-cache training and inference. For both models, we develop essential hyperbolic equivalents of rotary positional encodings and RMS normalization. We are the first to train fully hyperbolic LLMs at billion-parameter scale, and evaluate them on well-known benchmarks such as MMLU and ARC, spanning STEM problem-solving, general knowledge, and commonsense reasoning. Our results show consistent gains from our HELM architectures -- up to 4% -- over popular Euclidean architectures used in LLaMA and DeepSeek, highlighting the efficacy and enhanced reasoning afforded by hyperbolic geometry in large-scale LM pretraining.

Hypergraphs are marked by complex topology, expressing higher-order interactions among multiple nodes with hyperedges, and better capturing the topology is essential for effective representation learning. Recent advances in generative self-supervised learning (SSL) suggest that hypergraph neural networks learned from generative self supervision have the potential to effectively encode the complex hypergraph topology. Designing a generative SSL strategy for hypergraphs, however, is not straightforward. Questions remain with regard to its generative SSL task, connection to downstream tasks, and empirical properties of learned representations. In light of the promises and challenges, we propose a novel generative SSL strategy for hypergraphs. We first formulate a generative SSL task on hypergraphs, hyperedge filling, and highlight its theoretical connection to node classification. Based on the generative SSL task, we propose a hypergraph SSL method, HypeBoy. HypeBoy learns effective general-purpose hypergraph representations, outperforming 16 baseline methods across 11 benchmark datasets.

Recent advances in representation learning have shown that hyperbolic geometry can offer a more expressive alternative to the Euclidean embeddings used in CLIP models, capturing hierarchical structures and leading to better-organized representations. However, current hyperbolic CLIP variants are trained entirely from scratch, which is computationally expensive and resource-intensive. In this work, we propose HAC (Hyperbolic Adaptation of CLIP), a parameter-efficient framework that enables pretrained CLIP models to transition into hyperbolic space via lightweight fine-tuning. We apply HAC to Visual Question Answering (VQA), where models must interpret visual elements and align them with textual queries. Notably, HAC's training is performed on a dataset with no overlap with any VQA benchmark, resulting in a strict zero-shot evaluation paradigm that underscores HAC's task-agnostic adaptability. We evaluate HAC across a diverse suite of VQA benchmarks spanning General, Reasoning, and OCR categories. Both HAC-S (small) and HAC-B (medium) consistently surpass Euclidean baselines and prior hyperbolic approaches, with HAC-B delivering up to a +1.9 point average improvement over CLIP-B on reasoning-intensive tasks. Our code is available at https://github.com/fdibiton/HAC

Learnable embedding vector is one of the most important applications in machine learning, and is widely used in various database-related domains. However, the high dimensionality of sparse data in recommendation tasks and the huge volume of corpus in retrieval-related tasks lead to a large memory consumption of the embedding table, which poses a great challenge to the training and deployment of models. Recent research has proposed various methods to compress the embeddings at the cost of a slight decrease in model quality or the introduction of other overheads. Nevertheless, the relative performance of these methods remains unclear. Existing experimental comparisons only cover a subset of these methods and focus on limited metrics. In this paper, we perform a comprehensive comparative analysis and experimental evaluation of embedding compression. We introduce a new taxonomy that categorizes these techniques based on their characteristics and methodologies, and further develop a modular benchmarking framework that integrates 14 representative methods. Under a uniform test environment, our benchmark fairly evaluates each approach, presents their strengths and weaknesses under different memory budgets, and recommends the best method based on the use case. In addition to providing useful guidelines, our study also uncovers the limitations of current methods and suggests potential directions for future research.

We introduce a generalized attention mechanism for spherical domains, enabling Transformer architectures to natively process data defined on the two-dimensional sphere - a critical need in fields such as atmospheric physics, cosmology, and robotics, where preserving spherical symmetries and topology is essential for physical accuracy. By integrating numerical quadrature weights into the attention mechanism, we obtain a geometrically faithful spherical attention that is approximately rotationally equivariant, providing strong inductive biases and leading to better performance than Cartesian approaches. To further enhance both scalability and model performance, we propose neighborhood attention on the sphere, which confines interactions to geodesic neighborhoods. This approach reduces computational complexity and introduces the additional inductive bias for locality, while retaining the symmetry properties of our method. We provide optimized CUDA kernels and memory-efficient implementations to ensure practical applicability. The method is validated on three diverse tasks: simulating shallow water equations on the rotating sphere, spherical image segmentation, and spherical depth estimation. Across all tasks, our spherical Transformers consistently outperform their planar counterparts, highlighting the advantage of geometric priors for learning on spherical domains.

Segmenting unseen objects from images is a critical perception skill that a robot needs to acquire. In robot manipulation, it can facilitate a robot to grasp and manipulate unseen objects. Mean shift clustering is a widely used method for image segmentation tasks. However, the traditional mean shift clustering algorithm is not differentiable, making it difficult to integrate it into an end-to-end neural network training framework. In this work, we propose the Mean Shift Mask Transformer (MSMFormer), a new transformer architecture that simulates the von Mises-Fisher (vMF) mean shift clustering algorithm, allowing for the joint training and inference of both the feature extractor and the clustering. Its central component is a hypersphere attention mechanism, which updates object queries on a hypersphere. To illustrate the effectiveness of our method, we apply MSMFormer to unseen object instance segmentation. Our experiments show that MSMFormer achieves competitive performance compared to state-of-the-art methods for unseen object instance segmentation. The project page, appendix, video, and code are available at https://irvlutd.github.io/MSMFormer

In an era where the volume of data drives the effectiveness of self-supervised learning, the specificity and clarity of data semantics play a crucial role in model training. Addressing this, we introduce HYPerbolic Entailment filtering (HYPE), a novel methodology designed to meticulously extract modality-wise meaningful and well-aligned data from extensive, noisy image-text pair datasets. Our approach leverages hyperbolic embeddings and the concept of entailment cones to evaluate and filter out samples with meaningless or underspecified semantics, focusing on enhancing the specificity of each data sample. HYPE not only demonstrates a significant improvement in filtering efficiency but also sets a new state-of-the-art in the DataComp benchmark when combined with existing filtering techniques. This breakthrough showcases the potential of HYPE to refine the data selection process, thereby contributing to the development of more accurate and efficient self-supervised learning models. Additionally, the image specificity epsilon_{i} can be independently applied to induce an image-only dataset from an image-text or image-only data pool for training image-only self-supervised models and showed superior performance when compared to the dataset induced by CLIP score.

Recent 3D content generation pipelines often leverage Variational Autoencoders (VAEs) to encode shapes into compact latent representations, facilitating diffusion-based generation. Efficiently compressing 3D shapes while preserving intricate geometric details remains a key challenge. Existing 3D shape VAEs often employ uniform point sampling and 1D/2D latent representations, such as vector sets or triplanes, leading to significant geometric detail loss due to inadequate surface coverage and the absence of explicit 3D representations in the latent space. Although recent work explores 3D latent representations, their large scale hinders high-resolution encoding and efficient training. Given these challenges, we introduce Hyper3D, which enhances VAE reconstruction through efficient 3D representation that integrates hybrid triplane and octree features. First, we adopt an octree-based feature representation to embed mesh information into the network, mitigating the limitations of uniform point sampling in capturing geometric distributions along the mesh surface. Furthermore, we propose a hybrid latent space representation that integrates a high-resolution triplane with a low-resolution 3D grid. This design not only compensates for the lack of explicit 3D representations but also leverages a triplane to preserve high-resolution details. Experimental results demonstrate that Hyper3D outperforms traditional representations by reconstructing 3D shapes with higher fidelity and finer details, making it well-suited for 3D generation pipelines.

A well-designed vectorized representation is crucial for the learning systems natively based on 3D Gaussian Splatting. While 3DGS enables efficient and explicit 3D reconstruction, its parameter-based representation remains hard to learn as features, especially for neural-network-based models. Directly feeding raw Gaussian parameters into learning frameworks fails to address the non-unique and heterogeneous nature of the Gaussian parameterization, yielding highly data-dependent models. This challenge motivates us to explore a more principled approach to represent 3D Gaussian Splatting in neural networks that preserves the underlying color and geometric structure while enforcing unique mapping and channel homogeneity. In this paper, we propose an embedding representation of 3DGS based on continuous submanifold fields that encapsulate the intrinsic information of Gaussian primitives, thereby benefiting the learning of 3DGS.

In this report, we introduce Gemini Embedding, a state-of-the-art embedding model leveraging the power of Gemini, Google's most capable large language model. Capitalizing on Gemini's inherent multilingual and code understanding capabilities, Gemini Embedding produces highly generalizable embeddings for text spanning numerous languages and textual modalities. The representations generated by Gemini Embedding can be precomputed and applied to a variety of downstream tasks including classification, similarity, clustering, ranking, and retrieval. Evaluated on the Massive Multilingual Text Embedding Benchmark (MMTEB), which includes over one hundred tasks across 250+ languages, Gemini Embedding substantially outperforms prior state-of-the-art models, demonstrating considerable improvements in embedding quality. Achieving state-of-the-art performance across MMTEB's multilingual, English, and code benchmarks, our unified model demonstrates strong capabilities across a broad selection of tasks and surpasses specialized domain-specific models.

Knowledge Graphs, such as Wikidata, comprise structural and textual knowledge in order to represent knowledge. For each of the two modalities dedicated approaches for graph embedding and language models learn patterns that allow for predicting novel structural knowledge. Few approaches have integrated learning and inference with both modalities and these existing ones could only partially exploit the interaction of structural and textual knowledge. In our approach, we build on existing strong representations of single modalities and we use hypercomplex algebra to represent both, (i), single-modality embedding as well as, (ii), the interaction between different modalities and their complementary means of knowledge representation. More specifically, we suggest Dihedron and Quaternion representations of 4D hypercomplex numbers to integrate four modalities namely structural knowledge graph embedding, word-level representations (e.g.\ Word2vec, Fasttext), sentence-level representations (Sentence transformer), and document-level representations (sentence transformer, Doc2vec). Our unified vector representation scores the plausibility of labelled edges via Hamilton and Dihedron products, thus modeling pairwise interactions between different modalities. Extensive experimental evaluation on standard benchmark datasets shows the superiority of our two new models using abundant textual information besides sparse structural knowledge to enhance performance in link prediction tasks.

Deep face recognition has achieved great success due to large-scale training databases and rapidly developing loss functions. The existing algorithms devote to realizing an ideal idea: minimizing the intra-class distance and maximizing the inter-class distance. However, they may neglect that there are also low quality training images which should not be optimized in this strict way. Considering the imperfection of training databases, we propose that intra-class and inter-class objectives can be optimized in a moderate way to mitigate overfitting problem, and further propose a novel loss function, named sigmoid-constrained hypersphere loss (SFace). Specifically, SFace imposes intra-class and inter-class constraints on a hypersphere manifold, which are controlled by two sigmoid gradient re-scale functions respectively. The sigmoid curves precisely re-scale the intra-class and inter-class gradients so that training samples can be optimized to some degree. Therefore, SFace can make a better balance between decreasing the intra-class distances for clean examples and preventing overfitting to the label noise, and contributes more robust deep face recognition models. Extensive experiments of models trained on CASIA-WebFace, VGGFace2, and MS-Celeb-1M databases, and evaluated on several face recognition benchmarks, such as LFW, MegaFace and IJB-C databases, have demonstrated the superiority of SFace.

Contrastive Language-Image Pre-Training (CLIP) is highly instrumental in machine learning applications within a large variety of domains. We investigate the geometry of this embedding, which is still not well understood. We examine the raw unnormalized embedding and show that text and image reside on linearly separable ellipsoid shells, not centered at the origin. We explain the benefits of having this structure, allowing to better embed instances according to their uncertainty during contrastive training. Frequent concepts in the dataset yield more false negatives, inducing greater uncertainty. A new notion of conformity is introduced, which measures the average cosine similarity of an instance to any other instance within a representative data set. We show this measure can be accurately estimated by simply computing the cosine similarity to the modality mean vector. Furthermore, we find that CLIP's modality gap optimizes the matching of the conformity distributions of image and text.

Despite significant advances in large-scale text-to-image models, achieving hyper-realistic human image generation remains a desirable yet unsolved task. Existing models like Stable Diffusion and DALL-E 2 tend to generate human images with incoherent parts or unnatural poses. To tackle these challenges, our key insight is that human image is inherently structural over multiple granularities, from the coarse-level body skeleton to fine-grained spatial geometry. Therefore, capturing such correlations between the explicit appearance and latent structure in one model is essential to generate coherent and natural human images. To this end, we propose a unified framework, HyperHuman, that generates in-the-wild human images of high realism and diverse layouts. Specifically, 1) we first build a large-scale human-centric dataset, named HumanVerse, which consists of 340M images with comprehensive annotations like human pose, depth, and surface normal. 2) Next, we propose a Latent Structural Diffusion Model that simultaneously denoises the depth and surface normal along with the synthesized RGB image. Our model enforces the joint learning of image appearance, spatial relationship, and geometry in a unified network, where each branch in the model complements to each other with both structural awareness and textural richness. 3) Finally, to further boost the visual quality, we propose a Structure-Guided Refiner to compose the predicted conditions for more detailed generation of higher resolution. Extensive experiments demonstrate that our framework yields the state-of-the-art performance, generating hyper-realistic human images under diverse scenarios. Project Page: https://snap-research.github.io/HyperHuman/

Despite high semantic alignment, modern text-to-image (T2I) generative models still struggle to synthesize diverse images from a given prompt. This lack of diversity not only restricts user choice, but also risks amplifying societal biases. In this work, we enhance the T2I diversity through a geometric lens. Unlike most existing methods that rely primarily on entropy-based guidance to increase sample dissimilarity, we introduce Geometry-Aware Spherical Sampling (GASS) to enhance diversity by explicitly controlling both prompt-dependent and prompt-independent sources of variation. Specifically, we decompose the diversity measure in CLIP embeddings using two orthogonal directions: the text embedding, which captures semantic variation related to the prompt, and an identified orthogonal direction that captures prompt-independent variation (e.g., backgrounds). Based on this decomposition, GASS increases the geometric projection spread of generated image embeddings along both axes and guides the T2I sampling process via expanded predictions along the generation trajectory. Our experiments on different frozen T2I backbones (U-Net and DiT, diffusion and flow) and benchmarks demonstrate the effectiveness of disentangled diversity enhancement with minimal impact on image fidelity and semantic alignment.

Rapid growth of high-dimensional datasets in fields such as single-cell RNA sequencing and spatial genomics has led to unprecedented opportunities for scientific discovery, but it also presents unique computational and statistical challenges. Traditional methods struggle with geometry-aware data generation, interpolation along meaningful trajectories, and transporting populations via feasible paths. To address these issues, we introduce Geometry-Aware Generative Autoencoder (GAGA), a novel framework that combines extensible manifold learning with generative modeling. GAGA constructs a neural network embedding space that respects the intrinsic geometries discovered by manifold learning and learns a novel warped Riemannian metric on the data space. This warped metric is derived from both the points on the data manifold and negative samples off the manifold, allowing it to characterize a meaningful geometry across the entire latent space. Using this metric, GAGA can uniformly sample points on the manifold, generate points along geodesics, and interpolate between populations across the learned manifold using geodesic-guided flows. GAGA shows competitive performance in simulated and real-world datasets, including a 30% improvement over the state-of-the-art methods in single-cell population-level trajectory inference.

Embedding words in high-dimensional vector spaces has proven valuable in many natural language applications. In this work, we investigate whether similarly-trained embeddings of integers can capture concepts that are useful for mathematical applications. We probe the integer embeddings for mathematical knowledge, apply them to a set of numerical reasoning tasks, and show that by learning the representations from mathematical sequence data, we can substantially improve over number embeddings learned from English text corpora.

Real-world visual data exhibit intrinsic hierarchical structures that can be represented effectively in hyperbolic spaces. Hyperbolic neural networks (HNNs) are a promising approach for learning feature representations in such spaces. However, current HNNs in computer vision rely on Euclidean backbones and only project features to the hyperbolic space in the task heads, limiting their ability to fully leverage the benefits of hyperbolic geometry. To address this, we present HCNN, a fully hyperbolic convolutional neural network (CNN) designed for computer vision tasks. Based on the Lorentz model, we generalize fundamental components of CNNs and propose novel formulations of the convolutional layer, batch normalization, and multinomial logistic regression. {Experiments on standard vision tasks demonstrate the promising performance of our HCNN framework in both hybrid and fully hyperbolic settings.} Overall, we believe our contributions provide a foundation for developing more powerful HNNs that can better represent complex structures found in image data. Our code is publicly available at https://github.com/kschwethelm/HyperbolicCV.

This work proposes a novel representation of injective deformations of 3D space, which overcomes existing limitations of injective methods: inaccuracy, lack of robustness, and incompatibility with general learning and optimization frameworks. The core idea is to reduce the problem to a deep composition of multiple 2D mesh-based piecewise-linear maps. Namely, we build differentiable layers that produce mesh deformations through Tutte's embedding (guaranteed to be injective in 2D), and compose these layers over different planes to create complex 3D injective deformations of the 3D volume. We show our method provides the ability to efficiently and accurately optimize and learn complex deformations, outperforming other injective approaches. As a main application, we produce complex and artifact-free NeRF and SDF deformations.

Geographic location is essential for modeling tasks in fields ranging from ecology to epidemiology to the Earth system sciences. However, extracting relevant and meaningful characteristics of a location can be challenging, often entailing expensive data fusion or data distillation from global imagery datasets. To address this challenge, we introduce Satellite Contrastive Location-Image Pretraining (SatCLIP), a global, general-purpose geographic location encoder that learns an implicit representation of locations from openly available satellite imagery. Trained location encoders provide vector embeddings summarizing the characteristics of any given location for convenient usage in diverse downstream tasks. We show that SatCLIP embeddings, pretrained on globally sampled multi-spectral Sentinel-2 satellite data, can be used in various predictive tasks that depend on location information but not necessarily satellite imagery, including temperature prediction, animal recognition in imagery, and population density estimation. Across tasks, SatCLIP embeddings consistently outperform embeddings from existing pretrained location encoders, ranging from models trained on natural images to models trained on semantic context. SatCLIP embeddings also help to improve geographic generalization. This demonstrates the potential of general-purpose location encoders and opens the door to learning meaningful representations of our planet from the vast, varied, and largely untapped modalities of geospatial data.

Machine learning models often learn latent embedding representations that capture the domain semantics of their training data. These embedding representations are valuable for interpreting trained models, building new models, and analyzing new datasets. However, interpreting and using embeddings can be challenging due to their opaqueness, high dimensionality, and the large size of modern datasets. To tackle these challenges, we present WizMap, an interactive visualization tool to help researchers and practitioners easily explore large embeddings. With a novel multi-resolution embedding summarization method and a familiar map-like interaction design, WizMap enables users to navigate and interpret embedding spaces with ease. Leveraging modern web technologies such as WebGL and Web Workers, WizMap scales to millions of embedding points directly in users' web browsers and computational notebooks without the need for dedicated backend servers. WizMap is open-source and available at the following public demo link: https://poloclub.github.io/wizmap.

Let T be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in R^d. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a rm polylog(Delta)-approximation algorithm for embedding n-point ultrametrics into R^d with minimum distortion, where Delta denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in n. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.

Few-shot models aim at making predictions using a minimal number of labeled examples from a given task. The main challenge in this area is the one-shot setting where only one element represents each class. We propose HyperShot - the fusion of kernels and hypernetwork paradigm. Compared to reference approaches that apply a gradient-based adjustment of the parameters, our model aims to switch the classification module parameters depending on the task's embedding. In practice, we utilize a hypernetwork, which takes the aggregated information from support data and returns the classifier's parameters handcrafted for the considered problem. Moreover, we introduce the kernel-based representation of the support examples delivered to hypernetwork to create the parameters of the classification module. Consequently, we rely on relations between embeddings of the support examples instead of direct feature values provided by the backbone models. Thanks to this approach, our model can adapt to highly different tasks.

As the most common representation for 3D shapes, mesh is often stored discretely with arrays of vertices and faces. However, 3D shapes in the real world are presented continuously. In this paper, we propose to learn a continuous representation for meshes with fixed topology, a common and practical setting in many faces-, hand-, and body-related applications. First, we split the template into multiple closed manifold genus-0 meshes so that each genus-0 mesh can be parameterized onto the unit sphere. Then we learn spherical implicit surface (SIS), which takes a spherical coordinate and a global feature or a set of local features around the coordinate as inputs, predicting the vertex corresponding to the coordinate as an output. Since the spherical coordinates are continuous, SIS can depict a mesh in an arbitrary resolution. SIS representation builds a bridge between discrete and continuous representation in 3D shapes. Specifically, we train SIS networks in a self-supervised manner for two tasks: a reconstruction task and a super-resolution task. Experiments show that our SIS representation is comparable with state-of-the-art methods that are specifically designed for meshes with a fixed resolution and significantly outperforms methods that work in arbitrary resolutions.

We investigate spaces of symplectic embeddings of nleq 4 balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of n points. We compute the rational homotopy type of these embedding spaces and their cohomology with rational coefficients. Our approach relies on the comparison of the action of PGL(3,C) on the configuration space of n ordered points in CP^2 with the action of the symplectomorphism group Symp(CP^2) on the space of n embedded symplectic balls.

A recurring challenge in self-supervised learning is preventing representation collapse. Existing solutions typically rely on global regularization, such as maximizing distances, decorrelating dimensions or enforcing certain distributions. We instead reinterpret representation learning as a discrete packing problem, where preserving information simplifies to maintaining injectivity. We operationalize this in Hypersolid, a method using short-range hard-ball repulsion to prevent local collisions. This constraint results in a high-separation geometric regime that preserves augmentation diversity, excelling on fine-grained and low-resolution classification tasks.

Models for image representation learning are typically designed for either recognition or generation. Various forms of contrastive learning help models learn to convert images to embeddings that are useful for classification, detection, and segmentation. On the other hand, models can be trained to reconstruct images with pixel-wise, perceptual, and adversarial losses in order to learn a latent space that is useful for image generation. We seek to unify these two directions with a first-of-its-kind model that learns representations which are simultaneously useful for recognition and generation. We train our model as a hyper-network for implicit neural representation, which learns to map images to model weights for fast, accurate reconstruction. We further integrate our INR hyper-network with knowledge distillation to improve its generalization and performance. Beyond the novel training design, the model also learns an unprecedented compressed embedding space with outstanding performance for various visual tasks. The complete model competes with state-of-the-art results for image representation learning, while also enabling generative capabilities with its high-quality tiny embeddings. The code is available at https://github.com/tiktok/huvr.

Collaborative filtering (CF) plays a critical role in the development of recommender systems. Most CF methods utilize an encoder to embed users and items into the same representation space, and the Bayesian personalized ranking (BPR) loss is usually adopted as the objective function to learn informative encoders. Existing studies mainly focus on designing more powerful encoders (e.g., graph neural network) to learn better representations. However, few efforts have been devoted to investigating the desired properties of representations in CF, which is important to understand the rationale of existing CF methods and design new learning objectives. In this paper, we measure the representation quality in CF from the perspective of alignment and uniformity on the hypersphere. We first theoretically reveal the connection between the BPR loss and these two properties. Then, we empirically analyze the learning dynamics of typical CF methods in terms of quantified alignment and uniformity, which shows that better alignment or uniformity both contribute to higher recommendation performance. Based on the analyses results, a learning objective that directly optimizes these two properties is proposed, named DirectAU. We conduct extensive experiments on three public datasets, and the proposed learning framework with a simple matrix factorization model leads to significant performance improvements compared to state-of-the-art CF methods. Our implementations are publicly available at https://github.com/THUwangcy/DirectAU.

Conventional urban indicators derived from censuses, surveys, and administrative records are often costly, spatially inconsistent, and slow to update. Recent geospatial foundation models enable Earth embeddings, compact satellite image representations transferable across downstream tasks, but their utility for neighborhood-scale urban monitoring remains unclear. Here, we benchmark three Earth embedding families, AlphaEarth, Prithvi, and Clay, for urban signal prediction across six U.S. metropolitan areas from 2020 to 2023. Using a unified supervised-learning framework, we predict 14 neighborhood-level indicators spanning crime, income, health, and travel behavior, and evaluate performance under four settings: global, city-wise, year-wise, and city-year. Results show that Earth embeddings capture substantial urban variation, with the highest predictive skill for outcomes more directly tied to built-environment structure, including chronic health burdens and dominant commuting modes. By contrast, indicators shaped more strongly by fine-scale behavior and local policy, such as cycling, remain difficult to infer. Predictive performance varies markedly across cities but remains comparatively stable across years, indicating strong spatial heterogeneity alongside temporal robustness. Exploratory analysis suggests that cross-city variation in predictive performance is associated with urban form in task-specific ways. Controlled dimensionality experiments show that representation efficiency is critical: compact 64-dimensional AlphaEarth embeddings remain more informative than 64-dimensional reductions of Prithvi and Clay. This study establishes a benchmark for evaluating Earth embeddings in urban remote sensing and demonstrates their potential as scalable, low-cost features for SDG-aligned neighborhood-scale urban monitoring.

Embedding-based methods for reasoning in knowledge hypergraphs learn a representation for each entity and relation. Current methods do not capture the procedural rules underlying the relations in the graph. We propose a simple embedding-based model called ReAlE that performs link prediction in knowledge hypergraphs (generalized knowledge graphs) and can represent high-level abstractions in terms of relational algebra operations. We show theoretically that ReAlE is fully expressive and provide proofs and empirical evidence that it can represent a large subset of the primitive relational algebra operations, namely renaming, projection, set union, selection, and set difference. We also verify experimentally that ReAlE outperforms state-of-the-art models in knowledge hypergraph completion, and in representing each of these primitive relational algebra operations. For the latter experiment, we generate a synthetic knowledge hypergraph, for which we design an algorithm based on the Erdos-R'enyi model for generating random graphs.

Given a finite poset mathcal P, the hypercube-height, denoted by h^*(mathcal P), is defined to be the largest h such that, for any natural number n, the subsets of [n] of size less than h do not contain an induced copy of mathcal P. The hypercube-width, denoted by w^*(mathcal P), is the smallest w such that the subsets of [w] of size at most h^*(mathcal P) contain an induced copy of mathcal P. In other words, h^*(mathcal P) asks how `low' can a poset be embedded, and w^*(mathcal P) asks for the first hypercube in which such an `optimal' embedding occurs. These notions were introduced by Bastide, Groenland, Ivan and Johnston in connection to upper bounds for the poset saturation numbers. While it is not hard to see that h^*(mathcal P)leq |mathcal P|-1 (and this bound can be tight), the hypercube-width has proved to be much more elusive. It was shown by the authors mentioned above that w^*(mathcal P)leq|mathcal P|^2/4, but they conjectured that in fact w^*(mathcal P)leq |mathcal P| for any finite poset mathcal P. In this paper we prove this conjecture. The proof uses Hall's theorem for bipartite graphs as a precision tool for modifing an existing copy of our poset.

Recent works on compression of large language models (LLM) using quantization considered reparameterizing the architecture such that weights are distributed on the sphere. This demonstratively improves the ability to quantize by increasing the mathematical notion of coherence, resulting in fewer weight outliers without affecting the network output. In this work, we aim to further exploit this spherical geometry of the weights when performing quantization by considering Pyramid Vector Quantization (PVQ) for large language models. Arranging points evenly on the sphere is notoriously difficult, especially in high dimensions, and in case approximate solutions exists, representing points explicitly in a codebook is typically not feasible due to its additional memory cost. Instead, PVQ uses a fixed integer lattice on the sphere by projecting points onto the 1-sphere, which allows for efficient encoding and decoding without requiring an explicit codebook in memory. To obtain a practical algorithm, we propose to combine PVQ with scale quantization for which we derive theoretically optimal quantizations, under empirically verified assumptions. Further, we extend pyramid vector quantization to use Hessian information to minimize quantization error under expected feature activations, instead of only relying on weight magnitudes. Experimentally, we achieves state-of-the-art quantization performance with pareto-optimal trade-off between performance and bits per weight and bits per activation, compared to compared methods. On weight-only, we find that we can quantize a Llama-3 70B model to 3.25 bits per weight and retain 98\% accuracy on downstream tasks.

While the Earth observation community has witnessed a surge in high-impact foundation models and global Earth embedding datasets, a significant barrier remains in translating these academic assets into freely accessible tools. This tutorial introduces EarthEmbeddingExplorer, an interactive web application designed to bridge this gap, transforming static research artifacts into dynamic, practical workflows for discovery. We will provide a comprehensive hands-on guide to the system, detailing its cloud-native software architecture, demonstrating cross-modal queries (natural language, visual, and geolocation), and showcasing how to derive scientific insights from retrieval results. By democratizing access to precomputed Earth embeddings, this tutorial empowers researchers to seamlessly transition from state-of-the-art models and data archives to real-world application and analysis. The web application is available at https://modelscope.ai/studios/Major-TOM/EarthEmbeddingExplorer.

We consider the problem of embedding point cloud data sampled from an underlying manifold with an associated flow or velocity. Such data arises in many contexts where static snapshots of dynamic entities are measured, including in high-throughput biology such as single-cell transcriptomics. Existing embedding techniques either do not utilize velocity information or embed the coordinates and velocities independently, i.e., they either impose velocities on top of an existing point embedding or embed points within a prescribed vector field. Here we present FlowArtist, a neural network that embeds points while jointly learning a vector field around the points. The combination allows FlowArtist to better separate and visualize velocity-informed structures. Our results, on toy datasets and single-cell RNA velocity data, illustrate the value of utilizing coordinate and velocity information in tandem for embedding and visualizing high-dimensional data.

The development of data-dependent heuristics and representations for biological sequences that reflect their evolutionary distance is critical for large-scale biological research. However, popular machine learning approaches, based on continuous Euclidean spaces, have struggled with the discrete combinatorial formulation of the edit distance that models evolution and the hierarchical relationship that characterises real-world datasets. We present Neural Distance Embeddings (NeuroSEED), a general framework to embed sequences in geometric vector spaces, and illustrate the effectiveness of the hyperbolic space that captures the hierarchical structure and provides an average 22% reduction in embedding RMSE against the best competing geometry. The capacity of the framework and the significance of these improvements are then demonstrated devising supervised and unsupervised NeuroSEED approaches to multiple core tasks in bioinformatics. Benchmarked with common baselines, the proposed approaches display significant accuracy and/or runtime improvements on real-world datasets. As an example for hierarchical clustering, the proposed pretrained and from-scratch methods match the quality of competing baselines with 30x and 15x runtime reduction, respectively.

In this paper, we present a hypergraph neural networks (HGNN) framework for data representation learning, which can encode high-order data correlation in a hypergraph structure. Confronting the challenges of learning representation for complex data in real practice, we propose to incorporate such data structure in a hypergraph, which is more flexible on data modeling, especially when dealing with complex data. In this method, a hyperedge convolution operation is designed to handle the data correlation during representation learning. In this way, traditional hypergraph learning procedure can be conducted using hyperedge convolution operations efficiently. HGNN is able to learn the hidden layer representation considering the high-order data structure, which is a general framework considering the complex data correlations. We have conducted experiments on citation network classification and visual object recognition tasks and compared HGNN with graph convolutional networks and other traditional methods. Experimental results demonstrate that the proposed HGNN method outperforms recent state-of-the-art methods. We can also reveal from the results that the proposed HGNN is superior when dealing with multi-modal data compared with existing methods.

Many pre-trained language models (PLMs) exhibit suboptimal performance on mid- and low-resource languages, largely due to limited exposure to these languages during pre-training. A common strategy to address this is to introduce new tokens specific to the target languages, initialize their embeddings, and apply continual pre-training on target-language data. Among such methods, OFA (Liu et al., 2024a) proposes a similarity-based subword embedding initialization heuristic that is both effective and efficient. However, OFA restricts target-language token embeddings to be convex combinations of a fixed number of source-language embeddings, which may limit expressiveness. To overcome this limitation, we propose HYPEROFA, a hypernetwork-based approach for more adaptive token embedding initialization. The hypernetwork is trained to map from an external multilingual word vector space to the PLMs token embedding space using source-language tokens. Once trained, it can generate flexible embeddings for target-language tokens, serving as a good starting point for continual pretraining. Experiments demonstrate that HYPEROFA consistently outperforms random initialization baseline and matches or exceeds the performance of OFA in both continual pre-training convergence and downstream task performance. We make the code publicly available.

We introduce the Sphere Encoder, an efficient generative framework capable of producing images in a single forward pass and competing with many-step diffusion models using fewer than five steps. Our approach works by learning an encoder that maps natural images uniformly onto a spherical latent space, and a decoder that maps random latent vectors back to the image space. Trained solely through image reconstruction losses, the model generates an image by simply decoding a random point on the sphere. Our architecture naturally supports conditional generation, and looping the encoder/decoder a few times can further enhance image quality. Across several datasets, the sphere encoder approach yields performance competitive with state of the art diffusions, but with a small fraction of the inference cost. Project page is available at https://sphere-encoder.github.io .

The use of relative representations for latent embeddings has shown potential in enabling latent space communication and zero-shot model stitching across a wide range of applications. Nevertheless, relative representations rely on a certain amount of parallel anchors to be given as input, which can be impractical to obtain in certain scenarios. To overcome this limitation, we propose an optimization-based method to discover new parallel anchors from a limited known set (seed). Our approach can be used to find semantic correspondence between different domains, align their relative spaces, and achieve competitive results in several tasks.

The increasing demand for AR/VR applications has highlighted the need for high-quality 360-degree panoramic content. However, generating high-quality 360-degree panoramic images and videos remains a challenging task due to the severe distortions introduced by equirectangular projection (ERP). Existing approaches either fine-tune pretrained diffusion models on limited ERP datasets or attempt tuning-free methods that still rely on ERP latent representations, leading to discontinuities near the poles. In this paper, we introduce SphereDiff, a novel approach for seamless 360-degree panoramic image and video generation using state-of-the-art diffusion models without additional tuning. We define a spherical latent representation that ensures uniform distribution across all perspectives, mitigating the distortions inherent in ERP. We extend MultiDiffusion to spherical latent space and propose a spherical latent sampling method to enable direct use of pretrained diffusion models. Moreover, we introduce distortion-aware weighted averaging to further improve the generation quality in the projection process. Our method outperforms existing approaches in generating 360-degree panoramic content while maintaining high fidelity, making it a robust solution for immersive AR/VR applications. The code is available here. https://github.com/pmh9960/SphereDiff

Word embeddings are an essential component in a wide range of natural language processing applications. However, distributional semantic models are known to struggle when only a small number of context sentences are available. Several methods have been proposed to obtain higher-quality vectors for these words, leveraging both this context information and sometimes the word forms themselves through a hybrid approach. We show that the current tasks do not suffice to evaluate models that use word-form information, as such models can easily leverage word forms in the training data that are related to word forms in the test data. We introduce 3 new tasks, allowing for a more balanced comparison between models. Furthermore, we show that hyperparameters that have largely been ignored in previous work can consistently improve the performance of both baseline and advanced models, achieving a new state of the art on 4 out of 6 tasks.

Learning 3D representations that generalize well to arbitrarily oriented inputs is a challenge of practical importance in applications varying from computer vision to physics and chemistry. We propose a novel multi-resolution convolutional architecture for learning over concentric spherical feature maps, of which the single sphere representation is a special case. Our hierarchical architecture is based on alternatively learning to incorporate both intra-sphere and inter-sphere information. We show the applicability of our method for two different types of 3D inputs, mesh objects, which can be regularly sampled, and point clouds, which are irregularly distributed. We also propose an efficient mapping of point clouds to concentric spherical images, thereby bridging spherical convolutions on grids with general point clouds. We demonstrate the effectiveness of our approach in improving state-of-the-art performance on 3D classification tasks with rotated data.

Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs. Prior work introduced the concept of entailment cones, which uses partial orders defined by nested cones in the Poincar\'e ball to model hierarchies. Here, we introduce the ``shadow cones" framework, a physics-inspired entailment cone construction. Specifically, we model partial orders as subset relations between shadows formed by a light source and opaque objects in hyperbolic space. The shadow cones framework generalizes entailment cones to a broad class of formulations and hyperbolic space models beyond the Poincar\'e ball. This results in clear advantages over existing constructions: for example, shadow cones possess better optimization properties over constructions limited to the Poincar\'e ball. Our experiments on datasets of various sizes and hierarchical structures show that shadow cones consistently and significantly outperform existing entailment cone constructions. These results indicate that shadow cones are an effective way to model partial orders in hyperbolic space, offering physically intuitive and novel insights about the nature of such structures.

Hyperbolic geometry is gaining traction in machine learning for its effectiveness at capturing hierarchical structures in real-world data. Hyperbolic spaces, where neighborhoods grow exponentially, offer substantial advantages and consistently deliver state-of-the-art results across diverse applications. However, hyperbolic classifiers often grapple with computational challenges. Methods reliant on Riemannian optimization frequently exhibit sluggishness, stemming from the increased computational demands of operations on Riemannian manifolds. In response to these challenges, we present hyperDT, a novel extension of decision tree algorithms into hyperbolic space. Crucially, hyperDT eliminates the need for computationally intensive Riemannian optimization, numerically unstable exponential and logarithmic maps, or pairwise comparisons between points by leveraging inner products to adapt Euclidean decision tree algorithms to hyperbolic space. Our approach is conceptually straightforward and maintains constant-time decision complexity while mitigating the scalability issues inherent in high-dimensional Euclidean spaces. Building upon hyperDT we introduce hyperRF, a hyperbolic random forest model. Extensive benchmarking across diverse datasets underscores the superior performance of these models, providing a swift, precise, accurate, and user-friendly toolkit for hyperbolic data analysis.

Neural networks embed the geometric structure of a data manifold lying in a high-dimensional space into latent representations. Ideally, the distribution of the data points in the latent space should depend only on the task, the data, the loss, and other architecture-specific constraints. However, factors such as the random weights initialization, training hyperparameters, or other sources of randomness in the training phase may induce incoherent latent spaces that hinder any form of reuse. Nevertheless, we empirically observe that, under the same data and modeling choices, the angles between the encodings within distinct latent spaces do not change. In this work, we propose the latent similarity between each sample and a fixed set of anchors as an alternative data representation, demonstrating that it can enforce the desired invariances without any additional training. We show how neural architectures can leverage these relative representations to guarantee, in practice, invariance to latent isometries and rescalings, effectively enabling latent space communication: from zero-shot model stitching to latent space comparison between diverse settings. We extensively validate the generalization capability of our approach on different datasets, spanning various modalities (images, text, graphs), tasks (e.g., classification, reconstruction) and architectures (e.g., CNNs, GCNs, transformers).

Recently, studies exemplified by Hyper-Connections (HC) have extended the ubiquitous residual connection paradigm established over the past decade by expanding the residual stream width and diversifying connectivity patterns. While yielding substantial performance gains, this diversification fundamentally compromises the identity mapping property intrinsic to the residual connection, which causes severe training instability and restricted scalability, and additionally incurs notable memory access overhead. To address these challenges, we propose Manifold-Constrained Hyper-Connections (mHC), a general framework that projects the residual connection space of HC onto a specific manifold to restore the identity mapping property, while incorporating rigorous infrastructure optimization to ensure efficiency. Empirical experiments demonstrate that mHC is effective for training at scale, offering tangible performance improvements and superior scalability. We anticipate that mHC, as a flexible and practical extension of HC, will contribute to a deeper understanding of topological architecture design and suggest promising directions for the evolution of foundational models.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

We introduce OpenShape, a method for learning multi-modal joint representations of text, image, and point clouds. We adopt the commonly used multi-modal contrastive learning framework for representation alignment, but with a specific focus on scaling up 3D representations to enable open-world 3D shape understanding. To achieve this, we scale up training data by ensembling multiple 3D datasets and propose several strategies to automatically filter and enrich noisy text descriptions. We also explore and compare strategies for scaling 3D backbone networks and introduce a novel hard negative mining module for more efficient training. We evaluate OpenShape on zero-shot 3D classification benchmarks and demonstrate its superior capabilities for open-world recognition. Specifically, OpenShape achieves a zero-shot accuracy of 46.8% on the 1,156-category Objaverse-LVIS benchmark, compared to less than 10% for existing methods. OpenShape also achieves an accuracy of 85.3% on ModelNet40, outperforming previous zero-shot baseline methods by 20% and performing on par with some fully-supervised methods. Furthermore, we show that our learned embeddings encode a wide range of visual and semantic concepts (e.g., subcategories, color, shape, style) and facilitate fine-grained text-3D and image-3D interactions. Due to their alignment with CLIP embeddings, our learned shape representations can also be integrated with off-the-shelf CLIP-based models for various applications, such as point cloud captioning and point cloud-conditioned image generation.

A crucial component of many deep learning applications (such as FAQ and RAG) is dense retrieval, in which embedding models are used to convert raw text to numerical vectors and then get the most similar text by MIPS (Maximum Inner Product Search). Some text embedding benchmarks (e.g. MTEB, BEIR, and AIR-Bench) have been established to evaluate embedding models accurately. Thanks to these benchmarks, we can use SOTA models; however, the deployment and application of these models in industry were hampered by their large vector dimensions and numerous parameters. To alleviate this problem, 1) we present a distillation technique that can enable a smaller student model to achieve good performance. 2) Inspired by MRL we present a training approach of reducing the vector dimensions based on its own vectors or its teacher vectors. 3) We do simple yet effective alignment training between images and text to make our model a multimodal encoder. We trained Stella and Jasper models using the technologies above and achieved high scores on the MTEB leaderboard. We release the model and data at Hugging Face Hub (https://huggingface.co/infgrad/jasper_en_vision_language_v1) and the training logs are at https://api.wandb.ai/links/dunnzhang0/z8jqoqpb.

We propose Hyper-Dimensional Function Encoding (HDFE). Given samples of a continuous object (e.g. a function), HDFE produces an explicit vector representation of the given object, invariant to the sample distribution and density. Sample distribution and density invariance enables HDFE to consistently encode continuous objects regardless of their sampling, and therefore allows neural networks to receive continuous objects as inputs for machine learning tasks, such as classification and regression. Besides, HDFE does not require any training and is proved to map the object into an organized embedding space, which facilitates the training of the downstream tasks. In addition, the encoding is decodable, which enables neural networks to regress continuous objects by regressing their encodings. Therefore, HDFE serves as an interface for processing continuous objects. We apply HDFE to function-to-function mapping, where vanilla HDFE achieves competitive performance as the state-of-the-art algorithm. We apply HDFE to point cloud surface normal estimation, where a simple replacement from PointNet to HDFE leads to immediate 12% and 15% error reductions in two benchmarks. In addition, by integrating HDFE into the PointNet-based SOTA network, we improve the SOTA baseline by 2.5% and 1.7% in the same benchmarks.

Drawing motivation from the manifold hypothesis, which posits that most high-dimensional data lies on or near low-dimensional manifolds, we apply manifold learning to the space of neural networks. We learn manifolds where datapoints are neural networks by introducing a distance between the hidden layer representations of the neural networks. These distances are then fed to the non-linear dimensionality reduction algorithm PHATE to create a manifold of neural networks. We characterize this manifold using features of the representation, including class separation, hierarchical cluster structure, spectral entropy, and topological structure. Our analysis reveals that high-performing networks cluster together in the manifold, displaying consistent embedding patterns across all these features. Finally, we demonstrate the utility of this approach for guiding hyperparameter optimization and neural architecture search by sampling from the manifold.

Context. The Multi-Order Coverage map (MOC) is a widely adopted standard promoted by the International Virtual Observatory Alliance (IVOA) to support data sharing and interoperability within the Virtual Observatory (VO) ecosystem. This hierarchical data structure efficiently encodes and visualizes irregularly shaped regions of the sky, enabling applications such as cross-matching large astronomical catalogs. Aims. This study aims to explore potential enhancements to the MOC data structure by encapsulating textual descriptions and semantic embeddings into sky regions. Specifically, we introduce "Textual MOCs", in which textual content is encapsulated, and "Semantic MOCs" that transform textual content into semantic embeddings. These enhancements are designed to enable advanced operations such as similarity searches and complex queries and to integrate with generative artificial intelligence (GenAI) tools. Method. We experimented with Textual MOCs by annotating detailed descriptions directly into the MOC sky regions, enriching the maps with contextual information suitable for interactive learning tools. For Semantic MOCs, we converted the textual content into semantic embeddings, numerical representations capturing textual meanings in multidimensional spaces, and stored them in high-dimensional vector databases optimized for efficient retrieval. Results. The implementation of Textual MOCs enhances user engagement by providing meaningful descriptions within sky regions. Semantic MOCs enable sophisticated query capabilities, such as similarity-based searches and context-aware data retrieval. Integration with multimodal generative AI systems allows for more accurate and contextually relevant interactions supporting both spatial, semantic and visual operations for advancing astronomical data analysis capabilities.

We introduce TetSphere Splatting, a Lagrangian geometry representation designed for high-quality 3D shape modeling. TetSphere splatting leverages an underused yet powerful geometric primitive -- volumetric tetrahedral meshes. It represents 3D shapes by deforming a collection of tetrahedral spheres, with geometric regularizations and constraints that effectively resolve common mesh issues such as irregular triangles, non-manifoldness, and floating artifacts. Experimental results on multi-view and single-view reconstruction highlight TetSphere splatting's superior mesh quality while maintaining competitive reconstruction accuracy compared to state-of-the-art methods. Additionally, TetSphere splatting demonstrates versatility by seamlessly integrating into generative modeling tasks, such as image-to-3D and text-to-3D generation.

Vietnam ranks among the top countries in terms of both internet traffic and online toxicity. As a result, implementing embedding models for recommendation and content control duties in applications is crucial. However, a lack of large-scale test datasets, both in volume and task diversity, makes it tricky for scientists to effectively evaluate AI models before deploying them in real-world, large-scale projects. To solve this important problem, we introduce a Vietnamese benchmark, VN-MTEB for embedding models, which we created by translating a large number of English samples from the Massive Text Embedding Benchmark using our new automated framework. We leverage the strengths of large language models (LLMs) and cutting-edge embedding models to conduct translation and filtering processes to retain high-quality samples, guaranteeing a natural flow of language and semantic fidelity while preserving named entity recognition (NER) and code snippets. Our comprehensive benchmark consists of 41 datasets from six tasks specifically designed for Vietnamese text embeddings. In our analysis, we find that bigger and more complex models using Rotary Positional Embedding outperform those using Absolute Positional Embedding in embedding tasks. Datasets are available at HuggingFace: https://huggingface.co/collections/GreenNode/vn-mteb-68871433f0f7573b8e1a6686

Effective approaches that can scale embedding model depth (i.e. layers) and embedding size allow for the creation of models that are highly scalable across different computational resources and task requirements. While the recently proposed 2D Matryoshka training approach can efficiently produce a single embedding model such that its sub-layers and sub-dimensions can measure text similarity, its effectiveness is significantly worse than if smaller models were trained separately. To address this issue, we propose Starbucks, a new training strategy for Matryoshka-like embedding models, which encompasses both the fine-tuning and pre-training phases. For the fine-tuning phase, we discover that, rather than sampling a random sub-layer and sub-dimensions for each training steps, providing a fixed list of layer-dimension pairs, from small size to large sizes, and computing the loss across all pairs significantly improves the effectiveness of 2D Matryoshka embedding models, bringing them on par with their separately trained counterparts. To further enhance performance, we introduce a new pre-training strategy, which applies masked autoencoder language modelling to sub-layers and sub-dimensions during pre-training, resulting in a stronger backbone for subsequent fine-tuning of the embedding model. Experimental results on both semantic text similarity and retrieval benchmarks demonstrate that the proposed pre-training and fine-tuning strategies significantly improved the effectiveness over 2D Matryoshka models, enabling Starbucks models to perform more efficiently and effectively than separately trained models.

Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology and physics. While it is thought that these methods preserve underlying manifold structure of data by learning a proxy for geodesic distances, no specific theoretical links have been established. Here, we establish such a link via results in Riemannian geometry explicitly connecting heat diffusion to manifold distances. In this process, we also formulate a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. This novel perspective makes clearer the choices available in manifold learning and denoising. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances, and preserving cluster structure in toy datasets. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure, where our method enables interpolation of withheld timepoints of data. Finally, we show that parameters of our more general method can be configured to give results similar to PHATE (a state-of-the-art diffusion based manifold learning method) as well as SNE (an attraction/repulsion neighborhood based method that forms the basis of t-SNE).

Pre-trained multi-modal models, such as CLIP, provide transferable embeddings and show promising results in diverse applications. However, the analysis of learned multi-modal embeddings is relatively unexplored, and the embedding transferability can be improved. In this work, we observe that CLIP holds separated embedding subspaces for two different modalities, and then we investigate it through the lens of uniformity-alignment to measure the quality of learned representation. Both theoretically and empirically, we show that CLIP retains poor uniformity and alignment even after fine-tuning. Such a lack of alignment and uniformity might restrict the transferability and robustness of embeddings. To this end, we devise a new fine-tuning method for robust representation equipping better alignment and uniformity. First, we propose a Geodesic Multi-Modal Mixup that mixes the embeddings of image and text to generate hard negative samples on the hypersphere. Then, we fine-tune the model on hard negatives as well as original negatives and positives with contrastive loss. Based on the theoretical analysis about hardness guarantee and limiting behavior, we justify the use of our method. Extensive experiments on retrieval, calibration, few- or zero-shot classification (under distribution shift), embedding arithmetic, and image captioning further show that our method provides transferable representations, enabling robust model adaptation on diverse tasks. Code: https://github.com/changdaeoh/multimodal-mixup

We study the problem of visualizing large-scale and high-dimensional data in a low-dimensional (typically 2D or 3D) space. Much success has been reported recently by techniques that first compute a similarity structure of the data points and then project them into a low-dimensional space with the structure preserved. These two steps suffer from considerable computational costs, preventing the state-of-the-art methods such as the t-SNE from scaling to large-scale and high-dimensional data (e.g., millions of data points and hundreds of dimensions). We propose the LargeVis, a technique that first constructs an accurately approximated K-nearest neighbor graph from the data and then layouts the graph in the low-dimensional space. Comparing to t-SNE, LargeVis significantly reduces the computational cost of the graph construction step and employs a principled probabilistic model for the visualization step, the objective of which can be effectively optimized through asynchronous stochastic gradient descent with a linear time complexity. The whole procedure thus easily scales to millions of high-dimensional data points. Experimental results on real-world data sets demonstrate that the LargeVis outperforms the state-of-the-art methods in both efficiency and effectiveness. The hyper-parameters of LargeVis are also much more stable over different data sets.

Few-step image generation has seen rapid progress, with consistency and meanflow-based methods significantly reducing the number of sampling steps. Despite their low inference cost, these approaches often suffer from training instability and limited scalability. Sphere Encoder is a recent alternative that produces high-quality images in only a few steps; however, it requires repeated transitions between the pixel space and latent space during inference while jointly optimizing reconstruction and generation within a single architecture. This design leads to computational inefficiency and objective conflict between reconstruction and generation. To address these limitations, we decouple the framework into a fixed pretrained image encoder and a separate latent denoising model trained entirely in a spherical latent space. Our approach eliminates repeated pixel-space operations during training and inference, improving efficiency and allowing reconstruction and generation to specialize independently. On Animal-Faces, Oxford-Flowers and ImageNet-1K datasets, our method significantly outperforms Sphere Encoder in both generation quality and inference speed, while achieving competitive results against strong few-step and multi-step baselines.

Hypercomplex neural networks have proven to reduce the overall number of parameters while ensuring valuable performance by leveraging the properties of Clifford algebras. Recently, hypercomplex linear layers have been further improved by involving efficient parameterized Kronecker products. In this paper, we define the parameterization of hypercomplex convolutional layers and introduce the family of parameterized hypercomplex neural networks (PHNNs) that are lightweight and efficient large-scale models. Our method grasps the convolution rules and the filter organization directly from data without requiring a rigidly predefined domain structure to follow. PHNNs are flexible to operate in any user-defined or tuned domain, from 1D to nD regardless of whether the algebra rules are preset. Such a malleability allows processing multidimensional inputs in their natural domain without annexing further dimensions, as done, instead, in quaternion neural networks for 3D inputs like color images. As a result, the proposed family of PHNNs operates with 1/n free parameters as regards its analog in the real domain. We demonstrate the versatility of this approach to multiple domains of application by performing experiments on various image datasets as well as audio datasets in which our method outperforms real and quaternion-valued counterparts. Full code is available at: https://github.com/eleGAN23/HyperNets.

Graph Domain Adaptation (GDA) typically uses adversarial learning to align graph embeddings in Euclidean space. However, this paradigm suffers from two critical challenges: Structural Degeneration, where hierarchical and semantic representations are entangled, and Optimization Instability, which arises from oscillatory dynamics of minimax adversarial training. To tackle these issues, we propose DisRFM, a geometry-aware GDA framework that unifies Riemannian embedding and flow-based transport. First, to overcome structural degeneration, we embed graphs into a Riemannian manifold. By adopting polar coordinates, we explicitly disentangle structure (radius) from semantics (angle). Then, we enforce topology preservation through radial Wasserstein alignment and semantic discrimination via angular clustering, thereby preventing feature entanglement and collapse. Second, we address the instability of adversarial alignment by using Riemannian flow matching. This method learns a smooth vector field to guide source features toward the target along geodesic paths, guaranteeing stable convergence. The geometric constraints further guide the flow to maintain the disentangled structure during transport. Theoretically, we prove the asymptotic stability of the flow matching and derive a tighter bound for the target risk. Extensive experiments demonstrate that DisRFM consistently outperforms state-of-the-art methods.