Spherical Latent Space

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6 papers in the last four weeks, against 2 the four weeks before. 0.1% of all new papers.

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Latest papers 47

Oct 1, 2026cs.CV

SmoothOperator: Enhancing Representations for Fine-grained Open-set Recognition via Modulated Label Smoothing

Open Set Recognition (OSR) aims to enable models to accurately classify known classes while rejecting samples from unseen classes. A key challenge in OSR lies in the inability to model the unbounded distribution of unknown classes during training, often leading to the misclassification of samples from these classes. Rather than modeling unknowns, recent work shapes the feature space so that known classes are compact and well separated, and spherical representation learning methods have achieved strong results this way. Label smoothing has been identified as one of the key drivers of this success, yet it applies the same coefficient to every training sample, regardless of how well each sample is already embedded. We show that the spherical representation learning objectives used in OSR share a single alignment--uniformity structure in which labels enter only through the alignment term. Label smoothing therefore acts as an alignment dial, and a fixed coefficient sets this dial to the same value for every sample. We propose a plug-in, SmoothOperator (SmoothOP), which sets the smoothing coefficient of each sample from its \textbf{prominence}, an embedding-space signal measuring how clearly the sample's own class stands out against its strongest competing class. Our method integrates into four existing spherical representation learning methods at minimal training overhead. SmoothOP assigns strong smoothing to samples with high prominence, which reduces their alignment and relaxes their pull. On the Semantic Shift Benchmark, SmoothOP-augmented variants generally outperform their base objectives across datasets, degrees of semantic shift, and OSR post-processors, with gains of up to 4.7% in AUROC, OSCR, and closed-set accuracy.

Thiru Thillai Nadarasar Bahavan, Yu Xia, Sachith Seneviratne +1

The University of Melbourne Parkville, VIC, Australia

Open-Set Recognition Smoothing Fine-Grained Perception Spherical Latent Space

Sep 30, 2026cs.LG

Group-Invariant Statistics Determine Embedding Geometry: Harmonic Analysis of Representations from Bach to the Night Sky

The representations that language models learn for concepts such as months, weekdays, and places display consistent geometric structure: circles and saddle-shaped "Pringle" manifolds. Recent work traced these structures to

\textit{translation symmetry}

in word co-occurrence statistics, deriving the observed Fourier geometry when co-occurrence depends only on distance on an abelian lattice of concepts. We demonstrate that more general notions of symmetry lead to equally structured predictions. Considering symmetries defined by arbitrary finite groups, compact groups, and homogeneous spaces, we prove that whenever the co-occurrence statistics of a word family are invariant under a group

G

, the learned word embeddings consist of matrix elements of the irreducible representations (irreps) of

G

. Circles and Pringles arise when

G

is cyclic, in which case the irreps are Fourier modes. We verify the irrep structure in three experimental settings. (i) The cyclic group

\mathbb{Z}_{12}

: for the months of the year we recover the known circular geometry. (ii) A dihedral group acting on the major and minor triads: we unify two classical observations -- that transposition and chord inversion form a group (

T/I

) acting on chords (music theory), which

\textit{implies}

that the well-known "circle of fifths" emerges in learned chord embeddings (machine learning). (iii) We explain and reproduce a recently discovered spherical representation of celestial objects in large language models (LLMs) as a spherical-harmonic embedding derived from our theory. Our results demonstrate that the geometry of learned representations is often a consequence of the statistical symmetry of underlying data.

Liam Storan, Andreas Tolias, Nina Miolane

Stanford University · University of California, Santa Barbara

Representation Space Symmetry Spherical Latent Space Contextual Embeddings Sphere Fourier+1

Sep 30, 2026cs.LG

A library for differentiable signal processing and machine learning on the sphere

The two-dimensional sphere embedded in three-dimensional Euclidean space S2, plays a central role in a variety of scientific and engineering domains, including geophysics, planetary science, geodesy, atmospheric physics, quantum chemistry, cosmology, and virtual reality, among many others. As machine learning increasingly permeates these fields, the demand grows for robust tools that process and model functions on the sphere, while respecting the inherent topological and symmetry properties of the domain. We present torch-harmonics, a comprehensive library that offers efficient, differentiable implementations of advanced signal processing and machine learning (ML) methods for spherical data. These include the spherical harmonic transform (SHT), the spherical analogue of the Fourier transform, vector spherical harmonics, discrete-continuous and spectral convolutions, as well as both global and neighborhood spherical attention mechanisms. Beyond traditional representations, torch-harmonics provides the building blocks for state-of-the-art spherical ML architectures such as spherical transformers in order to enable scalable, rotationally-aware learning and inference in modern scientific and engineering applications.

Thorsten Kurth, Max Rietmann, Mauro Bisson +6

NVIDIA Corporation · California Institute of Technology

Spherical Harmonics Sphere Spherical Latent Space Harmonization Graph Signal Processing

Sep 28, 2026cs.LG

HALO: Enhancing Time Series Generation via Hyperspherical Latents and Masked AutoregRessive Modeling

Most existing time series generators rely on a two-stage modeling paradigm: the first stage learns discrete latent representations of time series; the second stage performs autoregressive modeling on these discrete latents through next token prediction. However, this paradigm suffers from two stage-specific limitations: the first stage can lead to information loss when discretizing continuous time series, while the second stage is prone to error accumulation during autoregressive generation. To address these limitations, our core idea is to perform generative modeling in a continuous latent space with a more efficient autoregressive framework. We propose HALO, which enhances time series generation via Hyperspherical Latents and Masked Autoregressive modeling to achieve this goal by tackling two key bottlenecks: (1) variance and scale heterogeneity of continuous latent representations; (2) the difficulty of balancing generation efficiency with temporal correlation modeling. HALO first introduces a hyperspherical VAE that constrains continuous latents to a fixed-radius hyperspherical shell, effectively stabilizing the numerical fluctuations of continuous latent representations. Secondly, we develop a masked autoregressive model that balances parallel decoding and temporal correlation learning, substantially reducing the number of inference steps required for generation and improving generation stability. Our extensive experiments demonstrate that HALO achieves state-of-the-art generation performance while offering significantly improved inference efficiency over existing advanced baselines.

Chunyi Hou, Xiangfei Qiu, Hanyin Cheng +2

School of Data Science and Engineering, East China Normal University, Shanghai, China

Time-Series Generation Autoregressive Generation Autoregressive Model Generative Models Time Series Future Latent Representations+2

Sep 22, 2026cs.LG

Can We Predict Anomaly Detection Performance from Embedding-Space Geometry?

Anomaly detection systems are often trained using normal data alone, while model selection and evaluation typically require labeled anomalies. We study whether anomaly detection performance can be predicted without access to anomalous data. For kNN-based detectors, we derive a lower bound on the area under the ROC curve (AUC) that relates detection performance to the separation between inlier and outlier scores and to their respective variances. Under a local scaling model, we use this bound to characterize how density variation, intrinsic-dimensional heterogeneity, and cross-domain mismatch contribute to score variability. We then investigate anomaly-free model selection and show that inlier score variance alone does not reliably predict performance across different representations. To address this limitation, we introduce simple pseudo-anomaly probes that provide a reference for estimating relative score separation. Experiments on the DCASE 2022-2025 benchmarks, spanning four embedding models and 208 candidate systems, show that pseudo-anomaly-based estimators substantially improve anomaly-free model selection. In particular, diverse pseudo-anomalies enable anomaly-free model selection to outperform conventional development-set selection under domain shift. These results show that embedding-space geometry contains predictive information about anomaly detection performance while also highlighting the representation-dependent nature of inlier-only performance estimates.

Kevin Wilkinghoff, Zheng-Hua Tan

Department of Electronic Systems, Aalborg University, Denmark · Pioneer Centre for Artificial Intelligence, Denmark

Area Under The Receiver Operating Characteristic Curve Detection Performance Outliers Model Selection Spherical Latent Space

Sep 22, 2026cs.LG

When Riemann flows with Wasserstein: Generative Modeling of Probability Distributions on Manifolds

Many scientific datasets, such as molecular conformational ensembles or single-cell tissue measurements, are naturally modeled as meta-distributions: distributions over probability measures on non-Euclidean domains. Existing generative methods largely assume Euclidean geometry and fail to capture this structure. We introduce Riemannian Wasserstein Entropic Flow Matching (RWEFM), a generative framework on the Wasserstein space

\mathcal{P}_2(\mathcal{M})

of a Riemannian manifold

(\mathcal{M},g)

. RWEFM is trained by regressing a neural vector field onto Riemannian optimal transport velocities, using McCann displacement interpolations as conditional paths. We confirm theoretically that this construction leads to a valid flow matching approach on

\mathcal{P}_2(\mathcal{M})

and introduce the Riemannian Entropic Map, a GPU-efficient approximation of the optimal transport map on manifolds. Our experiments show that by respecting the intrinsic geometry of the data, RWEFM can generate whole single-cell samples in hyperspherical latent spaces and protein conformational ensembles on the torus. As RWEFM requires only a geodesic distance and a projection operator, it is not restricted to manifolds with closed-form geometry, which we demonstrate by generating distributions on a general triangulated mesh.

Doron Haviv, Edward De Brouwer, Rishabh Anand +4

Genentech Inc., South San Francisco, CA. · Yale University, New Haven, CT.

Riemannian Flow Matching Data Manifold Wasserstein Distance Riemannian Manifolds Manifolds Spherical Latent Space+2

Sep 16, 2026stat.ML

A General Kernel Framework for Non-CND Distance Measures Using |D|-Dimensional Sparse Landmark Embeddings

Kernel methods, and Gaussian Processes (GPs) in particular, require a Hilbertian distance measure---one whose square is conditionally negative definite (CND)---to guarantee positive semi-definiteness (PSD) of the kernel matrix; a condition that fails for many natural input spaces, including smooth manifolds and spaces of probability distributions. We propose the Sparse Landmark Embedding (SLE) kernel, which eliminates this requirement entirely. Each input is embedded into a sparse feature vector via compactly supported bump functions centered at all |D| training points; applying any standard PSD kernel in this embedding space yields a kernel that is provably PSD for arbitrary distance measures. The compact support automatically controls embedding sparsity, keeping kernel matrices well-conditioned and computationally tractable despite the high ambient dimension. We provide theoretical guarantees on PSD, sparsity, stability, and universal approximation, and demonstrate, using geodesic and Wasserstein distances, that the SLE kernel matches or substantially exceeds domain-specific baselines in both predictive accuracy and uncertainty quantification.

Marcus M. Noack, Maher B. Alghalayini, Mark D. Risser

Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory Berkeley, CA 94720, USA · Climate and Ecosystem Sciences Division, Lawrence Berkeley National Laboratory Berkeley, CA 94720, USA

Kernel Method Spherical Latent Space Gaussian Process Distance Measures

Sep 14, 2026cs.LG

Skeletal Prototypes on Iterative Nerve Expansions

Prototype reduction replaces a training set with a smaller representation, and the established methods return a finite set of points. We propose Skeletal Prototypes on Iterative Nerve Expansions (SPINE). The model for each class is an embedded 1-complex rather than a point set. Its initial edge set is a class-conditional Mapper graph, so the data decide which localized clusters are joined. Later phases fit the vertices under a classification objective, and an observation is assigned to the class whose complex is nearest. The segments therefore enter the decision rule and not only the fitting. We evaluate SPINE on seventeen benchmark datasets under stratified 10-fold cross validation, against seven other prototype reduction methods at a matched budget. SPINE attains the highest mean accuracy and the best average rank. It is significantly better than five of the seven competitors under Wilcoxon signed-rank tests with Holm correction. A budget sweep shows that the decision rule using the entire graph segments contribute most when prototypes are scarce, while the method as a whole competes best at moderate budgets. Construction cost places SPINE with the discriminative methods, and it is faster than generalized learning vector quantization on fourteen of the seventeen datasets.

Jordan Eckert, Henry Schenck

Department of Mathematics and Statistics Auburn University Alabama, USA

Learnable Prototypes Spherical Latent Space Approximate Nearest-Neighbor Search Support Vector Machines Skeleton Benchmark Datasets

Sep 1, 2026cs.LG

DK-GBMKKM: Dynamic Kernel-Space Granular-Ball Multiple Kernel $k$ -Means Clustering

Multiple kernel

k

-means integrates complementary nonlinear similarities by learning a combination of base kernels. Its pointwise optimization, however, is sensitive to noisy and boundary samples and repeatedly operates on sample-scale kernel matrices. Granular-ball representations organize local sample groups into mesoscopic units, but granular balls generated once in the input space may be inconsistent with the fused-kernel geometry that evolves during multiple kernel learning. We propose dynamic kernel-space granular-ball multiple kernel

k

-means (DK-GBMKKM). The method generates granular balls in the current fused kernel space and alternates kernel-weight learning with granular-ball membership updates, allowing the representation to adapt to changes in the fused-kernel geometry. A sample-size-weighted granular-ball kernel is further constructed to preserve the contributions of balls of different sizes, and its positive semidefiniteness and related equivalence properties are established. Experiments on 12 public datasets demonstrate the strong overall clustering performance of DK-GBMKKM. The code has been open-sourced for reproducibility: https://github.com/lianxiaoyu724/DK-GBMKKM.

Xiaoyu Lian, Yuchao Zhang, Shuyin Xia +2

Chongqing University of Posts and Telecommunications · Chongqing Open University

Granular-Ball Computing K-Means Kernel Method Spherical Latent Space

Sep 1, 2026cs.CV

Soft-Argmax for the Projective Plane via the Veronese Embedding

From horizon detection to fibre structures in X-ray imaging, many vision tasks recover lines via peak detection in Hough space

H=S^1\times\mathbb{R}

, the domain of orientation-offset pairs

(θ,ρ)

. Differentiable pipelines extract coordinates via \emph{soft-argmax}, a probability-weighted average that is only meaningful in a globally linear space. However,

(θ,ρ)

and

(θ+π,-ρ)

describe the same undirected line, so

H

double-covers the space of undirected lines

H/\mathbb{Z}_2

: a Möbius strip, obtained by identifying each pair under

\mathbb{Z}_2

action. Soft-argmax operates on the cover

H

, but since

H/\mathbb{Z}_2

admits no linear structure, it tears geometrically adjacent lines apart. Thus we need a

\mathbb{Z}_2

-invariant embedding of lines into a linear space, on which soft-argmax is well-defined. We achieve this by parametrising lines via unit-norm homogeneous vectors

\ell=(1+ρ^2)^{-1/2}(\cosθ,\sinθ,-ρ)^{\top}\in\mathbb{R}^3

and applying the Veronese map

v_2(\ell)=\ell\ell^{\top}

that satisfies

v_2(\ell)=v_2(-\ell)

. This descends continuously to an embedding of the quotient

H/\mathbb{Z}_2

into the linear space

\mathrm{Sym}^2(\mathbb{R}^3)

, where the antipodal ambiguity vanishes. Line extraction becomes a barycentre in

\mathrm{Sym}^2(\mathbb{R}^3)

, projected back via its leading eigenvector. We validate our \emph{Veronese soft-argmax} in a Hough transform-based network across all resolvable lines, confirming uniform and seam-free recovery. We further derive that the

L_2

-loss on isometrically weighted Veronese embeddings equals the squared chordal distance between lines in projective space, enabling a geometrically precise training objective.

Benjamin El-Zein, Dominik Eckert, Paul Zech +4

Siemens Healthineers AG, X-ray Products, Forchheim, Germany · Artificial Intelligence Lab, Otto-von-Guericke-University, Magdeburg, Germany

Projection Spherical Latent Space Homography Gumbel-Softmax Relaxation Horizon

Aug 12, 2026cs.LG

HYDRA: Hyperbolic Dynamic Representation Architecture for Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KANs) enhance nonlinear function approximation by replacing scalar weights with learnable univariate functions. However, assigning an independent function to every connection results in substantial parameter redundancy, limiting their scalability and efficiency. To reduce this redundancy, we introduce \textbf{HY}perbolic \textbf{D}ynamic \textbf{R}epresentation \textbf{A}rchitecture (HYDRA), a parameter-efficient hyperbolic extension of KAN that combines spline-based functional learning with representations in the Poincaré ball. HYDRA maps vector-valued inputs into a bounded hyperbolic latent space, performs KAN-style updates in tangent space, and employs a low-rank prototype block to share functional transformations across hidden dimensions. The resulting hyperbolic representations provide a structured radial coordinate for interpretation, while radius control improves training stability by preventing boundary saturation. Extensive experiments across eight benchmark datasets demonstrate that HYDRA consistently achieves competitive or superior predictive performance while improving parameter efficiency and representation interpretability.

Zhao Su, Yuxin Xia, Haoran Li +4

School of Information Science and Engineering, Lanzhou University · Department of Data Science and AI, Monash University · School of Computing and Information Technology, University of Wollongong +1

Kolmogorov-Arnold Networks Hyperbolic Learning Neural Network Representations Spherical Latent Space Splines Domain Adaptation

Aug 9, 2026cs.CV

DoRF++: Spherical Representation Learning over Doppler Radiance Fields for Robust Wi-Fi Sensing

Motivated by the IEEE 802.11bf effort to standardize advanced WLAN sensing, interest in Wi-Fi Channel State Information (CSI) for passive, device-free, and privacy-preserving activity and gesture recognition has grown rapidly. Recent studies have shown that Doppler velocity projections extracted from CSI, which directly reflect human-motion velocity, enable more robust human activity recognition (HAR) and stronger generalization across users and unseen conditions. Nevertheless, reliable generalization under real-world variability remains a major challenge, hindering the adoption of Wi-Fi sensing in real-world applications. To address this challenge, we introduce Doppler Radiance Fields (DoRF), bringing the concept of neural radiance fields (NeRF) from computer vision into Wi-Fi sensing. DoRF models Doppler velocity projections extracted from Wi-Fi CSI as sparse and diverse virtual-camera views of human motion. It then infers a latent 3D motion sequence whose projections along learned effective Doppler directions explain the CSI-derived Doppler observations. The recovered motion is subsequently projected onto an equiangular grid of directions on the unit sphere, producing a spherical representation of the underlying motion. Since DoRF naturally defines the Doppler representation on spheres, we further introduce DoRF++, a spherical-learning design that applies spherical Transformers for activity classification. Experiments on our collected hand-gesture dataset show that DoRF++ significantly outperforms state-of-the-art Wi-Fi-based HAR methods in cross-user generalization accuracy, especially for difficult gestures in settings with a single multi-antenna receiver access point (AP).

Navid Hasanzadeh, Shahrokh Valaee

Department of Electrical & Computer Engineering, University of Toronto, Toronto, ON, Canada.

Wifi Doppler Velocity Log Human Activity Recognition Wireless Radiance Field Collaborative Sensing Spherical Latent Space

Aug 2, 2026cs.LG

Training nGPT

The normalized Transformer (nGPT) realizes hyperspherical representation learning by constraining model parameter vectors and activation vectors to the unit hypersphere. In this paper, we describe a practical training recipe for nGPT and evaluate it on modern hybrid Mamba-2--Transformer Mixture-of-Experts (MoE) models. The recipe introduces Logit Gradient Preconditioning, Logarithmic Learning Rate Decay, GatedAdamW, angular update control, and optional exploration mechanisms. Compared with an unnormalized model of the same hybrid MoE architecture trained with AdamW, the 30B-total-parameter nGPT model reaches the same validation loss using approximately half as many training tokens. The recipe scales across the models considered, which contain up to 30B total parameters.

Ilya Loshchilov, Boris Ginsburg

NVIDIA

Generative Pretrained Transformers Transformer Architectures Spherical Latent Space Topomamba

Jul 24, 2026cs.LG

Unbiased Open World Regularization for Fair Self-Supervised Learning

Despite recent advances, self-supervised learning (SSL) models and Joint-Embedding Predictive Architectures (JEPAs) remain susceptible to learning spurious biases in the dataset. These techniques rely on regularization, which prevents representation collapse by enforcing a global target distribution such as a multivariate Gaussian or a uniform distribution on the sphere. However, these global constraints are insufficient to prevent bias entanglement, as task-irrelevant features can still segregate the latent space into distinct sub-regions. While recent approaches like Entangling and Disentangling (EnD) and Fair Supervised Contrastive Learning (FSCL) empirically debias the latent space, we show that they act as partial approximations of conditional distribution matching. To enforce this matching explicitly, we propose Unbiased Open World Regularization (UOWReg), an encoder-only framework. We show that this shift from a global to a conditional objective guarantees statistical independence between the learned representations and the targeted attributes, regardless of the chosen target distribution. We empirically validate this framework across both Gaussian and spherical latent spaces, using statistical measures to enforce these target distributions. While conditional matching successfully mitigates bias with both distributions, we demonstrate that enforcing conditional uniformity on the sphere yields a lower linearprobing classification error. Empirically, UOWReg reduces Equalized Odds violations on the CelebA benchmark while maintaining competitive classification accuracy compared to existing encoder-only baselines. Furthermore, we introduce the Synthetic Engraving Task-a novel setting in which a dominant macro-structure masks a fine-grained micro-signature. We show that UOWReg effectively prevents the subpopulation collapse observed in standard SSL, successfully isolating micro-signatures even when heavily entangled with the global structure.

L{é}o Nicollier, Marc Pic, Pablo Mus{é} +2

CB, ATT · Université Paris-Saclay, CNRS, ENS Paris-Saclay, Centre Borelli, France · ATT +4

Self-Supervised Learning Regularization Spherical Latent Space Entanglement

Jul 23, 2026cs.LG

Data eccentricity, asymptotics of Gaussian RBF reproducing kernel Hilbert space, and kernel PCA

We show that, up to isotropic scaling, the Gaussian RBF reproducing kernel Hilbert space (RKHS) is asymptotically isometric to Euclidean space in the large bandwidth limit. This strongly suggests that kernel-based constructions reliant on metric properties of the RKHS will yield results for Gaussian RBF kernels that similarly approach those of linear kernels for large bandwidths. The asymptotic behavior of Gaussian CKA can be understood in this light. We further consider kernel PCA, showing that Gaussian RBF eigenvalues, eigenprojections, and principal components all converge to those of classical (linear) PCA as bandwidth

σ\rightarrow \infty

. For a given data representation, both the RKHS feature embeddings and the orthogonal PCA eigenframes of the two kernel types differ asymptotically by a geometric similarity transformation, up to a residual of size

O \left (\fracρσ \right )^2

, where

ρ

is a measure of geometric eccentricity of the representation, equal to the ratio of maximum to median pairwise distance between data examples. Experiments over a diverse collection of data sets demonstrate that

ρ

provides a simple and reliable predictor of dataset-specific convergence behavior in the top principal directions.

Sergio A. Alvarez

Department of Computer Science, Boston College, Chestnut Hill, MA 02467 USA

Principal Component Analysis Kernel Hilbert Spaces Radial Basis Function Spherical Latent Space Finite Time Analysis Near-Earth Objects

Jul 16, 2026cs.CV

Quantifying Training Membership Information in the Hyperspherical Embedding Geometry of Face Recognition Models

Face recognition models represent each face as an embedding vector on the unit hypersphere by clustering embeddings of the same identity while pushing different identities apart through angular-margin losses. Because these losses act only on training identities, non-member identities may form clusters with different geometric properties. In this paper, we quantify the magnitude of this difference and what training-time factors control it. We compute four statistics based on cluster geometry across 180 face recognition models in a factorial design over IResNet backbone size, loss head, training duration, and the number of training identities, and evaluate each configuration on nine benchmarks. Our results indicate that the number of training identities has the largest effect on member/non-member separability, while backbone and loss head contribute far less, and that, on a same-domain held-out reference, the geometric membership signal decreases monotonically as more identities are added to training. We provide an analysis of cross-domain (pose, age, quality, ethnicity) non-member benchmarks and report that these inflate the apparent membership signal. Finally, we fuse all four statistics with a learned classifier to reveal additional membership information beyond the best individual statistic.

Ünsal Öztürk, Sébastien Marcel

Idiap Research Institute, Martigny, Switzerland · Universit´e de Lausanne (UNIL), Lausanne, Switzerland

Face Recognition Model Spherical Latent Space Membership Inference Discriminative Models Cluster

Jul 15, 2026cs.LG

The Hyperspherical Geometry of CLIP Latent Space: A Semantic Mixture Model

Contrastive Language-Image Pretraining (CLIP) representations form a semantic embedding space governed by cosine similarity, reflecting an intrinsic hyperspherical geometry. However, existing probabilistic interpretations typically rely on Gaussian assumptions, which fail to capture this directional and multimodal structure. We propose a principled density model for the CLIP latent space based on Mixtures of von Mises-Fisher (MovMF) distributions defined on the unit hypersphere. Using the Expectation-Maximization (EM) algorithm, we efficiently learn a probabilistic model in which each mixture component corresponds to a coherent semantic concept. This formulation yields a closed-form likelihood naturally aligned with hyperspherical geometry, enabling accurate and interpretable density estimation. Empirically, our model significantly improves long-tailed and out-of-distribution detection and provides a natural semantic decomposition, representing each embedding as a sparse probabilistic combination of interpretable concepts. These results suggest that CLIP latent space is more faithfully characterized as a hyperspherical semantic mixture rather than an isotropic Gaussian, establishing a simple and geometrically consistent probabilistic framework for modeling and understanding multimodal representations. Project page is available at https://xiaoyuzhizi.github.io/movmf-clip/.

Zijie Yu, Gaowen Liu, Ramana Rao Kompella +2

Tsinghua University · Cisco Research · University of Illinois Chicago

Contrastive Language-Image Pre-Training Model Spherical Latent Space Multimodal Representations Hyperbolic Space Mixture Models Empirical Bayes

Jul 14, 2026cs.LG

Contrastive-Collapsed Loss for Flexible and Geometrically Optimal Embeddings and Faster Convergence

In this work, we introduce CoCo, a loss function aimed at learning normalized and well-structured representations. The proposed loss encourages intra-class collapse and inter-class contrast while preserving sufficient flexibility for neural networks to approximate geometrically optimal embeddings with large angular separation between classes. We provide a theoretical analysis positioning CoCo with respect to related objectives such as dot regression and cross-entropy, showing that the new proposed loss benefits from closer initialization to the optimal configuration, more informative gradients, and stronger incentives for class-wise representation collapse. Extensive experiments on diverse tabular datasets from the OpenML-CC18 benchmark show that CoCo achieves competitive performance with state-of-the-art methods, including kernel SVM, Random Forest, dot regression, and cross-entropy-based neural networks. In addition, both theoretical arguments and empirical analyses demonstrate that the proposal promotes tighter class clustering and faster convergence. These results highlight CoCo loss as an effective objective for learning discriminative representations while maintaining competitive predictive performance.

Blanca Cano-Camarero, Ángela Fernández-Pascual, José R. Dorronsoro

1Departamento de Ingeniería Informática, Universidad Autónoma de Madrid, Calle Francisco Tomás y Valiente, 11, 28049, Madrid, España

Spherical Latent Space Loss Function Discriminative Models Neural Network Convergence

Jul 6, 2026cs.CV

DeGenseGS: Geometrically and Semantically Decoupled Surgical Scene Understanding in 4D Gaussian Splatting

Real-time, text-promptable 4D reconstruction is indispensable for autonomous surgical interaction. Severe misalignment between semantic meaning and physical anatomy still persists, largely because existing solutions integrate Vision-Language Models into deformable fields via a rigid coupling scheme that tightly binds semantic features to geometric warping. In this paper, we propose DeGenseGS, Geometrically and Semantically Decoupled Surgical Scene Understanding in 4D Gaussian Splatting, a novel framework that independently models semantic evolution and geometric deformation. Specifically, we propose a HexPlane-based spatiotemporal entanglement module that uses shared kinematic latents to synchronize semantic mutations with scene dynamics, while explicitly disentangling semantic updates from geometric deformation. To further ensure robustness against reconstruction artifacts, we devise a Rasterization-Native Semantic Extraction mechanism that infers semantics from topologically continuous feature maps. Additionally, we incorporate an angular-aligned optimization strategy that conforms to the native hyperspherical latent space, thereby preventing semantic distortion. Extensive evaluations on the CholecSeg8k and EndoVis18 datasets demonstrate that DeGenseGS achieves state-of-the-art performance. Our framework yields enhanced geometric completeness and robust semantic-anatomic alignment, enabling spatially continuous segmentation despite drastic tissue deformation and topological transitions.

Yimo Wang, Bin Kang, Shuojue Yang +1

School of Automation, Southeast University, Nanjing 210096, China · Department of Biomedical Engineering, National University of Singapore, Singapore · School of Internet of Things, Nanjing University of Posts and Telecommunications, Nanjing, China

4D Reconstruction Deformable Cnn Gaussian Splatting Spherical Latent Space Scene Understanding

Jul 1, 2026eess.IV

Group-invariant Coresets for Data-efficient Active Learning

Active learning reduces labeling cost by querying the most informative unlabeled samples, but standard coreset methods ignore known data symmetries and can waste budget on transformed versions of the same instance. We propose GRINCO, a group-invariant coreset framework that performs acquisition in the quotient space induced by a transformation group, so that selection operates on orbits rather than raw samples. The method uses either canonical representatives or learned orbit-separating invariant embeddings to define practical quotient metrics, and combines quotient-space k-center selection with invariant training through an orbit-averaged loss. We further derive a generalization bound that relates excess orbit-averaged risk to quotient-space coverage, label uncertainty, and intra-orbit variability. Experiments on synthetic scale-invariant data and image benchmarks with rotation-induced redundancy show that GRINCO improves orbit coverage and achieves stronger label efficiency than conventional coreset baselines, especially when group-induced redundancy is substantial.

L. C. Ayres, J. C. M. Bermudez, S. J. M. de Almeida +1

Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil · Universidade Católica de Pelotas, Pelotas, RS, Brazil · Université de Lorraine, CNRS, CRAN, Vandoeuvre-lès-Nancy, France

Geometry-Aware Uncertainty Coresets Pool-Based Active Learning Active Learning Covariate Shift Spherical Latent Space

Jul 1, 2026cs.CV

HieDG: A Hierarchical Discrete Geometry-Guided Framework for Multi-Animal Tracking

Multi-animal tracking (MAT) is critical for wildlife monitoring and behavioral analysis, yet remains challenging due to uniform appearance, high density, and irregular motion. Existing methods typically follow heuristic- or query-based paradigms: the former relies on handcrafted geometric associations without end-to-end optimization, whereas the latter enables joint optimization but relies heavily on appearance embeddings. In such conditions, continuous geometric embeddings can be unstable, as small coordinate perturbations may disproportionately alter cross-frame attention weights, degrading identity association performance. To address this limitation, we propose HieDG, a Hierarchical Discrete Geometry-guided tracking framework that reformulates geometric dynamics as structured discrete representations within a query-based tracker. Instead of directly using raw geometric signals, HieDG employs a two-stage residual codebook to discretize position, scale, and velocity cues, transforming unstable continuous geometry into structured, stable discrete tokens. These tokens are aligned with visual embeddings and integrated into the tracking queries to enhance identity consistency. Extensive experiments on animal-specific benchmarks (AnimalTrack, BFT, and BuckTales) demonstrate state-of-the-art association performance with significant improvements in HOTA, AssA, and IDF1. Additional evaluations on generic multi-object tracking benchmarks, including DanceTrack and SportsMOT, show competitive performance, indicating the broader applicability of discretized geometric modeling beyond animal-specific scenarios.

Chenxun Deng, Zhongde Zhang, Ye Yuan +7

State Key Laboratory of Brain Cognition and Brain-inspired Intelligence Technology, Institute of Automation, Chinese Academy of Sciences · School of Artificial Intelligence, University of Chinese Academy of Sciences · School of Technology, Beijing Forestry University +6

Multi-Object Tracking Spherical Latent Space Hierarchical

Jul 1, 2026cs.CV

GenSP: Consistent Spherical Parameterization via Learning Shape Generative Models

We introduce GenSP, a data-driven framework that learns consistent spherical parameterizations across a collection of genus-0 shapes. Instead of optimizing the parameterization of each shape independently, our method learns a neural generative model that predicts a continuous mapping from the unit sphere to shapes in a dataset. Under this formulation, spherical parameterizations are obtained through the inverse mappings of the learned generator, which encourages similar shapes to share consistent parameterizations. To make this formulation practical, we address several key challenges in learning such a generative model. First, we introduce a continuous neural deformation model that predicts surface points from sphere coordinates and latent shape codes, avoiding discretization artifacts common in mesh-based formulations. Second, we augment the training space with intermediate shapes that bridge the sphere and input shapes, allowing the model to learn meaningful deformations across a heterogeneous shape collection. Third, we compute reliable initial correspondences by propagating mappings along a spanning tree of training shapes in the latent space. Experiments on the ShapeNet dataset demonstrate that our approach significantly reduces geometric distortion and improves cross-shape consistency compared with state-of-the-art spherical parameterization methods.

Sai Karthikey Pentapati, Shashank Gupta, Rajesh Sureddi +3

The University of Texas at Austin · University of Colorado Boulder

Spherical Latent Space Deformable Cnn Generative Models

Jun 17, 2026stat.ML

Kernel of Partition Paths: A Unified Representation for Tree Ensembles

A recent line of work has reframed individual decision trees as linear models on engineered features associated with their splits, opening routes for oracle inequalities and feature-importance reinterpretation, but leaving open the question of what unified geometric object a forest induces when one indexes its feature map by nodes rather than by splits. The present paper studies that object. KPP indexes the feature map by the nodes of the forest, weighted by a path metric that turns each coordinate into a component of a squared-Euclidean path-isometric embedding. KPP unifies four pillars under a single node-indexed representation whose Gram is non-diagonal and carries a metric: prediction, exact additive attribution, deterministic Lipschitz robust radius in the KPP metric, and uniform Rademacher risk bounds for regression and classification under fixed, honest, or cross-fit conditioning. All probabilistic guarantees are conditional on the representation and are stated under three explicit conditioning regimes; the robust-radius guarantee is deterministic in the KPP metric rather than in a norm on the raw input. Conjectured fast-rate refinements for both regression and classification are stated as open problems and are not claimed as theorems.

Nicolas Mahler

Datapred SAS 23 rue Mirabeau 94300 Vincennes, France · Datapred SA EPFL Innovation Park – Bâtiment A 1015 Lausanne, Switzerland

Tree Ensembles Decision Trees Kernel Method Partitioning Concentration Bounds Spherical Latent Space+3

Jun 16, 2026cs.LG

Expanding SPHERE-JEPA: A Family of Statistical Regularizers for the Hypersphere

In Self-Supervised Learning (SSL), preventing representation collapse by explicitly enforcing a uniform distribution on the unit hypersphere has proven to be effective. However, current frameworks typically rely on sliced statistical regularizers such as SIGReg (used in LeJEPA) and SUSReg (used in SPHERE-JEPA), which approximate this continuous objective via Monte Carlo sampling along random 1D directions. This stochasticity injects projection variance into the training gradients, destabilizing optimization, and hindering convergence. In this work, we first show that analytically integrating out these random projections natively yields a deterministic Maximum Mean Discrepancy (MMD), bypassing the variance of sliced methods. Motivated by this equivalence, we formulate full-dimensional objectives for MMD, Kernel Stein Discrepancy (KSD), and Kullback-Leibler (KL) divergence directly on the sphere to enforce a uniform distribution. To prevent spatial bias, we equip these tests with rotationally invariant kernels constructed via spectral theory, systematically evaluating two canonical families: smooth exponential decay (Heat) and strict frequency cutoff (Bandlimited) filters. Empirically, removing projection-induced noise results in more stable optimization, faster convergence, and consistent improvements over stochastic sliced regularizers on ImageNet and Galaxy10. Furthermore, we reveal that the choice of the statistical test shapes the geometry of the learned latent space: MMD and KSD favor locally clustered organization suitable for object-centric domains, whereas the continuous KDE-based KL divergence promotes fine-grained instance separation, yielding the strongest results on unclustered procedural texture retrieval.

Léo Nicollier, Enric Meinhardt-Llopis, Max Dunitz +3

CB, ATT · Université Paris-Saclay, CNRS, ENS Paris-Saclay, Centre Borelli, France · CB +4

Spherical Latent Space Regularization Self-Supervised Learning Hyperbolic Space Kullback-Leibler Divergence Covariance+1

Jun 1, 2026cs.CV

From Extrinsic to Intrinsic: Geodesic-Guided Representation Learning for 3D Geometric Data

Geometric analysis fundamentally distinguishes between \textit{extrinsic} and \textit{intrinsic} perspectives. The dominant paradigm in current 3D representation learning relies on either extrinsic spatial structures or high-level semantics, struggling to capture the essence of shape identity and underlying manifold topology. To bridge this gap, we introduce a novel 3D representation learning paradigm, namely \textbf{PRISM}, for \textbf{P}re-training, which learns isometric embeddings by \textbf{R}ecovering the \textbf{I}ntrinsic \textbf{S}urface geodesic \textbf{M}etric. PRISM incorporates a topology-enforcing objective that explicitly constrains the structure of latent space, alongside a specialized two-stage training recipe mitigating sample imbalance inherent in the distribution of geodesic distances. Experiments demonstrate that our approach shows satisfactory accuracy, robustness, and high efficiency in geodesic distance prediction and achieves superior performance across diverse downstream tasks, including shape recognition, surface parameterization, and non-rigid correspondence. The code will be publicly available at https://github.com/AidenZhao/PRISM.

Yuming Zhao, Junhui Hou, Qijian Zhang +2

Department of Computer Science, City University of Hong Kong, Hong Kong, China · Bambu Lab, Shenzhen, China · Meshy AI, California, USA +1

Geometric Machine Learning 3D Geometry Spherical Latent Space Representation Learning 3D Generation Prism

May 29, 2026cs.LG

Learning Hyperspherical Time-Frequency Representations for Time-Series Out-of-Distribution Detection

Out-of-distribution (OOD) detection for time-series data remains comparatively underexplored compared to vision and language, with a limited principled understanding of how supervised time-series representations can be leveraged for reliable detection under distributional shifts. This work formulates time-series OOD detection as representation learning with hyperspherical embeddings, where class-conditional structure is induced by a von Mises-Fisher (vMF) likelihood-based objective on the unit sphere. The learned representation combines time- and frequency-domain views of the input signal via domain-specific encoders, integrating them into a joint embedding space for OOD detection. Detection uses distance-based scores over the learned embeddings, including k-nearest neighbors (k-NN) and Mahalanobis scores. We evaluate the approach at scale on the complete UCR and UEA time-series archives under a cross-dataset protocol. Empirical results show consistent improvements under both k-NN and Mahalanobis scoring over strong contrastive learning and post-hoc baselines in the same setting. Code is available at https://github.com/tiiuae/hypertf-time-series-ood.

Willian T. Lunardi, Samridha Shrestha, Martin Andreoni

Technology Innovation Institute, Abu Dhabi, United Arab Emirates · Khalifa University, Abu Dhabi, United Arab Emirates

Out-Of-Distribution Detection Time-Series Classification Time-Frequency Spherical Latent Space Time Series Spatiotemporal Representations+1

May 26, 2026cs.LG

SPHERE-JEPA: Spherical Prediction with Homogeneous Embeddings

A fundamental open question in self-supervised learning (SSL) is the explicit characterization of the optimal geometry of the learned representations. Recently, LeJEPA identified isotropic Gaussian embeddings as optimal for minimizing downstream prediction risk in Euclidean spaces. However, the corresponding problem for distributions supported on lower-dimensional manifolds, such as the hypersphere, remains unexplored. In this work, we demonstrate that extending this minimax analysis to smooth distributions on Riemannian manifolds fundamentally changes the optimal solution. We show that, under a worst-case formulation, both k-nearest neighbors and kernel ridge regression induce hyperspherical uniformity. More precisely, we show that uniform distributions on manifolds are optimal for k-nearest neighbors, and that the uniform distribution on the sphere is optimal for kernel ridge regression with both the exponential dot-product kernel and the linear kernel. This theoretical insight reveals a fundamental limitation of Gaussian embeddings: their non-uniform density induces anisotropic k-NN neighborhoods, severely biasing the estimator. To correct this, we introduce SPHERE-JEPA, a theoretically grounded SSL framework. We adapt LeJEPA's Cram{é}r-Wold projection mechanism to enforce hyperspherical uniformity rather than a Gaussian prior. Empirically, SPHERE-JEPA yields significant improvements, boosting texture retrieval mAP by over 6%, while consistently matching or outperforming LeJEPA on standard benchmarks-including a +1.8% linear probing gain on ImageNet-1K (ViT-B/14).

Léo Nicollier, Max Dunitz, Marc Pic +3

CB, ATT · Université Paris-Saclay, CNRS, ENS Paris-Saclay, Centre Borelli · Advanced Track and Trace +4

Spherical Latent Space Sphere Approximate Nearest-Neighbor Search Hyperbolic Space Kernel Method S-Jepa

May 15, 2026cs.CV

Efficient Image Synthesis with Sphere Latent Encoder

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.

Tung Do, Thuan Hoang Nguyen, Hao Li

Mohamed Bin Zayed University of Artificial Intelligence, UAE

Multi-Reference Image Generation Few-Step Generation Latent Diffusion Model Latent Space Imagenet Spherical Latent Space

May 13, 2026cs.LG

A Unified Geometric Framework for Weighted Contrastive Learning

Contrastive learning (CL) aims to preserve relational structure between samples by learning representations that reflect a similarity graph. Yet, the geometry of the resulting embeddings remains poorly understood. Here we show that weighted InfoNCE objectives can be interpreted as Distance Geometry Problems, where the weighting scheme specifies the target geometry to be realized by the representation. This viewpoint yields exact characterizations of the optimal embeddings for several supervised and weakly supervised objectives. In supervised classification, both SupCon and Soft SupCon (a dense relaxation of it where pairs from distinct classes have small non-zero similarity) collapse samples within each class to a single prototype. However, while balanced SupCon recovers the classical regular simplex geometry, class imbalance breaks this symmetry: SupCon induces non-uniform inter-class similarities depending on class sizes, whereas Soft SupCon preserves a regular simplex geometry regardless of class imbalance. In continuous-label settings, our framework reveals a different failure mode: y-Aware CL generally cannot attain its entropic optimum unless the labels lie on a hypersphere, exposing a mismatch between Euclidean label weights and spherical latent similarity. By contrast, geometrically consistent choices such as Euclidean-Euclidean weighting or X-CLR admit unique optimal embeddings. Our results show that the choice of weighting scheme determines whether contrastive learning is geometrically realizable, degenerate, or inconsistent, providing a principled framework for designing contrastive objectives.

Raphael Vock, Edouard Duchesnay, Benoit Dufumier

GAIA Lab, NeuroSpin, CEA, CNRS · Université Paris-Saclay, Gif-sur-Yvette, France

Contrastive Learning Spherical Latent Space Simplex

May 11, 2026cs.LG

CORE: Cyclic Orthotope Relation Embedding for Knowledge Graph Completion

Knowledge graph completion (KGC) aims to automatically infer missing facts in multi-relational data by mapping entities and relations into continuous representation spaces. Recent region-based embedding models have shown great promise in capturing complex logical patterns by representing relations as geometric regions. However, these models inevitably suffer from absolute boundary constraints during optimization. Conversely, without such constraints, relation regions expand indefinitely. To address the limitation, we propose \textbf{CORE} (Cyclic Orthotope Relation Embedding), a novel KGC model that embeds entities and relations onto a boundary-less torus manifold.CORE represents relations as cyclic orthotopes on the torus manifold, allowing regions to seamlessly wrap around spatial boundaries to ensure smooth gradient conduction. Furthermore, an adaptive width regularization is introduced to prevent unconditional region expansion. Theoretical analysis proves that CORE can capture various complex relation patterns such as subsumption and intersection. Extensive experiments on four benchmark datasets demonstrate that CORE achieves highly competitive performance, significantly improving link prediction accuracy in dense semantic environments.

Yingqi Zeng, Luying Wang, Huiling Zhu

Knowledge Graph Completion Zero-Shot Link Prediction Relational Databases Spherical Latent Space Region

May 11, 2026cs.CV

Geometry-aware Prototype Learning for Cross-domain Few-shot Medical Image Segmentation

Cross-domain few-shot medical image segmentation (CD-FSMIS) requires a model to generalise simultaneously to novel anatomical categories and unseen imaging domains from only a handful of annotated examples. Existing prototypical approaches inevitably entangle anatomical structure with domain-specific appearance variations, and thus lack a stable reference for reliable matching under domain shift. We observe that the geometric structure of human anatomy constitutes a reliable, domain-transferable prior that has been overlooked. Building on this insight, we propose GeoProto, a geometry-aware CD-FSMIS framework that enriches prototypical matching with explicit structural priors. The core component, Geometry-Aware Prototype Enrichment (GAPE), augments each local appearance prototype with a learned geometric offset encoding its ordinal position within the organ's interior topology. This offset is derived from an auxiliary Ordinal Shape Branch (OSB) trained under an ordinally consistent objective that enforces monotonic variation of geometric embeddings across interior strata, requiring no annotation beyond standard segmentation masks. Extensive experiments across seven datasets spanning three evaluation settings (cross-modality, cross-sequence, and cross-context) demonstrate that GeoProto achieves state-of-the-art performance.

Feifan Song, Yuntian Bo, Haofeng Zhang

School of Computer Science and Engineering, Nanjing University of Science and Technology

Few-Shot Segmentation Semi-Supervised Medical Image Segmentation Learnable Prototypes Anatomy Spherical Latent Space

May 11, 2026cs.LG

On periodic distributed representations using Fourier embeddings

Periodic signals are critical for representing physical and perceptual phenomena. Scalar, real angular measures, e.g., radians and degrees, result in difficulty processing and distinguishing nearby angles, especially when their absolute difference exceeds pi. We can avoid this problem by using real-valued, periodic embeddings in high-dimensional space. These representations also allow us to control the nature of their dot product similarities, allowing us to construct a variety of different kernel shapes. In this work, we aim of highlight how these representations can be constructed and focus on the formalization of Dirichlet and periodic Gaussian kernels using the neurally-plausible representation scheme of Spatial Semantic Pointers.

Jakeb Chouinard

Department of Systems Design Engineering, University of Waterloo · Centre for Theoretical Neuroscience, University of Waterloo

Spherical Latent Space Kernel Method Fourier Semantic Geometry Similarity and Metrics

May 9, 2026cs.LG

The Wristband Gaussian Loss: Deterministic, Composable Latents via a Sphere-Interval Decomposition

We present the Wristband Gaussian Loss, a deterministic batch loss for Gaussianizing point embeddings without sampling, KL terms, or iterative transport. Each

x \in \mathbb{R}^d

is mapped to a direction

u = x/\|x\|

and a CDF-transformed radius

t = F_{χ^2_d}(\|x\|^2)

on the wristband

S^{d-1} \times [0,1]

. We prove (and machine-verify in Lean~4) that for

d \ge 2

the pushforward wristband map equals

σ_{d-1} \otimes \mathrm{Unif}[0,1]

iff the source is

\mathcal{N}(0, I_d)

, and that the Neumann-reflected wristband repulsion energy is uniquely minimized at the uniform target. We compute this reflected-kernel objective in two ways: a nearest three-image pairwise truncation at

O(N^2 d)

, and a spectral Neumann path joining angular and radial Mercer modes (spherical-harmonic and cosine) at

O(N d K)

, with empirically matched gradients. A 1D Wasserstein radial term and a moment penalty serve as finite-sample accelerators with the same optimum, and Monte-Carlo null calibration turns the components into a single standardized statistic. We evaluate direct point-cloud Gaussianization with a calibrated barycentric

W_2

score: a deterministic Gaussian reference batch is built by recursive Hungarian averaging, with each method reported as a

z

-score against same-size Gaussian batches. On the axis-uniform X benchmark, Wristband is competitive in 2D and gives the best 10D score. On a harder radial--angular-copula impostor whose Gaussian radial and angular marginals are correct but dependent, Wristband gives the best 10D and 128D scores. Coupled with learnable-key Euclidean attention and exact invertible flows, the resulting Deterministic Gaussian Autoencoder delivers a Gaussian-latent interface for counterfactual sampling with independent factors and a context/residual construction for dependent factors.

Mikhail Parakhin, André M. Carvalho, Patrick Haluptzok

Shopify

Wasserstein Distance Wrist Spherical Latent Space Loss Function Video Latents

May 8, 2026cs.LG

Neurally-plausible radial basis kernels using distributed Fourier embeddings

Coherent, continuous spatial representations are critical for synthesizing physical and perceptual phenomena into a single representational space. Radial basis kernels provide a path forward for this type of distributed representation. In this work, we aim to characterize and analyze common radial basis kernels realizable in the neurally-plausible framework of spatial semantic pointers. Further, we analyze previous radial basis kernel work based on grid cell-like representations and demonstrate that such representations are both capable of and optimal for realizing radial basis kernels.

Jakeb Chouinard

Department of Systems Design Engineering, University of Waterloo · Centre for Theoretical Neuroscience, University of Waterloo

Radial Basis Function Semantic Geometry Spherical Latent Space Fourier Spatiotemporal Representations Neuro-Symbolic Framework+1

May 8, 2026cs.LG

bispectrum: Selective $G$ -Bispectra Made Practical

Many machine learning tasks are invariant under the action of a group

G

of transformations: signal classification can be invariant under translations, image classification under 2D rotations, and spherical-image classification under 3D rotations. The

G

-bispectrum is a principled complete invariant of a signal (retaining all all signal's information up to the group action) with proven benefits in machine learning and as a pooling layer in deep networks. However, its deployment has been hampered by high computational cost and a patchwork of group-specific implementations. We present bispectrum, an open-source, fully unit-tested PyTorch library that implements selective

G

-bispectra for seven different group actions, as differentiable modules that can be directly incorporated into machine learning pipelines and deep learning architectures. For finite groups

G

, selectivity reduces the computational cost from

O(|G|^2)

to

O(|G|)

. For planar rotations, we leverage the disk bispectrum. For spherical 3D rotations, we introduce an augmented selective bispectrum at band-limit

L

which reduces the cost from

O(L^3)

to

Θ(L^2)

coefficients. We profile the entire library (for which we implemented various compute optimizations), showing that it delivers near-exact

G

-invariance with its selective

G

-bispectra computed in sub-millisecond time on GPU (up to commonly used bandlimits). We evaluate the benefits of incorporating

G

-bispectra as pooling layers into deep learning architectures on three classical benchmark datasets --comparing against norm pooling, gated pooling, Fourier-ELU pooling, max pooling, and (non-equivariant) data-augmented convolutional baselines. Results show that

G

-bispectra consistently outperform alternatives in the low-data, moderate-capacity regime.

Johan Mathe, Adele Myers, Simon Mataigne +1

Atmo, Inc. San Francisco, CA · UC Santa Barbara Santa Barbara, CA · UCLouvain Louvain-la-Neuve, Belgium

Global Average Pooling Spherical Latent Space Spectral Deep Learning Selectivity Selective

May 7, 2026stat.ML

Gaussian mixture models in Hilbert spaces via kernel methods

Modern datasets across many disciplines increasingly consist of time-evolving, potentially infinite-dimensional random objects, such as dynamic functional data, which are naturally modeled in Hilbert spaces. In these settings, characterizing probability measures, for example, through densities, can be ill-defined or technically challenging. Motivated by clustering applications, we propose a Gaussian mixture framework for Hilbert-space-valued data based on kernel mean embeddings and develop efficient optimization algorithms for estimation. We establish theoretical guarantees showing that the proposed algorithm is well defined and that the model yields a dense class of approximations in infinite-dimensional spaces. We evaluate the framework through extensive experiments on diverse structures and data geometries, including

L^2

-functional data and random graphs in Laplacian spaces arising in modern medical applications.

Daniel López-Montero, Antonio Álvarez-López, Marcos Matabuena

Friedrich-Alexander-Universität Erlangen-Nürnberg · Universidad Autónoma de Madrid · Mohamed bin Zayed University of Artificial Intelligence

Kernel Hilbert Spaces Gaussian Mixture Models Kernel Method Spherical Latent Space Probability Measures Graph Laplacians

May 7, 2026cs.LG

Uncertainty Estimation via Hyperspherical Confidence Mapping

Quantifying uncertainty in neural network predictions is essential for high-stakes domains such as autonomous driving, healthcare, and manufacturing. While existing approaches often depend on costly sampling or restrictive distributional assumptions, we propose Hyperspherical Confidence Mapping (HCM), a simple yet principled framework for sampling-free and distribution-free uncertainty estimation. HCM decomposes outputs into a magnitude and a normalized direction vector constrained to lie on the unit hypersphere, enabling a novel interpretation of uncertainty as the degree of violation of this geometric constraint. This yields deterministic and interpretable estimates applicable to both regression and classification. Experiments across diverse benchmarks and real-world industrial tasks demonstrate that HCM matches or surpasses ensemble and evidential approaches, with far lower inference cost and stronger confidence-error alignment. Our results highlight the power of geometric structure in uncertainty estimation and position HCM as a versatile alternative to conventional techniques.

Eunseo Choi, Ho-Yeon Kim, Jaewon Lee +3

KAIST · Samsung Electronic Co., Ltd

Uncertainty Bayesian Neural Networks Spherical Latent Space

May 3, 2026cs.LG

DR-SNE: Density-Regularized Stochastic Neighbor Embedding

Dimensionality reduction methods such as t-SNE are designed to preserve local neighborhood structure but do not explicitly account for how probability mass is distributed, often leading to distortions of data density. We reformulate dimensionality reduction as the joint alignment of two components: (i) conditional structure, capturing local relationships, and (ii) relative density structure, captured via local density statistics. Based on this perspective, we introduce Density-Regularized SNE (DR-SNE), which augments the stochastic neighbor embedding objective with a density regularization term derived from normalized log-density estimates. Unlike prior approaches such as DensMAP and DenSNE, which rely on local scale consistency, DR-SNE directly aligns normalized density estimates, providing a simple and scale-invariant mechanism for preserving relative density variations. Empirically, DR-SNE improves density preservation while maintaining competitive neighborhood fidelity, and yields gains on density-sensitive tasks such as anomaly detection across multiple datasets. These results suggest that incorporating density information complements geometry-focused objectives in dimensionality reduction.

Maksim Kazanskii

Independent Researcher

Dimensionality Reduction Spherical Latent Space Regularization Neighbors

May 2, 2026cs.DS

New Bounds for Kernel Sums via Fast Spherical Embeddings

We study query time bounds for the fundamental problem of estimating the kernel mean

\frac1{|X|}\sum_{x\in X}\mathbf{k}(x,y)

of a query

y

in a finite dataset

X\subset\mathbb{R}^d

up to a prescribed additive error

\varepsilon

. The best known bounds for the Gaussian kernel are

O(d/\varepsilon^2)

,

\widetilde O(d+1/\varepsilon^4)

, and

\widetilde O(d+Δ^2/\varepsilon^2)

, where

Δ

is the diameter of a region containing the points. We prove the new bound

\tilde O(d+\varepsilonΔ^2+1/\varepsilon^3)

, which improves over the previous ones in regimes with small error

\varepsilon

and intermediate diameter

Δ

. At the center of our proof is a new fast spherical embedding theorem in the sense introduced by Bartal, Recht and Schulman (2011), which limits the embedded data diameter while preserving local Euclidean distances and avoiding ``distance collapse'' at larger scales. This fast embedding theorem may be of independent interest.

Tal Wagner

Tel Aviv University

Kernel Method Spherical Latent Space Optimal Sample Complexity Distance Cuts

Apr 30, 2026cs.CV

Beyond Gaussian Bottlenecks: Topologically Aligned Encoding of Vision-Transformer Feature Spaces

Modern visual world modeling systems increasingly rely on high-capacity architectures and large-scale data to produce plausible motion, yet they often fail to preserve underlying 3D geometry or physically consistent camera dynamics. A key limitation lies not only in model capacity, but in the latent representations used to encode geometric structure. We propose S

^2

VAE, a geometry-first latent learning framework that focuses on compressing and representing the latent 3D state of a scene, including camera motion, depth, and point-level structure, rather than modeling appearance alone. Building on representations from a Visual Geometry Grounded Transformer (VGGT), we introduce a novel type of variational autoencoder using a product of Power Spherical latent distributions, explicitly enforcing hyperspherical structure in the bottleneck to preserve directional and geometric semantics under strong compression. Across depth estimation, camera pose recovery, and point cloud reconstruction, we show that geometry-aligned hyperspherical latents consistently outperform conventional Gaussian bottlenecks, particularly in high-compression regimes. Our results highlight latent geometry as a first-class design choice for physically grounded visual and world models.

Andrew Bond, Ilkin Umut Melanlioglu, Erkut Erdem +1

Department of Computer Engineering, Kocı University, Istanbul, Turkey · KUIS AI Research Center, Istanbul, Turkey · Department of Electrical and Electronics Engineering, Kocı University, Istanbul, Turkey +1

Visual Geometry Grounded Transformer Discriminative Congruence Transform Spherical Latent Space Passage Encoding Bottlenecks

Apr 30, 2026cs.LG

A Unified Framework of Hyperbolic Graph Representation Learning Methods

Hyperbolic geometry has emerged as an effective latent space for representing complex networks, owing to its ability to capture hierarchical organization and heterogeneous connectivity patterns using low-dimensional embeddings. As a result, numerous hyperbolic graph representation learning methods have been proposed in recent years. However, their practical adoption and systematic comparison remain challenging, as implementations are fragmented and shared tools for reproducible and fair evaluation are lacking. In this work, we introduce a unified open-source framework for hyperbolic graph representation learning that integrates several widely used embedding methods under a common optimization interface. The novel framework enables consistent training, visualization, and evaluation of hyperbolic embeddings, and interfaces seamlessly with standard network analysis tools. Leveraging this unified setup, we conduct an experimental study of hyperbolic embedding methods on real-world networks, focusing on two canonical downstream tasks: link prediction and node classification. Beyond predictive accuracy, the study offers practical insights into the strengths and limitations of existing approaches, thereby facilitating informed method selection and fostering reproducible research in hyperbolic graph representation learning.

Sofía Pérez Casulo, Marcelo Fiori, Bernardo Marenco +1

Facultad de Ingenier´ıa, Universidad de la Rep´ublica, Uruguay · Centro Interdisciplinario en Ciencia de Datos y Aprendizaje Autom´atico (CICADA), Universidad de la Rep´ublica, Uruguay

Hyperbolic Learning Spherical Latent Space Graph Representations Representation Learning Unified Framework

Apr 30, 2026cs.LG

Hyperspherical Forward-Forward with Prototypical Representations

The Forward-Forward (FF) algorithm presents a compelling, bio-inspired alternative to backpropagation. However, while efficient in training, it has a computationally prohibitive inference process that requires a separate forward pass for every class that is evaluated. In this work, we introduce the Hyperspherical Forward-Forward (HFF), a novel reformulation that resolves this critical bottleneck. Our core innovation is to reframe the local objective of each layer from a binary goodness-of-fit task to a direct multi-class classification problem within a hyperspherical feature space. We achieve this by learning a set of class-specific, unit-norm prototypes that act as geometric anchors and implicit negatives. This architectural innovation preserves the benefits of local training while enabling weight update and inference in a single forward pass, making it >40x faster than the original FF algorithm. Our method is simple to implement, scales effectively to modern convolutional architectures, and achieves superior accuracy on standard image classification benchmarks, closing the gap with backpropagation. Most notably, we are among the first greedy local-learning methods to report over 25% top-1 accuracy on ImageNet-1k, and 65.96% with transfer learning.

Shalini Sarode, Brian Moser, Joachim Folz +4

German Research Center for Artificial Intelligence (DFKI) · RPTU Kaiserslautern-Landau

Forward Propagation Backpropagation Convolutional Neural Networks Spherical Latent Space Hyperbolic Space Transfer Learning

Apr 26, 2026cs.LG

Gromov-Wasserstein Methods for Multi-View Relational Embedding and Clustering

Learning low-dimensional representations from multi-view relational data is challenging when underlying geometries differ across views. We propose Bary-GWMDS, a Gromov-Wasserstein-based method that operates directly on distance matrices to learn a consensus embedding preserving shared relational structure. By leveraging intrinsic distances, the approach naturally handles nonlinear distortions across views. We also introduce Mean-GWMDS-C, a clustering-oriented formulation that averages distance matrices and learns reduced-support representations via a consensus Gromov-Wasserstein transport. Experiments on synthetic and real-world datasets show that the proposed framework yields stable and geometrically meaningful embeddings.

Rafael Pereira Eufrazio, Eduardo Fernandes Montesuma, Charles Casimiro Cavalcante

Instituto Federal de Educação, Ciência e Tecnologia do Ceará, Canindé, Ceará, Brazil · Federal University of Ceara, Fortaleza, Ceará, Brazil · Sigma Nova, Paris, France

Gromov--Wasserstein Multi-View Spherical Latent Space Clustering Distance

Apr 25, 2026cs.CV

Sphere-Depth: A Benchmark for Depth Estimation Methods with Varying Spherical Camera Orientations

Reliable depth estimation from spherical images is crucial for 360° vision in robotic navigation and immersive scene understanding. However, the onboard spherical camera can experience unintentional pose variations in real-world robotic platforms that, along with the geometric distortions inherent in equirectangular projections, significantly impact the effectiveness of depth estimation. To study this issue, a novel public benchmark, called Sphere-Depth, is introduced to systematically evaluate the robustness of monocular depth estimation models from equirectangular images in a reproducible way. Camera pose perturbations are simulated and used to assess the performance of a popular perspective-based model, Depth Anything, and of spherical-aware models such as Depth Anywhere, ACDNet, Bifuse++, and SliceNet. Furthermore, to ensure meaningful evaluation across models, a depth calibration-based error protocol is proposed to convert predicted relative depth values into metric depth values using supervised learned scaling factors for each model. Experiments show that even models explicitly designed to process spherical images exhibit substantial performance degradation when variations in the camera pose are observed with respect to the canonical pose. The full benchmark, evaluation protocol, and dataset splits are made publicly available at: https://github.com/sgazzeh/Sphere_depth

Soulayma Gazzeh, Giuseppe Mazzola, Liliana Lo Presti +1

Department of Engineering, University of Palermo, Palermo, Italy

Depth Estimation Spherical Latent Space

Jun 26, 2025stat.ML

Spherical Cauchy Variational Autoencoders: Heavy Angular Tails and Exact KL Evaluation

Heavy-tailed posteriors are routine in Euclidean variational autoencoders, where the Student family relaxes the Gaussian without new machinery. The sphere has had no comparable option. Von Mises-Fisher distribution needs modified Bessel functions and a rejection sampler, and Power Spherical buys its closed forms by forcing the density to vanish at the antipode. We develop the spherical Cauchy distribution as a hyperspherical posterior that needs neither compromise. Stereographic projection carries it to a multivariate Student law, and a Möbius transformation turns a uniform spherical draw into an exact posterior sample from inner products, norms, and scalar arithmetic. The same transformation settles the regularizer. Evaluating the density along the sampling map reduces the Kullback-Leibler (KL) divergence to the uniform prior to a scalar expectation whose expansion terminates in every even ambient dimension, leaving one logarithm and a polynomial with finitely many correction terms. Odd dimensions admit certified truncation of value and gradient, the KL is increasing and convex in concentration, and the same function gives the pairwise KL. At matched modal curvature it has broader angular tails and a smaller KL penalty than both alternatives, so equal local precision costs less regularization. In dimension 128 the fused evaluator runs 1.5 times faster per latent-layer step than Power Spherical and 4.2 times faster than robust von Mises-Fisher on CPU, with factors of 1.6 and 5.4 on CUDA. Across five paired seeds it attains the lowest MNIST reconstruction loss at every tested dimension and lowers held-out viewpoint-gap negative log-likelihood on smallNORB by 3.6 percent.

Lukas Sablica, Kurt Hornik

Institute for Statistics and Mathematics Vienna University of Economics and Business Austria

Spherical Latent Space Conditional Variational Autoencoder Kullback-Leibler Divergence Posterior Sampling Long-Tailed Distribution

Jan 15, 2025cs.CV

Human Pose-Constrained UV Map Estimation

UV map estimation is used in computer vision for detailed analysis of human posture or activity. Previous methods assign pixels to body model vertices by comparing pixel descriptors independently, without enforcing global coherence or plausibility in the UV map. We propose Pose-Constrained Continuous Surface Embeddings (PC-CSE), which integrates estimated 2D human pose into the pixel-to-vertex assignment process. The pose provides global anatomical constraints, ensuring that UV maps remain coherent while preserving local precision. Evaluation on DensePose COCO demonstrates consistent improvement, regardless of the chosen 2D human pose model. Whole-body poses offer better constraints by incorporating additional details about the hands and feet. Conditioning UV maps with human pose reduces invalid mappings and enhances anatomical plausibility. In addition, we highlight inconsistencies in the ground-truth annotations.

Matej Suchanek, Miroslav Purkrabek, Jiri Matas

Visual Recognition Group Department of Cybernetics Faculty of Electrical Engineering Czech Technical University in Prague

Extreme Ultraviolet Body Spherical Latent Space

Jun 10, 2022cs.LG

Anisotropic View Distance Metric for High-Dimensional Data: Theory, Geometry, and Fast Computation

K-Means clustering algorithm is one of the most commonly used clustering algorithms because of its simplicity and efficiency. K-Means clustering algorithm based on Euclidean distance only pays attention to the linear distance between Euclidean distance is an efficient and interpretable similarity measurement, but its effectiveness may deteriorate in sample spaces with anisotropic structures, redundant features, or complex feature interactions. In this paper, we propose a novel distance metric called View distance. Inspired by orthographic projection, the proposed metric projects the sample space onto

n(n-1)/2

two-dimensional planes and defines the final distance as the sum of the Euclidean distances across the projected planes. Theoretical derivations verify that the View distance strictly satisfies the metric axioms and norm constraints. Beyond that, the View distance achieves feature coupling through projection and not only enables constant features to indirectly participate in distance calculation, but also suppresses interference from redundant features while exhibiting anisotropic geometric properties. Furthermore, to address the high computational complexity and poor scalability of full-projection View distance, we propose a two-dimensional projection plane selection strategy based on iterative Maximum Weight Matching, which reduces the computational complexity of distance calculation from

\mathcal{O}(n^2)

to

\mathcal{O}(k)

. Extensive experiments on 12 diverse datasets demonstrate that View distance and the deterministic selection strategy provide competitive or superior performance compared with Euclidean distance and other

L_{p}

metrics while maintaining strong interpretability and computational efficiency. The View distance provides a new perspective and option for similarity measurements.

Yiqun Zhang, Hou-biao Li

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

K-Means Distance High-Dimensional Spherical Latent Space Maximum Mean Discrepancy Anisotropic+1