Riemannian Representation Learning

Latest papers

13 papers5 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Oct 29, 2026

ACAM-J-MARS: Multi-Atlas Riemannian Spectral Representations for Decoding Advanced Meditation States from fMRI

Inferring jhana-type advanced concentration absorption meditation (ACAM-J) states from fMRI data is challenging due to the spatially distributed nature of meditation-related effects, substantial inter-individual variability that can exceed within-subject state differences, and limited sample sizes. Conventional machine-learning approaches often formulate this problem as unimodal, Euclidean, atlas-based classification, thereby neglecting the non-Euclidean geometry of functional connectivity matrices and underrepresenting temporal information in functional magnetic resonance imaging (fMRI); these limitations can lead to suboptimal classification performance. We propose ACAM-J-MARS (Multi-Atlas Riemannian Spectral representation), which computes functional connectivity using multiple atlases, represents symmetric positive definite (SPD) connectivity matrices on a Riemannian manifold, projects them into a log-Euclidean tangent space, and augments connectivity-derived features with low-frequency BOLD spectral energy to better resolve subtle state differences under heterogeneous variability. Evaluated on 204 segments from 22 practitioners, ACAM-J-MARS improves performance across multiple backbone models, achieving a balanced accuracy of 0.757 for early versus late ACAM-J classification and 0.545 for three-class classification, corresponding to a 10.8-point absolute improvement over the strongest baseline. These findings suggest that the proposed representation improves the decoding of advanced meditative brain states and may enable downstream applications such as closed-loop neurofeedback and real-time psychophysiological brain-computer interfaces.

Jian Wang, Winson F. Z. Yang, Wugeng Zheng +1

Meditation Research Program, Psychiatry, MGH and Harvard Medical School, Boston, MA, USA · Martinos Center for Biomedical Imaging, Radiology, MGH and Harvard Medical School, Boston, MA, USA · College of Engineering, Northeastern University, Boston, MA, USA

Riemannian Representation Learning Neural Decoding fMRI Representation Learning Neuroimaging Functional Connectivity Geometric Representation Learning

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Oct 27, 2026

Euclidean Crowding-Driven Adaptive Euclidean-Hyperbolic Graph Learning

Hyperbolic graph representation learning has emerged as a powerful paradigm for modeling hierarchical data. Unlike Euclidean space, which exhibits polynomial volume growth, hyperbolic space grows exponentially with radius, naturally accommodating tree-like structures without the geometric distortion known as Euclidean crowding. Despite strong empirical performance, the theoretical mechanisms underlying hyperbolic embeddings remain poorly understood. In this work, we provide a geometric quantifcation research for Euclidean crowding, a generalization-theoretic explanation for the superiority of hyperbolic space on hierarchical data, and further develop an adaptive Euclidean-hyperbolic fusion learning framework toward bridging this gap. Specifically, we first establish a geometric quantification of Euclidean crowding by analyzing layer-wise minimum separation in node embeddings, proving that fixed-dimensional Euclidean embeddings of tree-like graphs inevitably exhibit layer-wise crowding that leads to weaker margin-based generalization bounds. Second, we provide a generalization-bound for the superiority of hyperbolic geometry on hierarchical data. Third, we propose EH-Fusion, a geometry-adaptive framework that lifts Euclidean GNN representations into hyperbolic space via exponential mapping and adaptively fuses both geometries based on node-level neighborhood expansion patterns. Extensive experiments validate our theoretical insights and demonstrate the effectiveness of EH-Fusion on benchmark datasets.

Hao Chen, Jian Sun, Chuan Shi +1

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China · Beijing Key Lab of Intelligent Telecommunications Software and Multimedia, Beijing University of Posts and Telecommunications, Beijing 100876, China

Riemannian Representation Learning Graph Representation Learning Hyperbolic Representation Learning Geometric Representation Learning Hierarchical Representation Learning

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Oct 5, 2026

Constant-Curvature Sliced Gromov-Wasserstein for Heterogeneous Cross-Curvature Alignment

Recent advances in representation learning have highlighted the utility of constant-curvature models, such as hyperbolic and spherical spaces, for modeling complex data. Mixed-curvature models further enhance this by integrating multiple constant-curvature components. However, these models typically learn each component space independently because spaces with different curvatures are inherently heterogeneous and lack a unified metric. Consequently, they lack explicit mechanisms to enforce geometric consistency across various spaces. Moreover, the problem of comparing probability distributions across mixed-curvature spaces remains unexplored. To compare distributions on heterogeneous spaces, Gromov-Wasserstein (GW) distances provide a principled framework by aligning their intra-space geometries. Building on this, we propose constant-curvature sliced Gromov-Wasserstein (CCSGW), a novel divergence for aligning distributions supported on heterogeneous constant-curvature spaces. We first introduce the missing geodesic-based one-dimensional projections for spherical spaces, and then extend sliced GW to constant-curvature spaces, enabling efficient and principled comparison across manifolds with different curvatures. This formulation preserves intrinsic geometric relationships while avoiding the high computational cost. We provide theoretical analysis showing that CCSGW controls intrinsic geometric discrepancy across heterogeneous spaces, promoting distribution-level geometric consistency. By integrating CCSGW into existing mixed-curvature learning tasks, including graph anomaly detection, graph node classification, and multimodal learning, we observe consistent performance gains across diverse settings.

Shanglin Li, Wenjing Lu, Muyang Li +2

BIFOLD, Germany · Technical University of Berlin, Germany · RIKEN AIP, Japan +4

Distribution Matching Geometric Representation Learning Optimal Transport Riemannian Representation Learning

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Oct 5, 2026

The Implicit Bias of Hyperbolic Representation Learning for Multiclass Data: A Busemann Risk Perspective

We study the implicit bias of Riemannian gradient flow for hyperbolic multiclass classification with fixed class prototypes in hyperbolic space

\mathbb{H}^n

. Our framework accommodates general permutation invariant relative margin (PERM) losses, a class that includes cross entropy and other standard multiclass losses. Our analysis is based on a decomposition: at large radius, the distance to each prototype splits into a radial term and a direction-dependent term described by the Busemann function. This yields two main results. First, we prove a radial dichotomy: the sign of a drift coefficient

μ

determines whether the radius is pushed toward the ideal boundary or back toward the interior; if the positive drift persists, then

r(t)=\frac{1}{2}\log t+O(1)

, while persistent negative drift returns the trajectory to the large-radius threshold in finite time. Second, we show that the boundary direction converges to a critical point of the Busemann risk on

\partial\mathbb{H}^n

. These results provide a rigorous asymptotic perspective on two phenomena we refer to as boundary saturation and near-boundary clustering in hyperbolic representation learning.

Xingrun Li, Sho Kuno, Yusuke Mukuta +2

The University of Tokyo · RIKEN

Riemannian Optimization Implicit Bias Multiclass Classification Hyperbolic Representation Learning Riemannian Representation Learning

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Oct 5, 2026

On the Geometry of Multimodal Saturation: Riemannian VICReg

In self-supervised learning, a third modality should improve, or at least preserve, performance. Across nine image-text-tabular datasets, we show that it instead harms performance: the trimodal model underperforms its own best bimodal subset in 55.6% of paired runs under VICReg. The same failure occurs in 51.1% of paired runs under SimSiam. We call this failure multimodal saturation. We propose that the failure lies in the alignment geometry. Riemannian VICReg (R-VICReg) generalizes classical VICReg: it aligns views by squared geodesic distance on learnable negative-curvature product factors and recovers VICReg exactly as curvature vanishes. Over the same 45 paired runs, R-VICReg raises the probability that the third modality helps from 44.4% to 64.4%, with gains concentrated where VICReg saturates.

Nessim Ben Abbes, Duc Han Le, Sabri Mtibaa +1

Protectline · Télécom Paris

Self-Supervised Learning Geometric Representation Learning Hyperbolic Representation Learning Multimodal Learning Riemannian Representation Learning Multimodal SSL

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Sep 29, 2026

Minkowski Attractor Networks: Closed-Form Hyperbolic Flows for Visual Representations

Geometric representation learning predominantly scaffolds representations onto flat Euclidean subspaces or compact product tori (

\mathbb{T}^K

). However, flat manifolds possess vanishing curvature and polynomial volume growth, inherently suffering from metric distortion when embedding multi-scale, tree-like visual hierarchies. While hyperbolic spaces (

\mathbb{H}^m

) circumvent this via constant negative curvature (

K<0

) and exponential volume expansion, prior hyperbolic deep architectures are hindered by computationally cumbersome Riemannian optimization, non-linear gyrovector calculus, and floating-point instabilities. In this work, we introduce Minkowski Attractor Networks (MAN), an operator-splitting-inspired framework that embeds representations within pseudo-Riemannian Minkowski spacetime (

\mathbb{R}^{1,m}

). By framing hyperbolic manifolds as quadric level sets, MAN resolves hyperbolic geometry by combining linear Lorentz group transport with non-linear cone lifting and closed-form radial rescaling, evaluating in a single forward pass without numerical ODE solvers or iterative retractions. We establish MAN-2D (

\mathbb{R}^{1,1} \to \mathbb{H}^1

) as our primary, high-throughput visual backbone, which maximizes channel factorization granularity into

D/2

independent two-dimensional Minkowski blocks. We further formulate MAN-4D (

\mathbb{R}^{1,3} \to \mathbb{H}^3

) as a spacetime extension, leveraging a commuting Cartan-subalgebra parameterization of

\mathrm{SO}^+(1,3)

to evaluate 4D Lorentz isometries via two commuting 2D planar maps without matrix-exponential overhead.

Zhongping Ji

Visual Representation Learning Hierarchical Representation Learning Geometric Representation Learning Hyperbolic Representation Learning Riemannian Representation Learning

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Sep 28, 2026

Building Transformation Layers for Riemannian Neural Networks

Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications. One recent focus is the generalization of Euclidean fully connected (FC) and convolutional layers to non-Euclidean geometries. However, previous approaches typically focus on a few selected manifolds and rely on specific properties of the target manifold. In contrast, this work proposes a framework for constructing FC and convolutional layers over computationally tractable Riemannian spaces. This framework incorporates several previous FC layers across different geometries as special cases and is instantiated on ten representative manifolds, including three hyperbolic models, five geometries of the symmetric positive definite (SPD) manifold, and two Grassmannian perspectives. Experiments on different manifolds demonstrate the effectiveness and applicability of our approach. Code can be found at https://github.com/GitZH-Chen/RieTrans.

Ziheng Chen

University of Trento · MPI for Intelligent Systems, Tübingen

Riemannian Neural Networks Geometric Representation Learning Riemannian Representation Learning

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Sep 14, 2026

$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition

Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the

\mathbb{SL}(n)

space, a representation geometry defined by the simple

\det(A)=1

constraint and a left invariant Schatten-

p

Finsler structure. Despite this minimal construction,

\mathbb{SL}(n)

exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically,

\mathbb{SL}(n)

consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by

44.3\%

on KEGG and

40.5\%

on HumanCyc, and improves Hits@20 by

42.8\%

on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation space.

Xingrun Li, Yusuke Mukuta, Xin Yang +2

The University of Tokyo · Stanford University

Finsler Geometry Graph Representation Learning Geometric Representation Learning Lie Group Methods Riemannian Representation Learning Latent Space Geometry

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Aug 6, 2026

Dynamic Graph Prompting via Topology-Routed Mixed-Curvature Experts

Dynamic graph prompting freezes a pre-trained temporal backbone and adapts it to label-scarce downstream tasks using lightweight prompts. However, existing methods operate within a single, fixed embedding space. In this work, we reveal that temporal shifts in local clustering and degree heterogeneity actively reorganize the edge curvature spectrum---indicating that the optimal representation geometry dynamically evolves with local topology over time. We formalize this unaddressed mismatch as geometry under-adaptation. To overcome this limitation, we propose CurvPrompt, a topology-routed geometry prompting framework for dynamic graphs. Instead of relying on a single space, CurvPrompt maintains a bank of curvature-diverse Riemannian experts, each paired with a learnable prompt. A topology-aware gate dynamically routes each node--time instance to a sparse subset of experts, constructing a personalized mixed-curvature representation. To ensure parameter efficiency and training stability under extreme label scarcity, CurvPrompt employs soft routing during pre-training to build a continuous topology--geometry mapping, and transitions to hard Top-K routing with uniform weights during downstream adaptation. Extensive experiments across four benchmark datasets show that CurvPrompt significantly advances few-shot link prediction while delivering strong, consistent performance on node classification tasks, validating the necessity of geometry-adaptive prompting.

Quanxin Wang, Xuanting Xie, Bingheng Li +4

University of Electronic Science and Technology of China · 2Michigan State University · 3Singapore Management University +2

Link Prediction Few-Shot Learning Geometric Representation Learning Dynamic Graph Learning Prompt Engineering Riemannian Representation Learning

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Jul 21, 2026

Riemannian Deep Learning: Modules, Networks, and Geometries

Deep neural networks on manifold-valued representations have attracted growing interest, but many basic components remain tied to specific manifolds, rely on Euclidean approximations, or require costly and numerically fragile geometric operations. This thesis develops a unified framework for Riemannian deep learning from three complementary perspectives: reusable neural modules, manifold-specific network architectures, and the design of underlying geometries. It generalizes batch normalization from Euclidean spaces and individual manifolds to broad classes of Lie groups and gyrogroups, and extends multinomial logistic regression from Euclidean space to SPD manifolds and then to general Riemannian manifolds. It further develops neural networks for several important geometric representations, including an unconstrained model of hyperbolic space, Busemann-based hyperbolic learning, and full-rank correlation matrices. Finally, it introduces adaptive and computationally efficient Riemannian metrics on SPD manifolds, including learnable Log-Euclidean geometries and fast, stable Cholesky-based geometries. The proposed methods are supported by theoretical analysis and validated through numerical experiments and applications in vision, signal processing, graph learning, and genomics.

Ziheng Chen

Doctoral School in Information and Communication Technology

Riemannian Neural Networks Geometric Representation Learning Riemannian Metric Learning Hyperbolic Representation Learning Riemannian Representation Learning

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Jul 1, 2026

I\textsuperscript{2}RiMA: Spectral Riemannian Representation with Temporal Attention for Mental Stress Detection based on EEG Signals

Cross-subject EEG stress detection remains challenging because discriminative stress-related patterns are both subject-dependent and frequency-specific. Conventional Riemannian methods model spatial covariance mainly in the time domain, overlooking neural oscillations that are critical for high-level cognitive state decoding, while standard temporal tokenization often fragments inter-slice temporal coherence. To address these limitations, we propose \method{}, an Intra-Inter Riemannian Manifold Attention Network for EEG-based stress detection. \method{} constructs spatial covariance matrices independently at each frequency point and maps them to the SPD tangent space, preserving channel-wise geometry together with frequency-specific discriminative cues. It further introduces frequency cluster aggregation to select informative spectral components and reduce redundancy by forming compact, data-driven frequency clusters aligned with EEG rhythms. Finally, an intra-inter slice attention module adaptively integrates local slice-level spectral dynamics and global temporal context across EEG sequences. Experiments on three datasets show that \method{} consistently outperforms five state-of-the-art baselines, achieving up to 82.78% balanced accuracy while remaining efficient with only 1.60M parameters and 31.95M FLOPs.

Cheng He, Kunyu Peng, Shangen Han +3

Laboratory of Neural Computing and Intelligent Perception, College of Information Engineering, Capital Normal University, Beijing, China · Karlsruhe Institute of Technology, Germany · School of Psychology, Capital Normal University, Beijing, China

Cross-Subject EEG Decoding Electroencephalography Frequency-Domain Feature Learning Geometric Representation Learning Riemannian Representation Learning

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May 6, 2026

Beyond Rigid Geometries: The Spline-Pullback Metric for Universal Diffeomorphic SPD Representation Learning

The integration of Symmetric Positive Definite (SPD) matrices into deep learning has historically relied on fixed algebraic Riemannian metrics. Analogous to hand-crafted features in classical machine learning, these static formulations impose rigid geometries limiting network expressivity and adaptability. Recent attempts to parameterize these geometries often violate the axioms of primary matrix functions through unconstrained powers or rank-dependent scaling, inviting spatial folding, loss of global surjectivity, and gradient collapse at spectral singularities. In this paper, we introduce the Spline-Pullback Metric (SPM), instantiated as Spectral-SPM and Cholesky-SPM, marking a paradigm shift from static metric selection to universal geometric approximation. By parameterizing the global diffeomorphism via a rank-invariant, monotonically constrained B-spline, SPM acts as a dense universal approximator for strictly increasing

C^1

diffeomorphisms and theoretically subsumes existing pullback metrics while enabling localized non-linear spectral modelling. Topologically, SPM provides a globally bijective pullback geometry precluding rank-swapping discontinuities and gradient instabilities. Empirically, SPM achieves a state-of-the-art performance across 3 datasets utilizing Linear Probes, SPDNets, and deep Riemannian ResNets.

Tushar Das, Subrata Dutta, Sarmistha Neogy +1

National Institute of Technology Jamshedpur · Jadavpur University · Machine Vision and Intelligence Lab National Institute of Technology Jamshedpur

Geometric Representation Learning Metric Learning Riemannian Geometry Riemannian Metric Learning Riemannian Representation Learning

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Mar 20, 2026

Geometric Mixture-of-Experts with Curvature-Guided Adaptive Routing for Graph Representation Learning

Graph-structured data typically exhibits complex topological heterogeneity, making it difficult to model accurately within a single Riemannian manifold. While emerging mixed-curvature methods attempt to capture such diversity, they often rely on implicit, task-driven routing that lacks fundamental geometric grounding. To address this challenge, we propose a Geometric Mixture-of-Experts framework (GeoMoE) that adaptively fuses node representations across diverse Riemannian spaces to better accommodate multi-scale topological structures. At its core, GeoMoE leverages Ollivier-Ricci Curvature (ORC) as an intrinsic geometric prior to orchestrate the collaboration of specialized experts. Specifically, we design a graph-aware gating network that assigns node-specific fusion weights, regularized by a curvature-guided alignment loss to ensure interpretable and geometry-consistent routing. Additionally, we introduce a curvature-aware contrastive objective that promotes geometric discriminability by constructing positive and negative pairs according to curvature consistency. Extensive experiments on six benchmark datasets demonstrate that GeoMoE outperforms state-of-the-art baselines across diverse graph types.

Haifang Cao, Yu Wang, Timing Li +2

College of Intelligence and Computing, Tianjin University, Tianjin, China

Ollivier-Ricci Curvature Riemannian Representation Learning Graph Representation Learning Geometric Representation Learning Mixture-of-Experts Models

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