Morphological variations and asymmetry

Latest papers

• Intrinsic Associative Memory on Riemannian Manifolds: Curvature, Capacity, and Emergent Modes

  Krishnakumar Balasubramanian, Zhaoyang Shi · 30 September 2026

  Geometry does more than constrain an associative memory: curvature determines what it remembers and which states it creates. We develop intrinsic dense associative memories on Riemannian manifolds by casting memory as Epanechnikov kernel-density mode seeking. We compare geodesic and volume-corrected…

• Building Transformation Layers for Riemannian Neural Networks

  Ziheng Chen · 29 September 2026

  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 approac…

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

  Doron Haviv, Edward De Brouwer, Rishabh Anand, Rex Ying, A\"icha Bentaieb, Gabriele Scalia, Hector Corrada Bravo · 23 September 2026

  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 th…

• Beyond Gaussian Worlds: Latent Geometry Matters for JEPAs

  L\'eo Nicollier (CB, ATT), Enric Meinhardt-Llopis (CB), Marc Pic (ATT), Pablo Mus\'e (CB, IFUMI), Gabriele Facciolo (CB) · 21 September 2026

  Recent Joint-Embedding Predictive Architectures (JEPAs) prevent representation collapse by constraining learned representations to follow a prescribed target distribution, such as an isotropic Gaussian or the uniform distribution on a hypersphere. Klindt et al. (2026) showed that, under their Euclid…

• Riemannian Neural Hamiltonian Flows: Geodesic Symplectic Transport and Interpretability

  Vincent Souveton · 21 September 2026

  Hamiltonian normalizing flows are attractive generative models because their phase-space maps are invertible and volume preserving, but most neural constructions are formulated in Euclidean space. We introduce Riemannian Neural Hamiltonian Flows, which combine the fixed kinetic energy of a Riemannia…

• Nested Inductive Bias Framework for SPD Manifold Learning

  Tushar Das · 7 September 2026

  In Geometric Deep Learning, inductive biases serve two primary functions: enforcing manifold constraints and embedding relational priors. Currently, representation learning on SPD manifolds frequently relies on pullback Euclidean metrics, such as the Log-Euclidean Metric, to satisfy the former. Whil…

• Hyperbolic Latent Geometry for Tree-Structured Prototype Networks: A Local-vs-Global Trade-off

  Peter Flo, Luca Grossmann · 27 August 2026

  We study a tree-structured regularizer over class-prototype layouts in a hierarchical-classification model and ask whether the choice of latent manifold for the prototypes (Euclidean R^d vs. the Poincare ball B^d_c) affects how well that regularizer can be satisfied without distorting the data likel…

• Learning Random Geometric Graphs Drawn in Probabilistic Metric Spaces

  Dalia Chakrabarty, Kangrui Wang, Chuqiao Zhang, Ye Liu · 20 August 2026

  We present a new data-driven learning of a Random Geometric Graph (RGG) of a multivariate dataset, where the graph is drawn in a probabilistic metric space. This graph learning works for generic datasets, irrespective of the type of the observables; their probability distributions; or size of the da…

• Shape Operator PCA: Curvature-Aware Projections for Geometric Machine Learning

  Alexandre L. M. Levada · 18 August 2026

  In this paper, we propose SHOPCA (Shape Operator-based Principal Component Analysis), a novel method for unsupervised metric learning and dimensionality reduction that incorporates differential geometric information into the covariance structure of classical PCA. SHOPCA regularizes the global covari…

• A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

  Pierre-Alexandre Murena, Marcelo Hartmann · 17 August 2026

  Analogies are quaternary relations of the form "a is to b as c is to d", usually denoted a : b :: c : d. This notion is formalized in particular with the notion of proportional analogy, which imposes some constraints on the valid analogies. Whereas proportional analogies have been studied mostly in …

• The Field Knows: Cross-Dimensional Geometry from Navigation to Black Holes

  Chenghao Xu · 11 August 2026

  We introduce a continuous metric field framework trained by a single causal contrastive loss. The framework encodes a scene into coefficients of a fixed symmetric matrix basis, assembles them into a Lie algebra element, and exponentiates the result to a Riemannian or Lorentzian metric. Across dimens…

• Mixture of Geodesic Factor Analyzers on Riemannian Homogeneous Spaces

  Hengchao Chen, Yuanyao Tan, Chao Huang, Hongtu Zhu, Qiang Sun · 10 August 2026

  This paper introduces Mixtures of Geodesic Factor Analyzers (MGFA) on Riemannian homogeneous spaces. MGFA uses a geodesic factor model within each mixture component, providing greater expressiveness than mixtures of Riemannian radial distributions and enabling clustering of manifold-valued data with…

• Sphere Retraction Normalizations

  Jie Zhang, Cheng-Fang Su, Yi-Jui Huang, Min-Te Sun · 5 August 2026

  Residual connections are the de facto mechanism for training deep neural networks stably. Geodesic Normalization (GeoNorm) recasts them on a Riemannian manifold, orthogonalizing each layer output against the current hidden state and applying the resulting update through the Riemannian exponential ma…

• Smooth Reparameterizations of Functions on Simplicial Product Spaces: Applications to Probabilistic Tensor Decomposition and Functional Data Registration

  Shashwat Kumar, Arafat Rahman, Anuj Srivastava, P. -A. Absil · 4 August 2026

  We consider optimization problems defined on product spaces of simplices. Examples of this class of problems include learning low-rank discrete multivariate probability distributions via simplex constrained tensor decomposition and performing functional data registration under the Square Root Veloci…

• Riemannian Attention Mechanisms for Transformers: A Theoretical Framework and Architecture Design

  Sen Song · 4 August 2026

  All Transformer-based large language models compute attention via the Euclidean inner product, an architectural choice that Dong et al. (2021) proved causes representational rank to decay doubly exponentially with depth in pure self-attention stacks. We develop a theoretical framework that targets t…

• Landmark shape spaces with induced metrics

  Sarang Joshi, Peter W. Michor, Stefan Sommer · 31 July 2026

  We present a unification of Kendall's landmark shape spaces, where rigid motions are factored out and scale fixed on landmark configurations equipped with Euclidean geometry, with landmark configuration spaces carrying Riemannian metrics descending from right-invariant Sobolev metrics on the diffeom…

• A Riemannian View on Active Subspaces

  Zachary Grey · 29 July 2026

  Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of int…

• Deep Shape Regression for Planar Curves with Multimodal Covariates

  Manuel Pfeuffer, Roshan Prakash Rane, Hadya Yassin, Kerstin Ritter, Sonja Greven · 23 July 2026

  The shape of a planar curve is the geometric information that remains once translation, rotation, scale and reparametrisation are removed and is of interest in many health applications, e.g. in neuroimaging. We propose a deep shape regression model for open planar curves that admits multimodal and h…

• Riemannian Deep Learning:Modules, Networks, and Geometries

  Chen Ziheng · 22 July 2026

  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 Riemann…

• TreeSRNF: Square-Root Normal Fields for Generative Modelling of the Geometric and Structural Variability in Tree-like 3D Objects

  Tahmina Khanam, Hamid Laga, Mohammed Bennamoun, Guanjin Wang, Ferdous Sohel, Farid Boussaid, Anuj Srivastava · 16 July 2026

  We introduce a novel mathematical framework for analyzing and generating complex tree-shaped 3D objects, such as botanical trees and plants, which deform both in their 3D geometry and branching structure. Unlike previous works, which either consider only the skeletal structure of tree-like objects o…

• Learning Lineage-guided Geodesics with Finsler Geometry

  Aaron Zweig, Mingxuan Zhang, David A. Knowles, Elham Azizi · 13 July 2026

  Trajectory inference investigates how to interpolate paths between observed timepoints of dynamical systems, such as temporally resolved population distributions, with the goal of inferring trajectories at unseen times and better understanding system dynamics. Previous work has focused on continuous…

• LieBN: Batch Normalization over Lie Groups

  Ziheng Chen, Yue Song, Rui Wang, Xiao-Jun Wu, Nicu Sebe · 13 July 2026

  Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds, accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization. However, mo…

• Heat-Kernel Entropy Profiles and Geometric Effective Sample Size for Weighted Measures on Manifolds

  Kisung You · 9 July 2026

  Weighted empirical measures on compact manifolds arise in importance sampling, particle approximations, posterior summaries, quadrature, and representation learning. Standard weight-only summaries, such as ordinary effective sample size, ignore the geometry of the support. We introduce heat-kernel e…

• EntroPath: Maximum Entropy Path Ensemble Embedding for Manifold Learning

  Przemys{\l}aw Rola · 8 July 2026

  We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths. Many existing graph-based embeddings rely either on locally normalised random walks or on shortest-path distances. The former can concentrate diffusion in densely…

• Missing Data Imputation under Manifold Hypothesis

  Zelong Bi, Amuchechukwu Ibenegbu, Sarat Moka · 7 July 2026

  The manifold hypothesis posits that high-dimensional data are concentrated near a low-dimensional embedded manifold. Recent advances in mixture variational autoencoders (VAEs) provide a powerful tool for extracting such underlying structure in a faithful manner. The resulting geometric structure nat…