arXiv Machine Learning
Jul 22
1-Lipschitz Neural Networks on Hadamard Manifolds
arXiv:2607. 19335v1 Announce Type: cross Abstract: Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability.
1-Lipschitz Neural Networks on Hadamard Manifolds
Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean spaces.
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arXiv AI
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Riemannian Deep Learning: Modules, Networks, and Geometries
arXiv:2607. 19305v2 Announce Type: replace-cross Abstract: 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.
arXiv AI
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Riemannian Deep Learning:Modules, Networks, and Geometries
arXiv:2607. 19305v1 Announce Type: cross Abstract: 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.
arXiv AI
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arXiv:2609.35436v2 Announce Type: replace Abstract: Recently, deep neural networks on manifold-valued representations have garnered significant attention across various machine learning applications....
arXiv Machine Learning
Jun 9
Generalization in Nonlinear Least Squares via Learned Feature Geometry
arXiv:2606. 08799v1 Announce Type: cross Abstract: We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual--curvature term.
arXiv AI
Jul 13
LieBN: Batch Normalization over Lie Groups
arXiv:2607. 08783v1 Announce Type: cross Abstract: Manifold-valued measurements are prevalent in various machine learning tasks.
arXiv Machine Learning
Jun 25
Towards Robust EEG Decoding Based on Riemannian Self-Attention
arXiv:2606. 25456v1 Announce Type: new Abstract: Brain-Computer Interface (BCI) based on electroencephalography (EEG) enables direct interaction between the brain and external environments and has significant applications in assistive technologies, medical rehabilitation, and entertainment.
arXiv Machine Learning
Sep 7
Nested Inductive Bias Framework for SPD Manifold Learning
The paper introduces a Nested Inductive Bias framework that uses a two‑stage diffeomorphic composition to pull back non‑Euclidean target geometries onto symmetric positive definite (SPD) manifolds. This approach allows the construction of curvature‑aligned Riemannian classifiers that respect both matrix constraints and the intrinsic relational geometry of data. Empirical results on kinematic, signal processing, and synthetic benchmarks show that class separability degrades when metric curvature does not match the data distribution, and the authors also propose the Rational Conformal Metric (RCM) for robust vectorized architectures.
arXiv AI
Aug 24
SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges
This review discusses how neuroimaging data can be represented as symmetric positive-definite (SPD) matrices and analyzed using the Riemannian geometry of the SPD manifold. It surveys the evolution from modality-specific SPD representations to geometric shallow and deep learning methods, emphasizing how these approaches maintain structural constraints while integrating modern AI techniques. The paper frames SPD matrix learning as a bridge between classical geometric statistics and contemporary machine learning in neuroimaging and brain‑computer interface research.
arXiv Machine Learning
Sep 15
Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds
arXiv:2609. 13646v1 Announce Type: new Abstract: This work addresses decentralized online Riemannian optimization on Hadamard manifolds.
arXiv Machine Learning
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Iso-Riemannian Optimization on Learned Data Manifolds
arXiv:2510. 21033v3 Announce Type: replace-cross Abstract: We develop a theory of iso-Riemannian optimization for problems constrained to learned data manifolds, a setting in which classical Riemannian optimization - and Riemannian gradient descent in particular - can be poorly suited.
arXiv Statistics ML
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Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold
arXiv:2609.30487v1 Announce Type: cross Abstract: We here develop a functional neural network, termed MatFAE, for learning trajectories on the Riemannian manifold of symmetric positive definite (SPD)...