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:2607. 22004v1 Announce Type: new Abstract: Energy natural gradient descent (ENGD) aligns parameter updates with the curvature of an underlying function-space energy, but existing formulations assume an unconstrained Euclidean parameter domain.
The paper introduces a new convergence framework for solving distributionally robust optimization problems formulated as nonconvex, nonconcave minimax problems over a Euclidean space and a Riemannian manifold. It defines a "basin saddle point"—a locally defined Nash equilibrium—and proves that a Riemannian gradient ascent–descent algorithm converges to such points under a local Łojasiewicz growth condition. The authors apply this theory to a statistical risk DRO problem over Gaussian measures, deriving explicit convergence rates and constants in terms of data dimension, loss moments, and reference covariance.
arXiv:2209. 15130v3 Announce Type: replace-cross Abstract: We study a general matrix optimization problem with a fixed-rank positive semidefinite (PSD) constraint.
arXiv:2608. 02576v1 Announce Type: new Abstract: We consider optimization problems defined on product spaces of simplices.
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:2503. 24075v4 Announce Type: replace-cross Abstract: Low-rank optimization problems with sparse simplex constraints involve variables that must satisfy nonnegativity, sparsity, and sum-to-1 conditions, making their optimization particularly challenging due to the interplay between low-rank structures and constraints.
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:2606. 07926v1 Announce Type: cross Abstract: Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps.
arXiv:2607. 06723v1 Announce Type: cross Abstract: Most gradient-based optimization methods move parameters through a fixed background geometry, even when their internal states implicitly define changing notions of length, curvature, and preconditioning.
arXiv:2509. 07779v2 Announce Type: replace-cross Abstract: We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting.
The paper presents a framework for retraction-based convex optimization on Riemannian manifolds, introducing retraction-specific convex sets and retraction-based gradient projection algorithms. It extends the standard theory of gradient projection algorithms to this setting and proves convergence results for various stepsize rules. The authors apply the framework to weighted low-rank approximation and validate the convergence results numerically on an image completion task.
arXiv:2501.14993v4 Announce Type: replace-cross Abstract: The proximal algorithm is a powerful tool to minimize nonlinear and nonsmooth functionals in a general metric space. Motivated by the recent...