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This research introduces a novel Riemannian optimization framework specifically designed to handle low-rank problems with sparse simplex constraints, addressing the geometric challenges of nonnegativity and sum-to-1 conditions. By leveraging oblique manifolds, the proposed method achieves superior efficiency over standard Euclidean approaches in applications ranging from machine learning to computational biology. The work advances the mathematical foundations required for solving complex constrained optimization tasks central to modern AI models.
Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates
arXiv:2503.24075v5 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. These problems arise in various applications, including machine learning, signal processing, environmental fields, and computational biology. In this work, we propose a novel manifold optimization approach to efficiently tackle these problems. Our method leverages the geometry of oblique manifolds to reformulate the problem and introduces a new Riemannian optimization method based on Riemannian gradient descent that strictly maintains the simplex constraints. By exploiting the underlying manifold structure, our approach improves optimization efficiency. Experiments on synthetic and real datasets demonstrate the effectiveness of the proposed method compared to standard Euclidean and Riemannian methods, paving the way for broader applications.
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. These problems arise in various applications, including machine learning, signal processing, environmental fields, and computational biology. In this work, we propose a novel manifold optimization approach to efficiently tackle these problems. Our method leverages the geometry of oblique manifolds to reformulate the problem and introduces a new Riemannian optimization method based on Riemannian gradient descent that strictly maintains the simplex constraints. By exploiting the underlying manifold structure, our approach improves optimization efficiency. Experiments on synthetic and real datasets demonstrate the effectiveness of the proposed method compared to standard Euclidean and Riemannian methods, paving the way for broader applications.
Optimization and Control (math.OC); Machine Learning (cs.LG)
arXiv:2503.24075 [math.OC]
(or arXiv:2503.24075v5 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2503.24075 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 225 words
