Riemannian Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates
This addresses optimization problems in machine learning and other fields where variables must satisfy nonnegativity, sparsity, and sum-to-one conditions, representing an incremental advancement in method efficiency.
The paper tackled the challenge of optimizing low-rank problems with sparse simplex constraints by proposing a novel manifold optimization approach on the oblique manifold, which improved efficiency in experiments on synthetic datasets compared to existing methods.
Low-rank optimization problems with sparse simplex constraints involve variables that must satisfy nonnegativity, sparsity, and sum-to-one 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 paper, we propose a novel manifold optimization approach to tackle these problems efficiently. Our method leverages the geometry of oblique rotation 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 datasets compared to standard Euclidean and Riemannian methods show the effectiveness of the proposed method.