Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912976251912192 |
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| author | Esposito, Flavia Ang, Andersen |
| author_facet | Esposito, Flavia Ang, Andersen |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_24075 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates Esposito, Flavia Ang, Andersen Optimization and Control Machine Learning 15A23, 65K10, 49Q99, 90C26, 90C30 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. |
| title | Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates |
| topic | Optimization and Control Machine Learning 15A23, 65K10, 49Q99, 90C26, 90C30 |
| url | https://arxiv.org/abs/2503.24075 |