针对稀疏单纯形约束的低秩优化，提出基于斜流形的乘法更新方法。
Optimization on the Oblique Manifold for Sparse Simplex Constraints via Multiplicative Updates
- 利用斜流形几何重构问题，保持单纯形约束
- 相比欧氏与黎曼方法，优化效率显著提升
- 适合需要稀疏性与归一化约束的机器学习任务
带有稀疏单纯形约束的低秩优化问题要求变量满足非负性、稀疏性和和为1的条件，由于低秩结构与约束之间的相互作用，优化极具挑战性。这类问题广泛存在于机器学习、信号处理、环境科学和计算生物学等领域。本文提出一种新颖的流形优化方法，通过利用斜流形的几何特性重构问题，并引入基于黎曼梯度下降的新优化算法，严格维持单纯形约束。借助底层流形结构，该方法显著提升了优化效率。在合成与真实数据集上的实验表明，所提方法优于标准欧氏与黎曼优化方法，为更广泛应用开辟了道路。
原文摘要 · Abstract (English)
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.
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