提出高效算法实现矩阵的哈达玛分解，提升压缩与分析效率。
Efficient algorithms for the Hadamard decomposition
- 用交替优化将非凸问题转为多个凸子问题求解。
- 在多数据集上优于梯度下降法和传统低秩近似方法。
- 支持多矩阵分解，兼顾高有效秩与计算效率，适合矩阵压缩场景。
哈达玛分解是一种强大的数据分析与矩阵压缩技术，将给定矩阵分解为两个或多个低秩矩阵的逐元素乘积。本文提出一种高效算法，通过交替优化将全局非凸问题分解为一系列凸子问题求解。为提升性能，借鉴奇异值分解（SVD）设计先进初始化策略，并引入基于动量的更新加速技术。除优化双矩阵情形外，还扩展框架支持多于两个低秩矩阵，实现更高有效秩的近似，同时保持计算效率。最后，通过大量实验对比了所提方法与现有基于梯度下降的哈达玛分解方法及传统低秩近似技术。结果表明，该方法在多种数据集上均表现出色。
原文摘要 · Abstract (English)
The Hadamard decomposition is a powerful technique for data analysis and matrix compression, which decomposes a given matrix into the element-wise product of two or more low-rank matrices. In this paper, we develop an efficient algorithm to solve this problem, leveraging an alternating optimization approach that decomposes the global non-convex problem into a series of convex sub-problems. To improve performance, we explore advanced initialization strategies inspired by the singular value decomposition (SVD) and incorporate acceleration techniques by introducing momentum-based updates. Beyond optimizing the two-matrix case, we also extend the Hadamard decomposition framework to support more than two low-rank matrices, enabling approximations with higher effective ranks while preserving computational efficiency. Finally, we conduct extensive experiments to compare our method with the existing gradient descent-based approaches for the Hadamard decomposition and with traditional low-rank approximation techniques. The results highlight the effectiveness of our proposed method across diverse datasets.
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