Efficient algorithms for the Hadamard decomposition

• URL: http://arxiv.org/abs/2504.13633v2
• Date: Tue, 22 Apr 2025 11:02:40 GMT
• Title: Efficient algorithms for the Hadamard decomposition
• Authors: Samuel Wertz, Arnaud Vandaele, Nicolas Gillis,
• Abstract summary: The Hadamard decomposition is a powerful technique for data analysis and matrix compression.<n>In this paper, we develop a framework that decomposes a given matrix into the product of two or more low-rank approximations.<n>We conduct experiments to compare our method with the existing descent-based approaches for the Hadamard decomposition.
• Score: 14.653207365119796
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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