The Hadamard decomposition problem

Martino Ciaperoni*, Aristides Gionis, Heikki Mannila

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We introduce the Hadamard decomposition problem in the context of data analysis. The problem is to represent exactly or approximately a given matrix as the Hadamard (or element-wise) product of two or more low-rank matrices. The motivation for this problem comes from situations where the input matrix has a multiplicative structure. The Hadamard decomposition has potential for giving more succint but equally accurate representations of matrices when compared with the gold-standard of singular value decomposition (svd). Namely, the Hadamard product of two rank-h matrices can have rank as high as h2. We study the computational properties of the Hadamard decomposition problem and give gradient-based algorithms for solving it approximately. We also introduce a mixed model that combines svd and Hadamard decomposition. We present extensive empirical results comparing the approximation accuracy of the Hadamard decomposition with that of the svd using the same number of basis vectors. The results demonstrate that the Hadamard decomposition is competitive with the svd and, for some datasets, it yields a clearly higher approximation accuracy, indicating the presence of multiplicative structure in the data.

Original languageEnglish
JournalData Mining and Knowledge Discovery
DOIs
Publication statusE-pub ahead of print - 21 May 2024
MoE publication typeA1 Journal article-refereed

Keywords

  • Hadamard product
  • Matrix decomposition

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