TY - JOUR

T1 - The Hadamard decomposition problem

AU - Ciaperoni, Martino

AU - Gionis, Aristides

AU - Mannila, Heikki

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/5/21

Y1 - 2024/5/21

N2 - 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.

AB - 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.

KW - Hadamard product

KW - Matrix decomposition

UR - http://www.scopus.com/inward/record.url?scp=85193700816&partnerID=8YFLogxK

U2 - 10.1007/s10618-024-01033-y

DO - 10.1007/s10618-024-01033-y

M3 - Article

AN - SCOPUS:85193700816

SN - 1384-5810

JO - Data Mining and Knowledge Discovery

JF - Data Mining and Knowledge Discovery

ER -