Approximate matrix completion based on cavity method
Abstract
In order to solve large matrix completion problems with practical computational cost, an approximate approach based on matrix factorization has been widely used. Alternating least squares (ALS) and stochastic gradient descent (SGD) are two major algorithms to this end. In this study, we propose a new algorithm, namely cavitybased matrix factorization (CBMF) and approximate cavitybased matrix factorization (ACBMF), which are developed based on the cavity method from statistical mechanics. ALS yields solutions with less iterations when compared to those of SGD. This is because its update rules are described in a closed form although it entails higher computational cost. CBMF can also write its update rules in a closed form, and its computational cost is lower than that of ALS. ACBMF is proposed to compensate a disadvantage of CBMF in terms of relatively high memory cost. We experimentally illustrate that the proposed methods outperform the two existing algorithms in terms of convergence speed per iteration, and it can work under the condition where observed entries are relatively fewer. Additionally, in contrast to SGD, (A)CBMF does not require scheduling of the learning rate.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 October 2019
 DOI:
 10.1088/17518121/ab40de
 arXiv:
 arXiv:1907.00138
 Bibcode:
 2019JPhA...52P4004N
 Keywords:

 matrix completion;
 matrix factorization;
 cavity method;
 Mathematics  Numerical Analysis;
 Condensed Matter  Statistical Mechanics;
 Computer Science  Machine Learning
 EPrint:
 20 pages, 11 figures