  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 cavity-based matrix factorization (CBMF) and approximate
cavity-based 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.
