Variational Bayesian matrix factorization (VBMF) efficiently approximates the posterior distribution of factorized matrices by assuming matrix-wise independence of the two factors. A recent study on fully-observed VBMF showed that, under a stronger assumption that the two factorized matrices are column-wise independent, the global optimal solution can be analytically computed. However, it was not clear how restrictive the column-wise independence assumption is. In this paper, we prove that the global solution under matrix-wise independence is actually column-wise independent, implying that the column-wise independence assumption is harmless. A practical consequence of our theoretical finding is that the global solution under matrix-wise independence (which is a standard setup) can be obtained analytically in a computationally very efficient way without any iterative algorithms. We experimentally illustrate advantages of using our analytic solution in probabilistic principal component analysis.

Variational Bayesian matrix factorization (VBMF) efficiently approximates the posterior distribution of factorized matrices by assuming matrix-wise independence of the two factors. A recent study on fully-observed VBMF showed that, under a stronger assumption that the two factorized matrices are column-wise independent, the global optimal solution can be analytically computed. However, it was not clear how restrictive the column-wise independence assumption is. In this paper, we prove that the global solution under matrix-wise independence is actually column-wise independent, implying that the column-wise independence assumption is harmless. A practical consequence of our theoretical finding is that the global solution under matrix-wise independence (which is a standard setup) can be obtained analytically in a computationally very efficient way without any iterative algorithms.

Variational Bayesian matrix factorization (VBMF) efficiently approximates the posterior distribution of factorized matrices by assuming matrix-wise independence of the two factors. A recent study on fully-observed VBMF showed that, under a stronger assumption that the two factorized matrices are column-wise independent, the global optimal solution can be analytically computed. However, it was not clear how restrictive the column-wise independence assumption is. In this paper, we prove that the global solution under matrix-wise independence is actually column-wise independent, implying that the column-wise independence assumption is harmless. A practical consequence of our theoretical finding is that the global solution under matrix-wise independence (which is a standard setup) can be obtained analytically in a computationally very efficient way without any iterative algorithms. We experimentally illustrate advantages of using our analytic solution in probabilistic principal component analysis.

Variational Bayesian matrix factorization (VBMF) efficiently approximates the posterior distribution of factorized matrices by assuming matrix-wise independence of the two factors. A recent study on fully-observed VBMF showed that, under a stronger assumption that the two factorized matrices are column-wise independent, the global optimal solution can be analytically computed. However, it was not clear how restrictive the column-wise independence assumption is. In this paper, we prove that the global solution under matrix-wise independence is actually column-wise independent, implying that the column-wise independence assumption is harmless. A practical consequence of our theoretical finding is that the global solution under matrix-wise independence (which is a standard setup) can be obtained analytically in a computationally very efficient way without any iterative algorithms. We experimentally illustrate advantages of using our analytic solution in probabilistic principal component analysis.