A column-wise update algorithm for nonnegative matrix factorization in Bregman divergence with an orthogonal constraint

Abstract

Recently orthogonal nonnegative matrix factorization (ONMF), imposing an orthogonal constraint into NMF, has been attracting a great deal of attention. ONMF is more appropriate than standard NMF for a clustering task because the constrained matrix can be considered as an indicator matrix. Several iterative ONMF algorithms have been proposed, but they suffer from slow convergence because of their matrix-wise updating. In this paper, therefore, a column-wise update algorithm is proposed for speeding up ONMF. To make the idea possible, we transform the matrix-based orthogonal constraint into a set of column-wise orthogonal constraints. The algorithm is stated first with the Frobenius norm and then with Bregman divergence, both for measuring the degree of approximation. Experiments on one artificial and six real-life datasets showed that the proposed algorithms converge faster than the other conventional ONMF algorithms, more than four times in the best cases, due to their smaller numbers of iterations.