An Algorithm for Randomized Nonnegative Matrix Factorization and Its Global Convergence

Nonnegative Matrix Factorization (NMF) is to decompose a given nonnegative matrix into two nonnegative factor matrices. Recently, randomized NMF has been proposed as an approach to fast NMF of large nonnegative matrices. The main idea of this approach is to perform NMF after reducing the dimensionality of the given nonnegative matrix by multiplying it by a random matrix. Since randomized NMF is formulated as a constrained optimization problem which is slightly different from the one for original NMF, it is necessary to develop suitable algorithms for solving it. However, the conventional algorithm has a serious drawback that the constraints are not satisfied. In addition, the convergence of the algorithm has not been analyzed. In this paper, in order to overcome these drawbacks, we propose to modify the optimization problem and design an algorithm based on the hierarchical alternating least squares method to solve the modified optimization problem. We also prove the global convergence of the designed algorithm.