Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization

Abstract

We introduce and analyze BPALM and A-BPALM, two multi-block proximal alternating linearized minimization algorithms using Bregman distances for solving structured nonconvex problems. The objective function is the sum of a multi-block relatively smooth function (i.e., relatively smooth by fixing all the blocks except one) and block separable (nonsmooth) nonconvex functions. The sequences generated by our algorithms are subsequentially convergent to critical points of the objective function, while they are globally convergent under the KL inequality assumption. Moreover, the rate of convergence is further analyzed for functions satisfying the Łojasiewicz’s gradient inequality. We apply this framework to orthogonal nonnegative matrix factorization (ONMF) that satisfies all of our assumptions and the related subproblems are solved in closed forms, where some preliminary numerical results are reported.