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

08/04/2019

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by   Masoud Ahookhosh, et al.

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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 bauschke2016descent,lu2018relatively) and block separable (nonsmooth) nonconvex functions. It turns out that the sequences generated by our algorithms are subsequentially convergent to critical points of the objective function, while they are globally convergent under KL inequality assumption. 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) and sparse ONMF (SONMF) that both satisfy all of our assumptions and the related subproblems are solved in closed forms.