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In the applications of signal processing and data analytics, there is a wideclass of non-convex problems whose objective function is freed from the commonglobal Lipschitz continuous gradient assumption (e.g., the nonnegative matrixfactorization (NMF) problem). Recently, this type of problem with some certainspecial structures has been solved by Bregman proximal gradient (BPG). Thisinspires us to propose a new Block-wise two-references Bregman proximalgradient (B2B) method, which adopts two reference functions so that aclosed-form solution in the Bregman projection is obtained. Based on therelative smoothness, we prove the global convergence of the proposed algorithmsfor various block selection rules. In particular, we establish the globalconvergence rate of $O(\frac{\sqrt{s}}{\sqrt{k}})$ for the greedy andrandomized block updating rule for B2B, which is $O(\sqrt{s})$ times fasterthan the cyclic variant, i.e., $O(\frac{s}{\sqrt{k}} )$, where $s$ is thenumber of blocks, and $k$ is the number of iterations. Multiple numericalresults are provided to illustrate the superiority of the proposed B2B comparedto the state-of-the-art works in solving NMF problems.

Leveraging Two Reference Functions in Block Bregman Proximal Gradient Descent for Non-convex and Non-Lipschitz Problems