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.

我们介绍和分析BPALM和A-BPALM, 两种使用 Bregman 距离解决结构化非凸问题的多块近端交替线性化最小化算法。目标函数是多块相对平滑函数（即通过固定除一个块之外的所有块而相对平滑）和块可分离（非光滑）非凸函数的总和。我们的算法生成的序列随后收敛到目标函数的临界点，而它们在 KL 不等式假设下全局收敛。此外，还进一步分析了满足 Łojasiewicz 梯度不等式的函数的收敛速度。我们将此框架应用于满足我们所有假设的正交非负矩阵分解（ONMF），并且相关子问题以封闭形式解决，