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

## TL;DR

This paper introduces BPALM and A-BPALM algorithms using Bregman distances for solving structured nonconvex problems, with convergence analysis and application to orthogonal nonnegative matrix factorization.

## Contribution

It proposes novel multi-block proximal algorithms with convergence guarantees for nonconvex problems, applied specifically to ONMF with closed-form solutions.

## Key findings

- Sequences converge to critical points of the objective
- Global convergence under KL inequality
- Preliminary numerical results on ONMF

## 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. 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. Further, the rate of convergence is further analyzed for functions satisfying the {\L}ojasiewicz'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 is reported.