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

arXiv: https://arxiv.org/abs/1908.01402 · HTML: https://mlsanity.com/paper/1908.01402/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Aug 2019.

## Built on by (most cited first, 7 of 7)

- [Outlier-Robust Gromov-Wasserstein for Graph Data](https://mlsanity.com/paper/2302.04610/) (2023)
- [Block Alternating Bregman Majorization Minimization with Extrapolation](https://mlsanity.com/paper/2107.04395/) (2021)
- [An Accelerated Alternating Partial Bregman Algorithm for ReLU-based Matrix Decomposition](https://mlsanity.com/paper/2503.02386/) (2025)
- [Spatially Coherent Clustering Based on Orthogonal Nonnegative Matrix Factorization](https://mlsanity.com/paper/2104.12289/) (2021)
- [A Variational Approach for Joint Image Recovery and Feature Extraction Based on Spatially-Varying Generalised Gaussian Models](https://mlsanity.com/paper/2209.01375/) (2022)
- [Matrix Completion via Nonsmooth Regularization of Fully Connected Neural Networks](https://mlsanity.com/paper/2403.10232/) (2024)
- [Generalized Geometry Block Proximal Linearized Method for Multiblock Nonconvex and Nonsmooth Optimization](https://mlsanity.com/paper/2609.39301/) (2026)

## Abstract (arXiv)

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 Lojasiewicz'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.