# Non-negative matrix factorisation

October 14, 2019 — October 26, 2023

An intersting special matrix factorisation, where the goal is to decode an *element-wise non-negative* matrix into a product of two smaller matrices, which looks a lot like adaptive sparse coding if you squint at it.

David Austin simple intro to Non-negative matrix factorization for the American Mathematical Society is a good inroads to the classic problem.

## 1 Loss functions

TBD

## 2 Optimisation matters

TBD.

## 3 Connection to optimal transport

Optimal transport turns out to have a connection to NNMF (Huynh, Zhao, and Phung 2020; Qian et al. 2016; Scetbon, Cuturi, and Peyré 2021; Schmitz et al. 2018; S. Y. Zhang 2021; Zhao et al. 2020).

## 4 Tools

NMF (R) 🏗

Matlab: Chih-Jen Lin’s nmf.m — “This tool solves NMF by alternative non-negative least squares using projected gradients. It converges faster than the popular multiplicative update approach.”

In this repository, we offer both MPI and OPENMP implementation for MU, HALS and ANLS/BPP based NMF algorithms. This can run off the shelf as well easy to integrate in other source code. These are very highly tuned NMF algorithms to work on super computers. We have tested this software in NERSC as well OLCF cluster. The openmp implementation is tested on many different Linux variants with intel processors. The library works well for both sparse and dense matrix. (Fairbanks et al. 2015; Kannan, Ballard, and Park 2016; Kannan 2016)

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