# The NMF problem and lattice-subspaces

## TL;DR

This paper introduces a mathematical method for nonnegative matrix factorization (NMF) using lattice-subspaces, providing a way to determine the intermediate dimension and compute factors with an accompanying MATLAB program.

## Contribution

It presents a novel mathematical approach for NMF based on lattice-subspaces, including a criterion for the intermediate dimension and a MATLAB implementation.

## Key findings

- A general method for NMF based on lattice-subspaces.
- A criterion for determining the intermediate dimension p.
- MATLAB code for computing the factors F and V.

## Abstract

Suppose that $A$ is a nonnegative $n\times m$ real matrix. The NMF problem is the determination of two nonnegative real matrices $F$, $V$ so that $A=FV$ with intermediate dimension $p$ smaller than $min\{ n,m\}$. In this article we present a general mathematical method for the determination of two nonnegative real factors $F,V$ of $A$. During the first steps of this process the intermediate dimension $p$ of $F,V$ is determined, therefore we have an easy criterion for $p$. This study is based on the theory of lattice-subspaces and positive bases. Also we give the matlab program for the computation of $F,V$ but the mathematical part is the main part of this article.