Non-Negative Matrix Factorization

Definition

${\mathbf{X}}_{n\times p}\approx {\mathbf{W}}_{n\times r}{\mathbf{H}}_{r\times p}$ where $r\le \max (n,p)$

Non-negative matrix factorization (NMF) is an algorithm where a matrix $\mathbf{X}$ is factorized into two matrices $\mathbf{W}$ and $\mathbf{H}$ have no negative elements.

The matrices $\mathbf{W}$ and $\mathbf{H}$ are found by maximizing $\sum _{i=1}^n\sum _{j=1}^p[x_{ij}\log (\mathbf{WH}{)}_{ij}-(\mathbf{WH}{)}_{ij}]$