# Nonnegative matrix factorization

### From Wikimization

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Current revision (16:40, 30 October 2016) (edit) (undo) |
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by solving | by solving | ||

- | <math>\begin{array}{cl} | + | <math>\begin{array}{cl}{\text find}_{A\in\mathbb{S}^3,\,B\in\mathbb{S}^3,\,W\in\mathbb{R}^{3\times2},\,H\in\mathbb{R}^{2\times3}}&W\,,\,H\\ |

- | + | {\text subject to}&Z=\left[\begin{array}{ccc}I&W^{\rm T}&H\\W&A&X \\H^{\rm T}&X^{\rm T}&B\end{array}\right]\succeq0\\ | |

&W\geq0\\ | &W\geq0\\ | ||

&H\geq0\\ | &H\geq0\\ | ||

- | & | + | &{\text rank}\,Z\leq2\end{array}</math> |

which follows from the fact, at optimality, | which follows from the fact, at optimality, | ||

Line 30: | Line 30: | ||

<math>\begin{array}{cl}\mbox{minimize}_{A\in\mathbb{S}^3,\,B\in\mathbb{S}^3,\,W\in\mathbb{R}^{3\times2},\,H\in\mathbb{R}^{2\times3}}&\langle Z\,,Y\rangle\\ | <math>\begin{array}{cl}\mbox{minimize}_{A\in\mathbb{S}^3,\,B\in\mathbb{S}^3,\,W\in\mathbb{R}^{3\times2},\,H\in\mathbb{R}^{2\times3}}&\langle Z\,,Y\rangle\\ | ||

- | \mbox{subject to}&Z=\left[\begin{array}{ccc}I&W^{\rm T}&H\\W&A&X\\H^{\rm T}&X^{\rm T}&B\end{array}\right]\succeq0\\ | + | \mbox{subject to}&Z=\left[\begin{array}{ccc}I&W^{\rm T}&H\\W&A&X \\H^{\rm T}&X^{\rm T}&B\end{array}\right]\succeq0\\ |

&W\geq0\\ | &W\geq0\\ | ||

&H\geq0\end{array}</math> | &H\geq0\end{array}</math> |

## Current revision

Exercise from Convex Optimization & Euclidean Distance Geometry, ch.4:

Given rank-2 nonnegative matrix find a nonnegative factorization by solving

which follows from the fact, at optimality,

Use the known closed-form solution for a direction vector to regulate rank (rank constraint is replaced) by Convex Iteration;

set to a nonincreasingly ordered diagonalization and , then

In summary, initialize then alternate solution of

with

Global convergence occurs, in this example, in only a few iterations.