# Nonnegative matrix factorization

### From Wikimization

(Difference between revisions)

Line 18: | Line 18: | ||

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

- | <math> Z^ | + | <math> Z^*=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right]\begin{array}{c}\textbf{[}\,I~~W^{\rm T}~H\,\textbf{]} |

\end{array}</math> | \end{array}</math> | ||

Use the known closed-form solution for a direction vector <math>Y\,</math> to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; | Use the known closed-form solution for a direction vector <math>Y\,</math> to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; | ||

- | set <math>_{}Z^ | + | set <math>_{}Z^*\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and |

- | <math>_{}U^ | + | <math>_{}U^*\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\mathbb{R}^{\mathbf{8}\times\mathbf{6}}</math>, |

- | then <math>Y\!=U^ | + | then <math>Y\!=U^* U^{*\rm T}.</math> |

<br> | <br> | ||

Line 37: | Line 37: | ||

with | with | ||

- | <math>Y\!=U^ | + | <math>Y\!=U^* U^{*\rm T}.</math> |

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

## Revision as of 12:41, 24 November 2011

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.