# Nonnegative matrix factorization

### From Wikimization

(Difference between revisions)

Line 4: | Line 4: | ||

<math>X=\!\left[\!\begin{array}{ccc}17&28&42\\ | <math>X=\!\left[\!\begin{array}{ccc}17&28&42\\ | ||

16&47&51\\ | 16&47&51\\ | ||

- | + | 17&82&72\end{array}\!\right],</math> | |

- | + | ||

find a nonnegative factorization | find a nonnegative factorization | ||

<math> X=WH\,</math> | <math> X=WH\,</math> | ||

Line 23: | Line 22: | ||

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

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

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

## Revision as of 12:44, 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.