Nonnegative matrix factorization
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(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 13:44, 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.