Nonnegative matrix factorization
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(Difference between revisions)
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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^\star\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to | + | set <math>_{}Z^\star\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and |
<math>_{}U^\star\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\reals^{\mathbf{8}\times\mathbf{6}}</math>, | <math>_{}U^\star\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\reals^{\mathbf{8}\times\mathbf{6}}</math>, | ||
then <math>Y\!=U^\star U^{\star\rm T}.</math> | then <math>Y\!=U^\star U^{\star\rm T}.</math> | ||
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<math>Y\!=U^\star U^{\star\rm T}.</math> | <math>Y\!=U^\star U^{\star\rm T}.</math> | ||
- | Global convergence | + | Global convergence occurs, in this example, in only a few iterations. |
Revision as of 15:16, 28 September 2009
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.