# Nonnegative matrix factorization

(Difference between revisions)
 Revision as of 14:16, 28 September 2009 (edit)← Previous diff Revision as of 02:44, 17 February 2010 (edit) (undo) (ruqKKcWrG)Next diff → Line 1: Line 1: - Exercise from [http://meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], ch.4: + xA3hHv fhanfeusylsr, [url=http://idhijwizkysp.com/]idhijwizkysp[/url], [link=http://cxgygcljwequ.com/]cxgygcljwequ[/link], http://ogpjkfcdsjji.com/ - + - Given rank-2 nonnegative matrix + - X=\!\left[\!\begin{array}{ccc}17&28&42\\ + - 16&47&51\\ + - 17&82&72\end{array}\!\right], + - + - find a nonnegative factorization + - X=WH\,[/itex] + - by solving + - + - $\begin{array}{cl}\mbox{find}_{A\in\mathbb{S}^3,\,B\in\mathbb{S}^3,\,W\in\mathbb{R}^{3\times2},\,H\in\mathbb{R}^{2\times3}}&W\,,\,H\\ + - \mbox{subject to}&Z=\left[\begin{array}{ccc}I&W^{\rm T}&H\\W&A&X\\H^{\rm T}&X^{\rm T}&B\end{array}\right]\succeq0\\ + - &W\geq0\\ + - &H\geq0\\ + - &\mbox{rank}\,Z\leq2\end{array} + - + - which follows from the fact, at optimality, + - + - [itex] Z^\star=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right]\begin{array}{c}\textbf{[}\,I~~W^{\rm T}~H\,\textbf{]} + - \end{array}$ + - + - Use the known closed-form solution for a direction vector $Y\,$ to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; + - + - set $_{}Z^\star\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}$ to a nonincreasingly ordered diagonalization and + - [itex]_{}U^\star\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\reals^{\mathbf{8}\times\mathbf{6}}, + - then [itex]Y\!=U^\star U^{\star\rm T}. + - + -