Nonnegative matrix factorization

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Given rank-2 nonnegative matrix
Given rank-2 nonnegative matrix
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<math>X=\!\left[\!\begin{array}{ccc}17&28&42\\
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<math>X=\!\left[\!\begin{array}{ccc}\textbf{17}&28&42\\
16&47&51\\
16&47&51\\
17&82&72
17&82&72

Current revision

Exercise from Convex Optimization & Euclidean Distance Geometry, ch.4:

Given rank-2 nonnegative matrix LaTeX: X=\!\left[\!\begin{array}{ccc}\textbf{17}&28&42\\ </p> <pre> 16&47&51\\ 17&82&72 \end{array}\!\right]\,,

find a nonnegative factorization LaTeX: \,X=WH\, by solving

LaTeX: \begin{array}{rl}{\text 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\\ {\text 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\\ &{\text rank}\,Z\leq2\end{array}

which follows from the fact, at optimality,

LaTeX: 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}

Use the known closed-form solution for a direction vector LaTeX: Y to regulate rank (rank constraint is replaced) by Convex Iteration;

set LaTeX: _{}Z^*\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^{\mathbf{8}} to a nonincreasingly ordered diagonalization and LaTeX: _{}U^*\!=_{\!}Q(:\,,\,3:8)\!\in_{\!}\mathbb{R}^{\mathbf{8}\times\mathbf{6}}, then LaTeX: Y\!=U^* U^{*\rm T}.


In summary, initialize LaTeX: Y=I then alternate solution of

LaTeX: \begin{array}{rl}\mbox{minimize}_{A\in\mathbb{S}^3,\,B\in\mathbb{S}^3,\,W\in\mathbb{R}^{3\times2},\,H\in\mathbb{R}^{2\times3}}&\langle Z\,,\;Y\rangle\\ \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\end{array}

with LaTeX: \,Y\!=U^* U^{*\rm T}.

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

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