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| - | Exercise from [http://meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], ch.4:
| + | xA3hHv <a href="http://fhanfeusylsr.com/">fhanfeusylsr</a>, [url=http://idhijwizkysp.com/]idhijwizkysp[/url], [link=http://cxgygcljwequ.com/]cxgygcljwequ[/link], http://ogpjkfcdsjji.com/ |
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| - | Given rank-2 nonnegative matrix
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| - | <math>X=\!\left[\!\begin{array}{ccc}17&28&42\\
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| - | 16&47&51\\
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| - | 17&82&72\end{array}\!\right],</math>
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| - | find a nonnegative factorization
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| - | <math> X=WH\,</math>
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| - | by solving
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| - | <math>\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\\
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| - | \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\\
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| - | &W\geq0\\
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| - | &H\geq0\\
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| - | &\mbox{rank}\,Z\leq2\end{array}</math>
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| - | which follows from the fact, at optimality,
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| - | <math> Z^\star=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right]\begin{array}{c}\textbf{[}\,I~~W^{\rm T}~H\,\textbf{]}
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| - | \end{array}</math>
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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]];
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| - | set <math>_{}Z^\star\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and
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| - | <math>_{}U^\star\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\reals^{\mathbf{8}\times\mathbf{6}}</math>,
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| - | then <math>Y\!=U^\star U^{\star\rm T}.</math>
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| - | <br>
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| - | In summary, initialize <math>Y=I\,</math> then alternate solution of
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| - | <math>\begin{array}{cl}\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\\
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| - | \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\\
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| - | &W\geq0\\
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| - | &H\geq0\end{array}</math>
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| - | with
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| - | <math>Y\!=U^\star U^{\star\rm T}.</math>
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| - | Global convergence occurs, in this example, in only a few iterations.
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