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Nonnegative matrix factorization - Revision history
2024-03-29T06:19:18Z
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http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=3112&oldid=prev
Ranjelin at 00:40, 31 October 2016
2016-10-31T00:40:44Z
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>by solving</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>by solving</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>\begin{array}{cl}<del style="color: red; font-weight: bold; text-decoration: none;">\mbox</del>{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\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\begin{array}{cl}{<ins style="color: red; font-weight: bold; text-decoration: none;">\text </ins>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\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\mbox</del>{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\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>{<ins style="color: red; font-weight: bold; text-decoration: none;">\text </ins>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\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&W\geq0\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&W\geq0\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&H\geq0\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&H\geq0\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>&<del style="color: red; font-weight: bold; text-decoration: none;">\mbox</del>{rank}\,Z\leq2\end{array}</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>&{<ins style="color: red; font-weight: bold; text-decoration: none;">\text </ins>rank}\,Z\leq2\end{array}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>which follows from the fact, at optimality,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>which follows from the fact, at optimality,</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><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\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><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\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\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\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\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\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&W\geq0\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&W\geq0\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&H\geq0\end{array}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>&H\geq0\end{array}</math></div></td></tr>
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Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=2918&oldid=prev
Ranjelin at 20:44, 24 November 2011
2011-11-24T20:44:28Z
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>X=\!\left[\!\begin{array}{ccc}17&28&42\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>X=\!\left[\!\begin{array}{ccc}17&28&42\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> 16&47&51\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> 16&47&51\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"> </del>17&82&72\end{array}\!\right],</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>17&82&72\end{array}\!\right],</math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>find a nonnegative factorization</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>find a nonnegative factorization</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math> X=WH\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math> X=WH\,</math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Use the known closed-form solution for a direction vector <math>Y\,</math> to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Use the known closed-form solution for a direction vector <math>Y\,</math> to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>set <math>_{}Z^*\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>set <math>_{}Z^*\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^<ins style="color: red; font-weight: bold; text-decoration: none;">{</ins>\mathbf{8<ins style="color: red; font-weight: bold; text-decoration: none;">}</ins>}</math> to a nonincreasingly ordered diagonalization and</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>_{}U^*\!=_{\!}Q(:\,,_{^{}}3<del style="color: red; font-weight: bold; text-decoration: none;">\!</del>:<del style="color: red; font-weight: bold; text-decoration: none;">\!</del>8)\!\in_{\!}\mathbb{R}^{\mathbf{8}\times\mathbf{6}}</math>,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>_{}U^*\!=_{\!}Q(:\,,_{^{}}3:8)\!\in_{\!}\mathbb{R}^{\mathbf{8}\times\mathbf{6}}</math>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>then <math>Y\!=U^* U^{*\rm T}.</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>then <math>Y\!=U^* U^{*\rm T}.</math></div></td></tr>
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Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=2917&oldid=prev
Ranjelin at 20:41, 24 November 2011
2011-11-24T20:41:11Z
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<td colspan='2' style="background-color: white; color:black;">Revision as of 20:41, 24 November 2011</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>which follows from the fact, at optimality,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>which follows from the fact, at optimality,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math> Z^<del style="color: red; font-weight: bold; text-decoration: none;">\star</del>=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right]\begin{array}{c}\textbf{[}\,I~~W^{\rm T}~H\,\textbf{]}</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math> Z^<ins style="color: red; font-weight: bold; text-decoration: none;">*</ins>=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right]\begin{array}{c}\textbf{[}\,I~~W^{\rm T}~H\,\textbf{]}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Use the known closed-form solution for a direction vector <math>Y\,</math> to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Use the known closed-form solution for a direction vector <math>Y\,</math> to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>set <math>_{}Z^<del style="color: red; font-weight: bold; text-decoration: none;">\star</del>\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>set <math>_{}Z^<ins style="color: red; font-weight: bold; text-decoration: none;">*</ins>\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>_{}U^<del style="color: red; font-weight: bold; text-decoration: none;">\star</del>\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\mathbb{R}^{\mathbf{8}\times\mathbf{6}}</math>,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>_{}U^<ins style="color: red; font-weight: bold; text-decoration: none;">*</ins>\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\mathbb{R}^{\mathbf{8}\times\mathbf{6}}</math>,</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>then <math>Y\!=U^<del style="color: red; font-weight: bold; text-decoration: none;">\star </del>U^{<del style="color: red; font-weight: bold; text-decoration: none;">\star</del>\rm T}.</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>then <math>Y\!=U^<ins style="color: red; font-weight: bold; text-decoration: none;">* </ins>U^{<ins style="color: red; font-weight: bold; text-decoration: none;">*</ins>\rm T}.</math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>with </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>with </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>Y\!=U^<del style="color: red; font-weight: bold; text-decoration: none;">\star </del>U^{<del style="color: red; font-weight: bold; text-decoration: none;">\star</del>\rm T}.</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>Y\!=U^<ins style="color: red; font-weight: bold; text-decoration: none;">* </ins>U^{<ins style="color: red; font-weight: bold; text-decoration: none;">*</ins>\rm T}.</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Global convergence occurs, in this example, in only a few iterations.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Global convergence occurs, in this example, in only a few iterations.</div></td></tr>
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Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=2916&oldid=prev
Ranjelin at 20:40, 24 November 2011
2011-11-24T20:40:03Z
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<td colspan='2' style="background-color: white; color:black;">Revision as of 20:40, 24 November 2011</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>set <math>_{}Z^\star\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>set <math>_{}Z^\star\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>_{}U^\star\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\<del style="color: red; font-weight: bold; text-decoration: none;">reals</del>^{\mathbf{8}\times\mathbf{6}}</math>,</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>_{}U^\star\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb{R}</ins>^{\mathbf{8}\times\mathbf{6}}</math>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>then <math>Y\!=U^\star U^{\star\rm T}.</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>then <math>Y\!=U^\star U^{\star\rm T}.</math></div></td></tr>
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Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=1942&oldid=prev
Ranjelin: Protected "Nonnegative matrix factorization" [edit=autoconfirmed:move=autoconfirmed]
2010-02-17T12:24:12Z
<p>Protected "<a href="/wikimization/index.php/Nonnegative_matrix_factorization" title="Nonnegative matrix factorization">Nonnegative matrix factorization</a>" [edit=autoconfirmed:move=autoconfirmed]</p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 12:24, 17 February 2010</td>
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Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=1941&oldid=prev
Ranjelin: Reverted edits by 41.190.16.17 (Talk); changed back to last version by Ranjelin
2010-02-17T12:24:01Z
<p>Reverted edits by <a href="/wikimization/index.php/Special:Contributions/41.190.16.17" title="Special:Contributions/41.190.16.17">41.190.16.17</a> (<a href="/wikimization/index.php?title=User_talk:41.190.16.17&action=edit" class="new" title="User talk:41.190.16.17">Talk</a>); changed back to last version by <a href="/wikimization/index.php/User:Ranjelin" title="User:Ranjelin">Ranjelin</a></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">xA3hHv <a href="</del>http://<del style="color: red; font-weight: bold; text-decoration: none;">fhanfeusylsr</del>.com/<del style="color: red; font-weight: bold; text-decoration: none;">"</del>><del style="color: red; font-weight: bold; text-decoration: none;">fhanfeusylsr</del></a>, <del style="color: red; font-weight: bold; text-decoration: none;">[url</del>=<del style="color: red; font-weight: bold; text-decoration: none;">http://idhijwizkysp.com</del>/]<del style="color: red; font-weight: bold; text-decoration: none;">idhijwizkysp</del>[<del style="color: red; font-weight: bold; text-decoration: none;">/url</del>], [<del style="color: red; font-weight: bold; text-decoration: none;">link</del>=<del style="color: red; font-weight: bold; text-decoration: none;">http</del>:/<del style="color: red; font-weight: bold; text-decoration: none;">/cxgygcljwequ</del>.<del style="color: red; font-weight: bold; text-decoration: none;">com</del>/<del style="color: red; font-weight: bold; text-decoration: none;">]cxgygcljwequ[</del>/<del style="color: red; font-weight: bold; text-decoration: none;">link]</del>, <del style="color: red; font-weight: bold; text-decoration: none;">http:</del>/<del style="color: red; font-weight: bold; text-decoration: none;">/ogpjkfcdsjji</del>.<del style="color: red; font-weight: bold; text-decoration: none;">com</del>/</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Exercise from [</ins>http://<ins style="color: red; font-weight: bold; text-decoration: none;">meboo.convexoptimization</ins>.com/<ins style="color: red; font-weight: bold; text-decoration: none;">Meboo.html Convex Optimization & Euclidean Distance Geometry], ch.4:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Given rank-2 nonnegative matrix </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math</ins>><ins style="color: red; font-weight: bold; text-decoration: none;">X=\!\left[\!\begin{array}{ccc}17&28&42\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> 16&47&51\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"> 17&82&72\end{array}\!\right],</ins></<ins style="color: red; font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">find </ins>a <ins style="color: red; font-weight: bold; text-decoration: none;">nonnegative factorization</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math</ins>> <ins style="color: red; font-weight: bold; text-decoration: none;">X=WH\</ins>,<ins style="color: red; font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">by solving</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><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\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\mbox{subject to}&Z</ins>=<ins style="color: red; font-weight: bold; text-decoration: none;">\left[\begin{array}{ccc}I&W^{\rm T}&H\\W&A&X\\H^{\rm T}&X^{\rm T}&B\end{array}\right]\succeq0\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&W\geq0\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&H\geq0\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&\mbox{rank}\,Z\leq2\end{array}<</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">which follows from the fact, at optimality,</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math> Z^\star=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right</ins>]<ins style="color: red; font-weight: bold; text-decoration: none;">\begin{array}{c}\textbf{</ins>[<ins style="color: red; font-weight: bold; text-decoration: none;">}\,I~~W^{\rm T}~H\,\textbf{</ins>]<ins style="color: red; font-weight: bold; text-decoration: none;">}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\end{array}</math></ins></div></td></tr>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Use the known closed-form solution for a direction vector <math>Y\</ins>,<ins style="color: red; font-weight: bold; text-decoration: none;"></math> to regulate rank (rank constraint is replaced) by </ins>[<ins style="color: red; font-weight: bold; text-decoration: none;">[Convex Iteration]]; </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">set <math>_{}Z^\star\!</ins>=<ins style="color: red; font-weight: bold; text-decoration: none;">Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math>_{}U^\star\!=_{\!}Q(</ins>:<ins style="color: red; font-weight: bold; text-decoration: none;">\,,_{^{}}3\!:\!8)\!\in_{\!}\reals^{\mathbf{8}\times\mathbf{6}}<</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">math>,</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">then <math>Y\!=U^\star U^{\star\rm T}</ins>.<ins style="color: red; font-weight: bold; text-decoration: none;"><</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><br></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">In summary, initialize <math>Y=I\,<</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">math> then alternate solution of </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math>\begin{array}{cl}\mbox{minimize}_{A\in\mathbb{S}^3</ins>,<ins style="color: red; font-weight: bold; text-decoration: none;">\,B\in\mathbb{S}^3,\,W\in\mathbb{R}^{3\times2},\,H\in\mathbb{R}^{2\times3}}&\langle Z\,,Y\rangle\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">\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\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&W\geq0\\</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">&H\geq0\end{array}<</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">with </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;"><math>Y\!=U^\star U^{\star\rm T}</ins>.<ins style="color: red; font-weight: bold; text-decoration: none;"><</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Global convergence occurs, in this example, in only a few iterations.</ins></div></td></tr>
</table>
Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=1928&oldid=prev
41.190.16.17: ruqKKcWrG
2010-02-17T10:44:52Z
<p>ruqKKcWrG</p>
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<td colspan='2' style="background-color: white; color:black;">←Older revision</td>
<td colspan='2' style="background-color: white; color:black;">Revision as of 10:44, 17 February 2010</td>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Exercise from [</del>http://<del style="color: red; font-weight: bold; text-decoration: none;">meboo.convexoptimization</del>.com/<del style="color: red; font-weight: bold; text-decoration: none;">Meboo.html Convex Optimization & Euclidean Distance Geometry], ch.4:</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">xA3hHv <a href="</ins>http://<ins style="color: red; font-weight: bold; text-decoration: none;">fhanfeusylsr</ins>.com/<ins style="color: red; font-weight: bold; text-decoration: none;">"</ins>><ins style="color: red; font-weight: bold; text-decoration: none;">fhanfeusylsr</ins></a>, <ins style="color: red; font-weight: bold; text-decoration: none;">[url</ins>=<ins style="color: red; font-weight: bold; text-decoration: none;">http://idhijwizkysp.com</ins>/]<ins style="color: red; font-weight: bold; text-decoration: none;">idhijwizkysp</ins>[<ins style="color: red; font-weight: bold; text-decoration: none;">/url</ins>], [<ins style="color: red; font-weight: bold; text-decoration: none;">link</ins>=<ins style="color: red; font-weight: bold; text-decoration: none;">http</ins>:/<ins style="color: red; font-weight: bold; text-decoration: none;">/cxgygcljwequ</ins>.<ins style="color: red; font-weight: bold; text-decoration: none;">com</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">]cxgygcljwequ[</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">link]</ins>, <ins style="color: red; font-weight: bold; text-decoration: none;">http:</ins>/<ins style="color: red; font-weight: bold; text-decoration: none;">/ogpjkfcdsjji</ins>.<ins style="color: red; font-weight: bold; text-decoration: none;">com</ins>/</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Given rank-2 nonnegative matrix </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><math</del>><del style="color: red; font-weight: bold; text-decoration: none;">X=\!\left[\!\begin{array}{ccc}17&28&42\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"> 16&47&51\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"> 17&82&72\end{array}\!\right],</del></<del style="color: red; font-weight: bold; text-decoration: none;">math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">find </del>a <del style="color: red; font-weight: bold; text-decoration: none;">nonnegative factorization</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><math</del>> <del style="color: red; font-weight: bold; text-decoration: none;">X=WH\</del>,<del style="color: red; font-weight: bold; text-decoration: none;"></math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">by solving</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><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\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\mbox{subject to}&Z</del>=<del style="color: red; font-weight: bold; text-decoration: none;">\left[\begin{array}{ccc}I&W^{\rm T}&H\\W&A&X\\H^{\rm T}&X^{\rm T}&B\end{array}\right]\succeq0\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">&W\geq0\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">&H\geq0\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">&\mbox{rank}\,Z\leq2\end{array}<</del>/<del style="color: red; font-weight: bold; text-decoration: none;">math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">which follows from the fact, at optimality,</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><math> Z^\star=\left[\!\begin{array}{c}I\\W\\H^{\rm T}\end{array}\!\right</del>]<del style="color: red; font-weight: bold; text-decoration: none;">\begin{array}{c}\textbf{</del>[<del style="color: red; font-weight: bold; text-decoration: none;">}\,I~~W^{\rm T}~H\,\textbf{</del>]<del style="color: red; font-weight: bold; text-decoration: none;">}</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\end{array}</math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Use the known closed-form solution for a direction vector <math>Y\</del>,<del style="color: red; font-weight: bold; text-decoration: none;"></math> to regulate rank (rank constraint is replaced) by </del>[<del style="color: red; font-weight: bold; text-decoration: none;">[Convex Iteration]]; </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">set <math>_{}Z^\star\!</del>=<del style="color: red; font-weight: bold; text-decoration: none;">Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to a nonincreasingly ordered diagonalization and</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><math>_{}U^\star\!=_{\!}Q(</del>:<del style="color: red; font-weight: bold; text-decoration: none;">\,,_{^{}}3\!:\!8)\!\in_{\!}\reals^{\mathbf{8}\times\mathbf{6}}<</del>/<del style="color: red; font-weight: bold; text-decoration: none;">math>,</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">then <math>Y\!=U^\star U^{\star\rm T}</del>.<del style="color: red; font-weight: bold; text-decoration: none;"><</del>/<del style="color: red; font-weight: bold; text-decoration: none;">math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><br></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">In summary, initialize <math>Y=I\,<</del>/<del style="color: red; font-weight: bold; text-decoration: none;">math> then alternate solution of </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><math>\begin{array}{cl}\mbox{minimize}_{A\in\mathbb{S}^3</del>,<del style="color: red; font-weight: bold; text-decoration: none;">\,B\in\mathbb{S}^3,\,W\in\mathbb{R}^{3\times2},\,H\in\mathbb{R}^{2\times3}}&\langle Z\,,Y\rangle\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">\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\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">&W\geq0\\</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">&H\geq0\end{array}<</del>/<del style="color: red; font-weight: bold; text-decoration: none;">math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">with </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;"><math>Y\!=U^\star U^{\star\rm T}</del>.<del style="color: red; font-weight: bold; text-decoration: none;"><</del>/<del style="color: red; font-weight: bold; text-decoration: none;">math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Global convergence occurs, in this example, in only a few iterations.</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
</table>
41.190.16.17
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=1660&oldid=prev
Ranjelin at 22:16, 28 September 2009
2009-09-28T22:16:23Z
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Use the known closed-form solution for a direction vector <math>Y\,</math> to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Use the known closed-form solution for a direction vector <math>Y\,</math> to regulate rank (rank constraint is replaced) by [[Convex Iteration]]; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>set <math>_{}Z^\star\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to <del style="color: red; font-weight: bold; text-decoration: none;">an </del>ordered diagonalization and</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>set <math>_{}Z^\star\!=Q\Lambda Q^{\rm T}\!\in\mathbb{S}^\mathbf{8}</math> to <ins style="color: red; font-weight: bold; text-decoration: none;">a nonincreasingly </ins>ordered diagonalization and</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>_{}U^\star\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\reals^{\mathbf{8}\times\mathbf{6}}</math>,</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>_{}U^\star\!=_{\!}Q(:\,,_{^{}}3\!:\!8)\!\in_{\!}\reals^{\mathbf{8}\times\mathbf{6}}</math>,</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>then <math>Y\!=U^\star U^{\star\rm T}.</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>then <math>Y\!=U^\star U^{\star\rm T}.</math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>Y\!=U^\star U^{\star\rm T}.</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>Y\!=U^\star U^{\star\rm T}.</math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Global convergence <del style="color: red; font-weight: bold; text-decoration: none;">should occur</del>, in this example, in only a few iterations.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Global convergence <ins style="color: red; font-weight: bold; text-decoration: none;">occurs</ins>, in this example, in only a few iterations.</div></td></tr>
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Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=1659&oldid=prev
Ranjelin at 21:48, 28 September 2009
2009-09-28T21:48:51Z
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">Example </del>from [http://meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], ch.4:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins style="color: red; font-weight: bold; text-decoration: none;">Exercise </ins>from [http://meboo.convexoptimization.com/Meboo.html Convex Optimization & Euclidean Distance Geometry], ch.4:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Given rank-2 nonnegative matrix </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Given rank-2 nonnegative matrix </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>Y\!=U^\star U^{\star\rm T}.</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>Y\!=U^\star U^{\star\rm T}.</math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Global convergence should occur, in this example, in only a few iterations.</div></td></tr>
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Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Nonnegative_matrix_factorization&diff=1658&oldid=prev
Ranjelin at 21:27, 28 September 2009
2009-09-28T21:27:50Z
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>then <math>Y\!=U^\star U^{\star\rm T}.</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>then <math>Y\!=U^\star U^{\star\rm T}.</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In summary, initialize <math>Y=I\,</math> then alternate solution of </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In summary, initialize <math>Y=I\,</math> then alternate solution of </div></td></tr>
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Ranjelin