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Ranjelin: /* Comparing strain and sstress */
2016-09-29T02:23:29Z
<p><span class="autocomment">Comparing strain and sstress</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In this section we recall a result (see [2]) that relate the strain and sstress criteria. The sstress criterion is given by:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In this section we recall a result (see [2]) that relate the strain and sstress criteria. The sstress criterion is given by:</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>\begin{array}{rl}\text{minimize}&S(D)=<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|\Delta-D<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|^2\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\begin{array}{rl}\text{minimize}&S(D)=<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|\Delta-D<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|^2\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{subject to}&D\in\mathbb{EDM}^n(p)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{subject to}&D\in\mathbb{EDM}^n(p)</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>'''Result.''' The following inequality holds: </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Result.''' The following inequality holds: </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Given <math>\,p\leq n-1\,</math>, for any <math>\,B\in\mathbb{S}_+^n(p)\,</math>, let <math>\,D=\text<del style="color: red; font-weight: bold; text-decoration: none;">{</del>diag}(B)e^{\rm T}+e\;\text<del style="color: red; font-weight: bold; text-decoration: none;">{</del>diag}(B)^{\rm T}-2B\,</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Given <math>\,p\leq n-1\,</math>, for any <math>\,B\in\mathbb{S}_+^n(p)\,</math>, let <math>\,D=<ins style="color: red; font-weight: bold; text-decoration: none;">{</ins>\text diag}(B)e^{\rm T}+e\;<ins style="color: red; font-weight: bold; text-decoration: none;">{</ins>\text diag}(B)^{\rm T}-2B\,</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then </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><del style="color: red; font-weight: bold; text-decoration: none;">\</del>|\Delta-D<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|^2 \geq 4<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|W-B<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|^2</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math><ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|\Delta-D<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|^2 \geq 4<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|W-B<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|^2</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; 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><br>'''Proof.''' Let <math>\,B\in\mathbb{S}_+^n(p)\,</math>; we have</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br>'''Proof.''' Let <math>\,B\in\mathbb{S}_+^n(p)\,</math>; we have</div></td></tr>
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Ranjelin
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Ranjelin: /* Classical Multidimensional Scaling */
2016-09-29T02:21:41Z
<p><span class="autocomment">Classical Multidimensional Scaling</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Given <math>\,p\leq n\,</math>, let <math>\,\mathbb{S}_+^n(p)\,</math> denote the closed set of symmetric <math>\,n\times n\,</math> matrices that are positive semidefinite and have rank no greater than <math>\,p\,</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Given <math>\,p\leq n\,</math>, let <math>\,\mathbb{S}_+^n(p)\,</math> denote the closed set of symmetric <math>\,n\times n\,</math> matrices that are positive semidefinite and have rank no greater than <math>\,p\,</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: #ffa; color:black; font-size: smaller;"><div>Let <math>\,<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|<del style="color: red; font-weight: bold; text-decoration: none;">~\</del>|_{\rm F}\,</math> denote the Frobenius norm and <math>\,\Delta\,</math> a given symmetric <math>\,n\times n\,</math> matrix of squared dissimilarities. Let <math>\,W=W(\Delta)\,</math> and <math>\,W_s=W_s\,(\Delta)</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Let <math>\,|<ins style="color: red; font-weight: bold; text-decoration: none;">|.|</ins>|_{\rm F}\,</math> denote the Frobenius norm and <math>\,\Delta\,</math> a given symmetric <math>\,n\times n\,</math> matrix of squared dissimilarities. Let <math>\,W=W(\Delta)\,</math> and <math>\,W_s=W_s\,(\Delta)</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>Classical MDS can be defined by the optimization problem </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Classical MDS can be defined by the optimization problem </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\text{minimize}_B&<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|W-B<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|_{\rm F}^2\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\text{minimize}_B&<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|W-B<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|_{\rm F}^2\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{subject to}&B\in\mathbb{S}_+^n(p)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{subject to}&B\in\mathbb{S}_+^n(p)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~\textbf{(P)}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~\textbf{(P)}</math></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}{rl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\mbox{minimize}_B&<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|W_s-B<del style="color: red; font-weight: bold; text-decoration: none;">\</del>|_{\rm F}^2\\</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\mbox{minimize}_B&<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|W_s-B<ins style="color: red; font-weight: bold; text-decoration: none;">|</ins>|_{\rm F}^2\\</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{subject to}&B\in\mathbb{S}_+^n(p)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{subject to}&B\in\mathbb{S}_+^n(p)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~(\textbf{P}_\textbf{s)}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~(\textbf{P}_\textbf{s)}</math></div></td></tr>
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Ranjelin
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Ranjelin: /* Mathematical preliminaries */
2016-09-29T02:18:17Z
<p><span class="autocomment">Mathematical preliminaries</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>W_s(D)=-\frac{1}{2}(I-es^{\rm T})D(I-se^{\rm T})\qquad(1)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>W_s(D)=-\frac{1}{2}(I-es^{\rm T})D(I-se^{\rm T})\qquad(1)</math></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>is positive semidefinite with <math>\,\text<del style="color: red; font-weight: bold; text-decoration: none;">{</del>rank}(W_s(D))\leq p\,</math>, where <math>\,e\,</math> is a vector of ones and <math>\,s\,</math> is any vector such that </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>is positive semidefinite with <math>\,<ins style="color: red; font-weight: bold; text-decoration: none;">{</ins>\text rank}(W_s(D))\leq p\,</math>, where <math>\,e\,</math> is a vector of ones and <math>\,s\,</math> is any vector such that </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\,s^{\rm T}e=1\,</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\,s^{\rm T}e=1\,</math>.</div></td></tr>
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Ranjelin
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Ranjelin at 05:57, 5 December 2011
2011-12-05T05:57:14Z
<p></p>
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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>D\,</math> is said to be Euclidean distance matrix of dimension <math>\,p\,</math> if there exists a list of points <math>\,\{z_1\ldots z_n\}\,</math> in </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>D\,</math> is said to be Euclidean distance matrix of dimension <math>\,p\,</math> if there exists a list of points <math>\,\{z_1\ldots z_n\}\,</math> in </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>\,\mathbb R^p\,</math> <math>\,(p\leq n-1)\,</math> such that</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\,\mathbb<ins style="color: red; font-weight: bold; text-decoration: none;">{</ins>R<ins style="color: red; font-weight: bold; text-decoration: none;">}</ins>^p\,</math> <math>\,(p\leq n-1)\,</math> such that</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>d_{ij}=\|z_i-z_j\|^2 \quad\forall\,i,j=1\ldots n</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>d_{ij}=\|z_i-z_j\|^2 \quad\forall\,i,j=1\ldots n</math></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>= Classical Multidimensional Scaling =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Classical Multidimensional Scaling =</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Given <math>\,p\leq n\,</math>, let <math>\,\mathbb <del style="color: red; font-weight: bold; text-decoration: none;">S_</del>+^n(p)\,</math> denote the closed set of symmetric <math>\,n\times n\,</math> matrices that are positive semidefinite and have rank no greater than <math>\,p\,</math>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Given <math>\,p\leq n\,</math>, let <math>\,\mathbb<ins style="color: red; font-weight: bold; text-decoration: none;">{S}_</ins>+^n(p)\,</math> denote the closed set of symmetric <math>\,n\times n\,</math> matrices that are positive semidefinite and have rank no greater than <math>\,p\,</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>Let <math>\,\|~\|_{\rm F}\,</math> denote the Frobenius norm and <math>\,\Delta\,</math> a given symmetric <math>\,n\times n\,</math> matrix of squared dissimilarities. Let <math>\,W=W(\Delta)\,</math> and <math>\,W_s=W_s\,(\Delta)</math>. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Let <math>\,\|~\|_{\rm F}\,</math> denote the Frobenius norm and <math>\,\Delta\,</math> a given symmetric <math>\,n\times n\,</math> matrix of squared dissimilarities. Let <math>\,W=W(\Delta)\,</math> and <math>\,W_s=W_s\,(\Delta)</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>\begin{array}{rl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{minimize}_B&\|W-B\|_{\rm F}^2\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{minimize}_B&\|W-B\|_{\rm F}^2\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\text{subject to}&B\in\mathbb <del style="color: red; font-weight: bold; text-decoration: none;">S_</del>+^n(p)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\text{subject to}&B\in\mathbb<ins style="color: red; font-weight: bold; text-decoration: none;">{S}_</ins>+^n(p)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~\textbf{(P)}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~\textbf{(P)}</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>\begin{array}{rl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\mbox{minimize}_B&\|W_s-B\|_{\rm F}^2\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\mbox{minimize}_B&\|W_s-B\|_{\rm F}^2\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\text{subject to}&B\in\mathbb <del style="color: red; font-weight: bold; text-decoration: none;">S_</del>+^n(p)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\text{subject to}&B\in\mathbb<ins style="color: red; font-weight: bold; text-decoration: none;">{S}_</ins>+^n(p)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~(\textbf{P}_\textbf{s)}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~(\textbf{P}_\textbf{s)}</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>
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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>'''Result.''' The following inequality holds: </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Result.''' The following inequality holds: </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Given <math>\,p\leq n-1\,</math>, for any <math>\,B\in\mathbb <del style="color: red; font-weight: bold; text-decoration: none;">S_</del>+^n(p)\,</math>, let <math>\,D=\text{diag}(B)e^{\rm T}+e\;\text{diag}(B)^{\rm T}-2B\,</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Given <math>\,p\leq n-1\,</math>, for any <math>\,B\in\mathbb<ins style="color: red; font-weight: bold; text-decoration: none;">{S}_</ins>+^n(p)\,</math>, let <math>\,D=\text{diag}(B)e^{\rm T}+e\;\text{diag}(B)^{\rm T}-2B\,</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then </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><math>\|\Delta-D\|^2 \geq 4\|W-B\|^2</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\|\Delta-D\|^2 \geq 4\|W-B\|^2</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; 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><br>'''Proof.''' Let <math>\,B\in\mathbb <del style="color: red; font-weight: bold; text-decoration: none;">S_</del>+^n(p)\,</math>; we have</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><br>'''Proof.''' Let <math>\,B\in\mathbb<ins style="color: red; font-weight: bold; text-decoration: none;">{S}_</ins>+^n(p)\,</math>; we have</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><math>\begin{array}{rcl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rcl}</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>;Theorem. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>;Theorem. </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>For any <math>\,s\in\mathbb R^n\,</math> such that <math>\,s^{\rm T}e=1\,</math> and for any <math>\,p\,</math> we have</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>For any <math>\,s\in\mathbb<ins style="color: red; font-weight: bold; text-decoration: none;">{</ins>R<ins style="color: red; font-weight: bold; text-decoration: none;">}</ins>^n\,</math> such that <math>\,s^{\rm T}e=1\,</math> and for any <math>\,p\,</math> we have</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><math>\,f \leq f_s\qquad(18)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\,f \leq f_s\qquad(18)</math></div></td></tr>
</table>
Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Proximity_Problems&diff=2907&oldid=prev
Ranjelin at 20:15, 24 November 2011
2011-11-24T20:15:31Z
<p></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 20:15, 24 November 2011</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Writing <math>\,a_{ij}=w_{ij}-b_{ij}\,</math> we get</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Writing <math>\,a_{ij}=w_{ij}-b_{ij}\,</math> we get</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>\sum_i\sum_j (\delta_{ij}-d_{ij})^2=2n\,\sum_ia_{ii}^2+4\sum_i\sum_j a_{ij}^2\geq 4\sum_i\sum_ja_{ij}^2\qquad\<del style="color: red; font-weight: bold; text-decoration: none;">square</del></math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\sum_i\sum_j (\delta_{ij}-d_{ij})^2=2n\,\sum_ia_{ii}^2+4\sum_i\sum_j a_{ij}^2\geq 4\sum_i\sum_ja_{ij}^2\qquad\<ins style="color: red; font-weight: bold; text-decoration: none;">diamond</ins></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>= Main result =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Main result =</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>because <math>\,J\,</math> and <math>\,W_s^2\,</math> are PSD we have</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>because <math>\,J\,</math> and <math>\,W_s^2\,</math> are PSD we have</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; 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><math>\lambda_i(W_sJW_s)\leq\lambda_i(W_s^2)\lambda_1(J)=\lambda_i(W_s^2)\qquad\<del style="color: red; font-weight: bold; text-decoration: none;">square</del></math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\lambda_i(W_sJW_s)\leq\lambda_i(W_s^2)\lambda_1(J)=\lambda_i(W_s^2)\qquad\<ins style="color: red; font-weight: bold; text-decoration: none;">diamond</ins></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>= Modified Gower problem =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Modified Gower problem =</div></td></tr>
</table>
Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Proximity_Problems&diff=2906&oldid=prev
Ranjelin at 20:09, 24 November 2011
2011-11-24T20:09:36Z
<p></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 20:09, 24 November 2011</td>
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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>;Lemma</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>;Lemma</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Let <math>\,\lambda(C)\in\<del style="color: red; font-weight: bold; text-decoration: none;">reals</del>^n\,</math> denote the eigenvalues of any symmetric <math>\,n\times n\,</math> matrix in nonincreasing order. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Let <math>\,\lambda(C)\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb{R}</ins>^n\,</math> denote the eigenvalues of any symmetric <math>\,n\times n\,</math> matrix in nonincreasing order. </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>* For all <math>\,A,B\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* For all <math>\,A,B\,</math></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>where <math>\,\lambda_i(W_s)\,</math> denotes the <math>\,i^{\rm th}\,</math> eigenvalue of <math>\,W_s\,</math>. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where <math>\,\lambda_i(W_s)\,</math> denotes the <math>\,i^{\rm th}\,</math> eigenvalue of <math>\,W_s\,</math>. </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>But by a well known result... we have, for <math>\,X\!\in\<del style="color: red; font-weight: bold; text-decoration: none;">reals</del>^{n\times p}\,</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>But by a well known result... we have, for <math>\,X\!\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb{R}</ins>^{n\times p}\,</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><math>\sum_{i=1}^p\lambda_i(W_s)=\max_{X^{\rm T}X=I}\;\mathrm{tr}(X^{\rm T}W_sX)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\sum_{i=1}^p\lambda_i(W_s)=\max_{X^{\rm T}X=I}\;\mathrm{tr}(X^{\rm T}W_sX)</math></div></td></tr>
</table>
Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Proximity_Problems&diff=2905&oldid=prev
Ranjelin at 20:07, 24 November 2011
2011-11-24T20:07:43Z
<p></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 20:07, 24 November 2011</td>
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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>= Introduction =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Introduction =</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>We consider an <math>\,n\times n\,</math> matrix <math>\,D=[d_{ij}]\,</math> defined as a real symmetric matrix that is ''hollow'' <math>\,d_{ii}=0\,</math> for <math>\,i=1\ldots n\,</math> and nonnegative <math>\,d_{ij}\<del style="color: red; font-weight: bold; text-decoration: none;">geqslant </del>0\,</math> for all <math>\,i,j\,</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>We consider an <math>\,n\times n\,</math> matrix <math>\,D=[d_{ij}]\,</math> defined as a real symmetric matrix that is ''hollow'' <math>\,d_{ii}=0\,</math> for <math>\,i=1\ldots n\,</math> and nonnegative <math>\,d_{ij}\<ins style="color: red; font-weight: bold; text-decoration: none;">geq </ins>0\,</math> for all <math>\,i,j\,</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><math>D\,</math> is said to be Euclidean distance matrix of dimension <math>\,p\,</math> if there exists a list of points <math>\,\{z_1\ldots z_n\}\,</math> in </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>D\,</math> is said to be Euclidean distance matrix of dimension <math>\,p\,</math> if there exists a list of points <math>\,\{z_1\ldots z_n\}\,</math> in </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>\,\mathbb R^p\,</math> <math>\,(p\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>n-1)\,</math> such that</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\,\mathbb R^p\,</math> <math>\,(p\<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>n-1)\,</math> such that</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><math>d_{ij}=\|z_i-z_j\|^2 \quad\forall\,i,j=1\ldots n</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>d_{ij}=\|z_i-z_j\|^2 \quad\forall\,i,j=1\ldots n</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>W_s(D)=-\frac{1}{2}(I-es^{\rm T})D(I-se^{\rm T})\qquad(1)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>W_s(D)=-\frac{1}{2}(I-es^{\rm T})D(I-se^{\rm T})\qquad(1)</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: #ffa; color:black; font-size: smaller;"><div>is positive semidefinite with <math>\,\text{rank}(W_s(D))\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>p\,</math>, where <math>\,e\,</math> is a vector of ones and <math>\,s\,</math> is any vector such that </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>is positive semidefinite with <math>\,\text{rank}(W_s(D))\<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>p\,</math>, where <math>\,e\,</math> is a vector of ones and <math>\,s\,</math> is any vector such that </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\,s^{\rm T}e=1\,</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\,s^{\rm T}e=1\,</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>
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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>= Classical Multidimensional Scaling =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Classical Multidimensional Scaling =</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Given <math>\,p\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>n\,</math>, let <math>\,\mathbb S_+^n(p)\,</math> denote the closed set of symmetric <math>\,n\times n\,</math> matrices that are positive semidefinite and have rank no greater than <math>\,p\,</math>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Given <math>\,p\<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>n\,</math>, let <math>\,\mathbb S_+^n(p)\,</math> denote the closed set of symmetric <math>\,n\times n\,</math> matrices that are positive semidefinite and have rank no greater than <math>\,p\,</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>Let <math>\,\|~\|_{\rm F}\,</math> denote the Frobenius norm and <math>\,\Delta\,</math> a given symmetric <math>\,n\times n\,</math> matrix of squared dissimilarities. Let <math>\,W=W(\Delta)\,</math> and <math>\,W_s=W_s\,(\Delta)</math>. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Let <math>\,\|~\|_{\rm F}\,</math> denote the Frobenius norm and <math>\,\Delta\,</math> a given symmetric <math>\,n\times n\,</math> matrix of squared dissimilarities. Let <math>\,W=W(\Delta)\,</math> and <math>\,W_s=W_s\,(\Delta)</math>. </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The following explicit solution to problem '''(P)''' (respectively problem '''(P<sub>s</sub>)''') is well known: </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The following explicit solution to problem '''(P)''' (respectively problem '''(P<sub>s</sub>)''') is well known: </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>let <math>\,\lambda_1\<del style="color: red; font-weight: bold; text-decoration: none;">geqslant</del>\ldots\<del style="color: red; font-weight: bold; text-decoration: none;">geqslant</del>\lambda_n\,</math> denote the eigenvalues of <math>\,W\,</math> (respectively of <math>\,W_s\,</math>) </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>let <math>\,\lambda_1\<ins style="color: red; font-weight: bold; text-decoration: none;">geq</ins>\ldots\<ins style="color: red; font-weight: bold; text-decoration: none;">geq</ins>\lambda_n\,</math> denote the eigenvalues of <math>\,W\,</math> (respectively of <math>\,W_s\,</math>) </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>and <math>\,v_1\ldots v_n\,</math> denote the corresponding eigenvectors. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>and <math>\,v_1\ldots v_n\,</math> denote the corresponding eigenvectors. </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In Section '''5''', we will prove that for any squared dissimilarity matrix <math>\,\Delta\,</math> we have</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In Section '''5''', we will prove that for any squared dissimilarity matrix <math>\,\Delta\,</math> we have</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; 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><math>f\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>f_s</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>f\<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>f_s</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>that is, at the minimum, the strain criterion always gives smaller value than criterion '''(P<sub>s</sub>)'''. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>that is, at the minimum, the strain criterion always gives smaller value than criterion '''(P<sub>s</sub>)'''. </div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* For all <math>\,A,B\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* For all <math>\,A,B\,</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: #ffa; color:black; font-size: smaller;"><div><math>\lambda_i(A+B) \<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>\lambda_i(A)+\lambda_1(B)</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\lambda_i(A+B) \<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>\lambda_i(A)+\lambda_1(B)</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>* For all positive semidefinite <math>\,A,B\,</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* For all positive semidefinite <math>\,A,B\,</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: #ffa; color:black; font-size: smaller;"><div><math>\lambda_i(A\,B)\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>\lambda_i(A)\lambda_1(B)\qquad\diamond</math> </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\lambda_i(A\,B)\<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>\lambda_i(A)\lambda_1(B)\qquad\diamond</math> </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>
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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>'''Result.''' The following inequality holds: </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Result.''' The following inequality holds: </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Given <math>\,p\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>n-1\,</math>, for any <math>\,B\in\mathbb S_+^n(p)\,</math>, let <math>\,D=\text{diag}(B)e^{\rm T}+e\;\text{diag}(B)^{\rm T}-2B\,</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Given <math>\,p\<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>n-1\,</math>, for any <math>\,B\in\mathbb S_+^n(p)\,</math>, let <math>\,D=\text{diag}(B)e^{\rm T}+e\;\text{diag}(B)^{\rm T}-2B\,</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then </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>\|\Delta-D\|^2 \<del style="color: red; font-weight: bold; text-decoration: none;">geqslant </del>4\|W-B\|^2</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\|\Delta-D\|^2 \<ins style="color: red; font-weight: bold; text-decoration: none;">geq </ins>4\|W-B\|^2</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; 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><br>'''Proof.''' Let <math>\,B\in\mathbb S_+^n(p)\,</math>; we have</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><br>'''Proof.''' Let <math>\,B\in\mathbb S_+^n(p)\,</math>; we have</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Writing <math>\,a_{ij}=w_{ij}-b_{ij}\,</math> we get</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Writing <math>\,a_{ij}=w_{ij}-b_{ij}\,</math> we get</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>\sum_i\sum_j (\delta_{ij}-d_{ij})^2=2n\,\sum_ia_{ii}^2+4\sum_i\sum_j a_{ij}^2\<del style="color: red; font-weight: bold; text-decoration: none;">geqslant </del>4\sum_i\sum_ja_{ij}^2\qquad\square</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\sum_i\sum_j (\delta_{ij}-d_{ij})^2=2n\,\sum_ia_{ii}^2+4\sum_i\sum_j a_{ij}^2\<ins style="color: red; font-weight: bold; text-decoration: none;">geq </ins>4\sum_i\sum_ja_{ij}^2\qquad\square</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>= Main result =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Main result =</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For any <math>\,s\in\mathbb R^n\,</math> such that <math>\,s^{\rm T}e=1\,</math> and for any <math>\,p\,</math> we have</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For any <math>\,s\in\mathbb R^n\,</math> such that <math>\,s^{\rm T}e=1\,</math> and for any <math>\,p\,</math> we have</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>\,f \<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>f_s\qquad(18)</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\,f \<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>f_s\qquad(18)</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;"></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>'''Proof.''' We show, for all <math>\,i\,</math>, that <math>\,|\lambda_i(W)|\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant</del>|\lambda_i(W_s)|\,</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>'''Proof.''' We show, for all <math>\,i\,</math>, that <math>\,|\lambda_i(W)|\<ins style="color: red; font-weight: bold; text-decoration: none;">leq</ins>|\lambda_i(W_s)|\,</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Toward that end, we consider two cases:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Toward that end, we consider two cases:</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* If <math>\,W\,</math> is PSD then <math>\,W_s\,</math> is PSD and the inequality becomes <math>\,\lambda_i(W)\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>\lambda_i(W_s)\,</math>. But</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* If <math>\,W\,</math> is PSD then <math>\,W_s\,</math> is PSD and the inequality becomes <math>\,\lambda_i(W)\<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>\lambda_i(W_s)\,</math>. But</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>\lambda_i(W)=\lambda_i(JW_sJ)\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant </del>\lambda_i(W_s)\lambda_1(J)=\lambda_i(W_s)</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\lambda_i(W)=\lambda_i(JW_sJ)\<ins style="color: red; font-weight: bold; text-decoration: none;">leq </ins>\lambda_i(W_s)\lambda_1(J)=\lambda_i(W_s)</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>because <math>\,\lambda_1(J)=1\,</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>because <math>\,\lambda_1(J)=1\,</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>\begin{array}{rcl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rcl}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\lambda_i(W^2) &=& \lambda_i(W_sJW_s-\frac{1}{n}ee^{\rm T}W_sJW_s) \\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\lambda_i(W^2) &=& \lambda_i(W_sJW_s-\frac{1}{n}ee^{\rm T}W_sJW_s) \\</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;">leqslant</del>& \lambda_i(W_sJW_s)+\lambda_1(-\frac{1}{n}ee^{\rm T}W_sJW_s)</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;">leq</ins>& \lambda_i(W_sJW_s)+\lambda_1(-\frac{1}{n}ee^{\rm T}W_sJW_s)</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>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>because <math>\,J\,</math> and <math>\,W_s^2\,</math> are PSD we have</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>because <math>\,J\,</math> and <math>\,W_s^2\,</math> are PSD we have</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; 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><math>\lambda_i(W_sJW_s)\<del style="color: red; font-weight: bold; text-decoration: none;">leqslant</del>\lambda_i(W_s^2)\lambda_1(J)=\lambda_i(W_s^2)\qquad\square</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\lambda_i(W_sJW_s)\<ins style="color: red; font-weight: bold; text-decoration: none;">leq</ins>\lambda_i(W_s^2)\lambda_1(J)=\lambda_i(W_s^2)\qquad\square</math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Modified Gower problem =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Modified Gower problem =</div></td></tr>
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Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Proximity_Problems&diff=2904&oldid=prev
Ranjelin at 20:04, 24 November 2011
2011-11-24T20:04:15Z
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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 20:04, 24 November 2011</td>
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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>D\,</math> is said to be Euclidean distance matrix of dimension <math>\,p\,</math> if there exists a list of points <math>\,\{z_1\ldots z_n\}\,</math> in </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>D\,</math> is said to be Euclidean distance matrix of dimension <math>\,p\,</math> if there exists a list of points <math>\,\{z_1\ldots z_n\}\,</math> in </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>\,\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb </del>R^p\,</math> <math>\,(p\leqslant n-1)\,</math> such that</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\,\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb </ins>R^p\,</math> <math>\,(p\leqslant n-1)\,</math> such that</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><math>d_{ij}=\|z_i-z_j\|^2 \quad\forall\,i,j=1\ldots n</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>d_{ij}=\|z_i-z_j\|^2 \quad\forall\,i,j=1\ldots n</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>where <math>\,\|~\|\,</math> denotes Euclidean norm.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>where <math>\,\|~\|\,</math> denotes Euclidean norm.</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Denote by <math>\,\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb</del>{EDM}^n(p)\,</math> the set of all <math>\,n\times n\,</math> Euclidean distance matrices of dimension <math>\,p\,</math>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Denote by <math>\,\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb</ins>{EDM}^n(p)\,</math> the set of all <math>\,n\times n\,</math> Euclidean distance matrices of dimension <math>\,p\,</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: #ffa; color:black; font-size: smaller;"><div>A problem common to various sciences is to find the Euclidean distance matrix <math>\,D\in\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb</del>{EDM}^n(p)\,</math> closest, in some sense, to a given ''predistance matrix'' <math>\,\Delta=[\delta_{ij}]\,</math> defined to be any symmetric hollow nonnegative real matrix. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>A problem common to various sciences is to find the Euclidean distance matrix <math>\,D\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb</ins>{EDM}^n(p)\,</math> closest, in some sense, to a given ''predistance matrix'' <math>\,\Delta=[\delta_{ij}]\,</math> defined to be any symmetric hollow nonnegative real matrix. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>There are three statements of the closest-EDM problem prevalent in the literature, the multiplicity due primarily to choice of projection on the EDM versus positive semidefinite (PSD) cone and vacillation between the distance-square variable <math>\,d_{ij}\,</math> versus absolute distance <math>\,\sqrt{d_{ij}}\,</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>There are three statements of the closest-EDM problem prevalent in the literature, the multiplicity due primarily to choice of projection on the EDM versus positive semidefinite (PSD) cone and vacillation between the distance-square variable <math>\,d_{ij}\,</math> versus absolute distance <math>\,\sqrt{d_{ij}}\,</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>During the past two decades a large amount of work has been devoted to Euclidean distance matrices and approximation of predistances by an </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>During the past two decades a large amount of work has been devoted to Euclidean distance matrices and approximation of predistances by an </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><math>\,\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb</del>{EDM}^n(p)\,</math> in a series of works including Gower[6-8], Mathar..., Critchley..., Hayden et al..., etc.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><math>\,\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb</ins>{EDM}^n(p)\,</math> in a series of works including Gower[6-8], Mathar..., Critchley..., Hayden et al..., etc.</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>= Mathematical preliminaries = </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Mathematical preliminaries = </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>It is well known that <math>\,D\in\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb</del>{EDM}^n(p)\,</math> if and only if the symmetric <math>\,n\times n\,</math> matrix</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>It is well known that <math>\,D\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb</ins>{EDM}^n(p)\,</math> if and only if the symmetric <math>\,n\times n\,</math> matrix</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><math>W_s(D)=-\frac{1}{2}(I-es^{\rm T})D(I-se^{\rm T})\qquad(1)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>W_s(D)=-\frac{1}{2}(I-es^{\rm T})D(I-se^{\rm T})\qquad(1)</math></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>= Classical Multidimensional Scaling =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= Classical Multidimensional Scaling =</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Given <math>\,p\leqslant n\,</math>, let <math>\,\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb </del>S_+^n(p)\,</math> denote the closed set of symmetric <math>\,n\times n\,</math> matrices that are positive semidefinite and have rank no greater than <math>\,p\,</math>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Given <math>\,p\leqslant n\,</math>, let <math>\,\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb </ins>S_+^n(p)\,</math> denote the closed set of symmetric <math>\,n\times n\,</math> matrices that are positive semidefinite and have rank no greater than <math>\,p\,</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>Let <math>\,\|~\|_{\rm F}\,</math> denote the Frobenius norm and <math>\,\Delta\,</math> a given symmetric <math>\,n\times n\,</math> matrix of squared dissimilarities. Let <math>\,W=W(\Delta)\,</math> and <math>\,W_s=W_s\,(\Delta)</math>. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Let <math>\,\|~\|_{\rm F}\,</math> denote the Frobenius norm and <math>\,\Delta\,</math> a given symmetric <math>\,n\times n\,</math> matrix of squared dissimilarities. Let <math>\,W=W(\Delta)\,</math> and <math>\,W_s=W_s\,(\Delta)</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>\begin{array}{rl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{minimize}_B&\|W-B\|_{\rm F}^2\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\text{minimize}_B&\|W-B\|_{\rm F}^2\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\text{subject to}&B\in\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb </del>S_+^n(p)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\text{subject to}&B\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb </ins>S_+^n(p)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~\textbf{(P)}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~\textbf{(P)}</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>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\mbox{minimize}_B&\|W_s-B\|_{\rm F}^2\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\mbox{minimize}_B&\|W_s-B\|_{\rm F}^2\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\text{subject to}&B\in\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb </del>S_+^n(p)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\text{subject to}&B\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb </ins>S_+^n(p)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~(\textbf{P}_\textbf{s)}</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>\end{array}~~~~~~~~~~~~~~~~~~~~~~~~(\textbf{P}_\textbf{s)}</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>
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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}{rl}\text{minimize}&S(D)=\|\Delta-D\|^2\\</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rl}\text{minimize}&S(D)=\|\Delta-D\|^2\\</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>\text{subject to}&D\in\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb</del>{EDM}^n(p)</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>\text{subject to}&D\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb</ins>{EDM}^n(p)</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>'''Result.''' The following inequality holds: </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Result.''' The following inequality holds: </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Given <math>\,p\leqslant n-1\,</math>, for any <math>\,B\in\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb </del>S_+^n(p)\,</math>, let <math>\,D=\text{diag}(B)e^{\rm T}+e\;\text{diag}(B)^{\rm T}-2B\,</math>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Given <math>\,p\leqslant n-1\,</math>, for any <math>\,B\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb </ins>S_+^n(p)\,</math>, let <math>\,D=\text{diag}(B)e^{\rm T}+e\;\text{diag}(B)^{\rm T}-2B\,</math>. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Then </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><math>\|\Delta-D\|^2 \geqslant 4\|W-B\|^2</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\|\Delta-D\|^2 \geqslant 4\|W-B\|^2</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div> </div></td><td class='diff-marker'> </td><td style="background: #eee; 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><br>'''Proof.''' Let <math>\,B\in\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb </del>S_+^n(p)\,</math>; we have</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><br>'''Proof.''' Let <math>\,B\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb </ins>S_+^n(p)\,</math>; we have</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><math>\begin{array}{rcl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rcl}</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>;Theorem. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>;Theorem. </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>For any <math>\,s\in\<del style="color: red; font-weight: bold; text-decoration: none;">Bbb </del>R^n\,</math> such that <math>\,s^{\rm T}e=1\,</math> and for any <math>\,p\,</math> we have</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>For any <math>\,s\in\<ins style="color: red; font-weight: bold; text-decoration: none;">mathbb </ins>R^n\,</math> such that <math>\,s^{\rm T}e=1\,</math> and for any <math>\,p\,</math> we have</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><math>\,f \leqslant f_s\qquad(18)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\,f \leqslant f_s\qquad(18)</math></div></td></tr>
</table>
Ranjelin
http://www.convexoptimization.com/wikimization/index.php?title=Proximity_Problems&diff=1112&oldid=prev
Cslaw: /* References */
2009-04-16T02:54:43Z
<p><span class="autocomment">References</span></p>
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<td colspan='2' style="background-color: white; color:black;">Revision as of 02:54, 16 April 2009</td>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= References =</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>= References =</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[1] Critchley, F., 1986. On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra Appl. 105, 91-107.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[1] Critchley, F., 1986. On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra Appl. 105, 91-107.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[2] De Leeuw, J., Heiser, W., 1982. Theory of multidimensional scaling. Krishnaiah, P.R., Kanal, I.N.(Eds.), Handbook of Statistics, vol. 2. North-Holland, Amsterdam, pp. 285-316 (chapter 13).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[2] De Leeuw, J., Heiser, W., 1982. Theory of multidimensional scaling. Krishnaiah, P.R., Kanal, I.N.(Eds.), Handbook of Statistics, vol. 2. North-Holland, Amsterdam, pp. 285-316 (chapter 13).</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[3] Gower, J.C., 1966. Some distance properties of latent root and vector methods in multivariate analysis. Biometrika 53, 315-328.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[3] Gower, J.C., 1966. Some distance properties of latent root and vector methods in multivariate analysis. Biometrika 53, 315-328.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[4] Gower, J.C., 1982. Euclidean distance geometry, Math. Scientist 7, 1-14.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[4] Gower, J.C., 1982. Euclidean distance geometry, Math. Scientist 7, 1-14.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[5] Schoenberg, I.J., 1935. Remarks to Maurice Fréchet's article Sur la définition axiomatique d'une classe d'espaces distanciés vectoriellement applicable sur l'espace de Hilbert. Ann. of Math. 38, 724-738.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[5] Schoenberg, I.J., 1935. Remarks to Maurice Fréchet's article Sur la définition axiomatique d'une classe d'espaces distanciés vectoriellement applicable sur l'espace de Hilbert. Ann. of Math. 38, 724-738.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[6] Torgerson, W.S., 1952. Multidimensional scaling: I. Theory and method. Psychometrika 17, 401-419.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[6] Torgerson, W.S., 1952. Multidimensional scaling: I. Theory and method. Psychometrika 17, 401-419.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>[7] Trosset, M.W., 1997. Numerical algorithms for multidimensional scaling. In: Klar, R., Opitz, P. (Eds.), Classification and Knowledge Organization. Springer, Berlin, pp. 80-92.</div></td><td colspan="2"> </td></tr>
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Cslaw
http://www.convexoptimization.com/wikimization/index.php?title=Proximity_Problems&diff=1064&oldid=prev
Ranjelin: /* Mathematical preliminaries */
2009-03-06T00:19:52Z
<p><span class="autocomment">Mathematical preliminaries</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The aim of this short paper is to clarify that point...</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>The aim of this short paper is to clarify that point...</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>We shall also use <del style="color: red; font-weight: bold; text-decoration: none;">the </del>notation</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>We shall also use <ins style="color: red; font-weight: bold; text-decoration: none;">[http://orion.uwaterloo.ca/~hwolkowi Wolkowicz'] </ins>notation<ins style="color: red; font-weight: bold; text-decoration: none;">:</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rcl}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><math>\begin{array}{rcl}</div></td></tr>
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Ranjelin