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inductive Kernel Low-rank Decomposition with Priors: A Generalized Nystrom Method - Revision history
2024-03-29T07:42:42Z
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Conversion script: Conversion script moved page Inductive Kernel Low-rank Decomposition with Priors: A Generalized Nystrom Method to inductive Kernel Low-rank Decomposition with Priors: A Generalized Nystrom Method: Converting page titles to lowercase
2017-08-30T13:46:11Z
<p>Conversion script moved page <a href="/statwiki/index.php?title=Inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method" class="mw-redirect" title="Inductive Kernel Low-rank Decomposition with Priors: A Generalized Nystrom Method">Inductive Kernel Low-rank Decomposition with Priors: A Generalized Nystrom Method</a> to <a href="/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method" title="inductive Kernel Low-rank Decomposition with Priors: A Generalized Nystrom Method">inductive Kernel Low-rank Decomposition with Priors: A Generalized Nystrom Method</a>: Converting page titles to lowercase</p>
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<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 09:46, 30 August 2017</td>
</tr><tr><td colspan="2" class="diff-notice" lang="us"><div class="mw-diff-empty">(No difference)</div>
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http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=23020&oldid=prev
Z47xu: /* Optimization */
2013-08-25T00:32:08Z
<p><span dir="auto"><span class="autocomment">Optimization</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:32, 24 August 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l45">Line 45:</td>
<td colspan="2" class="diff-lineno">Line 45:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>while <math>J(B_A^*)>J(S^{(t)})+tr({\nabla}_{S^(t)}(B_A^*-S^{(t)}))+{\frac A 2} \| B_A^*-S^{(t)} \|_F^2 </math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>while <math>J(B_A^*)>J(S^{(t)})+tr({\nabla}_{S^(t)}(B_A^*-S^{(t)}))+{\frac A 2} \| B_A^*-S^{(t)} \|_F^2 </math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">----</del>Increase A by a constant times;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>\ \qquad</math> <math>\ \qquad</math> </ins>Increase A by a constant times<ins style="font-weight: bold; text-decoration: none;">;</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>\ \qquad</math> <math>\ \qquad</math> <math>B_{A}^*={\underset{B \succeq 0}{\text{arg min}}} \ tr({\nabla}_{S^{(t)}}B)+{\frac A 2} \| B-S^{(t)} \|_F^2</math></ins>;</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><math>B_{A}^*={\underset{B \succeq 0}{\text{arg min}}} \ tr({\nabla}_{S^{(t)}}B)+{\frac A 2} \| B-S^{(t)} \|_F^2</math>;</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">----</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>end while</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>end while</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
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Z47xu
http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=23019&oldid=prev
Z47xu: /* Optimization */
2013-08-25T00:28:25Z
<p><span dir="auto"><span class="autocomment">Optimization</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:28, 24 August 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l47">Line 47:</td>
<td colspan="2" class="diff-lineno">Line 47:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>----Increase A by a constant times;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>----Increase A by a constant times;</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">----</del><math>B_{A}^*={\underset{B \succeq 0}{\text{arg min}}} \ tr({\nabla}_{S^{(t)}}B)+{\frac A 2} \| B-S^{(t)} \|_F^2</math>;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>B_{A}^*={\underset{B \succeq 0}{\text{arg min}}} \ tr({\nabla}_{S^{(t)}}B)+{\frac A 2} \| B-S^{(t)} \|_F^2</math>;</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">----</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>end while</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>end while</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>
Z47xu
http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=23018&oldid=prev
Z47xu: /* Optimization */
2013-08-25T00:27:43Z
<p><span dir="auto"><span class="autocomment">Optimization</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:27, 24 August 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l45">Line 45:</td>
<td colspan="2" class="diff-lineno">Line 45:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>while <math>J(B_A^*)>J(S^{(t)})+tr({\nabla}_{S^(t)}(B_A^*-S^{(t)}))+{\frac A 2} \| B_A^*-S^{(t)} \|_F^2 </math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>while <math>J(B_A^*)>J(S^{(t)})+tr({\nabla}_{S^(t)}(B_A^*-S^{(t)}))+{\frac A 2} \| B_A^*-S^{(t)} \|_F^2 </math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">****</del>Increase A by a constant times;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">----</ins>Increase A by a constant times;</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">****</del><math>B_{A}^*={\underset{B \succeq 0}{\text{arg min}}} \ tr({\nabla}_{S^{(t)}}B)+{\frac A 2} \| B-S^{(t)} \|_F^2</math>;</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">----</ins><math>B_{A}^*={\underset{B \succeq 0}{\text{arg min}}} \ tr({\nabla}_{S^{(t)}}B)+{\frac A 2} \| B-S^{(t)} \|_F^2</math>;</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>end while</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>end while</div></td></tr>
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Z47xu
http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=23014&oldid=prev
Z47xu: /* Optimization */
2013-08-24T05:51:42Z
<p><span dir="auto"><span class="autocomment">Optimization</span></span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:51, 24 August 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l34">Line 34:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Since both the objective function and the positive semidefinite constraints are convex, there exists a global optimal. The paper uses a gradient mapping strategy <ref name="Nemi1994"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Since both the objective function and the positive semidefinite constraints are convex, there exists a global optimal. The paper uses a gradient mapping strategy <ref name="Nemi1994"></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Nemirovski, A. Efficient method in convex programming. Lecture Notes 1994. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Nemirovski, A. Efficient method in convex programming. Lecture Notes 1994. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div></ref> to find the optimal solution. The method has a iterative gradient descent step and projection step. Given an initial solution <math>S^{(t)}</math> we update it by <math>S^{(t+1)} = S^{(t)}+ \eta^{(t)}\<del style="font-weight: bold; text-decoration: none;">nabla </del>S^{(t)}</math> where <math>\<del style="font-weight: bold; text-decoration: none;">nabla </del>S ^{(t)}</math> is the gradient of the objective function at <math>S</math>. <center><math> \nabla S = 2\lambda(S_0-S) +2E_l^T(E_lSE_l^T-K^*)E_l </math> </center>The step length <math>\eta^{(t)}</math>is determined by the Armijo Goldstein rule<ref name="Nemi1994"></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></ref> to find the optimal solution. The method has a iterative gradient descent step and projection step. Given an initial solution <math>S^{(t)}</math> we update it by <math>S^{(t+1)} = S^{(t)}+ \eta^{(t)}\<ins style="font-weight: bold; text-decoration: none;">nabla_{</ins>S^{(t)<ins style="font-weight: bold; text-decoration: none;">}</ins>}</math> where <math>\<ins style="font-weight: bold; text-decoration: none;">nabla_{</ins>S ^{(t)<ins style="font-weight: bold; text-decoration: none;">}</ins>}</math> is the gradient of the objective function at <math>S</math>. <center><math> \nabla S = 2\lambda(S_0-S) +2E_l^T(E_lSE_l^T-K^*)E_l </math> </center>The step length <math>\eta^{(t)}</math>is determined by the Armijo Goldstein rule<ref name="Nemi1994"></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Reference</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Reference</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div></ref>. After the descent step, we project the iterate <math>S^{t+1)}</math> onto the set of positive semi-definite cones as follows <center><math>S^{(t+1)} = U^{(t+1)}\Lambda_{+}^{(t+1)}(U^{(t+1)})^T</math></center> where <math>U^{(t+1)}</math> and <math>\Lambda^{(t+1)}</math> are the eigenvectors and eigenvalues of <math>S^{(t+1)}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></ref>. </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''The descent step:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>B_{A}^*={\underset{B \succeq 0}{\text{arg min}}} \ tr({\nabla}_{S^{(t)}}B)+{\frac A 2} \| B-S^{(t)} \|_F^2</math>;</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">while <math>J(B_A^*)>J(S^{(t)})+tr({\nabla}_{S^(t)}(B_A^*-S^{(t)}))+{\frac A 2} \| B_A^*-S^{(t)} \|_F^2 </math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">****Increase A by a constant times;</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">****<math>B_{A}^*={\underset{B \succeq 0}{\text{arg min}}} \ tr({\nabla}_{S^{(t)}}B)+{\frac A 2} \| B-S^{(t)} \|_F^2</math>;</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">end while</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''finish descent step.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>After the descent step, we project the iterate <math>S^{t+1)}</math> onto the set of positive semi-definite cones as follows <center><math>S^{(t+1)} = U^{(t+1)}\Lambda_{+}^{(t+1)}(U^{(t+1)})^T</math></center> where <math>U^{(t+1)}</math> and <math>\Lambda^{(t+1)}</math> are the eigenvectors and eigenvalues of <math>S^{(t+1)}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Initialization==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Initialization==</div></td></tr>
</table>
Z47xu
http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=23013&oldid=prev
Z47xu: /* Side Information as Grouping Constraints */
2013-08-24T04:20:34Z
<p><span dir="auto"><span class="autocomment">Side Information as Grouping Constraints</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:20, 24 August 2013</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Side Information as Grouping Constraints==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Side Information as Grouping Constraints==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The side information can used as grouping constraint and then be incorporated into the objective function. Given a set of grouping constraints denoted by <math>I</math>. Let <math>\Chi_I</math> be the subset of samples with such constraints. Then define <math>T\in R^{\Chi_I\times\Chi_I}</math> with <math>T_{ij} = 1 </math> if <math>x_i<del style="font-weight: bold; text-decoration: none;"></math> and <math></del>x_j<del style="font-weight: bold; text-decoration: none;"></math> belongs to <math></del>\Chi_I</math> and 0 otherwise. Then the objective function becomes, <center><math>min_{S\in R^{m \times m}} \lambda \|S-S_0\|^2_F+\|T\cdot (E_lSE_l^T) - K^*_l\|^2_F </math> s.t. <math>S \succeq 0.</math></center></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The side information can used as grouping constraint and then be incorporated into the objective function. Given a set of grouping constraints denoted by <math>I</math>. Let <math>\Chi_I</math> be the subset of samples with such constraints. Then define <math>T\in R^{\Chi_I\times\Chi_I}</math> with <math>T_{ij} = 1 </math> if <math><ins style="font-weight: bold; text-decoration: none;">(</ins>x_i<ins style="font-weight: bold; text-decoration: none;">,</ins>x_j<ins style="font-weight: bold; text-decoration: none;">) \in </ins>\Chi_I</math> and 0 otherwise. Then the objective function becomes, <center><math>min_{S\in R^{m \times m}} \lambda \|S-S_0\|^2_F+\|T\cdot (E_lSE_l^T) - K^*_l\|^2_F </math> s.t. <math>S \succeq 0.</math></center></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Optimization==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Optimization==</div></td></tr>
</table>
Z47xu
http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=20153&oldid=prev
HanShengSun: /* Including Side Information */
2013-07-08T14:27:12Z
<p><span dir="auto"><span class="autocomment">Including Side Information</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:27, 8 July 2013</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To satisfy both criteria the paper arrives at the following optimization problem: <center><math>min_{S\in R^{m \times m}} \lambda \|S-S_0\|^2_F+\|E_lSE_l^T - K^*_l\|^2_F </math> s.t. <math>S \succeq 0.</math></center></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To satisfy both criteria the paper arrives at the following optimization problem: <center><math>min_{S\in R^{m \times m}} \lambda \|S-S_0\|^2_F+\|E_lSE_l^T - K^*_l\|^2_F </math> s.t. <math>S \succeq 0.</math></center></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Here, <math>S_0 = W^{\dagger}</math> and <math>W^{\dagger}</math> is from the standard Nystrom method. <math>K_l^*</math> is the ideal kernel, it equals to 1 when <math>x_i</math> and <math>x_j</math> are in the same class, and 0 otherwise. The first term of the objective function corresponds to the first consideration and the second term of the objective corresponds to the second. The reason for choosing the euclidean norm is that minimizing the Euclidian distance is related to maximizing the alignment. We can then use the normalized kernel alignment score afterwards as an in- dependent measure to choose the hyper-parameter <del style="font-weight: bold; text-decoration: none;">λ</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Here, <math>S_0 = W^{\dagger}</math> and <math>W^{\dagger}</math> is from the standard Nystrom method. <math>K_l^*</math> is the ideal kernel, it equals to 1 when <math>x_i</math> and <math>x_j</math> are in the same class, and 0 otherwise. The first term of the objective function corresponds to the first consideration and the second term of the objective corresponds to the second. The reason for choosing the euclidean norm is that minimizing the Euclidian distance is related to maximizing the alignment. We can then use the normalized kernel alignment score afterwards as an in- dependent measure to choose the hyper-parameter <ins style="font-weight: bold; text-decoration: none;"><math>\lambda</math></ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Side Information as Grouping Constraints==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Side Information as Grouping Constraints==</div></td></tr>
</table>
HanShengSun
http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=20149&oldid=prev
HanShengSun: /* Bilateral Extrapolation of Dictionary Kernel */
2013-07-08T03:21:42Z
<p><span dir="auto"><span class="autocomment">Bilateral Extrapolation of Dictionary Kernel</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:21, 7 July 2013</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Bilateral Extrapolation of Dictionary Kernel==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Bilateral Extrapolation of Dictionary Kernel==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The ijth component of the kernel matrix is constructed as<center><math>K_{ij} = E_iW^{\dagger}E_j</math></center> Where, <math>E_i</math> is the ith row of the extrapolation matrix <math>E</math>, i.e. the similarity between any two points <math>x_i</math> and <math>x_j</math> is constructed by first computing their respective similarities to the landmark set nd then modulated by the inverse of the similarities among the landmark points <math>W^{\dagger}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The ijth component of the kernel matrix is constructed as<center><math>K_{ij} = E_iW^{\dagger}E_j<ins style="font-weight: bold; text-decoration: none;">^T</ins></math></center> Where, <math>E_i</math> is the ith row of the extrapolation matrix <math>E</math>, i.e. the similarity between any two points <math>x_i</math> and <math>x_j</math> is constructed by first computing their respective similarities to the landmark set nd then modulated by the inverse of the similarities among the landmark points <math>W^{\dagger}</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
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</table>
HanShengSun
http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=20148&oldid=prev
HanShengSun: /* Bilateral Extrapolation of Dictionary Kernel */
2013-07-08T03:05:40Z
<p><span dir="auto"><span class="autocomment">Bilateral Extrapolation of Dictionary Kernel</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:05, 7 July 2013</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Bilateral Extrapolation of Dictionary Kernel==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Bilateral Extrapolation of Dictionary Kernel==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The ijth component of the kernel matrix is constructed as<center><math>K_{ij} = E_iW^{\dagger}E_j</math></center> Where, <math>E_i</math> is the ith row of the extrapolation matrix <math>E</math>, i.e. the similarity between any two points <math>x_i</math> and <math>x_j</math> is constructed by first <del style="font-weight: bold; text-decoration: none;">comput- ing </del>their respective similarities to the landmark set nd then modulated by the inverse of the similarities among the landmark points <math>W^{\dagger}</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The ijth component of the kernel matrix is constructed as<center><math>K_{ij} = E_iW^{\dagger}E_j</math></center> Where, <math>E_i</math> is the ith row of the extrapolation matrix <math>E</math>, i.e. the similarity between any two points <math>x_i</math> and <math>x_j</math> is constructed by first <ins style="font-weight: bold; text-decoration: none;">computing </ins>their respective similarities to the landmark set nd then modulated by the inverse of the similarities among the landmark points <math>W^{\dagger}</math>.</div></td></tr>
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</table>
HanShengSun
http://wiki.math.uwaterloo.ca/statwiki/index.php?title=inductive_Kernel_Low-rank_Decomposition_with_Priors:_A_Generalized_Nystrom_Method&diff=20126&oldid=prev
J34liang: /* Selecting Hyper-parameter */
2013-07-07T18:58:32Z
<p><span dir="auto"><span class="autocomment">Selecting Hyper-parameter</span></span></p>
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<tr class="diff-title" lang="us">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:58, 7 July 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l67">Line 67:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Cortes, C., Mohri, M. and Rostamizadeh, A. Two stage learning kernel algorithm. International Conference on Machine Learning, 2010. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Cortes, C., Mohri, M. and Rostamizadeh, A. Two stage learning kernel algorithm. International Conference on Machine Learning, 2010. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></ref> between kernel matrices, <center><math>\rho[K_1, K_2] = \frac { \langle K_{1c} K_{2c}^T\rangle_F}{ \|K_{1c}\|_F \|K_{2c}\|_F }</math></center></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></ref> between kernel matrices, <center><math>\rho[K_1, K_2] = \frac { \langle K_{1c} K_{2c}^T\rangle_F}{ \|K_{1c}\|_F \|K_{2c}\|_F }</math></center></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <math>K_{1c}</math> is double-centralized <math>K_{1}</math>. The NKA score alway has magnitude that is smaller than 1 and it is independent of the scale of the solution is multiplicative by nature. Let <math>S(\lambda)</math> be the optimum of the objective function for a fixed <math>\lambda</math> then choose the best <math>\lambda</math> <del style="font-weight: bold; text-decoration: none;">as follows </del><center><math>\lambda^* = \underset{\lambda\in G} {arg\,max}\rho[S(\lambda),S_0]\times\rho[E_lS(\lambda)E^T_l, K^*_l ]</math></center> G is the set of candidate <math>\lambda</math> 's. The first terms measures the closeness between <math>S</math> and <math>S_0</math>, related to unsupervised structures of kernel matrix; the second term is on the closeness between <math>E_lSE^T_l</math> and <math>K_l^*</math>, related to side information. This criteria faithfully reflects what the objective function optimizes but numerically different. <del style="font-weight: bold; text-decoration: none;">This is an information criterion to measure </del>the <del style="font-weight: bold; text-decoration: none;">quality </del>of <del style="font-weight: bold; text-decoration: none;">solution</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <math>K_{1c}</math> is double-centralized <math>K_{1}</math>. The NKA score alway has magnitude that is smaller than 1 and it is independent of the scale of the solution is multiplicative by nature. Let <math>S(\lambda)</math> be the optimum of the objective function for a fixed <math>\lambda</math> then choose the best <math>\lambda</math> <ins style="font-weight: bold; text-decoration: none;">that maximize the following criterion: </ins><center><math>\lambda^* = \underset{\lambda\in G} {arg\,max}\rho[S(\lambda),S_0]\times\rho[E_lS(\lambda)E^T_l, K^*_l ]</math></center> G is the set of candidate <math>\lambda</math> 's. <ins style="font-weight: bold; text-decoration: none;">The above criterion is an information criterion to measure the quality of solution, and it has several nice properties: </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(1) It is scale invariant, so it does not need any additional tuning parameters. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(2). </ins>The first terms measures the closeness between <math>S</math> and <math>S_0</math>, related to unsupervised structures of kernel matrix; the second term is on the closeness between <math>E_lSE^T_l</math> and <math>K_l^*</math>, related to side information. This criteria faithfully reflects what the objective function optimizes but numerically different.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(3). If </ins>the <ins style="font-weight: bold; text-decoration: none;">value </ins>of <ins style="font-weight: bold; text-decoration: none;">second term (the NKA) is high, it suggests it is a big chance that there exists a good predictor <ref name="Cortes2010"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Cortes, C., Mohri, M. and Rostamizadeh, A. Two stage learning kernel algorithm. International Conference on Machine Learning, 2010. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ref>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">(4)</ins>. <ins style="font-weight: bold; text-decoration: none;">To obtain the best <math>\lambda</math> does not require the validation (or cross-validation), which is good for small sample problems</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Experiments==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Experiments==</div></td></tr>
</table>
J34liang