http://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&feed=atom&action=historya New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization - Revision history2024-03-28T19:57:34ZRevision history for this page on the wikiMediaWiki 1.41.0http://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=27542&oldid=prevConversion script: Conversion script moved page A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization to a New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization: Converting page titles to l...2017-08-30T13:45:42Z<p>Conversion script moved page <a href="/statwiki/index.php?title=A_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization" class="mw-redirect" title="A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization">A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization</a> to <a href="/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization" title="a New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization">a New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization</a>: Converting page titles to l...</p>
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</td></tr></table>Conversion scripthttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11363&oldid=prevRjcase at 22:49, 21 December 20102010-12-21T22:49:30Z<p></p>
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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>The second series of experiments were performed on the [http://www.grouplens.org/node/73 MovieLens] data-set which includes rating of <math>\,1,682</math> movies by <math>\,943</math> users. Movie Rating have been selected from the set of numbers <math>\,1</math> to <math>\,5</math>. The data-set also includes attribute of genre for movies and attributes of age, gender, and occupation for users. Root mean squared error (RMSE) has been measured in 10-fold [http://en.wikipedia.org/wiki/Cross-validation_%28statistics%29 cross-validation] for the experiments on this data-set. Again, the best balance between the attribute and the Dirac kernels was obtainable inside the mentioned square.</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>The second series of experiments were performed on the [http://www.grouplens.org/node/73 MovieLens] data-set which includes rating of <math>\,1,682</math> movies by <math>\,943</math> users. Movie Rating have been selected from the set of numbers <math>\,1</math> to <math>\,5</math>. The data-set also includes attribute of genre for movies and attributes of age, gender, and occupation for users. Root mean squared error (RMSE) has been measured in 10-fold [http://en.wikipedia.org/wiki/Cross-validation_%28statistics%29 cross-validation] for the experiments on this data-set. Again, the best balance between the attribute and the Dirac kernels was obtainable inside the mentioned square.</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><ins style="font-weight: bold; text-decoration: none;">==Discussion==</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;">One particularly interesting application of collaborative filtering was in the [http://en.wikipedia.org/wiki/Netflix_Prize Netflix contest].</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;">Netflix, an online movie rental company, proposed a prize of $1,000,000 to any team who could significantly best their own preference prediction algorithm.</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;">Interestingly, the [http://www.netflixprize.com/assets/GrandPrize2009_BPC_BellKor.pdf best-performing] algorithms were those that were finely-tuned ensemble approaches.</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;">So, while techniques for including side-information such as those described in this paper <ref name="self" /> are important, particularly when few or no observations have been made of users preferences, developing highly accurate prediction engines requires a lot of tuning for a specific dataset.</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>==Conclusion==</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>==Conclusion==</div></td></tr>
</table>Rjcasehttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11362&oldid=prevRjcase: /* Introduction */2010-12-21T22:33:39Z<p><span dir="auto"><span class="autocomment">Introduction</span></span></p>
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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>[http://en.wikipedia.org/wiki/Regularization_%28mathematics%29 Regularization]-based CF methods have been proposed <ref name="srebro03">N. Srebro and T. Jaakkola. Weighted low-rank approximations. In T. Fawcett and N.Mishra, editors,</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>[http://en.wikipedia.org/wiki/Regularization_%28mathematics%29 Regularization]-based CF methods have been proposed <ref name="srebro03">N. Srebro and T. Jaakkola. Weighted low-rank approximations. In T. Fawcett and N.Mishra, editors,</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>Proceedings of the Twentieth International Conference on Machine Learning, pages 720–727.</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>Proceedings of the Twentieth International Conference on Machine Learning, pages 720–727.</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>AAAI Press, 2003. </ref><del style="font-weight: bold; text-decoration: none;">. These methods only incorporate </del>the <del style="font-weight: bold; text-decoration: none;">users </del>preferences <del style="font-weight: bold; text-decoration: none;">which can be given </del>as a <del style="font-weight: bold; text-decoration: none;">preference </del>matrix in <del style="font-weight: bold; text-decoration: none;">which </del>the <del style="font-weight: bold; text-decoration: none;">rows </del>and <del style="font-weight: bold; text-decoration: none;">columns represent </del>users <del style="font-weight: bold; text-decoration: none;">and objects, respectively</del>. However, a major drawback of the currently-used regularization-based CF methods is that they do not take advantage of additional information, such as known attributes of each user, such as the users' gender and age, as well as the attributes of the object that are associated with the users, such as the authors and genres of books recently purchased by users. These additional information are often directly available at no extra cost and, from an intuitive standpoint, they might be very useful for guiding the inferencing of user preferences, in particular for users and their associated objects that have very few known ratings. <del style="font-weight: bold; text-decoration: none;">In regularization-based CF methods, the problem is to infer the unknown entries based on the known ones. </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>AAAI Press, 2003. </ref><ins style="font-weight: bold; text-decoration: none;">, which view </ins>the <ins style="font-weight: bold; text-decoration: none;">data recorded about a user's </ins>preferences as a <ins style="font-weight: bold; text-decoration: none;">partially observed </ins>matrix <ins style="font-weight: bold; text-decoration: none;">of the user's preferences of all items available.</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;">Our goal, then, is to predict or infer the other preferences---</ins>in <ins style="font-weight: bold; text-decoration: none;">a sense, completing </ins>the <ins style="font-weight: bold; text-decoration: none;">matrix.</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;">To do this, we assume that preferences are related in some way, </ins>and <ins style="font-weight: bold; text-decoration: none;">search for the hidden variables which explain </ins>users<ins style="font-weight: bold; text-decoration: none;">' preferences</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>However, a major drawback of the currently-used regularization-based CF methods is that they do not take advantage of additional information, such as known attributes of each user, such as the users' gender and age, as well as the attributes of the object that are associated with the users, such as the authors and genres of books recently purchased by users. </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;">These methods only incorporate the users preferences which can be given as a preference matrix in which the rows and columns represent users and objects, respectively. </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>These additional information are often directly available at no extra cost and, from an intuitive standpoint, they might be very useful for guiding the inferencing of user preferences, in particular for users and their associated objects that have very few known ratings.</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>In order to solve the problem, a low-rank matrix should be found that approximates the partially observed given preference matrix. This rank constraint is a regularization. Heuristics <ref name="srebro03"> </ref> and penalizing the predicted matrix by its [http://en.wikipedia.org/wiki/Matrix_norm trace norm] <ref name="srebro05">N. Srebro, J. D. M. Rennie, and T. S. Jaakkola. Maximum-margin matrix factorization. In L. K.</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>In order to solve the problem, a low-rank matrix should be found that approximates the partially observed given preference matrix. This rank constraint is a regularization. Heuristics <ref name="srebro03"> </ref> and penalizing the predicted matrix by its [http://en.wikipedia.org/wiki/Matrix_norm trace norm] <ref name="srebro05">N. Srebro, J. D. M. Rennie, and T. S. Jaakkola. Maximum-margin matrix factorization. In L. K.</div></td></tr>
</table>Rjcasehttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11361&oldid=prevRjcase: /* Introduction */2010-12-21T22:19:52Z<p><span dir="auto"><span class="autocomment">Introduction</span></span></p>
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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>[http://en.wikipedia.org/wiki/Collaborative_filtering Collaborative filtering (CF)] aims to predict preferences of a set of users for objects such as movies, music, or any object based on previously purchased or rated objects by these users. The goal is to offer new objects to the users based on the predicted preferences. A good example of an application of collaborative filtering in the market place is the use of CF by the popular online shopping website Amazon.com for recommending related products to users of Amazon.com based on what these users have recently purchased from this website. Details of the use of CF by Amazon.com for this purpose is available [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1167344&userType=inst here].</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>[http://en.wikipedia.org/wiki/Collaborative_filtering Collaborative filtering (CF)] aims to predict preferences of a set of users for objects such as movies, music, or any object based on previously purchased or rated objects by these users. The goal is to offer new objects to the users based on the predicted preferences. A good example of an application of collaborative filtering in the market place is the use of CF by the popular online shopping website Amazon.com for recommending related products to users of Amazon.com based on what these users have recently purchased from this website. Details of the use of CF by Amazon.com for this purpose is available [http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1167344&userType=inst here].</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>[http://en.wikipedia.org/wiki/Regularization_%28mathematics%29 Regularization]-based CF methods have <del style="font-weight: bold; text-decoration: none;">recently </del>been <del style="font-weight: bold; text-decoration: none;">attracting </del><ref name="srebro03">N. Srebro and T. Jaakkola. Weighted low-rank approximations. In T. Fawcett and N.Mishra, editors,</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>[http://en.wikipedia.org/wiki/Regularization_%28mathematics%29 Regularization]-based CF methods have been <ins style="font-weight: bold; text-decoration: none;">proposed </ins><ref name="srebro03">N. Srebro and T. Jaakkola. Weighted low-rank approximations. In T. Fawcett and N.Mishra, editors,</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>Proceedings of the Twentieth International Conference on Machine Learning, pages 720–727.</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>Proceedings of the Twentieth International Conference on Machine Learning, pages 720–727.</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>AAAI Press, 2003. </ref>. These methods only incorporate the users preferences which can be given as a preference matrix in which the rows and columns represent users and objects, respectively. However, a major drawback of the currently-used regularization-based CF methods is that they do not take advantage of additional information, such as known attributes of each user, such as the users' gender and age, as well as the attributes of the object that are associated with the users, such as the authors and genres of books recently purchased by users. These additional information are often directly available at no extra cost and, from an intuitive standpoint, they might be very useful for guiding the inferencing of user preferences, in particular for users and their associated objects that have very few known ratings. In regularization-based CF methods, the problem is to infer the unknown entries based on the known ones. </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>AAAI Press, 2003. </ref>. These methods only incorporate the users preferences which can be given as a preference matrix in which the rows and columns represent users and objects, respectively. However, a major drawback of the currently-used regularization-based CF methods is that they do not take advantage of additional information, such as known attributes of each user, such as the users' gender and age, as well as the attributes of the object that are associated with the users, such as the authors and genres of books recently purchased by users. These additional information are often directly available at no extra cost and, from an intuitive standpoint, they might be very useful for guiding the inferencing of user preferences, in particular for users and their associated objects that have very few known ratings. In regularization-based CF methods, the problem is to infer the unknown entries based on the known ones. </div></td></tr>
</table>Rjcasehttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11331&oldid=prevY24Sun: /* Experiments */2010-12-19T22:30:40Z<p><span dir="auto"><span class="autocomment">Experiments</span></span></p>
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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>[[File:Abernathy.jpg|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>[[File:Abernathy.jpg|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>The second series of experiments were performed on the [http://www.grouplens.org/node/73 MovieLens data-set<del style="font-weight: bold; text-decoration: none;">] </del>which includes rating of <math>\,1,682</math> movies by <math>\,943</math> users. Movie Rating have been selected from the set of numbers <math>\,1</math> to <math>\,5</math>. The data-set also includes attribute of genre for movies and attributes of age, gender, and occupation for users. Root mean squared error (RMSE) has been measured in 10-fold [http://en.wikipedia.org/wiki/Cross-validation_%28statistics%29 cross-validation] for the experiments on this data-set. Again, the best balance between the attribute and the Dirac kernels was obtainable inside the mentioned square.</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 second series of experiments were performed on the [http://www.grouplens.org/node/73 MovieLens<ins style="font-weight: bold; text-decoration: none;">] </ins>data-set which includes rating of <math>\,1,682</math> movies by <math>\,943</math> users. Movie Rating have been selected from the set of numbers <math>\,1</math> to <math>\,5</math>. The data-set also includes attribute of genre for movies and attributes of age, gender, and occupation for users. Root mean squared error (RMSE) has been measured in 10-fold [http://en.wikipedia.org/wiki/Cross-validation_%28statistics%29 cross-validation] for the experiments on this data-set. Again, the best balance between the attribute and the Dirac kernels was obtainable inside the mentioned square.</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>==Conclusion==</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>==Conclusion==</div></td></tr>
</table>Y24Sunhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11330&oldid=prevY24Sun: /* Conclusion */2010-12-19T04:15:29Z<p><span dir="auto"><span class="autocomment">Conclusion</span></span></p>
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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>==Conclusion==</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>==Conclusion==</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><del style="font-weight: bold; text-decoration: none;">Authors </del>presented a generalized formulation for the matrix completion problem. This formulation aims to find a linear compact operator between two Hilbert Schmidt spaces. Also, they showed that their generalized notion includes different approaches as its special cases. The experiments focused on the CF application and indicated that incorporating attribute information improves the accuracy of the predictions.</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;">The authors </ins>presented a generalized formulation for the matrix completion problem. This formulation aims to find a linear compact operator between two Hilbert<ins style="font-weight: bold; text-decoration: none;">-</ins>Schmidt spaces. Also, they showed that their generalized notion includes different approaches as its special cases. The experiments focused on the CF application and indicated that incorporating attribute information improves the accuracy of the predictions.</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;">An </del>future research could be to explore <del style="font-weight: bold; text-decoration: none;">further </del>the multi-task learning algorithms<del style="font-weight: bold; text-decoration: none;">. To </del>study the possibility to derive real-time implementations that may better fit the need for large-scale applications where training data is growing larger and larger. On the theoretical side, a better understanding of the effects of norm and rank regularizations and their interaction would be of an interesting topic.</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;">A </ins>future research could be to <ins style="font-weight: bold; text-decoration: none;">further </ins>explore the multi-task learning algorithms<ins style="font-weight: bold; text-decoration: none;">, such as to </ins>study the possibility to derive real-time implementations that may better fit the need for large-scale applications where training data is growing larger and larger. On the theoretical side, a better understanding of the effects of <ins style="font-weight: bold; text-decoration: none;">the </ins>norm and <ins style="font-weight: bold; text-decoration: none;">the </ins>rank regularizations and their interaction would be of an interesting topic.</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>==References==</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>==References==</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><references /></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><references /></div></td></tr>
</table>Y24Sunhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11329&oldid=prevY24Sun: /* Experiments */2010-12-19T03:21:59Z<p><span dir="auto"><span class="autocomment">Experiments</span></span></p>
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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>Experiments have been performed to show that incorporating the extra information from users or objects to the CF approach can lead to more accurate predictions. The first series of experiments were done on synthetic data which was created by first sampling i.i.d. multivariate features for <math>\,x</math> and <math>\,y</math> (both having 6 dimensions). Next, they sampled <math>\,z</math> from a bilinear form in <math>\,x</math> and <math>\,y</math> plus noise, and finally, restricting the observed features to only <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>Experiments have been performed to show that incorporating the extra information from users or objects to the CF approach can lead to more accurate predictions. The first series of experiments were done on synthetic data which was created by first sampling i.i.d. multivariate features for <math>\,x</math> and <math>\,y</math> (both having 6 dimensions). Next, they sampled <math>\,z</math> from a bilinear form in <math>\,x</math> and <math>\,y</math> plus noise, and finally, restricting the observed features to only <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>\,3</math> for both <math>\,x</math> and <math>\,y</math>. The comparison results between two cases of applying the trace norm and the [http://mathworld.wolfram.com/FrobeniusNorm.html Frobenius norm] spectral penalties (in the same rank-constrained problem) showed that the trace norm penalty performed slightly better. Also, the best performance in both cases was achieved when there has been a balance between using the Dirac and the <del style="font-weight: bold; text-decoration: none;">attributes </del>kernels, where the pair of values for <math>\,\eta</math> and <math>\,\zeta</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>\,3</math> for both <math>\,x</math> and <math>\,y</math>. The comparison results between two cases of applying the trace norm and the [http://mathworld.wolfram.com/FrobeniusNorm.html Frobenius norm] spectral penalties (in the same rank-constrained problem) showed that the trace norm penalty performed slightly better. Also, the best performance in both cases was achieved when there has been a balance between using the Dirac and the <ins style="font-weight: bold; text-decoration: none;">attribute </ins>kernels, where the pair of values for <math>\,\eta</math> and <math>\,\zeta</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;"><div>that define this combined usage of the Dirac and the attributes kernels would be in the middle of the square for which the corners are <math>(\,\eta=0,\,\zeta=0)</math>, <math>(\,\eta=0,\,\,\zeta=1)</math>,<math>(\,\eta=1,\,\zeta=0)</math> , and <math>(\,\eta=1,\,\zeta=1)</math>. This is shown in the figure below taken from Abernathy ''et al''.'s paper. </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>that define this combined usage of the Dirac and the attributes kernels would be in the middle of the square for which the corners are <math>(\,\eta=0,\,\zeta=0)</math>, <math>(\,\eta=0,\,\,\zeta=1)</math>,<math>(\,\eta=1,\,\zeta=0)</math> , and <math>(\,\eta=1,\,\zeta=1)</math>. This is shown in the figure below taken from Abernathy ''et al''.'s paper. </div></td></tr>
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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>[[File:Abernathy.jpg|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>[[File:Abernathy.jpg|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>The second series of experiments were performed on the [http://www.grouplens.org/node/73 MovieLens data-set] which includes rating of <math>\,1,682</math> movies by <math>\,943</math> users. <del style="font-weight: bold; text-decoration: none;">Rates </del>have been selected from the set of numbers 1 to 5. The <del style="font-weight: bold; text-decoration: none;">dataset </del>also includes attribute of genre for movies and attributes of age, gender, and occupation for users. Root mean squared error (RMSE) has been measured in 10-fold <del style="font-weight: bold; text-decoration: none;">croos </del>validation for the <del style="font-weight: bold; text-decoration: none;">expriments </del>on this <del style="font-weight: bold; text-decoration: none;">dataset</del>. Again the best balance between the attribute and Dirac kernels <del style="font-weight: bold; text-decoration: none;">is </del>obtainable inside the mentioned square.</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 second series of experiments were performed on the [http://www.grouplens.org/node/73 MovieLens data-set] which includes rating of <math>\,1,682</math> movies by <math>\,943</math> users. <ins style="font-weight: bold; text-decoration: none;">Movie Rating </ins>have been selected from the set of numbers <ins style="font-weight: bold; text-decoration: none;"><math>\,</ins>1<ins style="font-weight: bold; text-decoration: none;"></math> </ins>to <ins style="font-weight: bold; text-decoration: none;"><math>\,</ins>5<ins style="font-weight: bold; text-decoration: none;"></math></ins>. The <ins style="font-weight: bold; text-decoration: none;">data-set </ins>also includes attribute of genre for movies and attributes of age, gender, and occupation for users. Root mean squared error (RMSE) has been measured in 10-fold <ins style="font-weight: bold; text-decoration: none;">[http://en.wikipedia.org/wiki/Cross-validation_%28statistics%29 cross-</ins>validation<ins style="font-weight: bold; text-decoration: none;">] </ins>for the <ins style="font-weight: bold; text-decoration: none;">experiments </ins>on this <ins style="font-weight: bold; text-decoration: none;">data-set</ins>. Again<ins style="font-weight: bold; text-decoration: none;">, </ins>the best balance between the attribute and <ins style="font-weight: bold; text-decoration: none;">the </ins>Dirac kernels <ins style="font-weight: bold; text-decoration: none;">was </ins>obtainable inside the mentioned square.</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>==Conclusion==</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>==Conclusion==</div></td></tr>
</table>Y24Sunhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11328&oldid=prevY24Sun: /* Experiments */2010-12-19T03:08:26Z<p><span dir="auto"><span class="autocomment">Experiments</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:08, 18 December 2010</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>[[File:Abernathy.jpg|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>[[File:Abernathy.jpg|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>The second series of experiments were performed on the MovieLens <del style="font-weight: bold; text-decoration: none;">dataset </del>which includes rating of <del style="font-weight: bold; text-decoration: none;">1682 </del>movies by 943 users. Rates have been selected from the set of numbers 1 to 5. The dataset also includes attribute of genre for movies and attributes of age, gender, and occupation for users. Root mean squared error (RMSE) has been measured in 10-fold croos validation for the expriments on this dataset. Again the best balance between the attribute and Dirac kernels is obtainable inside the mentioned square.</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 second series of experiments were performed on the <ins style="font-weight: bold; text-decoration: none;">[http://www.grouplens.org/node/73 </ins>MovieLens <ins style="font-weight: bold; text-decoration: none;">data-set] </ins>which includes rating of <ins style="font-weight: bold; text-decoration: none;"><math>\,1,682</math> </ins>movies by <ins style="font-weight: bold; text-decoration: none;"><math>\,</ins>943<ins style="font-weight: bold; text-decoration: none;"></math> </ins>users. Rates have been selected from the set of numbers 1 to 5. The dataset also includes attribute of genre for movies and attributes of age, gender, and occupation for users. Root mean squared error (RMSE) has been measured in 10-fold croos validation for the expriments on this dataset. Again the best balance between the attribute and Dirac kernels is obtainable inside the mentioned square.</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>==Conclusion==</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>==Conclusion==</div></td></tr>
</table>Y24Sunhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11327&oldid=prevY24Sun: /* Experiments */2010-12-19T02:52:24Z<p><span dir="auto"><span class="autocomment">Experiments</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:52, 18 December 2010</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>==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>
<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>Experiments have been performed to show that <del style="font-weight: bold; text-decoration: none;">the </del>incorporating the extra information from users or objects to the CF approach can lead to more accurate predictions. The first series of experiments were done on synthetic data which was created by first sampling i.i.d. multivariate features for <math>x</math> and <math>y</math> (both <del style="font-weight: bold; text-decoration: none;">of dimension </del>6). Next, they sampled <math>z</math> from a bilinear form in <math>x</math> and <math>y</math> plus noise, and finally, restricting the observed features to only 3 for both <math>x</math> and <math>y</math>. The comparison results between two cases of applying trace and Frobenius norm spectral <del style="font-weight: bold; text-decoration: none;">penalty </del>(in the same rank constrained problem) showed that the trace norm <del style="font-weight: bold; text-decoration: none;">perform </del>slightly better. Also, the best <del style="font-weight: bold; text-decoration: none;">performace </del>in both cases <del style="font-weight: bold; text-decoration: none;">is </del>achieved when there has been a balance between using the Dirac and attributes kernels<del style="font-weight: bold; text-decoration: none;">. It </del>would be in the middle of the square for which the corners are <math>(\eta=0,\zeta=0)</math>, <math>(\eta=0,\zeta=1)</math>,<math>(\eta=1,\zeta=0)</math> , and <math>(\eta=1,\zeta=1)</math>. This is shown in the figure below taken from Abernathy et al. <del style="font-weight: bold; text-decoration: none;"> </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>Experiments have been performed to show that incorporating the extra information from users or objects to the CF approach can lead to more accurate predictions. The first series of experiments were done on synthetic data which was created by first sampling i.i.d. multivariate features for <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>x</math> and <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>y</math> (both <ins style="font-weight: bold; text-decoration: none;">having </ins>6 <ins style="font-weight: bold; text-decoration: none;">dimensions</ins>). Next, they sampled <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>z</math> from a bilinear form in <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>x</math> and <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>y</math> plus noise, and finally, restricting the observed features to only <ins style="font-weight: bold; text-decoration: none;"><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><ins style="font-weight: bold; text-decoration: none;">\,</ins>3<ins style="font-weight: bold; text-decoration: none;"></math> </ins>for both <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>x</math> and <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>y</math>. The comparison results between two cases of applying <ins style="font-weight: bold; text-decoration: none;">the </ins>trace <ins style="font-weight: bold; text-decoration: none;">norm </ins>and <ins style="font-weight: bold; text-decoration: none;">the [http://mathworld.wolfram.com/FrobeniusNorm.html </ins>Frobenius norm<ins style="font-weight: bold; text-decoration: none;">] </ins>spectral <ins style="font-weight: bold; text-decoration: none;">penalties </ins>(in the same rank<ins style="font-weight: bold; text-decoration: none;">-</ins>constrained problem) showed that the trace norm <ins style="font-weight: bold; text-decoration: none;">penalty performed </ins>slightly better. Also, the best <ins style="font-weight: bold; text-decoration: none;">performance </ins>in both cases <ins style="font-weight: bold; text-decoration: none;">was </ins>achieved when there has been a balance between using the Dirac and <ins style="font-weight: bold; text-decoration: none;">the attributes kernels, where the pair of values for <math>\,\eta</math> and <math>\,\zeta</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><ins style="font-weight: bold; text-decoration: none;">that define this combined usage of the Dirac and the </ins>attributes kernels would be in the middle of the square for which the corners are <math>(<ins style="font-weight: bold; text-decoration: none;">\,</ins>\eta=0<ins style="font-weight: bold; text-decoration: none;">,\</ins>,\zeta=0)</math>, <math>(<ins style="font-weight: bold; text-decoration: none;">\,</ins>\eta=0<ins style="font-weight: bold; text-decoration: none;">,\,\</ins>,\zeta=1)</math>,<math>(<ins style="font-weight: bold; text-decoration: none;">\,</ins>\eta=1<ins style="font-weight: bold; text-decoration: none;">,\</ins>,\zeta=0)</math> , and <math>(<ins style="font-weight: bold; text-decoration: none;">\,</ins>\eta=1<ins style="font-weight: bold; text-decoration: none;">,\</ins>,\zeta=1)</math>. This is shown in the figure below taken from Abernathy <ins style="font-weight: bold; text-decoration: none;">''</ins>et al<ins style="font-weight: bold; text-decoration: none;">''.'s paper</ins>. <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;"><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>[[File:Abernathy.jpg|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>[[File:Abernathy.jpg|center]]</div></td></tr>
</table>Y24Sunhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=a_New_Approach_to_Collaborative_Filtering:_Operator_Estimation_with_Spectral_Regularization&diff=11326&oldid=prevY24Sun: /* Optimization Problem */2010-12-19T01:49:58Z<p><span dir="auto"><span class="autocomment">Optimization Problem</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:49, 18 December 2010</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>For finding <math>\,\alpha</math>, first the optimal dual variable <math>\,\beta</math> is found and then <math>\,\alpha</math> is among the Fenchel duals of <math>-\frac{1}{\lambda}X^{T}\textrm{Diag}(\beta)Y</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>For finding <math>\,\alpha</math>, first the optimal dual variable <math>\,\beta</math> is found and then <math>\,\alpha</math> is among the Fenchel duals of <math>-\frac{1}{\lambda}X^{T}\textrm{Diag}(\beta)Y</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>Choosing either the primal or the dual formulation depends on the problem. For instance, in CF problem, the number of variables of the primal and dual formulation is equal to <math> m_{\mathcal{X}} m_{\mathcal{Y}}</math> and <math>N</math>, respectively, and based on the available ratings <math>N</math> compared to <math>m_{\mathcal{X}} m_{\mathcal{Y}}</math>, one of the primal or dual programs will be chosen.</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>Choosing either the primal or the dual formulation depends on the problem. For instance, in CF problem, the number of variables of the primal and dual formulation is equal to <math> m_{\mathcal{X}} m_{\mathcal{Y}}</math> and <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>N</math>, respectively, and based on the available ratings <math><ins style="font-weight: bold; text-decoration: none;">\,</ins>N</math> compared to <math>m_{\mathcal{X}} m_{\mathcal{Y}}</math>, one of the primal or dual programs will be chosen.</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>Y24Sun