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 <div class="header">
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 <div class="title">Low-rank Matrix Factorization<div class="ingroups"><a class="el" href="group__grp__matrix__factorization.html">Matrix Factorization</a></div></div>  </div>
 </div><!--header-->
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 <div id="dynsection-0" onclick="return toggleVisibility(this)" class="dynheader closed" style="cursor:pointer;">
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 <dl class="section user"><dt>About:</dt><dd></dd></dl>
 <p>This module implements "factor model" for representing an incomplete matrix using a low-rank approximation [1]. Mathematically, this model seeks to find matrices U and V (also referred as factors) that, for any given incomplete matrix A, minimizes: </p>
 <p class="formulaDsp">
 \[ \|\boldsymbol A - \boldsymbol UV^{T} \|_2 \]
 </p>
 <p> subject to \(rank(\boldsymbol UV^{T}) \leq r\), where \(\|\cdot\|_2\) denotes the Frobenius norm. Let \(A\) be a \(m \times n\) matrix, then \(U\) will be \(m \times r\) and \(V\) will be \(n \times r\), in dimension, and \(1 \leq r \ll \min(m, n)\). This model is not intended to do the full decomposition, or to be used as part of inverse procedure. This model has been widely used in recommendation systems (e.g., Netflix [2]) and feature selection (e.g., image processing [3]).</p>
 <dl class="section user"><dt>Input:</dt><dd></dd></dl>
 <p>The <b>input matrix</b> is expected to be of the following form: </p>
 <pre>{TABLE|VIEW} <em>input_table</em> (
     <em>row</em>    INTEGER,
     <em>col</em>    INTEGER,
     <em>value</em>  DOUBLE PRECISION
 )</pre><p>Input is contained in a table that describes an incomplete matrix, by having available entries specified as (row, column, value). The input matrix is expected to be based 1, which means row &gt;= 1, and col &gt;= 1. NULL values are not expected.</p>
 <dl class="section user"><dt>Usage:</dt><dd></dd></dl>
 <p>Please find descriptions of SQL functions in <a class="el" href="lmf_8sql__in.html" title="SQL functions for low-rank matrix factorization. ">lmf.sql_in</a></p>
 <p>Output factors matrix U and V are in flatten format. </p>
 <pre>RESULT AS (
         matrix_u    DOUBLE PRECISION[],
         matrix_v    DOUBLE PRECISION[],
         rmse        DOUBLE PRECISION
 );</pre><p>Features correspond to row i is <code>matrix_u[i:i][1:r]</code>. Features correspond to column j is <code>matrix_v[j:j][1:r]</code>.</p>
 <dl class="section user"><dt>Examples:</dt><dd><ol type="1">
 <li>Prepare an input table/view: <div class="fragment"><div class="line">CREATE TABLE lmf_data (</div>
 <div class="line"> column INT,</div>
 <div class="line"> row    INT,</div>
 <div class="line"> value  FLOAT8</div>
 <div class="line">);</div>
 </div><!-- fragment --></li>
 <li>Populate the input table with some data. e.g.: <div class="fragment"><div class="line">INSERT INTO lmf_data VALUES (1, 1, 5.0);</div>
 <div class="line">INSERT INTO lmf_data VALUES (3, 100, 1.0);</div>
 <div class="line">INSERT INTO lmf_data VALUES (999, 10000, 2.0);</div>
 </div><!-- fragment --></li>
 <li>Call <a class="el" href="lmf_8sql__in.html#ac1acb1f0e1f7008118f21c83546a4602" title="Low-rank matrix factorization of a incomplete matrix into two factors. ">lmf_igd_run()</a> stored procedure, e.g.: <div class="fragment"><div class="line">SELECT madlib.lmf_igd_run(</div>
 <div class="line"><span class="stringliteral">&#39;lmf_model&#39;</span>,                 -- result table</div>
 <div class="line"><span class="stringliteral">&#39;lmf_data&#39;</span>,                  -- input table</div>
 <div class="line"><span class="stringliteral">&#39;row&#39;</span>, <span class="stringliteral">&#39;col&#39;</span>, <span class="stringliteral">&#39;value&#39;</span>,       -- table column names</div>
 <div class="line">999,                         -- row dimension</div>
 <div class="line">10000,                       -- column dimension</div>
 <div class="line">3,                           -- rank (number of features)</div>
 <div class="line">0.1,                         -- stepsize</div>
 <div class="line">2,                           -- initial value scale factor</div>
 <div class="line">10,                          -- maximal number of iterations</div>
 <div class="line">1e-9);                       -- error tolerance</div>
 </div><!-- fragment --> Example output (the exact result may not be the same): <div class="fragment"><div class="line">NOTICE:</div>
 <div class="line">Finished low-rank matrix factorization <span class="keyword">using</span> incremental gradient</div>
 <div class="line">DETAIL:</div>
 <div class="line">   table : lmf_data (row, col, value)</div>
 <div class="line">Results:</div>
 <div class="line">   RMSE = 4.31144557397543e-05</div>
 <div class="line">Output:</div>
 <div class="line">   view : SELECT * FROM lmf_model WHERE <span class="keywordtype">id</span> = 1</div>
 <div class="line"> <a class="code" href="lmf_8sql__in.html#ac1acb1f0e1f7008118f21c83546a4602" title="Low-rank matrix factorization of a incomplete matrix into two factors. ">lmf_igd_run</a></div>
 <div class="line">-------------</div>
 <div class="line">           1</div>
 <div class="line">(1 row)</div>
 </div><!-- fragment --></li>
 <li>Sanity check of the result. You may need a model id returned and also indicated by the function <a class="el" href="lmf_8sql__in.html#ac1acb1f0e1f7008118f21c83546a4602" title="Low-rank matrix factorization of a incomplete matrix into two factors. ">lmf_igd_run()</a>, assuming 1 here, e.g.: <div class="fragment"><div class="line">SELECT array_dims(matrix_u), array_dims(matrix_v) FROM lmf_model WHERE <span class="keywordtype">id</span> = 1;</div>
 </div><!-- fragment --> Example output: <div class="fragment"><div class="line">  array_dims  |   array_dims</div>
 <div class="line">--------------+----------------</div>
 <div class="line"> [1:999][1:3] | [1:10000][1:3]</div>
 <div class="line">(1 row)</div>
 </div><!-- fragment --></li>
 <li>Query the result value, e.g.: <div class="fragment"><div class="line">SELECT matrix_u[2:2][1:3] AS row_2_features FROM lmf_model WHERE <span class="keywordtype">id</span> = 1;</div>
 </div><!-- fragment --> Example output (the exact result may not be the same): <div class="fragment"><div class="line">                      row_2_features</div>
 <div class="line">----------------------------------------------------------</div>
 <div class="line"> {{0.51117920037359,0.169582297094166,0.837417622096837}}</div>
 <div class="line">(1 row)</div>
 </div><!-- fragment --></li>
 </ol>
 </dd></dl>
 <dl class="section user"><dt>Literature:</dt><dd></dd></dl>
 <p>[1] N. Srebro and T. Jaakkola. “Weighted Low-Rank Approximations.” In: ICML. Ed. by T. Fawcett and N. Mishra. AAAI Press, 2003, pp. 720–727. isbn: 1-57735-189-4.</p>
 <p>[2] Simon Funk, Netflix Update: Try This at Home, December 11 2006, <a href="http://sifter.org/~simon/journal/20061211.html">http://sifter.org/~simon/journal/20061211.html</a></p>
 <p>[3] J. Wright, A. Ganesh, S. Rao, Y. Peng, and Y. Ma. “Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization.” In: NIPS. Ed. by Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta. Curran Associates, Inc., 2009, pp. 2080–2088. isbn: 9781615679119. </p>
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	<div class="header">
	<div class="headertitle">
	<div class="title">Low-rank Matrix Factorization<div class="ingroups"><a class="el" href="group__grp__matrix__factorization.html">Matrix Factorization</a></div></div> </div>
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	<dl class="section user"><dt>About:</dt><dd></dd></dl>
	<p>This module implements "factor model" for representing an incomplete matrix using a low-rank approximation [1]. Mathematically, this model seeks to find matrices U and V (also referred as factors) that, for any given incomplete matrix A, minimizes: </p>
	<p class="formulaDsp">
	\[ \\|\boldsymbol A - \boldsymbol UV^{T} \\|_2 \]
	</p>
	<p> subject to \(rank(\boldsymbol UV^{T}) \leq r\), where \(\\|\cdot\\|_2\) denotes the Frobenius norm. Let \(A\) be a \(m \times n\) matrix, then \(U\) will be \(m \times r\) and \(V\) will be \(n \times r\), in dimension, and \(1 \leq r \ll \min(m, n)\). This model is not intended to do the full decomposition, or to be used as part of inverse procedure. This model has been widely used in recommendation systems (e.g., Netflix [2]) and feature selection (e.g., image processing [3]).</p>
	<dl class="section user"><dt>Input:</dt><dd></dd></dl>
	<p>The <b>input matrix</b> is expected to be of the following form: </p>
	<pre>{TABLE\|VIEW} <em>input_table</em> (
	<em>row</em> INTEGER,
	<em>col</em> INTEGER,
	<em>value</em> DOUBLE PRECISION
	)</pre><p>Input is contained in a table that describes an incomplete matrix, by having available entries specified as (row, column, value). The input matrix is expected to be based 1, which means row >= 1, and col >= 1. NULL values are not expected.</p>
	<dl class="section user"><dt>Usage:</dt><dd></dd></dl>
	<p>Please find descriptions of SQL functions in <a class="el" href="lmf_8sql__in.html" title="SQL functions for low-rank matrix factorization. ">lmf.sql_in</a></p>
	<p>Output factors matrix U and V are in flatten format. </p>
	<pre>RESULT AS (
	matrix_u DOUBLE PRECISION[],
	matrix_v DOUBLE PRECISION[],
	rmse DOUBLE PRECISION
	);</pre><p>Features correspond to row i is <code>matrix_u[i:i][1:r]</code>. Features correspond to column j is <code>matrix_v[j:j][1:r]</code>.</p>
	<dl class="section user"><dt>Examples:</dt><dd><ol type="1">
	<li>Prepare an input table/view: <div class="fragment"><div class="line">CREATE TABLE lmf_data (</div>
	<div class="line"> column INT,</div>
	<div class="line"> row INT,</div>
	<div class="line"> value FLOAT8</div>
	<div class="line">);</div>
	</div><!-- fragment --></li>
	<li>Populate the input table with some data. e.g.: <div class="fragment"><div class="line">INSERT INTO lmf_data VALUES (1, 1, 5.0);</div>
	<div class="line">INSERT INTO lmf_data VALUES (3, 100, 1.0);</div>
	<div class="line">INSERT INTO lmf_data VALUES (999, 10000, 2.0);</div>
	</div><!-- fragment --></li>
	<li>Call <a class="el" href="lmf_8sql__in.html#ac1acb1f0e1f7008118f21c83546a4602" title="Low-rank matrix factorization of a incomplete matrix into two factors. ">lmf_igd_run()</a> stored procedure, e.g.: <div class="fragment"><div class="line">SELECT madlib.lmf_igd_run(</div>
	<div class="line"><span class="stringliteral">'lmf_model'</span>, -- result table</div>
	<div class="line"><span class="stringliteral">'lmf_data'</span>, -- input table</div>
	<div class="line"><span class="stringliteral">'row'</span>, <span class="stringliteral">'col'</span>, <span class="stringliteral">'value'</span>, -- table column names</div>
	<div class="line">999, -- row dimension</div>
	<div class="line">10000, -- column dimension</div>
	<div class="line">3, -- rank (number of features)</div>
	<div class="line">0.1, -- stepsize</div>
	<div class="line">2, -- initial value scale factor</div>
	<div class="line">10, -- maximal number of iterations</div>
	<div class="line">1e-9); -- error tolerance</div>
	</div><!-- fragment --> Example output (the exact result may not be the same): <div class="fragment"><div class="line">NOTICE:</div>
	<div class="line">Finished low-rank matrix factorization <span class="keyword">using</span> incremental gradient</div>
	<div class="line">DETAIL:</div>
	<div class="line"> table : lmf_data (row, col, value)</div>
	<div class="line">Results:</div>
	<div class="line"> RMSE = 4.31144557397543e-05</div>
	<div class="line">Output:</div>
	<div class="line"> view : SELECT * FROM lmf_model WHERE <span class="keywordtype">id</span> = 1</div>
	<div class="line"> <a class="code" href="lmf_8sql__in.html#ac1acb1f0e1f7008118f21c83546a4602" title="Low-rank matrix factorization of a incomplete matrix into two factors. ">lmf_igd_run</a></div>
	<div class="line">-------------</div>
	<div class="line"> 1</div>
	<div class="line">(1 row)</div>
	</div><!-- fragment --></li>
	<li>Sanity check of the result. You may need a model id returned and also indicated by the function <a class="el" href="lmf_8sql__in.html#ac1acb1f0e1f7008118f21c83546a4602" title="Low-rank matrix factorization of a incomplete matrix into two factors. ">lmf_igd_run()</a>, assuming 1 here, e.g.: <div class="fragment"><div class="line">SELECT array_dims(matrix_u), array_dims(matrix_v) FROM lmf_model WHERE <span class="keywordtype">id</span> = 1;</div>
	</div><!-- fragment --> Example output: <div class="fragment"><div class="line"> array_dims \| array_dims</div>
	<div class="line">--------------+----------------</div>
	<div class="line"> [1:999][1:3] \| [1:10000][1:3]</div>
	<div class="line">(1 row)</div>
	</div><!-- fragment --></li>
	<li>Query the result value, e.g.: <div class="fragment"><div class="line">SELECT matrix_u[2:2][1:3] AS row_2_features FROM lmf_model WHERE <span class="keywordtype">id</span> = 1;</div>
	</div><!-- fragment --> Example output (the exact result may not be the same): <div class="fragment"><div class="line"> row_2_features</div>
	<div class="line">----------------------------------------------------------</div>
	<div class="line"> {{0.51117920037359,0.169582297094166,0.837417622096837}}</div>
	<div class="line">(1 row)</div>
	</div><!-- fragment --></li>
	</ol>
	</dd></dl>
	<dl class="section user"><dt>Literature:</dt><dd></dd></dl>
	<p>[1] N. Srebro and T. Jaakkola. “Weighted Low-Rank Approximations.” In: ICML. Ed. by T. Fawcett and N. Mishra. AAAI Press, 2003, pp. 720–727. isbn: 1-57735-189-4.</p>
	<p>[2] Simon Funk, Netflix Update: Try This at Home, December 11 2006, <a href="http://sifter.org/~simon/journal/20061211.html">http://sifter.org/~simon/journal/20061211.html</a></p>
	<p>[3] J. Wright, A. Ganesh, S. Rao, Y. Peng, and Y. Ma. “Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization.” In: NIPS. Ed. by Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta. Curran Associates, Inc., 2009, pp. 2080–2088. isbn: 9781615679119. </p>
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