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 <div class="title">Low-rank Matrix Factorization</div>  </div>
 <div class="ingroups"><a class="el" href="group__grp__unsuplearn.html">Unsupervised Learning</a></div></div>
 <div class="contents">
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 <dl class="user"><dt><b>About:</b></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="user"><dt><b>Input:</b></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="user"><dt><b>Usage:</b></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="user"><dt><b>Examples:</b></dt><dd></dd></dl>
 <ol type="1">
 <li>Prepare an input table/view: <div class="fragment"><pre class="fragment">CREATE TABLE lmf_data (
  column INT,
  row    INT,
  value  FLOAT8
 );
 </pre></div></li>
 <li>Populate the input table with some data. e.g.: <div class="fragment"><pre class="fragment">INSERT INTO lmf_data VALUES (1, 1, 5.0);
 INSERT INTO lmf_data VALUES (3, 100, 1.0);
 INSERT INTO lmf_data VALUES (999, 10000, 2.0);
 </pre></div></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"><pre class="fragment">SELECT madlib.lmf_igd_run(
 <span class="stringliteral">&#39;lmf_model&#39;</span>,                 -- result table
 <span class="stringliteral">&#39;lmf_data&#39;</span>,                  -- input table
 <span class="stringliteral">&#39;row&#39;</span>, <span class="stringliteral">&#39;col&#39;</span>, <span class="stringliteral">&#39;value&#39;</span>,       -- table column names
 999,                         -- row dimension
 10000,                       -- column dimension
 3,                           -- rank (number of features)
 0.1,                         -- stepsize
 2,                           -- initial value scale factor
 10,                          -- maximal number of iterations
 1e-9);                       -- error tolerance
 </pre></div> Example output (the exact result may not be the same): <div class="fragment"><pre class="fragment">NOTICE:
 Finished low-rank matrix factorization <span class="keyword">using</span> incremental gradient
 DETAIL:
  table : lmf_data (row, col, value)
 Results:
  RMSE = 4.31144557397543e-05
 Output:
  view : SELECT * FROM lmf_model WHERE <span class="keywordtype">id</span> = 1
  <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>
 -------------
            1
 (1 row)
 </pre></div></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"><pre class="fragment">SELECT array_dims(matrix_u), array_dims(matrix_v) FROM lmf_model WHERE <span class="keywordtype">id</span> = 1;
 </pre></div> Example output: <div class="fragment"><pre class="fragment">  array_dims  |   array_dims
 --------------+----------------
  [1:999][1:3] | [1:10000][1:3]
 (1 row)
 </pre></div></li>
 <li>Query the result value, e.g.: <div class="fragment"><pre class="fragment">SELECT matrix_u[2:2][1:3] AS row_2_features FROM lmf_model WHERE <span class="keywordtype">id</span> = 1;
 </pre></div> Example output (the exact result may not be the same): <div class="fragment"><pre class="fragment">                      row_2_features
 ----------------------------------------------------------
  {{0.51117920037359,0.169582297094166,0.837417622096837}}
 (1 row)
 </pre></div></li>
 </ol>
 <dl class="user"><dt><b>Literature:</b></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="ingroups"><a class="el" href="group__grp__unsuplearn.html">Unsupervised Learning</a></div></div>
	<div class="contents">
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	<dl class="user"><dt><b>About:</b></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="user"><dt><b>Input:</b></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="user"><dt><b>Usage:</b></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="user"><dt><b>Examples:</b></dt><dd></dd></dl>
	<ol type="1">
	<li>Prepare an input table/view: <div class="fragment"><pre class="fragment">CREATE TABLE lmf_data (
	column INT,
	row INT,
	value FLOAT8
	);
	</pre></div></li>
	<li>Populate the input table with some data. e.g.: <div class="fragment"><pre class="fragment">INSERT INTO lmf_data VALUES (1, 1, 5.0);
	INSERT INTO lmf_data VALUES (3, 100, 1.0);
	INSERT INTO lmf_data VALUES (999, 10000, 2.0);
	</pre></div></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"><pre class="fragment">SELECT madlib.lmf_igd_run(
	<span class="stringliteral">'lmf_model'</span>, -- result table
	<span class="stringliteral">'lmf_data'</span>, -- input table
	<span class="stringliteral">'row'</span>, <span class="stringliteral">'col'</span>, <span class="stringliteral">'value'</span>, -- table column names
	999, -- row dimension
	10000, -- column dimension
	3, -- rank (number of features)
	0.1, -- stepsize
	2, -- initial value scale factor
	10, -- maximal number of iterations
	1e-9); -- error tolerance
	</pre></div> Example output (the exact result may not be the same): <div class="fragment"><pre class="fragment">NOTICE:
	Finished low-rank matrix factorization <span class="keyword">using</span> incremental gradient
	DETAIL:
	table : lmf_data (row, col, value)
	Results:
	RMSE = 4.31144557397543e-05
	Output:
	view : SELECT * FROM lmf_model WHERE <span class="keywordtype">id</span> = 1
	<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>
	-------------
	1
	(1 row)
	</pre></div></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"><pre class="fragment">SELECT array_dims(matrix_u), array_dims(matrix_v) FROM lmf_model WHERE <span class="keywordtype">id</span> = 1;
	</pre></div> Example output: <div class="fragment"><pre class="fragment"> array_dims \| array_dims
	--------------+----------------
	[1:999][1:3] \| [1:10000][1:3]
	(1 row)
	</pre></div></li>
	<li>Query the result value, e.g.: <div class="fragment"><pre class="fragment">SELECT matrix_u[2:2][1:3] AS row_2_features FROM lmf_model WHERE <span class="keywordtype">id</span> = 1;
	</pre></div> Example output (the exact result may not be the same): <div class="fragment"><pre class="fragment"> row_2_features
	----------------------------------------------------------
	{{0.51117920037359,0.169582297094166,0.837417622096837}}
	(1 row)
	</pre></div></li>
	</ol>
	<dl class="user"><dt><b>Literature:</b></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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