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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__datatrans.html">Data Types and Transformations</a> &raquo; <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a> &raquo; <a class="el" href="group__grp__matrix__factorization.html">Matrix Factorization</a></div></div>  </div>
 </div><!--header-->
 <div class="contents">
 <div class="toc"><b>Contents</b> <ul>
 <li>
 <a href="#syntax">Function Syntax</a> </li>
 <li>
 <a href="#examples">Examples</a> </li>
 <li>
 <a href="#literature">Literature</a> </li>
 </ul>
 </div><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>
 <p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Function Syntax</dt><dd></dd></dl>
 <p>Low-rank matrix factorization of an incomplete matrix into two factors.</p>
 <pre class="syntax">
 lmf_igd_run( rel_output,
              rel_source,
              col_row,
              col_column,
              col_value,
              row_dim,
              column_dim,
              max_rank,
              stepsize,
              scale_factor,
              num_iterations,
              tolerance
            )
 </pre><p> <b>Arguments</b> </p><dl class="arglist">
 <dt>rel_output </dt>
 <dd><p class="startdd">TEXT. The name of the table to receive the output.</p>
 <p>Output factors matrix U and V are in a flattened format. </p><pre>RESULT AS (
         matrix_u    DOUBLE PRECISION[],
         matrix_v    DOUBLE PRECISION[],
         rmse        DOUBLE PRECISION
 );</pre><p class="enddd">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>
 </dd>
 <dt>rel_source </dt>
 <dd><p class="startdd">TEXT. The name of the table containing the input data.</p>
 <p>The input matrix&gt; 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 class="enddd">Input is contained in a table that describes an incomplete matrix, with available entries specified as (row, column, value). The input matrix should be 1-based, which means row &gt;= 1, and col &gt;= 1. NULL values are not expected.  </p>
 </dd>
 <dt>col_row </dt>
 <dd>TEXT. The name of the column containing the row number. </dd>
 <dt>col_column </dt>
 <dd>TEXT. The name of the column containing the column number. </dd>
 <dt>col_value </dt>
 <dd>DOUBLE PRECISION. The value at (row, col). </dd>
 <dt>row_dim (optional) </dt>
 <dd>INTEGER, default: "SELECT max(col_row) FROM rel_source". The number of columns in the matrix. </dd>
 <dt>column_dim (optional) </dt>
 <dd>INTEGER, default: "SELECT max(col_col) FROM rel_source". The number of rows in the matrix. </dd>
 <dt>max_rank </dt>
 <dd>INTEGER, default: 20. The rank of desired approximation. </dd>
 <dt>stepsize (optional) </dt>
 <dd>DOUBLE PRECISION, default: 0.01. Hyper-parameter that decides how aggressive the gradient steps are.  </dd>
 <dt>scale_factor (optional) </dt>
 <dd>DOUBLE PRECISION, default: 0.1. Hyper-parameter that decides scale of initial factors. </dd>
 <dt>num_iterations (optional) </dt>
 <dd>INTEGER, default: 10. Maximum number if iterations to perform regardless of convergence. </dd>
 <dt>tolerance (optional) </dt>
 <dd>DOUBLE PRECISION, default: 0.0001. Acceptable level of error in convergence. </dd>
 </dl>
 <p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
 <ol type="1">
 <li>Prepare an input table/view: <pre class="example">
 DROP TABLE IF EXISTS lmf_data;
 CREATE TABLE lmf_data (
  row INT,
  col INT,
  val FLOAT8
 );
 </pre></li>
 <li>Populate the input table with some data. <pre class="example">
 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></li>
 <li>Call the <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. <pre class="example">
 DROP TABLE IF EXISTS lmf_model;
 SELECT madlib.lmf_igd_run( 'lmf_model',
                            'lmf_data',
                            'row',
                            'col',
                            'val',
                            999,
                            10000,
                            3,
                            0.1,
                            2,
                            10,
                            1e-9
                          );
 </pre> Example result (the exact result may not be the same). <pre class="result">
 NOTICE:
 Finished low-rank matrix factorization using incremental gradient
 DETAIL:
    table : lmf_data (row, col, val)
 Results:
    RMSE = 0.0145966345300041
 Output:
    view : SELECT * FROM lmf_model WHERE id = 1
  lmf_igd_run
 &#160;-----------
            1
  (1 row)
 </pre></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: <pre class="example">
 SELECT array_dims(matrix_u) AS u_dims, array_dims(matrix_v) AS v_dims
 FROM lmf_model
 WHERE id = 1;
 </pre> Result: <pre class="result">
      u_dims    |     v_dims
  --------------+----------------
   [1:999][1:3] | [1:10000][1:3]
  (1 row)
 </pre></li>
 <li>Query the result value. <pre class="example">
 SELECT matrix_u[2:2][1:3] AS row_2_features
 FROM lmf_model
 WHERE id = 1;
 </pre> Example output (the exact result may not be the same): <pre class="result">
                        row_2_features
 &#160;---------------------------------------------------------
   {{1.12030523084104,0.522217971272767,0.0264869043603539}}
  (1 row)
 </pre></li>
 <li>Make prediction of a missing entry (row=2, col=7654). <pre class="example">
 SELECT madlib.array_dot(
     matrix_u[2:2][1:3],
     matrix_v[7654:7654][1:3]
     ) AS row_2_col_7654
 FROM lmf_model
 WHERE id = 1;
 </pre> Example output (the exact result may not be the same due the randomness of the algorithm): <pre class="result">
    row_2_col_7654
 &#160;------------------
   1.3201582940851
  (1 row)
 </pre></li>
 </ol>
 <p><a class="anchor" id="literature"></a></p><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__datatrans.html">Data Types and Transformations</a> » <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a> » <a class="el" href="group__grp__matrix__factorization.html">Matrix Factorization</a></div></div> </div>
	</div><!--header-->
	<div class="contents">
	<div class="toc"><b>Contents</b> <ul>
	<li>
	<a href="#syntax">Function Syntax</a> </li>
	<li>
	<a href="#examples">Examples</a> </li>
	<li>
	<a href="#literature">Literature</a> </li>
	</ul>
	</div><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>
	<p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Function Syntax</dt><dd></dd></dl>
	<p>Low-rank matrix factorization of an incomplete matrix into two factors.</p>
	<pre class="syntax">
	lmf_igd_run( rel_output,
	rel_source,
	col_row,
	col_column,
	col_value,
	row_dim,
	column_dim,
	max_rank,
	stepsize,
	scale_factor,
	num_iterations,
	tolerance
	)
	</pre><p> <b>Arguments</b> </p><dl class="arglist">
	<dt>rel_output </dt>
	<dd><p class="startdd">TEXT. The name of the table to receive the output.</p>
	<p>Output factors matrix U and V are in a flattened format. </p><pre>RESULT AS (
	matrix_u DOUBLE PRECISION[],
	matrix_v DOUBLE PRECISION[],
	rmse DOUBLE PRECISION
	);</pre><p class="enddd">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>
	</dd>
	<dt>rel_source </dt>
	<dd><p class="startdd">TEXT. The name of the table containing the input data.</p>
	<p>The input matrix> 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 class="enddd">Input is contained in a table that describes an incomplete matrix, with available entries specified as (row, column, value). The input matrix should be 1-based, which means row >= 1, and col >= 1. NULL values are not expected. </p>
	</dd>
	<dt>col_row </dt>
	<dd>TEXT. The name of the column containing the row number. </dd>
	<dt>col_column </dt>
	<dd>TEXT. The name of the column containing the column number. </dd>
	<dt>col_value </dt>
	<dd>DOUBLE PRECISION. The value at (row, col). </dd>
	<dt>row_dim (optional) </dt>
	<dd>INTEGER, default: "SELECT max(col_row) FROM rel_source". The number of columns in the matrix. </dd>
	<dt>column_dim (optional) </dt>
	<dd>INTEGER, default: "SELECT max(col_col) FROM rel_source". The number of rows in the matrix. </dd>
	<dt>max_rank </dt>
	<dd>INTEGER, default: 20. The rank of desired approximation. </dd>
	<dt>stepsize (optional) </dt>
	<dd>DOUBLE PRECISION, default: 0.01. Hyper-parameter that decides how aggressive the gradient steps are. </dd>
	<dt>scale_factor (optional) </dt>
	<dd>DOUBLE PRECISION, default: 0.1. Hyper-parameter that decides scale of initial factors. </dd>
	<dt>num_iterations (optional) </dt>
	<dd>INTEGER, default: 10. Maximum number if iterations to perform regardless of convergence. </dd>
	<dt>tolerance (optional) </dt>
	<dd>DOUBLE PRECISION, default: 0.0001. Acceptable level of error in convergence. </dd>
	</dl>
	<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
	<ol type="1">
	<li>Prepare an input table/view: <pre class="example">
	DROP TABLE IF EXISTS lmf_data;
	CREATE TABLE lmf_data (
	row INT,
	col INT,
	val FLOAT8
	);
	</pre></li>
	<li>Populate the input table with some data. <pre class="example">
	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></li>
	<li>Call the <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. <pre class="example">
	DROP TABLE IF EXISTS lmf_model;
	SELECT madlib.lmf_igd_run( 'lmf_model',
	'lmf_data',
	'row',
	'col',
	'val',
	999,
	10000,
	3,
	0.1,
	2,
	10,
	1e-9
	);
	</pre> Example result (the exact result may not be the same). <pre class="result">
	NOTICE:
	Finished low-rank matrix factorization using incremental gradient
	DETAIL:
	table : lmf_data (row, col, val)
	Results:
	RMSE = 0.0145966345300041
	Output:
	view : SELECT * FROM lmf_model WHERE id = 1
	lmf_igd_run
	-----------
	1
	(1 row)
	</pre></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: <pre class="example">
	SELECT array_dims(matrix_u) AS u_dims, array_dims(matrix_v) AS v_dims
	FROM lmf_model
	WHERE id = 1;
	</pre> Result: <pre class="result">
	u_dims \| v_dims
	--------------+----------------
	[1:999][1:3] \| [1:10000][1:3]
	(1 row)
	</pre></li>
	<li>Query the result value. <pre class="example">
	SELECT matrix_u[2:2][1:3] AS row_2_features
	FROM lmf_model
	WHERE id = 1;
	</pre> Example output (the exact result may not be the same): <pre class="result">
	row_2_features
	---------------------------------------------------------
	{{1.12030523084104,0.522217971272767,0.0264869043603539}}
	(1 row)
	</pre></li>
	<li>Make prediction of a missing entry (row=2, col=7654). <pre class="example">
	SELECT madlib.array_dot(
	matrix_u[2:2][1:3],
	matrix_v[7654:7654][1:3]
	) AS row_2_col_7654
	FROM lmf_model
	WHERE id = 1;
	</pre> Example output (the exact result may not be the same due the randomness of the algorithm): <pre class="result">
	row_2_col_7654
	------------------
	1.3201582940851
	(1 row)
	</pre></li>
	</ol>
	<p><a class="anchor" id="literature"></a></p><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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