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<title>MADlib: Matrix Factorization</title>
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<div class="header">
  <div class="headertitle">
<div class="title">Matrix Factorization<div class="ingroups"><a class="el" href="group__grp__deprecated.html">Deprecated Modules</a></div></div>  </div>
</div><!--header-->
<div class="contents">
<dl class="section warning"><dt>Warning</dt><dd><em> This is an old implementation of Matrix Decomposition and has been deprecated. For SVD decomposition, please see <a class="el" href="group__grp__svd.html">Singular Value Decomposition</a>; for the latest version of low-rank approximation, please see <a class="el" href="group__grp__lmf.html">Low-rank Matrix Factorization</a></em></dd></dl>
<div class="toc"><b>Contents</b> </p><ul>
<li>
<a href="#syntax">SVD Function Syntax</a> </li>
<li>
<a href="#xamples">Examples</a> </li>
<li>
<a href="#literature">Literature</a> </li>
<li>
<a href="#related">Related Topics</a> </li>
</ul>
</div><p>This module implements "partial SVD decomposition" method for representing a sparse matrix using a low-rank approximation. Mathematically, this algorithm seeks to find matrices U and V that, for any given A, minimizes:<br />
 </p><p class="formulaDsp">
\[ ||\boldsymbol A - \boldsymbol UV ||_2 \]
</p>
<p> subject to \( rank(\boldsymbol UV) \leq k \), where \( ||\cdot||_2 \) denotes the Frobenius norm and \( k \leq rank(\boldsymbol A)\). If A is \( m \times n \), then U will be \( m \times k \) and V will be \( k \times n \).</p>
<p>This algorithm is not intended to do the full decomposition, or to be used as part of inverse procedure. It effectively computes the SVD of a low-rank approximation of A (preferably sparse), with the singular values absorbed in U and V. Code is based on the write-up as appears at [1], with some modifications.</p>
<p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Function Syntax</dt><dd></dd></dl>
<p>The SVD function is called as follows: </p><pre class="syntax">
svdmf_run( input_table,
           col_name,
           row_name,
           value, num_features)
</pre><p>The <b>input matrix</b> is expected to be of the following form: </p><pre>{TABLE|VIEW} <em>input_table</em> (
    <em>col_num</em> INTEGER,
    <em>row_num</em> INTEGER,
  <em>value</em> FLOAT
)</pre><p>Input is contained in a table where column number and row number for each cell are sequential; that is to say that if the data was written as a matrix, those values would be the actual row and column numbers and not some random identifiers. All rows and columns must be associated with a value. There should not be any missing row, columns or values.</p>
<p>The function returns two tables <code>matrix_u</code> and <code>matrix_v</code>, which represent the matrices U and V in table format.</p>
<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">
CREATE TABLE svd_test ( col INT,
                        row INT,
                        val FLOAT
                      );
</pre></li>
<li>Populate the input table with some data. <pre class="example">
INSERT INTO svd_test SELECT ( g.a%1000)+1, g.a/1000+1, random()
                     FROM generate_series(1,1000) AS g(a);
</pre></li>
<li><p class="startli">Call the <a class="el" href="svdmf_8sql__in.html#a6cff34415cca23aa0a826cc08a6283f5" title="Partial SVD decomposition of a sparse matrix into U and V components. ">svdmf_run()</a> stored procedure. </p><pre class="example">
SELECT madlib.svdmf_run( 'svd_test',
                         'col',
                         'row',
                         'val',
                         3);
</pre><p> Example result: </p><pre class="result">
INFO:  ('Started <a class="el" href="svdmf_8sql__in.html#a6cff34415cca23aa0a826cc08a6283f5" title="Partial SVD decomposition of a sparse matrix into U and V components. ">svdmf_run()</a> with parameters:',)
INFO:  (' * input_matrix = madlib_svdsparse_test.test',)
INFO:  (' * col_name = col_num',)
INFO:  (' * row_name = row_num',)
INFO:  (' * value = val',)
INFO:  (' * num_features = 3',)
INFO:  ('Copying the source data into a temporary table...',)
INFO:  ('Estimating feature: 1',)
INFO:  ('...Iteration 1: residual_error = 33345014611.1, step_size = 4.9997500125e-10, min_improvement = 1.0',)
INFO:  ('...Iteration 2: residual_error = 33345014557.6, step_size = 5.49972501375e-10, min_improvement = 1.0',)
INFO:  ('...Iteration 3: residual_error = 33345014054.3, step_size = 6.04969751512e-10, min_improvement = 1.0',)
...
INFO:  ('...Iteration 78: residual_error = 2.02512133868, step_size = 5.78105354457e-10, min_improvement = 1.0',)
INFO:  ('...Iteration 79: residual_error = 0.893810181282, step_size = 6.35915889903e-10, min_improvement = 1.0',)
INFO:  ('...Iteration 80: residual_error = 0.34496773222, step_size = 6.99507478893e-10, min_improvement = 1.0',)
INFO:  ('Swapping residual error matrix...',)
                                         svdmf_run
&#160;-------------------------------------------------------------------------------------------</pre><pre class="result"> Finished SVD matrix factorisation for madlib_svdsparse_test.test (row_num, col_num, val).
 Results:
    total error = 0.34496773222
    number of estimated features = 1
 Output:
    table : madlib.matrix_u
    table : madlib.matrix_v
 Time elapsed: 4 minutes 47.86839 seconds.
</pre></li>
</ol>
<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
<p>[1] 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><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="svdmf_8sql__in.html" title="SQL functions for SVD Matrix Factorization. ">svdmf.sql_in</a> documenting the SQL functions.</dd></dl>
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