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<title>MADlib: Low-rank Matrix Factorization</title>
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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> </p><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">
<img class="formulaDsp" alt="\[ \|\boldsymbol A - \boldsymbol UV^{T} \|_2 \]" src="form_46.png"/>
</p>
<p>subject to <img class="formulaInl" alt="$rank(\boldsymbol UV^{T}) \leq r$" src="form_47.png"/>, where <img class="formulaInl" alt="$\|\cdot\|_2$" src="form_48.png"/> denotes the Frobenius norm. Let <img class="formulaInl" alt="$A$" src="form_41.png"/> be a <img class="formulaInl" alt="$m \times n$" src="form_49.png"/> matrix, then <img class="formulaInl" alt="$U$" src="form_50.png"/> will be <img class="formulaInl" alt="$m \times r$" src="form_51.png"/> and <img class="formulaInl" alt="$V$" src="form_52.png"/> will be <img class="formulaInl" alt="$n \times r$" src="form_53.png"/>, in dimension, and <img class="formulaInl" alt="$1 \leq r \ll \min(m, n)$" src="form_54.png"/>. 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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