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| <div class="title">Singular Value Decomposition<div class="ingroups"><a class="el" href="group__grp__datatrans.html">Data Types and Transforms</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> |
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| <div class="toc"><b>Contents</b> </p><ul> |
| <li> |
| <a href="#syntax">SVD Functions</a> </li> |
| <li> |
| <a href="#output">Output Tables</a> </li> |
| <li> |
| <a href="#examples">Examples</a></li> |
| <li> |
| </li> |
| <li> |
| <a href="#background">Technical Background</a> </li> |
| </ul> |
| </div><p>In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics.</p> |
| <p>Let \(A\) be a \(mxn\) matrix, where \(m \ge n\). Then \(A\) can be decomposed as follows: </p><p class="formulaDsp"> |
| \[ A = U \Sigma V^T, \] |
| </p> |
| <p> where \(U\) is a \(m \times n\) orthonormal matrix, \(\Sigma\) is a \(n \times n\) diagonal matrix, and \(V\) is an \(n \times n\) orthonormal matrix. The diagonal elements of \(\Sigma\) are called the <em>{singular</em> values}.</p> |
| <p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>SVD Functions</dt><dd></dd></dl> |
| <p>SVD factorizations are provided for dense matrices, sparse matrices, and block matrices. In addition, a native implementation is provided for sparse matrices for improved performance.</p> |
| <p><b>SVD Function for Dense Matrices</b></p> |
| <pre class="syntax"> |
| svd( source_table, |
| output_table_prefix, |
| row_id, |
| k, |
| n_iterations, |
| result_summary_table |
| ); |
| </pre><p> <b>Arguments</b> </p><dl class="arglist"> |
| <dt>source_table </dt> |
| <dd><p class="startdd">TEXT. Source table name (dense matrix).</p> |
| <p class="enddd">The table contains a <code>row_id</code> column that identifies each row. Further, the other columns are assumed to be the data for the matrix represented in two possible forms, illustrated by the following 2x2 matrix example:</p><ol type="1"> |
| <li><pre class="example"> |
| row_id col1 col2 |
| row1 1 1 0 |
| row2 2 0 1 |
| </pre></li> |
| <li><pre class="example"> |
| row_id row_vec |
| row1 1 {1, 0} |
| row2 2 {0, 1} |
| </pre> </li> |
| </ol> |
| </dd> |
| <dt>output_table_prefix </dt> |
| <dd>TEXT. Prefix for output tables. See <a href="#output">Output Tables</a>. </dd> |
| <dt>row_id </dt> |
| <dd>TEXT. ID for each row. </dd> |
| <dt>k </dt> |
| <dd>INTEGER. Number of singular vectors to compute. </dd> |
| <dt>n_iterations (optional) </dt> |
| <dd>INTEGER. Number of iterations to run. </dd> |
| <dt>result_summary_table (optional) </dt> |
| <dd>TEXT. The name of the table to store result summary. </dd> |
| </dl> |
| <hr/> |
| <p> <b>SVD Function for Sparse Matrix input</b></p> |
| <p>Use this function for matrices that are represented in the sparse-matrix format (example below). <b>The input matrix is converted to a dense matrix before the SVD operation.</b></p> |
| <pre class="syntax"> |
| svd_sparse( source_table, |
| output_table_prefix, |
| row_id, |
| col_id, |
| value, |
| row_dim, |
| col_dim, |
| k, |
| n_iterations, |
| result_summary_table |
| ); |
| </pre><p> <b>Arguments</b> </p><dl class="arglist"> |
| <dt>source_table </dt> |
| <dd><p class="startdd">TEXT. Source table name (sparse matrix).</p> |
| <p>An example sparse matrix representation is given below: </p><pre class="example"> |
| row_id col_id value |
| row1 1 1 2 |
| row2 2 1 1 |
| row3 3 2 1 |
| </pre><p> The <code>row_id</code> represents the row number, <code>col_id</code> represents the column number and the <code>value</code> represents the matrix value at [<code>row_id</code>, <code>col_id</code>]. The <code>row_id</code> and <code>col_id</code> values are indexed starting from 0. Thus the <code>row_id</code> ranges from 1 to <code>row_dim</code>, while the <code>col_id</code> ranges from 1 to <code>col_dim</code> </p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>output_table_prefix </dt> |
| <dd>TEXT. Prefix for output tables. See <a href="#output">Output Tables</a>. </dd> |
| <dt>row_id </dt> |
| <dd>TEXT. Name of the column containing the row index for each entry in sparse matrix. </dd> |
| <dt>col_id </dt> |
| <dd>TEXT. Name of the column containing the column index for each entry in sparse matrix. </dd> |
| <dt>value </dt> |
| <dd>TEXT. Name of column containing the non-zero values of the sparse matrix. </dd> |
| <dt>row_dim </dt> |
| <dd>INTEGER. Number of rows in matrix. </dd> |
| <dt>col_dim </dt> |
| <dd>INTEGER. Number of columns in matrix. </dd> |
| <dt>k </dt> |
| <dd>INTEGER. Number of singular vectors to compute. </dd> |
| <dt>n_iterations (optional) </dt> |
| <dd>INTEGER. Number of iterations to run. </dd> |
| <dt>result_summary_table (optional) </dt> |
| <dd>TEXT, default: NULL. The name of the table to store a summary of the results. </dd> |
| </dl> |
| <hr/> |
| <p> <b>Native implementation for sparse matrix</b></p> |
| <p>Use this function for matrices that are represented in the sparse-matrix format (example below). This function use the native sparse representation while computing the SVD. <b>This function should be favored if the matrix is highly sparse.</b></p> |
| <pre class="syntax"> |
| svd_sparse_native( source_table, |
| output_table_prefix, |
| row_id, |
| col_id, |
| value, |
| row_dim, |
| col_dim, |
| k, |
| n_iterations, |
| result_summary_table |
| ); |
| </pre><p> <b>Arguments</b> </p><dl class="arglist"> |
| <dt>source_table </dt> |
| <dd>TEXT. Source table name (sparse matrix - see example above). </dd> |
| <dt>output_table_prefix </dt> |
| <dd>TEXT. Prefix for output tables. See <a href="#output">Output Tables</a>. </dd> |
| <dt>row_id </dt> |
| <dd>TEXT. ID for each row. </dd> |
| <dt>col_id </dt> |
| <dd>TEXT. ID for each column. </dd> |
| <dt>value </dt> |
| <dd>TEXT. Non-zero values of the sparse matrix. </dd> |
| <dt>row_dim </dt> |
| <dd>INTEGER. Row dimension of sparse matrix. </dd> |
| <dt>col_dim </dt> |
| <dd>INTEGER. Col dimension of sparse matrix. </dd> |
| <dt>k </dt> |
| <dd>INTEGER. Number of singular vectors to compute. </dd> |
| <dt>n_iterations (optional) </dt> |
| <dd>INTEGER. Number of iterations to run. </dd> |
| <dt>result_summary_table (optional) </dt> |
| <dd>TEXT. Table name to store result summary. </dd> |
| </dl> |
| <hr/> |
| <p> <b>Block matrices</b></p> |
| <pre class="syntax"> |
| svd_block( source_table, |
| output_table_prefix, |
| k, |
| n_iterations, |
| result_summary_table |
| ); |
| </pre><p> <b>Arguments</b> </p><dl class="arglist"> |
| <dt>source_table </dt> |
| <dd>TEXT. Source table name (block matrix). </dd> |
| <dt>output_table_prefix </dt> |
| <dd>TEXT. Prefix for output tables. See <a href="#output">Output Tables</a>. </dd> |
| <dt>k </dt> |
| <dd>INTEGER. Number of singular vectors to compute. </dd> |
| <dt>n_iterations (optional) </dt> |
| <dd>INTEGER. Number of iterations to run. </dd> |
| <dt>result_summary_table (optional) </dt> |
| <dd>TEXT. Table name to store result summary. </dd> |
| </dl> |
| <p><a class="anchor" id="output"></a></p><dl class="section user"><dt>Output Tables</dt><dd></dd></dl> |
| <p>Output for eigen vectors/values is in the following three tables:</p><ul> |
| <li>Left singular matrix: Table named <output_table_prefix>_left (e.g. ‘netflix_u’)</li> |
| <li>Right singular matrix: Table named <output_table_prefix>_right (e.g. ‘netflix_v’)</li> |
| <li>Singular values: Table named <output_table_prefix>_s (e.g. ‘netflix_s’)</li> |
| </ul> |
| <p>The singular vector tables are of the format: </p><table class="output"> |
| <tr> |
| <th>row_id </th><td>INTEGER. The ID corresponding to each eigen value (in decreasing order). </td></tr> |
| <tr> |
| <th>row_vec </th><td>FLOAT8[]. Singular vector elements for this row_id. Each array is of size k. </td></tr> |
| </table> |
| <p>The singular values table is in a sparse table format, since only the diagonal elements of the matrix are non-zero: </p><table class="output"> |
| <tr> |
| <th>row_id </th><td>INTEGER. <em>i</em> for <em>ith</em> eigen value. </td></tr> |
| <tr> |
| <th>col_id </th><td>INTEGER. <em>i</em> for <em>ith</em> eigen value (same as row_id). </td></tr> |
| <tr> |
| <th>value </th><td>FLOAT8. Eigen Value. </td></tr> |
| </table> |
| <p>All <code>row_id</code> (and <code>col_id</code>) in the above tables start from 0.</p> |
| <p>The result summary table has the following columns: </p><table class="output"> |
| <tr> |
| <th>rows_used </th><td>INTEGER. Number of rows used for SVD calculation. </td></tr> |
| <tr> |
| <th>exec_time </th><td>FLOAT8. Total time for executing SVD. </td></tr> |
| <tr> |
| <th>iter </th><td>INTEGER. Total number of iterations run. </td></tr> |
| <tr> |
| <th>recon_error </th><td>FLOAT8. Total quality score (i.e. approximation quality) for this set of orthonormal basis. </td></tr> |
| <tr> |
| <th>relative_recon_error </th><td>FLOAT8. relative quality score. </td></tr> |
| </table> |
| <p>In the result summary table, the reconstruction error is computed as \( \sqrt{mean((X - USV^T)_{ij}^2)} \), where the average is over all elements of the matrices. The relative reconstruction error is then computed as ratio of the reconstruction error and \( \sqrt{mean(X_{ij}^2)} \).</p> |
| <p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl> |
| <ol type="1"> |
| <li>View online help for the SVD function. <pre class="example"> |
| SELECT madlib.svd(); |
| </pre></li> |
| <li>Create an input dataset (dense matrix). <pre class="example"> |
| CREATE TABLE mat ( |
| row_id integer, |
| row_vec double precision[] |
| ); |
| COPY mat (row_id, row_vec) FROM stdin delimiter '|'; |
| 1|{396,840,353,446,318,886,15,584,159,383} |
| 2|{691,58,899,163,159,533,604,582,269,390} |
| 3|{293,742,298,75,404,857,941,662,846,2} |
| 4|{462,532,787,265,982,306,600,608,212,885} |
| 5|{304,151,337,387,643,753,603,531,459,652} |
| 6|{327,946,368,943,7,516,272,24,591,204} |
| 7|{877,59,260,302,891,498,710,286,864,675} |
| 8|{458,959,774,376,228,354,300,669,718,565} |
| 9|{824,390,818,844,180,943,424,520,65,913} |
| 10|{882,761,398,688,761,405,125,484,222,873} |
| 11|{528,1,860,18,814,242,314,965,935,809} |
| 12|{492,220,576,289,321,261,173,1,44,241} |
| 13|{415,701,221,503,67,393,479,218,219,916} |
| 14|{350,192,211,633,53,783,30,444,176,932} |
| 15|{909,472,871,695,930,455,398,893,693,838} |
| 16|{739,651,678,577,273,935,661,47,373,618} |
| \. |
| </pre></li> |
| <li>Run SVD function for a dense matrix. <pre class="example"> |
| DROP TABLE IF EXISTS svd_u; |
| DROP TABLE IF EXISTS svd_v; |
| DROP TABLE IF EXISTS svd_s; |
| SELECT madlib.svd( 'mat', |
| 'svd', |
| 'row_id', |
| 10 |
| ); |
| </pre></li> |
| <li>Create a sparse matrix by running the <a class="el" href="matrix__ops_8sql__in.html#a9ed8df5fc43740c00bfdfd3f934429ef">matrix_sparsify()</a> utility function on the dense matrix. <pre class="example"> |
| DROP TABLE IF EXISTS mat_sparse; |
| SELECT madlib.matrix_sparsify( 'mat', |
| 'mat_sparse', |
| FALSE |
| ); |
| DROP TABLE IF EXISTS svd_u; |
| DROP TABLE IF EXISTS svd_v; |
| DROP TABLE IF EXISTS svd_s; |
| </pre></li> |
| <li>Run the SVD function for a sparse matrix. <pre class="example"> |
| SELECT madlib.svd_sparse( 'mat_sparse', |
| 'svd', |
| 'row_id', |
| 'col_id', |
| 'value', |
| 10 |
| ); |
| </pre></li> |
| </ol> |
| <p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd>In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics. Let \(A\) be a \(m \times n\) matrix, where \(m \ge n\). Then \(A\) can be decomposed as follows: <p class="formulaDsp"> |
| \[ A = U \Sigma V^T, \] |
| </p> |
| where \(U\) is a \(m \times n\) orthonormal matrix, \(\Sigma\) is a \(n \times n\) diagonal matrix, and \(V\) is an \(n \times n\) orthonormal matrix. The diagonal elements of \(\Sigma\) are called the <em>{singular</em> values}. It is possible to formulate the problem of computing the singular triplets ( \(\sigma_i, u_i, v_i\)) of \(A\) as an eigenvalue problem involving a Hermitian matrix related to \(A\). There are two possible ways of achieving this:<ol type="1"> |
| <li>With the cross product matrix, \(A^TA\) and \(AA^T\)</li> |
| <li>With the cyclic matrix <p class="formulaDsp"> |
| \[ H(A) = \begin{bmatrix} 0 & A\\ A^* & 0 \end{bmatrix} \] |
| </p> |
| The singular values are the nonnegative square roots of the eigenvalues of the cross product matrix. This approach may imply a severe loss of accuracy in the smallest singular values. The cyclic matrix approach is an alternative that avoids this problem, but at the expense of significantly increasing the cost of the computation. Computing the cross product matrix explicitly is not recommended, especially in the case of sparse A. Bidiagonalization was proposed by Golub and Kahan [citation?] as a way of tridiagonalizing the cross product matrix without forming it explicitly. Consider the following decomposition <p class="formulaDsp"> |
| \[ A = P B Q^T, \] |
| </p> |
| where \(P\) and \(Q\) are unitary matrices and \(B\) is an \(m \times n\) upper bidiagonal matrix. Then the tridiagonal matrix \(B*B\) is unitarily similar to \(A*A\). Additionally, specific methods exist that compute the singular values of \(B\) without forming \(B*B\). Therefore, after computing the SVD of B, <p class="formulaDsp"> |
| \[ B = X\Sigma Y^T, \] |
| </p> |
| it only remains to compute the SVD of the original matrix with \(U = PX\) and \(V = QY\). </li> |
| </ol> |
| </dd></dl> |
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