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Posted to commits@madlib.apache.org by ok...@apache.org on 2017/12/28 22:52:00 UTC
[26/51] [abbrv] [partial] madlib-site git commit: Additional updates
for 1.13 release
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+<title>MADlib: k-Nearest Neighbors</title>
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+ <div id="projectbrief">User Documentation for MADlib</div>
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+<div class="title">k-Nearest Neighbors<div class="ingroups"><a class="el" href="group__grp__early__stage.html">Early Stage Development</a> » <a class="el" href="group__grp__nene.html">Nearest Neighbors</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li class="level1">
+<a href="#knn">K-Nearest Neighbors</a> </li>
+<li class="level1">
+<a href="#usage">Usage</a> </li>
+<li class="level1">
+<a href="#output">Output Format</a> </li>
+<li class="level1">
+<a href="#examples">Examples</a> </li>
+<li class="level1">
+<a href="#background">Technical Background</a> </li>
+<li class="level1">
+<a href="#literature">Literature</a> </li>
+</ul>
+</div><dl class="section warning"><dt>Warning</dt><dd><em> This MADlib method is still in early stage development. There may be some issues that will be addressed in a future version. Interface and implementation are subject to change. </em></dd></dl>
+<p><a class="anchor" id="knn"></a> K-nearest neighbors is a method for finding the k closest points to a given data point in terms of a given metric. Its input consists of data points as features from testing examples, and it looks for k closest points in the training set for each of the data points in test set. The output of KNN depends on the type of task. For classification, the output is the majority vote of the classes of the k nearest data points. That is, the testing example gets assigned the most popular class from the nearest neighbors. For regression, the output is the average of the values of k nearest neighbors of the given test point.</p>
+<p><a class="anchor" id="usage"></a></p><dl class="section user"><dt>Usage</dt><dd><pre class="syntax">
+knn( point_source,
+ point_column_name,
+ point_id,
+ label_column_name,
+ test_source,
+ test_column_name,
+ test_id,
+ output_table,
+ k,
+ output_neighbors,
+ fn_dist
+ )
+</pre></dd></dl>
+<p><b>Arguments</b> </p><dl class="arglist">
+<dt>point_source </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the training data points. Training data points are expected to be stored row-wise in a column of type <code>DOUBLE PRECISION[]</code>. </p>
+<p class="enddd"></p>
+</dd>
+<dt>point_column_name </dt>
+<dd><p class="startdd">TEXT. Name of the column with training data points.</p>
+<p class="enddd"></p>
+</dd>
+<dt>point_id </dt>
+<dd><p class="startdd">TEXT. Name of the column in 'point_source’ containing source data ids. The ids are of type INTEGER with no duplicates. They do not need to be contiguous. This parameter must be used if the list of nearest neighbors are to be output, i.e., if the parameter 'output_neighbors' below is TRUE or if 'label_column_name' is NULL.</p>
+<p class="enddd"></p>
+</dd>
+<dt>label_column_name </dt>
+<dd><p class="startdd">TEXT. Name of the column with labels/values of training data points. If this column is a Boolean, integer or text, it will run KNN classification, else if it is double precision values will run KNN regression. If you set this to NULL, it will only return the set of neighbors without actually doing classification or regression.</p>
+<p class="enddd"></p>
+</dd>
+<dt>test_source </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the test data points. Testing data points are expected to be stored row-wise in a column of type <code>DOUBLE PRECISION[]</code>. </p>
+<p class="enddd"></p>
+</dd>
+<dt>test_column_name </dt>
+<dd><p class="startdd">TEXT. Name of the column with testing data points.</p>
+<p class="enddd"></p>
+</dd>
+<dt>test_id </dt>
+<dd><p class="startdd">TEXT. Name of the column having ids of data points in test data table.</p>
+<p class="enddd"></p>
+</dd>
+<dt>output_table </dt>
+<dd><p class="startdd">TEXT. Name of the table to store final results.</p>
+<p class="enddd"></p>
+</dd>
+<dt>k (optional) </dt>
+<dd><p class="startdd">INTEGER. default: 1. Number of nearest neighbors to consider. For classification, should be an odd number to break ties, otherwise the result may depend on ordering of the input data.</p>
+<p class="enddd"></p>
+</dd>
+<dt>output_neighbors (optional) </dt>
+<dd><p class="startdd">BOOLEAN default: TRUE. Outputs the list of k-nearest neighbors that were used in the voting/averaging, sorted from closest to furthest.</p>
+<p class="enddd"></p>
+</dd>
+<dt>fn_dist (optional) </dt>
+<dd><p class="startdd">TEXT, default: 'squared_dist_norm2'. The name of the function used to calculate the distance between data points.</p>
+<p>The following distance functions can be used: </p><ul>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#aad193850e79c4b9d811ca9bc53e13476">dist_norm1</a></b>: 1-norm/Manhattan </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#aa58e51526edea6ea98db30b6f250adb4">dist_norm2</a></b>: 2-norm/Euclidean </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#a00a08e69f27524f2096032214e15b668">squared_dist_norm2</a></b>: squared Euclidean distance </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#a8c7b9281a72ff22caf06161701b27e84">dist_angle</a></b>: angle </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#afa13b4c6122b99422d666dedea136c18">dist_tanimoto</a></b>: tanimoto </li>
+<li>
+<b>user defined function</b> with signature <code>DOUBLE PRECISION[] x, DOUBLE PRECISION[] y -> DOUBLE PRECISION</code></li>
+</ul>
+<p class="enddd"></p>
+</dd>
+</dl>
+<p><a class="anchor" id="output"></a></p><dl class="section user"><dt>Output Format</dt><dd></dd></dl>
+<p>The output of the KNN module is a table with the following columns: </p><table class="output">
+<tr>
+<th>id </th><td>INTEGER. The ids of test data points. </td></tr>
+<tr>
+<th>test_column_name </th><td>DOUBLE PRECISION[]. The test data points. </td></tr>
+<tr>
+<th>prediction </th><td>INTEGER. Label in case of classification, average value in case of regression. </td></tr>
+<tr>
+<th>k_nearest_neighbours </th><td>INTEGER[]. List of nearest neighbors, sorted closest to furthest from the corresponding test point. </td></tr>
+</table>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Prepare some training data for classification: <pre class="example">
+DROP TABLE IF EXISTS knn_train_data;
+CREATE TABLE knn_train_data (
+ id integer,
+ data integer[],
+ label integer -- Integer label means for classification
+ );
+INSERT INTO knn_train_data VALUES
+(1, '{1,1}', 1),
+(2, '{2,2}', 1),
+(3, '{3,3}', 1),
+(4, '{4,4}', 1),
+(5, '{4,5}', 1),
+(6, '{20,50}', 0),
+(7, '{10,31}', 0),
+(8, '{81,13}', 0),
+(9, '{1,111}', 0);
+</pre></li>
+<li>Prepare some training data for regression: <pre class="example">
+DROP TABLE IF EXISTS knn_train_data_reg;
+CREATE TABLE knn_train_data_reg (
+ id integer,
+ data integer[],
+ label float -- Float label means for regression
+ );
+INSERT INTO knn_train_data_reg VALUES
+(1, '{1,1}', 1.0),
+(2, '{2,2}', 1.0),
+(3, '{3,3}', 1.0),
+(4, '{4,4}', 1.0),
+(5, '{4,5}', 1.0),
+(6, '{20,50}', 0.0),
+(7, '{10,31}', 0.0),
+(8, '{81,13}', 0.0),
+(9, '{1,111}', 0.0);
+</pre></li>
+<li>Prepare some testing data: <pre class="example">
+DROP TABLE IF EXISTS knn_test_data;
+CREATE TABLE knn_test_data (
+ id integer,
+ data integer[]
+ );
+INSERT INTO knn_test_data VALUES
+(1, '{2,1}'),
+(2, '{2,6}'),
+(3, '{15,40}'),
+(4, '{12,1}'),
+(5, '{2,90}'),
+(6, '{50,45}');
+</pre></li>
+<li>Run KNN for classification: <pre class="example">
+DROP TABLE IF EXISTS knn_result_classification;
+SELECT * FROM madlib.knn(
+ 'knn_train_data', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col name of id in train data
+ 'label', -- Training labels
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_classification', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'madlib.squared_dist_norm2' -- Distance function
+ );
+SELECT * from knn_result_classification ORDER BY id;
+</pre> Result: <pre class="result">
+ id | data | prediction | k_nearest_neighbours
+----+---------+------------+----------------------
+ 1 | {2,1} | 1 | {2,1,3}
+ 2 | {2,6} | 1 | {5,4,3}
+ 3 | {15,40} | 0 | {7,6,5}
+ 4 | {12,1} | 1 | {4,5,3}
+ 5 | {2,90} | 0 | {9,6,7}
+ 6 | {50,45} | 0 | {6,7,8}
+(6 rows)
+</pre> Note that the nearest neighbors are sorted from closest to furthest from the corresponding test point.</li>
+<li>Run KNN for regression: <pre class="example">
+DROP TABLE IF EXISTS knn_result_regression;
+SELECT * FROM madlib.knn(
+ 'knn_train_data_reg', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col Name of id in train data
+ 'label', -- Training labels
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_regression', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'madlib.dist_norm2' -- Distance function
+ );
+SELECT * FROM knn_result_regression ORDER BY id;
+</pre> Result: <pre class="result">
+ id | data | prediction | k_nearest_neighbours
+----+---------+-------------------+----------------------
+ 1 | {2,1} | 1 | {2,1,3}
+ 2 | {2,6} | 1 | {5,4,3}
+ 3 | {15,40} | 0.333333333333333 | {7,6,5}
+ 4 | {12,1} | 1 | {4,5,3}
+ 5 | {2,90} | 0 | {9,6,7}
+ 6 | {50,45} | 0 | {6,7,8}
+(6 rows)
+</pre></li>
+<li>List nearest neighbors only, without doing classification or regression: <pre class="example">
+DROP TABLE IF EXISTS knn_result_list_neighbors;
+SELECT * FROM madlib.knn(
+ 'knn_train_data_reg', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col Name of id in train data
+ NULL, -- NULL training labels means just list neighbors
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_list_neighbors', -- Output table
+ 3 -- Number of nearest neighbors
+ );
+SELECT * FROM knn_result_list_neighbors ORDER BY id;
+</pre> Result, with neighbors sorted from closest to furthest: <pre class="result">
+ id | data | k_nearest_neighbours
+----+---------+----------------------
+ 1 | {2,1} | {2,1,3}
+ 2 | {2,6} | {5,4,3}
+ 3 | {15,40} | {7,6,5}
+ 4 | {12,1} | {4,5,3}
+ 5 | {2,90} | {9,6,7}
+ 6 | {50,45} | {6,7,8}
+(6 rows)
+</pre></li>
+</ol>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
+<p>The training data points are vectors in a multidimensional feature space, each with a class label. The training phase of the algorithm consists only of storing the feature vectors and class labels of the training points.</p>
+<p>In the classification phase, k is a user-defined constant, and an unlabeled vector (a test point) is classified by assigning the label which is most frequent among the k training samples nearest to that test point. In case of regression, average of the values of these k training samples is assigned to the test point.</p>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p><a class="anchor" id="knn-lit-1"></a>[1] Wikipedia, k-nearest neighbors algorithm, <a href="https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm">https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm</a></p>
+<p><a class="anchor" id="knn-lit-2"></a>[2] N. S. Altman: An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression <a href="http://www.stat.washington.edu/courses/stat527/s13/readings/Altman_AmStat_1992.pdf">http://www.stat.washington.edu/courses/stat527/s13/readings/Altman_AmStat_1992.pdf</a></p>
+<p><a class="anchor" id="knn-lit-3"></a>[3] Gongde Guo1, Hui Wang, David Bell, Yaxin Bi, Kieran Greer: KNN Model-Based Approach in Classification, <a href="https://ai2-s2-pdfs.s3.amazonaws.com/a7e2/814ec5db800d2f8c4313fd436e9cf8273821.pdf">https://ai2-s2-pdfs.s3.amazonaws.com/a7e2/814ec5db800d2f8c4313fd436e9cf8273821.pdf</a></p>
+</div><!-- contents -->
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+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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+<div class="title">Latent Dirichlet Allocation<div class="ingroups"><a class="el" href="group__grp__unsupervised.html">Unsupervised Learning</a> » <a class="el" href="group__grp__topic__modelling.html">Topic Modelling</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#vocabulary">Vocabulary Format</a> </li>
+<li>
+<a href="#train">Training Function</a> </li>
+<li>
+<a href="#predict">Prediction Function</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#notes">Notes</a> </li>
+<li>
+<a href="#literature">Literature</a> </li>
+<li>
+<a href="#related">Related Topics</a></li>
+<li>
+</li>
+</ul>
+</div><p>Latent Dirichlet Allocation (LDA) is an interesting generative probabilistic model for natural texts and has received a lot of attention in recent years. The model is quite versatile, having found uses in problems like automated topic discovery, collaborative filtering, and document classification.</p>
+<p>The LDA model posits that each document is associated with a mixture of various topics (e.g. a document is related to Topic 1 with probability 0.7, and Topic 2 with probability 0.3), and that each word in the document is attributable to one of the document's topics. There is a (symmetric) Dirichlet prior with parameter \( \alpha \) on each document's topic mixture. In addition, there is another (symmetric) Dirichlet prior with parameter \( \beta \) on the distribution of words for each topic.</p>
+<p>The following generative process then defines a distribution over a corpus of documents.</p>
+<ul>
+<li>Sample for each topic \( i \), a per-topic word distribution \( \phi_i \) from the Dirichlet( \(\beta\)) prior.</li>
+<li>For each document:<ul>
+<li>Sample a document length N from a suitable distribution, say, Poisson.</li>
+<li>Sample a topic mixture \( \theta \) for the document from the Dirichlet( \(\alpha\)) distribution.</li>
+<li>For each of the N words:<ul>
+<li>Sample a topic \( z_n \) from the multinomial topic distribution \( \theta \).</li>
+<li>Sample a word \( w_n \) from the multinomial word distribution \( \phi_{z_n} \) associated with topic \( z_n \).</li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+<p>In practice, only the words in each document are observable. The topic mixture of each document and the topic for each word in each document are latent unobservable variables that need to be inferred from the observables, and this is the problem people refer to when they talk about the inference problem for LDA. Exact inference is intractable, but several approximate inference algorithms for LDA have been developed. The simple and effective Gibbs sampling algorithm described in Griffiths and Steyvers [2] appears to be the current algorithm of choice.</p>
+<p>This implementation provides a parallel and scalable in-database solution for LDA based on Gibbs sampling. Different with the implementations based on MPI or Hadoop Map/Reduce, this implementation builds upon the shared-nothing MPP databases and enables high-performance in-database analytics.</p>
+<p><a class="anchor" id="vocabulary"></a></p><dl class="section user"><dt>Vocabulary Format</dt><dd></dd></dl>
+<p>The vocabulary, or dictionary, indexes all the words found in the corpus and has the following format: </p><pre>{TABLE|VIEW} <em>vocab_table</em> (
+ <em>wordid</em> INTEGER,
+ <em>word</em> TEXT
+)</pre><p> where <code>wordid</code> refers the word ID (the index of a word in the vocabulary) and <code>word</code> is the actual word.</p>
+<dl class="section user"><dt>Usage</dt><dd><ul>
+<li><p class="startli">The training (i.e. topic inference) can be done with the following function: </p><pre>
+ SELECT <a class="el" href="lda_8sql__in.html#aeb7593251a4dedb695494f65dc2d1f80">lda_train</a>(
+ <em>'data_table'</em>,
+ <em>'model_table'</em>,
+ <em>'output_data_table'</em>,
+ <em>voc_size</em>,
+ <em>topic_num</em>,
+ <em>iter_num</em>,
+ <em>alpha</em>,
+ <em>beta</em>)
+ </pre><p class="startli">This function stores the resulting model in <code><em>model_table</em></code>. The table has only 1 row and is in the following form: </p><pre>{TABLE} <em>model_table</em> (
+ <em>voc_size</em> INTEGER,
+ <em>topic_num</em> INTEGER,
+ <em>alpha</em> FLOAT,
+ <em>beta</em> FLOAT,
+ <em>model</em> BIGINT[])
+ </pre><p class="startli">This function also stores the topic counts and the topic assignments in each document in <code><em>output_data_table</em></code>. The table is in the following form: </p><pre>{TABLE} <em>output_data_table</em> (
+ <em>docid</em> INTEGER,
+ <em>wordcount</em> INTEGER,
+ <em>words</em> INTEGER[],
+ <em>counts</em> INTEGER[],
+ <em>topic_count</em> INTEGER[],
+ <em>topic_assignment</em> INTEGER[])
+ </pre></li>
+<li><p class="startli">The prediction (i.e. labelling of test documents using a learned LDA model) can be done with the following function: </p><pre>
+ SELECT <a class="el" href="lda_8sql__in.html#aaa89e30c8fd0ba41b6feee01ee195330">lda_predict</a>(
+ <em>'data_table'</em>,
+ <em>'model_table'</em>,
+ <em>'output_table'</em>);
+ </pre><p class="startli">This function stores the prediction results in <em>output_table</em>. Each row in the table stores the topic distribution and the topic assignments for a docuemnt in the dataset. The table is in the following form: </p><pre>{TABLE} <em>output_table</em> (
+ <em>docid</em> INTEGER,
+ <em>wordcount</em> INTEGER,
+ <em>words</em> INTEGER,
+ <em>counts</em> INTEGER,
+ <em>topic_count</em> INTEGER[],
+ <em>topic_assignment</em> INTEGER[])
+ </pre></li>
+<li>This module also provides a function for computing the perplexity: <pre>
+ SELECT <a class="el" href="lda_8sql__in.html#a25c3ef12d9808d8a38c5fd2630f3b5a9">lda_get_perplexity</a>(
+ <em>'model_table'</em>,
+ <em>'output_data_table'</em>);
+ </pre></li>
+</ul>
+</dd></dl>
+<dl class="section user"><dt>Implementation Notes</dt><dd>The input format requires the user to tokenize each document into an array of words. This process involves tokenizing and filtering documents - a process out-of-scope for this module. Internally, the input data will be validated and then converted to the following format for efficiency: <pre>{TABLE} <em>__internal_data_table__</em> (
+ <em>docid</em> INTEGER,
+ <em>wordcount</em> INTEGER,
+ <em>words</em> INTEGER[],
+ <em>counts</em> INTEGER[])
+</pre> where <code>docid</code> is the document ID, <code>wordcount</code> is the number of words in the document, <code>words</code> is the list of unique words in the document, and <code>counts</code> is a list of the number of occurrences of each unique word in the document.</dd></dl>
+<p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Training Function</dt><dd>The LDA training function has the following syntax. <pre class="syntax">
+lda_train( data_table,
+ model_table,
+ output_data_table,
+ voc_size,
+ topic_num,
+ iter_num,
+ alpha,
+ beta
+ )
+</pre> <b>Arguments</b> <dl class="arglist">
+<dt>data_table </dt>
+<dd><p class="startdd">TEXT. The name of the table storing the training dataset. Each row is in the form <code><docid, wordid, count></code> where <code>docid</code>, <code>wordid</code>, and <code>count</code> are non-negative integers.</p>
+<p>The <code>docid</code> column refers to the document ID, the <code>wordid</code> column is the word ID (the index of a word in the vocabulary), and <code>count</code> is the number of occurrences of the word in the document.</p>
+<p>Please note that column names for <code>docid</code>, <code>wordid</code>, and <code>count</code> are currently fixed, so you must use these exact names in the data_table.</p>
+<p class="enddd"></p>
+</dd>
+<dt>model_table </dt>
+<dd>TEXT. The name of the table storing the learned models. This table has one row and the following columns. <table class="output">
+<tr>
+<th>voc_size </th><td>INTEGER. Size of the vocabulary. Note that the <code>wordid</code> should be continous integers starting from 0 to <code>voc_size</code> − <code>1</code>. A data validation routine is called to validate the dataset. </td></tr>
+<tr>
+<th>topic_num </th><td>INTEGER. Number of topics. </td></tr>
+<tr>
+<th>alpha </th><td>DOUBLE PRECISION. Dirichlet parameter for the per-doc topic multinomial (e.g. 50/topic_num). </td></tr>
+<tr>
+<th>beta </th><td>DOUBLE PRECISION. Dirichlet parameter for the per-topic word multinomial (e.g. 0.01). </td></tr>
+<tr>
+<th>model </th><td>BIGINT[]. </td></tr>
+</table>
+</dd>
+<dt>output_data_table </dt>
+<dd>TEXT. The name of the table to store the output data. It has the following columns: <table class="output">
+<tr>
+<th>docid </th><td>INTEGER. </td></tr>
+<tr>
+<th>wordcount </th><td>INTEGER. </td></tr>
+<tr>
+<th>words </th><td>INTEGER[]. </td></tr>
+<tr>
+<th>counts </th><td>INTEGER[]. </td></tr>
+<tr>
+<th>topic_count </th><td>INTEGER[]. </td></tr>
+<tr>
+<th>topic_assignment </th><td>INTEGER[]. </td></tr>
+</table>
+</dd>
+<dt>voc_size </dt>
+<dd>INTEGER. Size of the vocabulary. Note that the <code>wordid</code> should be continous integers starting from 0 to <code>voc_size</code> − <code>1</code>. A data validation routine is called to validate the dataset. </dd>
+<dt>topic_num </dt>
+<dd>INTEGER. Number of topics. </dd>
+<dt>iter_num </dt>
+<dd>INTEGER. Number of iterations (e.g. 60). </dd>
+<dt>alpha </dt>
+<dd>DOUBLE PRECISION. Dirichlet parameter for the per-doc topic multinomial (e.g. 50/topic_num). </dd>
+<dt>beta </dt>
+<dd>DOUBLE PRECISION. Dirichlet parameter for the per-topic word multinomial (e.g. 0.01). </dd>
+</dl>
+</dd></dl>
+<p><a class="anchor" id="predict"></a></p><dl class="section user"><dt>Prediction Function</dt><dd></dd></dl>
+<p>Prediction—labelling test documents using a learned LDA model—is accomplished with the following function: </p><pre class="syntax">
+lda_predict( data_table,
+ model_table,
+ output_table
+ );
+</pre><p>This function stores the prediction results in <code><em>output_table</em></code>. Each row in the table stores the topic distribution and the topic assignments for a document in the dataset. The table has the following columns: </p><table class="output">
+<tr>
+<th>docid </th><td>INTEGER. </td></tr>
+<tr>
+<th>wordcount </th><td>INTEGER. </td></tr>
+<tr>
+<th>words </th><td>INTEGER[]. List of word IDs in this document. </td></tr>
+<tr>
+<th>counts </th><td>INTEGER[]. List of word counts in this document. </td></tr>
+<tr>
+<th>topic_count </th><td>INTEGER[]. Of length topic_num, list of topic counts in this document. </td></tr>
+<tr>
+<th>topic_assignment </th><td>INTEGER[]. Of length wordcount, list of topic index for each word. </td></tr>
+</table>
+<p><a class="anchor" id="perplexity"></a></p><dl class="section user"><dt>Perplexity Function</dt><dd>This module provides a function for computing the perplexity. <pre class="syntax">
+lda_get_perplexity( model_table,
+ output_data_table
+ );
+</pre></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 a training dataset for LDA. The examples below are small strings extracted from various Wikipedia documents . <pre class="example">
+DROP TABLE IF EXISTS documents;
+CREATE TABLE documents(docid INT4, contents TEXT);
+INSERT INTO documents VALUES
+(0, 'Statistical topic models are a class of Bayesian latent variable models, originally developed for analyzing the semantic content of large document corpora.'),
+(1, 'By the late 1960s, the balance between pitching and hitting had swung in favor of the pitchers. In 1968 Carl Yastrzemski won the American League batting title with an average of just .301, the lowest in history.'),
+(2, 'Machine learning is closely related to and often overlaps with computational statistics; a discipline that also specializes in prediction-making. It has strong ties to mathematical optimization, which deliver methods, theory and application domains to the field.'),
+(3, 'California''s diverse geography ranges from the Sierra Nevada in the east to the Pacific Coast in the west, from the Redwood–Douglas fir forests of the northwest, to the Mojave Desert areas in the southeast. The center of the state is dominated by the Central Valley, a major agricultural area. ');
+</pre></li>
+<li>Build a word count table by extracting the words and building a histogram for each document using the <code>term_frequency</code> function (<a class="el" href="group__grp__text__utilities.html">Term Frequency</a>). <pre class="example">
+-- Convert a string to a list of words
+ALTER TABLE documents ADD COLUMN words TEXT[];
+UPDATE documents SET words = regexp_split_to_array(lower(contents), E'[\\s+\\.\\,]');
+
+-- Create the term frequency table
+DROP TABLE IF EXISTS my_training, my_training_vocabulary;
+SELECT madlib.term_frequency('documents', 'docid', 'words', 'my_training', TRUE);
+SELECT * FROM my_training order by docid limit 20;
+</pre> <pre class="result">
+ docid | wordid | count
+-------+--------+-------
+ 0 | 57 | 1
+ 0 | 86 | 1
+ 0 | 4 | 1
+ 0 | 55 | 1
+ 0 | 69 | 2
+ 0 | 81 | 1
+ 0 | 30 | 1
+ 0 | 33 | 1
+ 0 | 36 | 1
+ 0 | 43 | 1
+ 0 | 25 | 1
+ 0 | 65 | 2
+ 0 | 72 | 1
+ 0 | 9 | 1
+ 0 | 0 | 2
+ 0 | 29 | 1
+ 0 | 18 | 1
+ 0 | 12 | 1
+ 0 | 96 | 1
+ 0 | 91 | 1
+(20 rows)
+</pre> <pre class="example">
+SELECT * FROM my_training_vocabulary order by wordid limit 20;
+</pre> <pre class="result">
+ wordid | word
+--------+--------------
+ 0 |
+ 1 | 1960s
+ 2 | 1968
+ 3 | 301
+ 4 | a
+ 5 | agricultural
+ 6 | also
+ 7 | american
+ 8 | an
+ 9 | analyzing
+ 10 | and
+ 11 | application
+ 12 | are
+ 13 | area
+ 14 | areas
+ 15 | average
+ 16 | balance
+ 17 | batting
+ 18 | bayesian
+ 19 | between
+(20 rows)
+</pre></li>
+<li>Create an LDA model using the <code><a class="el" href="lda_8sql__in.html#aeb7593251a4dedb695494f65dc2d1f80" title="This UDF provides an entry for the lda training process. ">lda_train()</a></code> function. <pre class="example">
+DROP TABLE IF EXISTS my_model, my_outdata;
+SELECT madlib.lda_train( 'my_training',
+ 'my_model',
+ 'my_outdata',
+ 104,
+ 5,
+ 10,
+ 5,
+ 0.01
+ );
+</pre> Reminder that column names for <code>docid</code>, <code>wordid</code>, and <code>count</code> are currently fixed, so you must use these exact names in the input table. After a successful run of the <a class="el" href="lda_8sql__in.html#aeb7593251a4dedb695494f65dc2d1f80" title="This UDF provides an entry for the lda training process. ">lda_train()</a> function two tables are generated, one for storing the learned model and the other for storing the output data table.</li>
+<li>To get the detailed information about the learned model, run these commands: <pre class="example">
+-- The topic description by top-k words
+DROP TABLE IF EXISTS my_topic_desc;
+SELECT madlib.lda_get_topic_desc( 'my_model',
+ 'my_training_vocabulary',
+ 'my_topic_desc',
+ 15);
+select * from my_topic_desc order by topicid, prob DESC;
+</pre> <pre class="result">
+ topicid | wordid | prob | word
+---------+--------+--------------------+-------------------
+ 1 | 69 | 0.181900726392252 | of
+ 1 | 52 | 0.0608353510895884 | is
+ 1 | 65 | 0.0608353510895884 | models
+ 1 | 30 | 0.0305690072639225 | corpora
+ 1 | 1 | 0.0305690072639225 | 1960s
+ 1 | 57 | 0.0305690072639225 | latent
+ 1 | 35 | 0.0305690072639225 | diverse
+ 1 | 81 | 0.0305690072639225 | semantic
+ 1 | 19 | 0.0305690072639225 | between
+ 1 | 75 | 0.0305690072639225 | pitchers
+ 1 | 43 | 0.0305690072639225 | for
+ 1 | 6 | 0.0305690072639225 | also
+ 1 | 40 | 0.0305690072639225 | favor
+ 1 | 47 | 0.0305690072639225 | had
+ 1 | 28 | 0.0305690072639225 | computational
+ ....
+</pre> <pre class="example">
+-- The per-word topic counts (sorted by topic id)
+DROP TABLE IF EXISTS my_word_topic_count;
+SELECT madlib.lda_get_word_topic_count( 'my_model',
+ 'my_word_topic_count');
+SELECT * FROM my_word_topic_count ORDER BY wordid;
+</pre> <pre class="result">
+ wordid | topic_count
+--------+--------------
+ 0 | {0,17,0,0,0}
+ 1 | {1,0,0,0,0}
+ 2 | {0,0,0,0,1}
+ 3 | {0,0,0,0,1}
+ 4 | {0,0,0,0,3}
+ 5 | {0,1,0,0,0}
+ 6 | {1,0,0,0,0}
+ 7 | {1,0,0,0,0}
+ 8 | {0,0,0,1,0}
+ 9 | {1,0,0,0,0}
+ 10 | {0,0,0,0,3}
+ 11 | {0,0,1,0,0}
+ ....
+</pre></li>
+<li>To get the topic counts and the topic assignments for each doucment, run the following commands: <pre class="example">
+-- The per-document topic assignments and counts:
+SELECT docid, topic_assignment, topic_count FROM my_outdata;
+</pre> <pre class="result">
+ docid | topic_assignment | topic_count
+-------+-----------------------------------------------------------------------------------------------------------------+----------------
+ 1 | {1,1,1,1,1,1,2,4,1,4,4,4,1,0,2,1,0,2,2,3,4,2,1,1,4,2,4,3,0,0,2,4,4,3,3,3,3,3,0,1,0,4} | {6,12,7,7,10}
+ 3 | {1,1,1,1,1,1,4,0,2,3,1,2,0,0,0,1,2,2,1,3,3,2,2,1,2,2,2,0,3,0,4,1,0,0,1,4,3,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,3} | {8,12,10,21,4}
+ 0 | {1,1,4,2,1,4,4,4,1,3,1,0,0,0,0,0,0,0,0,1,1,3,0,1} | {9,8,1,2,4}
+ 2 | {1,1,1,1,4,1,4,4,2,0,2,4,1,1,4,1,2,0,1,3,1,2,4,3,2,4,4,3,1,2,0,3,3,1,4,3,3,3,2,1} | {3,13,7,8,9}
+(4 rows)
+</pre></li>
+<li>To use a learned LDA model for prediction (that is, to label new documents), use the following command: <pre class="example">
+SELECT madlib.lda_predict( 'my_testing',
+ 'my_model',
+ 'my_pred'
+ );
+</pre> The test table (<em>my_testing</em>) is expected to be in the same form as the training table (<em>my_training</em>) and can be created with the same process. After a successful run of the <a class="el" href="lda_8sql__in.html#af1fde06c39dd12bb9e5544997f815323" title="This UDF provides an entry for the lda predicton process. ">lda_predict()</a> function, the prediction results are generated and stored in <em>my_pred</em>. This table has the same schema as the <em>my_outdata</em> table generated by the <a class="el" href="lda_8sql__in.html#aeb7593251a4dedb695494f65dc2d1f80" title="This UDF provides an entry for the lda training process. ">lda_train()</a> function.</li>
+<li>Use the following command to compute the perplexity of the result. <pre class="example">
+SELECT madlib.lda_get_perplexity( 'my_model',
+ 'my_pred'
+ );
+</pre></li>
+</ol>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] D.M. Blei, A.Y. Ng, M.I. Jordan, <em>Latent Dirichlet Allocation</em>, Journal of Machine Learning Research, vol. 3, pp. 993-1022, 2003.</p>
+<p>[2] T. Griffiths and M. Steyvers, <em>Finding scientific topics</em>, PNAS, vol. 101, pp. 5228-5235, 2004.</p>
+<p>[3] Y. Wang, H. Bai, M. Stanton, W-Y. Chen, and E.Y. Chang, <em>lda: Parallel Dirichlet Allocation for Large-scale Applications</em>, AAIM, 2009.</p>
+<p>[4] <a href="http://en.wikipedia.org/wiki/Latent_Dirichlet_allocation">http://en.wikipedia.org/wiki/Latent_Dirichlet_allocation</a></p>
+<p>[5] J. Chang, Collapsed Gibbs sampling methods for topic models, R manual, 2010.</p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="lda_8sql__in.html" title="SQL functions for Latent Dirichlet Allocation. ">lda.sql_in</a> documenting the SQL functions. </dd></dl>
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+<title>MADlib: Norms and Distance functions</title>
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+ <td style="padding-left: 0.5em;">
+ <div id="projectname">
+ <span id="projectnumber">1.13</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
+ </td>
+ <td> <div id="MSearchBox" class="MSearchBoxInactive">
+ <span class="left">
+ <img id="MSearchSelect" src="search/mag_sel.png"
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+ onmouseover="return searchBox.OnSearchSelectShow()"
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+ name="MSearchResults" id="MSearchResults">
+</iframe>
+</div>
+
+<div class="header">
+ <div class="headertitle">
+<div class="title">Norms and Distance functions<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></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li class="level1">
+<a href="#functions">Linear Algebra Utility Functions</a> </li>
+<li class="level1">
+<a href="#examples">Examples</a> </li>
+<li class="level1">
+<a href="#literature">Literature</a> </li>
+<li class="level1">
+<a href="#related">Related Functions</a> </li>
+</ul>
+</div><p>The linalg module consists of useful utility functions for basic linear algebra operations. Several of these functions can be used while implementing new algorithms. These functions operate on vectors (1-D FLOAT8 array) and matrices (2-D FLOAT8 array). Note that other array types may need to be casted into FLOAT8[] before calling the functions.</p>
+<p>Refer to the <a class="el" href="linalg_8sql__in.html" title="SQL functions for linear algebra. ">linalg.sql_in</a> file for documentation on each of the utility functions.</p>
+<p><a class="anchor" id="functions"></a></p><dl class="section user"><dt>Linear Algebra Utility Functions</dt><dd><table class="output">
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a300300fe4b8576ba0b97b95d8dea3057" title="1-norm of a vector ">norm1()</a> </th><td><p class="starttd">1-norm of a vector, \(\|\vec{a}\|_1\).</p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a50fdfe30cc0edc6888a909dbb4b4c239" title="2-norm of a vector ">norm2()</a> </th><td><p class="starttd">2-norm of a vector, \(\|\vec{a}\|_2\). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#aad193850e79c4b9d811ca9bc53e13476" title="1-norm of the difference between two vectors ">dist_norm1()</a> </th><td><p class="starttd">1-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_1\). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#aa58e51526edea6ea98db30b6f250adb4" title="2-norm of the difference between two vectors ">dist_norm2()</a> </th><td><p class="starttd">2-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_2\). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#ad9cc156ae57bf7c0a2fe90798259105a" title="p-norm of the difference between two vectors ">dist_pnorm()</a> </th><td><p class="starttd">Generic p-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_p, p > 0\). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a31fa9f2f5b45507c09f136464fdad1db" title="Infinity-norm of the difference between two vectors. ">dist_inf_norm()</a> </th><td><p class="starttd">Infinity-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_\infty\). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a00a08e69f27524f2096032214e15b668" title="Squared 2-norm of the difference between two vectors. ">squared_dist_norm2()</a> </th><td><p class="starttd">Squared 2-norm of the difference between two vectors, \(\|\vec{a} - \vec{b}\|_2^2\). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a1782f2ba00d9f9fab20894a576079f87" title="cosine similarity score between two vectors ">cosine_similarity()</a> </th><td><p class="starttd">Cosine score between two vectors, \(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2}\). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a8c7b9281a72ff22caf06161701b27e84" title="Angle between two vectors. ">dist_angle()</a> </th><td><p class="starttd">Angle between two vectors in an Euclidean space, \(\cos^{-1}(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2})\). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#afa13b4c6122b99422d666dedea136c18" title="Tanimoto distance between two vectors. ">dist_tanimoto()</a> </th><td><p class="starttd">Tanimoto distance between two vectors. [1] </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#ac1397ac9f4a35b3b67c3be05b5e1a828" title="Jaccard distance between two vectors (treated as sets) ">dist_jaccard()</a> </th><td><p class="starttd">Jaccard distance between two varchar vectors treated as sets. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#af6b905fcac7746ef0ed0c36df4a1e070" title="Get an indexed row of the given matrix (2-D array) ">get_row()</a> </th><td><p class="starttd">Return the indexed row of a matrix (2-D array). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a20f34c9e661191e5225cca7bc23252c5" title="Get an indexed col of the given matrix (2-D array) ">get_col()</a> </th><td><p class="starttd">Return the indexed col of a matrix (2-D array). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a1aa37f73fb1cd8d7d106aa518dd8c0b4" title="Compute the average of vectors. ">avg()</a> </th><td><p class="starttd">Compute the average of vectors. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a0b04663ca206f03e66aed5ea2b4cc461" title="Compute the normalized average of vectors. ">normalized_avg()</a> </th><td><p class="starttd">Compute the normalized average of vectors (unit vector in an Euclidean space). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a9c439706f35d6cac89f151d553a5f111" title="Combine vectors to a matrix. ">matrix_agg()</a> </th><td><p class="starttd">Combine vectors to a matrix. </p>
+<p class="endtd"></p>
+</td></tr>
+</table>
+</dd></dl>
+<p><a class="anchor" id="examples"></a></p>
+<p><b>Vector Norms and Distances</b></p>
+<ol type="1">
+<li>Create a database table with two vector columns and add some data. <pre class="example">
+CREATE TABLE two_vectors(
+ id integer,
+ a float8[],
+ b float8[]);
+</pre> <pre class="example">
+INSERT INTO two_vectors VALUES
+(1, '{3,4}', '{4,5}'),
+(2, '{1,1,0,-4,5,3,4,106,14}', '{1,1,0,6,-3,1,2,92,2}');
+</pre></li>
+<li>Invoke norm functions. <pre class="example">
+SELECT
+ id,
+ madlib.norm1(a),
+ madlib.norm2(a)
+FROM two_vectors;
+</pre> Result: <pre class="result">
+ id | norm1 | norm2
+----+-------+------------------
+ 1 | 7 | 5
+ 2 | 138 | 107.238052947636
+(2 rows)
+</pre></li>
+<li>Invoke distance functions. <pre class="example">
+SELECT
+ id,
+ madlib.dist_norm1(a, b),
+ madlib.dist_norm2(a, b),
+ madlib.dist_pnorm(a, b, 5) AS norm5,
+ madlib.dist_inf_norm(a, b),
+ madlib.squared_dist_norm2(a, b) AS sq_dist_norm2,
+ madlib.cosine_similarity(a, b),
+ madlib.dist_angle(a, b),
+ madlib.dist_tanimoto(a, b),
+ madlib.dist_jaccard(a::text[], b::text[])
+FROM two_vectors;
+</pre> Result: <pre class="result">
+ id | dist_norm1 | dist_norm2 | norm5 | dist_inf_norm | sq_dist_norm2 | cosine_similarity | dist_angle | dist_tanimoto | dist_jaccard
+----+------------+------------------+------------------+---------------+---------------+-------------------+--------------------+--------------------+-------------------
+ 1 | 2 | 1.4142135623731 | 1.14869835499704 | 1 | 2 | 0.999512076087079 | 0.0312398334302684 | 0.0588235294117647 | 0.666666666666667
+ 2 | 48 | 22.6274169979695 | 15.585086360695 | 14 | 512 | 0.985403348449008 | 0.17106899659286 | 0.0498733684005455 | 0.833333333333333
+(2 rows)
+</pre></li>
+</ol>
+<p><b>Matrix Functions</b></p>
+<ol type="1">
+<li>Create a database table with a matrix column. <pre class="example">
+CREATE TABLE matrix(
+ id integer,
+ m float8[]);
+</pre> <pre class="example">
+INSERT INTO matrix VALUES
+(1, '{{4,5},{3,5},{9,0}}');
+</pre></li>
+<li>Invoke matrix functions. <pre class="example">
+SELECT
+ madlib.get_row(m, 1) AS row_1,
+ madlib.get_row(m, 2) AS row_2,
+ madlib.get_row(m, 3) AS row_3,
+ madlib.get_col(m, 1) AS col_1,
+ madlib.get_col(m, 2) AS col_2
+FROM matrix;
+</pre> Result: <pre class="result">
+ row_1 | row_2 | row_3 | col_1 | col_2
+-------+-------+-------+---------+---------
+ {4,5} | {3,5} | {9,0} | {4,3,9} | {5,5,0}
+(1 row)
+</pre></li>
+</ol>
+<p><b>Aggregate Functions</b></p>
+<ol type="1">
+<li>Create a database table with a vector column. <pre class="example">
+CREATE TABLE vector(
+ id integer,
+ v float8[]);
+</pre> <pre class="example">
+INSERT INTO vector VALUES
+(1, '{4,3}'),
+(2, '{8,6}'),
+(3, '{12,9}');
+</pre></li>
+<li>Invoke aggregate functions. <pre class="example">
+SELECT
+ madlib.avg(v),
+ madlib.normalized_avg(v),
+ madlib.matrix_agg(v)
+FROM vector;
+</pre> Result: <pre class="result">
+ avg | normalized_avg | matrix_agg
+-------+----------------+----------------------
+ {8,6} | {0.8,0.6} | {{4,3},{8,6},{12,9}}
+(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] <a href="http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_similarity_and_distance">http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_similarity_and_distance</a></p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="linalg_8sql__in.html" title="SQL functions for linear algebra. ">linalg.sql_in</a> documenting the SQL functions. </dd></dl>
+</div><!-- contents -->
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+ <li class="footer">Generated on Wed Dec 27 2017 19:05:57 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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+ </div>
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+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<p>A collection of methods that implement solutions for systems of consistent linear equations. </p>
+<table class="memberdecls">
+<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="groups"></a>
+Modules</h2></td></tr>
+<tr class="memitem:group__grp__dense__linear__solver"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__dense__linear__solver.html">Dense Linear Systems</a></td></tr>
+<tr class="memdesc:group__grp__dense__linear__solver"><td class="mdescLeft"> </td><td class="mdescRight">Implements solution methods for large dense linear systems. Currently, restricted to problems that fit in memory. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__sparse__linear__solver"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__sparse__linear__solver.html">Sparse Linear Systems</a></td></tr>
+<tr class="memdesc:group__grp__sparse__linear__solver"><td class="mdescLeft"> </td><td class="mdescRight">Implements solution methods for linear systems with sparse matrix input. Currently, restricted to problems that fit in memory. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+</table>
+</div><!-- contents -->
+</div><!-- doc-content -->
+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
+ <ul>
+ <li class="footer">Generated on Wed Dec 27 2017 19:05:57 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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+</body>
+</html>
http://git-wip-us.apache.org/repos/asf/madlib-site/blob/6c103d3e/docs/v1.13/group__grp__linear__solver.js
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+var group__grp__linear__solver =
+[
+ [ "Dense Linear Systems", "group__grp__dense__linear__solver.html", null ],
+ [ "Sparse Linear Systems", "group__grp__sparse__linear__solver.html", null ]
+];
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