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Project: http://git-wip-us.apache.org/repos/asf/mahout/repo Commit: http://git-wip-us.apache.org/repos/asf/mahout/commit/9b7db9c6 Tree: http://git-wip-us.apache.org/repos/asf/mahout/tree/9b7db9c6 Diff: http://git-wip-us.apache.org/repos/asf/mahout/diff/9b7db9c6 Branch: refs/heads/asf-site Commit: 9b7db9c6d6f6f5fda74118179efd8032bb8e5d59 Parents: bdaf56d Author: jenkins <[email protected]> Authored: Mon Jun 18 15:40:30 2018 +0000 Committer: jenkins <[email protected]> Committed: Mon Jun 18 15:40:30 2018 +0000 ---------------------------------------------------------------------- developers/buildingmahout.html | 2 +- developers/developer-resources.html | 2 +- developers/github.html | 2 +- developers/githubPRs.html | 2 +- developers/gsoc.html | 2 +- developers/how-to-become-a-committer.html | 2 +- developers/how-to-contribute.html | 2 +- developers/how-to-release.html | 2 +- developers/how-to-update-the-website.html | 4 +- developers/issue-tracker.html | 2 +- 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| 2 +- .../samsara/classify-a-doc-from-the-shell.html | 2 +- .../tutorials/samsara/play-with-shell.html | 40 ++++++++-------- .../playing-with-samsara-flink-batch.html | 2 +- .../tutorials/samsara/spark-naive-bayes.html | 36 +++++++------- feed.xml | 2 +- general/books-tutorials-and-talks.html | 2 +- general/downloads.html | 2 +- general/faq.html | 2 +- general/glossary.html | 2 +- general/mahout-benchmarks.html | 2 +- general/mahout-wiki.html | 2 +- general/mailing-lists.html | 2 +- general/powered-by-mahout.html | 2 +- general/privacy-policy.html | 2 +- general/professional-support.html | 2 +- general/reference-reading.html | 2 +- general/release-notes.html | 2 +- general/who-we-are.html | 2 +- index.html | 2 +- news/2017/12/20/new-website.html | 2 +- posts/new-website.html | 2 +- users/algorithms/d-als.html | 2 +- users/algorithms/d-qr.html | 2 +- users/algorithms/d-spca.html | 2 +- users/algorithms/d-ssvd.html | 2 +- users/algorithms/intro-cooccurrence-spark.html | 2 +- users/algorithms/recommender-overview.html | 2 +- users/algorithms/spark-naive-bayes.html | 2 +- users/basics/algorithms.html | 2 +- users/basics/collections.html | 2 +- users/basics/collocations.html | 2 +- users/basics/creating-vectors-from-text.html | 2 +- users/basics/creating-vectors.html | 2 +- .../gaussian-discriminative-analysis.html | 2 +- .../basics/independent-component-analysis.html | 2 +- users/basics/mahout-collections.html | 2 +- users/basics/mahoutintegration.html | 2 +- users/basics/matrix-and-vector-needs.html | 2 +- users/basics/principal-components-analysis.html | 2 +- users/basics/quickstart.html | 2 +- .../svd---singular-value-decomposition.html | 2 +- users/basics/system-requirements.html | 2 +- ...rm-frequency-inverse-document-frequency.html | 2 +- users/classification/bankmarketing-example.html | 2 +- users/classification/bayesian-commandline.html | 2 +- users/classification/bayesian.html | 2 +- users/classification/breiman-example.html | 2 +- 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Nominations often occur internal to the PMC as well.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/developers/how-to-contribute.html ---------------------------------------------------------------------- diff --git a/developers/how-to-contribute.html b/developers/how-to-contribute.html index 399914b..85ba4f0 100644 --- a/developers/how-to-contribute.html +++ b/developers/how-to-contribute.html @@ -352,7 +352,7 @@ effort to improve the docs/tests for an issue, that helps even more.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/developers/how-to-release.html ---------------------------------------------------------------------- diff --git a/developers/how-to-release.html b/developers/how-to-release.html index 50d0e18..0cb41c3 100644 --- a/developers/how-to-release.html +++ b/developers/how-to-release.html @@ -386,7 +386,7 @@ details.</li> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/developers/how-to-update-the-website.html ---------------------------------------------------------------------- diff --git a/developers/how-to-update-the-website.html b/developers/how-to-update-the-website.html index c52399c..e6749d8 100644 --- a/developers/how-to-update-the-website.html +++ b/developers/how-to-update-the-website.html @@ -179,7 +179,7 @@ LaTeX portions are signalled with enclosing <code class="highlighter-rouge">$</c <p>In the terminal,</p> -<div class="language-bash highlighter-rouge"><pre class="highlight"><code><span class="nb">cd </span>website +<div class="language-bash highlighter-rouge"><pre class="highlight"><code><span class="nb">cd </span>wesite bundle <span class="nb">exec </span>jekyll serve </code></pre> </div> @@ -255,7 +255,7 @@ copy it to the <code class="highlighter-rouge">asf-site</code> branch, where <a <footer class="footer bg-light"> <div class="container text-center small"> - 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Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/clustering/canopy/index.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/clustering/canopy/index.html b/docs/latest/algorithms/clustering/canopy/index.html index 8ecdc70..7329e3b 100644 --- a/docs/latest/algorithms/clustering/canopy/index.html +++ b/docs/latest/algorithms/clustering/canopy/index.html @@ -303,7 +303,7 @@ model.cluster(drmA).collect <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/clustering/distance-metrics.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/clustering/distance-metrics.html b/docs/latest/algorithms/clustering/distance-metrics.html index f09b754..7a3fe78 100644 --- a/docs/latest/algorithms/clustering/distance-metrics.html +++ b/docs/latest/algorithms/clustering/distance-metrics.html @@ -254,7 +254,7 @@ val model = new CanopyClustering().fit(drmA, 'distanceMeasure -> 'Cosine) <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/clustering/index.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/clustering/index.html b/docs/latest/algorithms/clustering/index.html index 654d207..2bb146e 100644 --- a/docs/latest/algorithms/clustering/index.html +++ b/docs/latest/algorithms/clustering/index.html @@ -179,7 +179,7 @@ <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/linear-algebra/d-qr.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/linear-algebra/d-qr.html b/docs/latest/algorithms/linear-algebra/d-qr.html index 2ae3861..4875fe8 100644 --- a/docs/latest/algorithms/linear-algebra/d-qr.html +++ b/docs/latest/algorithms/linear-algebra/d-qr.html @@ -229,7 +229,7 @@ val(drmQ, inCoreR) = dqrThin(drma) <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/linear-algebra/d-spca.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/linear-algebra/d-spca.html b/docs/latest/algorithms/linear-algebra/d-spca.html index 5489626..8b340a5 100644 --- a/docs/latest/algorithms/linear-algebra/d-spca.html +++ b/docs/latest/algorithms/linear-algebra/d-spca.html @@ -167,36 +167,36 @@ <div class="col-lg-8"> <h2 id="intro">Intro</h2> -<p>Mahout has a distributed implementation of Stochastic PCA<a href="Lyubimov and Palumbo, ["Apache Mahout: Beyond MapReduce; Distributed Algorithm Design"](https://www.amazon.com/Apache-Mahout-MapReduce-Dmitriy-Lyubimov/dp/1523775785)">1</a>. This algorithm computes the exact equivalent of Mahoutâs dssvd(<code class="highlighter-rouge">$$\mathbf{A-1\mu^\top}$$</code>) by modifying the <code class="highlighter-rouge">dssvd</code> algorithm so as to avoid forming <code class="highlighter-rouge">$$\mathbf{A-1\mu^\top}$$</code>, which would densify a sparse input. Thus, it is suitable for work with both dense and sparse inputs.</p> +<p>Mahout has a distributed implementation of Stochastic PCA<a href="Lyubimov and Palumbo, ["Apache Mahout: Beyond MapReduce; Distributed Algorithm Design"](https://www.amazon.com/Apache-Mahout-MapReduce-Dmitriy-Lyubimov/dp/1523775785)">1</a>. This algorithm computes the exact equivalent of Mahoutâs dssvd(<code class="highlighter-rouge">\(\mathbf{A-1\mu^\top}\)</code>) by modifying the <code class="highlighter-rouge">dssvd</code> algorithm so as to avoid forming <code class="highlighter-rouge">\(\mathbf{A-1\mu^\top}\)</code>, which would densify a sparse input. Thus, it is suitable for work with both dense and sparse inputs.</p> <h2 id="algorithm">Algorithm</h2> -<p>Given an <em>m</em> <code class="highlighter-rouge">$$\times$$</code> <em>n</em> matrix <code class="highlighter-rouge">$$\mathbf{A}$$</code>, a target rank <em>k</em>, and an oversampling parameter <em>p</em>, this procedure computes a <em>k</em>-rank PCA by finding the unknowns in <code class="highlighter-rouge">$$\mathbf{Aâ1\mu^\top \approx U\Sigma V^\top}$$</code>:</p> +<p>Given an <em>m</em> <code class="highlighter-rouge">\(\times\)</code> <em>n</em> matrix <code class="highlighter-rouge">\(\mathbf{A}\)</code>, a target rank <em>k</em>, and an oversampling parameter <em>p</em>, this procedure computes a <em>k</em>-rank PCA by finding the unknowns in <code class="highlighter-rouge">\(\mathbf{Aâ1\mu^\top \approx U\Sigma V^\top}\)</code>:</p> <ol> - <li>Create seed for random <em>n</em> <code class="highlighter-rouge">$$\times$$</code> <em>(k+p)</em> matrix <code class="highlighter-rouge">$$\Omega$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{s_\Omega \leftarrow \Omega^\top \mu}$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{Y_0 \leftarrow A\Omega â 1 {s_\Omega}^\top, Y \in \mathbb{R}^{m\times(k+p)}}$$</code>.</li> - <li>Column-orthonormalize <code class="highlighter-rouge">$$\mathbf{Y_0} \rightarrow \mathbf{Q}$$</code> by computing thin decomposition <code class="highlighter-rouge">$$\mathbf{Y_0} = \mathbf{QR}$$</code>. Also, <code class="highlighter-rouge">$$\mathbf{Q}\in\mathbb{R}^{m\times(k+p)}, \mathbf{R}\in\mathbb{R}^{(k+p)\times(k+p)}$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{s_Q \leftarrow Q^\top 1}$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{B_0 \leftarrow Q^\top A: B \in \mathbb{R}^{(k+p)\times n}}$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{s_B \leftarrow {B_0}^\top \mu}$$</code>.</li> + <li>Create seed for random <em>n</em> <code class="highlighter-rouge">\(\times\)</code> <em>(k+p)</em> matrix <code class="highlighter-rouge">\(\Omega\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_\Omega \leftarrow \Omega^\top \mu}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{Y_0 \leftarrow A\Omega â 1 {s_\Omega}^\top, Y \in \mathbb{R}^{m\times(k+p)}}\)</code>.</li> + <li>Column-orthonormalize <code class="highlighter-rouge">\(\mathbf{Y_0} \rightarrow \mathbf{Q}\)</code> by computing thin decomposition <code class="highlighter-rouge">\(\mathbf{Y_0} = \mathbf{QR}\)</code>. Also, <code class="highlighter-rouge">\(\mathbf{Q}\in\mathbb{R}^{m\times(k+p)}, \mathbf{R}\in\mathbb{R}^{(k+p)\times(k+p)}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_Q \leftarrow Q^\top 1}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{B_0 \leftarrow Q^\top A: B \in \mathbb{R}^{(k+p)\times n}}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_B \leftarrow {B_0}^\top \mu}\)</code>.</li> <li>For <em>i</em> in 1..<em>q</em> repeat (power iterations): <ul> - <li>For <em>j</em> in 1..<em>n</em> apply <code class="highlighter-rouge">$$\mathbf{(B_{iâ1})_{âj} \leftarrow (B_{iâ1})_{âj}â\mu_j s_Q}$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{Y_i \leftarrow A{B_{iâ1}}^\topâ1(s_Bâ\mu^\top \mu s_Q)^\top}$$</code>.</li> - <li>Column-orthonormalize <code class="highlighter-rouge">$$\mathbf{Y_i} \rightarrow \mathbf{Q}$$</code> by computing thin decomposition <code class="highlighter-rouge">$$\mathbf{Y_i = QR}$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{s_Q \leftarrow Q^\top 1}$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{B_i \leftarrow Q^\top A}$$</code>.</li> - <li><code class="highlighter-rouge">$$\mathbf{s_B \leftarrow {B_i}^\top \mu}$$</code>.</li> + <li>For <em>j</em> in 1..<em>n</em> apply <code class="highlighter-rouge">\(\mathbf{(B_{iâ1})_{âj} \leftarrow (B_{iâ1})_{âj}â\mu_j s_Q}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{Y_i \leftarrow A{B_{iâ1}}^\topâ1(s_Bâ\mu^\top \mu s_Q)^\top}\)</code>.</li> + <li>Column-orthonormalize <code class="highlighter-rouge">\(\mathbf{Y_i} \rightarrow \mathbf{Q}\)</code> by computing thin decomposition <code class="highlighter-rouge">\(\mathbf{Y_i = QR}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_Q \leftarrow Q^\top 1}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{B_i \leftarrow Q^\top A}\)</code>.</li> + <li><code class="highlighter-rouge">\(\mathbf{s_B \leftarrow {B_i}^\top \mu}\)</code>.</li> </ul> </li> - <li>Let <code class="highlighter-rouge">$$\mathbf{C \triangleq s_Q {s_B}^\top}$$</code>. <code class="highlighter-rouge">$$\mathbf{M \leftarrow B_q {B_q}^\top â C â C^\top + \mu^\top \mu s_Q {s_Q}^\top}$$</code>.</li> - <li>Compute an eigensolution of the small symmetric <code class="highlighter-rouge">$$\mathbf{M = \hat{U} \Lambda \hat{U}^\top: M \in \mathbb{R}^{(k+p)\times(k+p)}}$$</code>.</li> - <li>The singular values <code class="highlighter-rouge">$$\Sigma = \Lambda^{\circ 0.5}$$</code>, or, in other words, <code class="highlighter-rouge">$$\mathbf{\sigma_i= \sqrt{\lambda_i}}$$</code>.</li> - <li>If needed, compute <code class="highlighter-rouge">$$\mathbf{U = Q\hat{U}}$$</code>.</li> - <li>If needed, compute <code class="highlighter-rouge">$$\mathbf{V = B^\top \hat{U} \Sigma^{â1}}$$</code>.</li> - <li>If needed, items converted to the PCA space can be computed as <code class="highlighter-rouge">$$\mathbf{U\Sigma}$$</code>.</li> + <li>Let <code class="highlighter-rouge">\(\mathbf{C \triangleq s_Q {s_B}^\top}\)</code>. <code class="highlighter-rouge">\(\mathbf{M \leftarrow B_q {B_q}^\top â C â C^\top + \mu^\top \mu s_Q {s_Q}^\top}\)</code>.</li> + <li>Compute an eigensolution of the small symmetric <code class="highlighter-rouge">\(\mathbf{M = \hat{U} \Lambda \hat{U}^\top: M \in \mathbb{R}^{(k+p)\times(k+p)}}\)</code>.</li> + <li>The singular values <code class="highlighter-rouge">\(\Sigma = \Lambda^{\circ 0.5}\)</code>, or, in other words, <code class="highlighter-rouge">\(\mathbf{\sigma_i= \sqrt{\lambda_i}}\)</code>.</li> + <li>If needed, compute <code class="highlighter-rouge">\(\mathbf{U = Q\hat{U}}\)</code>.</li> + <li>If needed, compute <code class="highlighter-rouge">\(\mathbf{V = B^\top \hat{U} \Sigma^{â1}}\)</code>.</li> + <li>If needed, items converted to the PCA space can be computed as <code class="highlighter-rouge">\(\mathbf{U\Sigma}\)</code>.</li> </ol> <h2 id="implementation">Implementation</h2> @@ -350,7 +350,7 @@ val (drmU, drmV, s) = dspca(drmA, k=200, q=1) <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/linear-algebra/d-ssvd.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/linear-algebra/d-ssvd.html b/docs/latest/algorithms/linear-algebra/d-ssvd.html index ac339ca..1839dbc 100644 --- a/docs/latest/algorithms/linear-algebra/d-ssvd.html +++ b/docs/latest/algorithms/linear-algebra/d-ssvd.html @@ -171,50 +171,50 @@ <h2 id="modified-ssvd-algorithm">Modified SSVD Algorithm</h2> -<p>Given an <code class="highlighter-rouge">$$m\times n$$</code> -matrix <code class="highlighter-rouge">$$\mathbf{A}$$</code>, a target rank <code class="highlighter-rouge">$$k\in\mathbb{N}_{1}$$</code> -, an oversampling parameter <code class="highlighter-rouge">$$p\in\mathbb{N}_{1}$$</code>, -and the number of additional power iterations <code class="highlighter-rouge">$$q\in\mathbb{N}_{0}$$</code>, -this procedure computes an <code class="highlighter-rouge">$$m\times\left(k+p\right)$$</code> -SVD <code class="highlighter-rouge">$$\mathbf{A\approx U}\boldsymbol{\Sigma}\mathbf{V}^{\top}$$</code>:</p> +<p>Given an <code class="highlighter-rouge">\(m\times n\)</code> +matrix <code class="highlighter-rouge">\(\mathbf{A}\)</code>, a target rank <code class="highlighter-rouge">\(k\in\mathbb{N}_{1}\)</code> +, an oversampling parameter <code class="highlighter-rouge">\(p\in\mathbb{N}_{1}\)</code>, +and the number of additional power iterations <code class="highlighter-rouge">\(q\in\mathbb{N}_{0}\)</code>, +this procedure computes an <code class="highlighter-rouge">\(m\times\left(k+p\right)\)</code> +SVD <code class="highlighter-rouge">\(\mathbf{A\approx U}\boldsymbol{\Sigma}\mathbf{V}^{\top}\)</code>:</p> <ol> <li> - <p>Create seed for random <code class="highlighter-rouge">$$n\times\left(k+p\right)$$</code> - matrix <code class="highlighter-rouge">$$\boldsymbol{\Omega}$$</code>. The seed defines matrix <code class="highlighter-rouge">$$\mathbf{\Omega}$$</code> + <p>Create seed for random <code class="highlighter-rouge">\(n\times\left(k+p\right)\)</code> + matrix <code class="highlighter-rouge">\(\boldsymbol{\Omega}\)</code>. The seed defines matrix <code class="highlighter-rouge">\(\mathbf{\Omega}\)</code> using Gaussian unit vectors per one of suggestions in [Halko, Martinsson, Tropp].</p> </li> <li> - <p><code class="highlighter-rouge">$$\mathbf{Y=A\boldsymbol{\Omega}},\,\mathbf{Y}\in\mathbb{R}^{m\times\left(k+p\right)}$$</code></p> + <p><code class="highlighter-rouge">\(\mathbf{Y=A\boldsymbol{\Omega}},\,\mathbf{Y}\in\mathbb{R}^{m\times\left(k+p\right)}\)</code></p> </li> <li> - <p>Column-orthonormalize <code class="highlighter-rouge">$$\mathbf{Y}\rightarrow\mathbf{Q}$$</code> - by computing thin decomposition <code class="highlighter-rouge">$$\mathbf{Y}=\mathbf{Q}\mathbf{R}$$</code>. - Also, <code class="highlighter-rouge">$$\mathbf{Q}\in\mathbb{R}^{m\times\left(k+p\right)},\,\mathbf{R}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}$$</code>; denoted as <code class="highlighter-rouge">$$\mathbf{Q}=\mbox{qr}\left(\mathbf{Y}\right).\mathbf{Q}$$</code></p> + <p>Column-orthonormalize <code class="highlighter-rouge">\(\mathbf{Y}\rightarrow\mathbf{Q}\)</code> + by computing thin decomposition <code class="highlighter-rouge">\(\mathbf{Y}=\mathbf{Q}\mathbf{R}\)</code>. + Also, <code class="highlighter-rouge">\(\mathbf{Q}\in\mathbb{R}^{m\times\left(k+p\right)},\,\mathbf{R}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\)</code>; denoted as <code class="highlighter-rouge">\(\mathbf{Q}=\mbox{qr}\left(\mathbf{Y}\right).\mathbf{Q}\)</code></p> </li> <li> - <p><code class="highlighter-rouge">$$\mathbf{B}_{0}=\mathbf{Q}^{\top}\mathbf{A}:\,\,\mathbf{B}\in\mathbb{R}^{\left(k+p\right)\times n}$$</code>.</p> + <p><code class="highlighter-rouge">\(\mathbf{B}_{0}=\mathbf{Q}^{\top}\mathbf{A}:\,\,\mathbf{B}\in\mathbb{R}^{\left(k+p\right)\times n}\)</code>.</p> </li> <li> - <p>If <code class="highlighter-rouge">$$q>0$$</code> - repeat: for <code class="highlighter-rouge">$$i=1..q$$</code>: - <code class="highlighter-rouge">$$\mathbf{B}_{i}^{\top}=\mathbf{A}^{\top}\mbox{qr}\left(\mathbf{A}\mathbf{B}_{i-1}^{\top}\right).\mathbf{Q}$$</code> + <p>If <code class="highlighter-rouge">\(q>0\)</code> + repeat: for <code class="highlighter-rouge">\(i=1..q\)</code>: + <code class="highlighter-rouge">\(\mathbf{B}_{i}^{\top}=\mathbf{A}^{\top}\mbox{qr}\left(\mathbf{A}\mathbf{B}_{i-1}^{\top}\right).\mathbf{Q}\)</code> (power iterations step).</p> </li> <li> - <p>Compute Eigensolution of a small Hermitian <code class="highlighter-rouge">$$\mathbf{B}_{q}\mathbf{B}_{q}^{\top}=\mathbf{\hat{U}}\boldsymbol{\Lambda}\mathbf{\hat{U}}^{\top}$$</code>, - <code class="highlighter-rouge">$$\mathbf{B}_{q}\mathbf{B}_{q}^{\top}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}$$</code>.</p> + <p>Compute Eigensolution of a small Hermitian <code class="highlighter-rouge">\(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}=\mathbf{\hat{U}}\boldsymbol{\Lambda}\mathbf{\hat{U}}^{\top}\)</code>, + <code class="highlighter-rouge">\(\mathbf{B}_{q}\mathbf{B}_{q}^{\top}\in\mathbb{R}^{\left(k+p\right)\times\left(k+p\right)}\)</code>.</p> </li> <li> - <p>Singular values <code class="highlighter-rouge">$$\mathbf{\boldsymbol{\Sigma}}=\boldsymbol{\Lambda}^{0.5}$$</code>, - or, in other words, <code class="highlighter-rouge">$$s_{i}=\sqrt{\sigma_{i}}$$</code>.</p> + <p>Singular values <code class="highlighter-rouge">\(\mathbf{\boldsymbol{\Sigma}}=\boldsymbol{\Lambda}^{0.5}\)</code>, + or, in other words, <code class="highlighter-rouge">\(s_{i}=\sqrt{\sigma_{i}}\)</code>.</p> </li> <li> - <p>If needed, compute <code class="highlighter-rouge">$$\mathbf{U}=\mathbf{Q}\hat{\mathbf{U}}$$</code>.</p> + <p>If needed, compute <code class="highlighter-rouge">\(\mathbf{U}=\mathbf{Q}\hat{\mathbf{U}}\)</code>.</p> </li> <li> - <p>If needed, compute <code class="highlighter-rouge">$$\mathbf{V}=\mathbf{B}_{q}^{\top}\hat{\mathbf{U}}\boldsymbol{\Sigma}^{-1}$$</code>. -Another way is <code class="highlighter-rouge">$$\mathbf{V}=\mathbf{A}^{\top}\mathbf{U}\boldsymbol{\Sigma}^{-1}$$</code>.</p> + <p>If needed, compute <code class="highlighter-rouge">\(\mathbf{V}=\mathbf{B}_{q}^{\top}\hat{\mathbf{U}}\boldsymbol{\Sigma}^{-1}\)</code>. +Another way is <code class="highlighter-rouge">\(\mathbf{V}=\mathbf{A}^{\top}\mathbf{U}\boldsymbol{\Sigma}^{-1}\)</code>.</p> </li> </ol> @@ -281,7 +281,7 @@ Another way is <code class="highlighter-rouge">$$\mathbf{V}=\mathbf{A}^{\top}\ma </code></pre> </div> -<p>Note: As a side effect of checkpointing, U and V values are returned as logical operators (i.e. they are neither checkpointed nor computed). Therefore there is no physical work actually done to compute <code class="highlighter-rouge">$$\mathbf{U}$$</code> or <code class="highlighter-rouge">$$\mathbf{V}$$</code> until they are used in a subsequent expression.</p> +<p>Note: As a side effect of checkpointing, U and V values are returned as logical operators (i.e. they are neither checkpointed nor computed). Therefore there is no physical work actually done to compute <code class="highlighter-rouge">\(\mathbf{U}\)</code> or <code class="highlighter-rouge">\(\mathbf{V}\)</code> until they are used in a subsequent expression.</p> <h2 id="usage">Usage</h2> @@ -311,7 +311,7 @@ val(drmU, drmV, s) = dssvd(drma, k = 40, q = 1) <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/linear-algebra/index.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/linear-algebra/index.html b/docs/latest/algorithms/linear-algebra/index.html index 1bdc4c5..328398d 100644 --- a/docs/latest/algorithms/linear-algebra/index.html +++ b/docs/latest/algorithms/linear-algebra/index.html @@ -184,7 +184,7 @@ <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/bayesian.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/bayesian.html b/docs/latest/algorithms/map-reduce/classification/bayesian.html index 5d11c37..b246409 100644 --- a/docs/latest/algorithms/map-reduce/classification/bayesian.html +++ b/docs/latest/algorithms/map-reduce/classification/bayesian.html @@ -181,38 +181,38 @@ <p>As described in <a href="http://people.csail.mit.edu/jrennie/papers/icml03-nb.pdf";>[1]</a> Mahout Naive Bayes is broken down into the following steps (assignments are over all possible index values):</p> <ul> - <li>Let <code class="highlighter-rouge">$$\vec{d}=(\vec{d_1},...,\vec{d_n})$$</code> be a set of documents; <code class="highlighter-rouge">$$d_{ij}$$</code> is the count of word <code class="highlighter-rouge">$$i$$</code> in document <code class="highlighter-rouge">$$j$$</code>.</li> - <li>Let <code class="highlighter-rouge">$$\vec{y}=(y_1,...,y_n)$$</code> be their labels.</li> - <li>Let <code class="highlighter-rouge">$$\alpha_i$$</code> be a smoothing parameter for all words in the vocabulary; let <code class="highlighter-rouge">$$\alpha=\sum_i{\alpha_i}$$</code>.</li> - <li><strong>Preprocessing</strong>(via seq2Sparse) TF-IDF transformation and L2 length normalization of <code class="highlighter-rouge">$$\vec{d}$$</code> + <li>Let <code class="highlighter-rouge">\(\vec{d}=(\vec{d_1},...,\vec{d_n})\)</code> be a set of documents; <code class="highlighter-rouge">\(d_{ij}\)</code> is the count of word <code class="highlighter-rouge">\(i\)</code> in document <code class="highlighter-rouge">\(j\)</code>.</li> + <li>Let <code class="highlighter-rouge">\(\vec{y}=(y_1,...,y_n)\)</code> be their labels.</li> + <li>Let <code class="highlighter-rouge">\(\alpha_i\)</code> be a smoothing parameter for all words in the vocabulary; let <code class="highlighter-rouge">\(\alpha=\sum_i{\alpha_i}\)</code>.</li> + <li><strong>Preprocessing</strong>(via seq2Sparse) TF-IDF transformation and L2 length normalization of <code class="highlighter-rouge">\(\vec{d}\)</code> <ol> - <li><code class="highlighter-rouge">$$d_{ij} = \sqrt{d_{ij}}$$</code></li> - <li><code class="highlighter-rouge">$$d_{ij} = d_{ij}\left(\log{\frac{\sum_k1}{\sum_k\delta_{ik}+1}}+1\right)$$</code></li> - <li><code class="highlighter-rouge">$$d_{ij} =\frac{d_{ij}}{\sqrt{\sum_k{d_{kj}^2}}}$$</code></li> + <li><code class="highlighter-rouge">\(d_{ij} = \sqrt{d_{ij}}\)</code></li> + <li><code class="highlighter-rouge">\(d_{ij} = d_{ij}\left(\log{\frac{\sum_k1}{\sum_k\delta_{ik}+1}}+1\right)\)</code></li> + <li><code class="highlighter-rouge">\(d_{ij} =\frac{d_{ij}}{\sqrt{\sum_k{d_{kj}^2}}}\)</code></li> </ol> </li> - <li><strong>Training: Bayes</strong><code class="highlighter-rouge">$$(\vec{d},\vec{y})$$</code> calculate term weights <code class="highlighter-rouge">$$w_{ci}$$</code> as: + <li><strong>Training: Bayes</strong><code class="highlighter-rouge">\((\vec{d},\vec{y})\)</code> calculate term weights <code class="highlighter-rouge">\(w_{ci}\)</code> as: <ol> - <li><code class="highlighter-rouge">$$\hat\theta_{ci}=\frac{d_{ic}+\alpha_i}{\sum_k{d_{kc}}+\alpha}$$</code></li> - <li><code class="highlighter-rouge">$$w_{ci}=\log{\hat\theta_{ci}}$$</code></li> + <li><code class="highlighter-rouge">\(\hat\theta_{ci}=\frac{d_{ic}+\alpha_i}{\sum_k{d_{kc}}+\alpha}\)</code></li> + <li><code class="highlighter-rouge">\(w_{ci}=\log{\hat\theta_{ci}}\)</code></li> </ol> </li> - <li><strong>Training: CBayes</strong><code class="highlighter-rouge">$$(\vec{d},\vec{y})$$</code> calculate term weights <code class="highlighter-rouge">$$w_{ci}$$</code> as: + <li><strong>Training: CBayes</strong><code class="highlighter-rouge">\((\vec{d},\vec{y})\)</code> calculate term weights <code class="highlighter-rouge">\(w_{ci}\)</code> as: <ol> - <li><code class="highlighter-rouge">$$\hat\theta_{ci} = \frac{\sum_{j:y_j\neq c}d_{ij}+\alpha_i}{\sum_{j:y_j\neq c}{\sum_k{d_{kj}}}+\alpha}$$</code></li> - <li><code class="highlighter-rouge">$$w_{ci}=-\log{\hat\theta_{ci}}$$</code></li> - <li><code class="highlighter-rouge">$$w_{ci}=\frac{w_{ci}}{\sum_i \lvert w_{ci}\rvert}$$</code></li> + <li><code class="highlighter-rouge">\(\hat\theta_{ci} = \frac{\sum_{j:y_j\neq c}d_{ij}+\alpha_i}{\sum_{j:y_j\neq c}{\sum_k{d_{kj}}}+\alpha}\)</code></li> + <li><code class="highlighter-rouge">\(w_{ci}=-\log{\hat\theta_{ci}}\)</code></li> + <li><code class="highlighter-rouge">\(w_{ci}=\frac{w_{ci}}{\sum_i \lvert w_{ci}\rvert}\)</code></li> </ol> </li> <li><strong>Label Assignment/Testing:</strong> <ol> - <li>Let <code class="highlighter-rouge">$$\vec{t}= (t_1,...,t_n)$$</code> be a test document; let <code class="highlighter-rouge">$$t_i$$</code> be the count of the word <code class="highlighter-rouge">$$t$$</code>.</li> - <li>Label the document according to <code class="highlighter-rouge">$$l(t)=\arg\max_c \sum\limits_{i} t_i w_{ci}$$</code></li> + <li>Let <code class="highlighter-rouge">\(\vec{t}= (t_1,...,t_n)\)</code> be a test document; let <code class="highlighter-rouge">\(t_i\)</code> be the count of the word <code class="highlighter-rouge">\(t\)</code>.</li> + <li>Label the document according to <code class="highlighter-rouge">\(l(t)=\arg\max_c \sum\limits_{i} t_i w_{ci}\)</code></li> </ol> </li> </ul> -<p>As we can see, the main difference between Bayes and CBayes is the weight calculation step. Where Bayes weighs terms more heavily based on the likelihood that they belong to class <code class="highlighter-rouge">$$c$$</code>, CBayes seeks to maximize term weights on the likelihood that they do not belong to any other class.</p> +<p>As we can see, the main difference between Bayes and CBayes is the weight calculation step. Where Bayes weighs terms more heavily based on the likelihood that they belong to class <code class="highlighter-rouge">\(c\)</code>, CBayes seeks to maximize term weights on the likelihood that they do not belong to any other class.</p> <h2 id="running-from-the-command-line">Running from the command line</h2> @@ -350,7 +350,7 @@ Classification and testing on a holdout set can then be performed via <code clas <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/class-discovery.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/class-discovery.html b/docs/latest/algorithms/map-reduce/classification/class-discovery.html index 50a9d62..ee9bcd6 100644 --- a/docs/latest/algorithms/map-reduce/classification/class-discovery.html +++ b/docs/latest/algorithms/map-reduce/classification/class-discovery.html @@ -328,7 +328,7 @@ format needed by CDGA. <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/classifyingyourdata.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/classifyingyourdata.html b/docs/latest/algorithms/map-reduce/classification/classifyingyourdata.html index 679b33b..79e8066 100644 --- a/docs/latest/algorithms/map-reduce/classification/classifyingyourdata.html +++ b/docs/latest/algorithms/map-reduce/classification/classifyingyourdata.html @@ -197,7 +197,7 @@ s</a> blog.</li> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/collocations.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/collocations.html b/docs/latest/algorithms/map-reduce/classification/collocations.html index ffa1540..f4644a8 100644 --- a/docs/latest/algorithms/map-reduce/classification/collocations.html +++ b/docs/latest/algorithms/map-reduce/classification/collocations.html @@ -552,7 +552,7 @@ subject to filtering with minSupport and minLLR.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/gaussian-discriminative-analysis.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/gaussian-discriminative-analysis.html b/docs/latest/algorithms/map-reduce/classification/gaussian-discriminative-analysis.html index 347802c..14b6b14 100644 --- a/docs/latest/algorithms/map-reduce/classification/gaussian-discriminative-analysis.html +++ b/docs/latest/algorithms/map-reduce/classification/gaussian-discriminative-analysis.html @@ -189,7 +189,7 @@ the paper is backwards, in that page 1 is at the end)</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/hidden-markov-models.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/hidden-markov-models.html b/docs/latest/algorithms/map-reduce/classification/hidden-markov-models.html index 125de02..dedc6f8 100644 --- a/docs/latest/algorithms/map-reduce/classification/hidden-markov-models.html +++ b/docs/latest/algorithms/map-reduce/classification/hidden-markov-models.html @@ -284,7 +284,7 @@ and selected applications in speech recognitionâ. Proceedings of the IEEE <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/independent-component-analysis.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/independent-component-analysis.html b/docs/latest/algorithms/map-reduce/classification/independent-component-analysis.html index 89f5dc4..1b65398 100644 --- a/docs/latest/algorithms/map-reduce/classification/independent-component-analysis.html +++ b/docs/latest/algorithms/map-reduce/classification/independent-component-analysis.html @@ -186,7 +186,7 @@ <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/locally-weighted-linear-regression.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/locally-weighted-linear-regression.html b/docs/latest/algorithms/map-reduce/classification/locally-weighted-linear-regression.html index 962d32c..d4ed47b 100644 --- a/docs/latest/algorithms/map-reduce/classification/locally-weighted-linear-regression.html +++ b/docs/latest/algorithms/map-reduce/classification/locally-weighted-linear-regression.html @@ -194,7 +194,7 @@ http://www.cs.cmu.edu/afs/cs/project/jair/pub/volume4/cohn96a-html/node7.html</p <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/logistic-regression.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/logistic-regression.html b/docs/latest/algorithms/map-reduce/classification/logistic-regression.html index 582f6d5..e3afc3a 100644 --- a/docs/latest/algorithms/map-reduce/classification/logistic-regression.html +++ b/docs/latest/algorithms/map-reduce/classification/logistic-regression.html @@ -304,7 +304,7 @@ Paul Komarek</a></p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/mahout-collections.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/mahout-collections.html b/docs/latest/algorithms/map-reduce/classification/mahout-collections.html index 7a5cb68..24f92b8 100644 --- a/docs/latest/algorithms/map-reduce/classification/mahout-collections.html +++ b/docs/latest/algorithms/map-reduce/classification/mahout-collections.html @@ -224,7 +224,7 @@ have any additional storage usage for unused keys.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/mlp.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/mlp.html b/docs/latest/algorithms/map-reduce/classification/mlp.html index ed20f60..5f742b5 100644 --- a/docs/latest/algorithms/map-reduce/classification/mlp.html +++ b/docs/latest/algorithms/map-reduce/classification/mlp.html @@ -415,7 +415,7 @@ This computes the activation <em>a</em> for neuron <em>j</em> where <em>w</em> i <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/naivebayes.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/naivebayes.html b/docs/latest/algorithms/map-reduce/classification/naivebayes.html index de8f53e..333708c 100644 --- a/docs/latest/algorithms/map-reduce/classification/naivebayes.html +++ b/docs/latest/algorithms/map-reduce/classification/naivebayes.html @@ -215,7 +215,7 @@ using the 20 Newsgroups data available at [http://people.csail.mit.edu/jrennie/2 <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/neural-network.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/neural-network.html b/docs/latest/algorithms/map-reduce/classification/neural-network.html index be07c41..8bf8443 100644 --- a/docs/latest/algorithms/map-reduce/classification/neural-network.html +++ b/docs/latest/algorithms/map-reduce/classification/neural-network.html @@ -190,7 +190,7 @@ planes but arbitrary decision boundaries.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/partial-implementation.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/partial-implementation.html b/docs/latest/algorithms/map-reduce/classification/partial-implementation.html index a69f13e..8c1e4fd 100644 --- a/docs/latest/algorithms/map-reduce/classification/partial-implementation.html +++ b/docs/latest/algorithms/map-reduce/classification/partial-implementation.html @@ -318,7 +318,7 @@ dataset is big and you try to build a large number of trees.</li> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/random-forests.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/random-forests.html b/docs/latest/algorithms/map-reduce/classification/random-forests.html index f77d83f..f083bfd 100644 --- a/docs/latest/algorithms/map-reduce/classification/random-forests.html +++ b/docs/latest/algorithms/map-reduce/classification/random-forests.html @@ -412,7 +412,7 @@ Business Administration. <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/restricted-boltzmann-machines.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/restricted-boltzmann-machines.html b/docs/latest/algorithms/map-reduce/classification/restricted-boltzmann-machines.html index 5323260..4baa457 100644 --- a/docs/latest/algorithms/map-reduce/classification/restricted-boltzmann-machines.html +++ b/docs/latest/algorithms/map-reduce/classification/restricted-boltzmann-machines.html @@ -222,7 +222,7 @@ systems and the size of your dataset</li> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/classification/support-vector-machines.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/classification/support-vector-machines.html b/docs/latest/algorithms/map-reduce/classification/support-vector-machines.html index de7bb19..562bc97 100644 --- a/docs/latest/algorithms/map-reduce/classification/support-vector-machines.html +++ b/docs/latest/algorithms/map-reduce/classification/support-vector-machines.html @@ -212,7 +212,7 @@ information encoded in only very few examples.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/canopy-clustering.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/canopy-clustering.html b/docs/latest/algorithms/map-reduce/clustering/canopy-clustering.html index f69f980..ceab3cd 100644 --- a/docs/latest/algorithms/map-reduce/clustering/canopy-clustering.html +++ b/docs/latest/algorithms/map-reduce/clustering/canopy-clustering.html @@ -367,7 +367,7 @@ thresholds.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/cluster-dumper.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/cluster-dumper.html b/docs/latest/algorithms/map-reduce/clustering/cluster-dumper.html index c4c22f8..17365d1 100644 --- a/docs/latest/algorithms/map-reduce/clustering/cluster-dumper.html +++ b/docs/latest/algorithms/map-reduce/clustering/cluster-dumper.html @@ -284,7 +284,7 @@ the radius of the cluster as a vector.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/expectation-maximization.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/expectation-maximization.html b/docs/latest/algorithms/map-reduce/clustering/expectation-maximization.html index fcf5bbc..af2adc9 100644 --- a/docs/latest/algorithms/map-reduce/clustering/expectation-maximization.html +++ b/docs/latest/algorithms/map-reduce/clustering/expectation-maximization.html @@ -232,7 +232,7 @@ exponentially small influence.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/fuzzy-k-means.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/fuzzy-k-means.html b/docs/latest/algorithms/map-reduce/clustering/fuzzy-k-means.html index 471984c..7947444 100644 --- a/docs/latest/algorithms/map-reduce/clustering/fuzzy-k-means.html +++ b/docs/latest/algorithms/map-reduce/clustering/fuzzy-k-means.html @@ -365,7 +365,7 @@ Fuzzy k-Means does a fair job handling this data set as well.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/hierarchical-clustering.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/hierarchical-clustering.html b/docs/latest/algorithms/map-reduce/clustering/hierarchical-clustering.html index 1ea643b..ff08526 100644 --- a/docs/latest/algorithms/map-reduce/clustering/hierarchical-clustering.html +++ b/docs/latest/algorithms/map-reduce/clustering/hierarchical-clustering.html @@ -185,7 +185,7 @@ different levels.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/k-means-clustering.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/k-means-clustering.html b/docs/latest/algorithms/map-reduce/clustering/k-means-clustering.html index 567633b..c48f391 100644 --- a/docs/latest/algorithms/map-reduce/clustering/k-means-clustering.html +++ b/docs/latest/algorithms/map-reduce/clustering/k-means-clustering.html @@ -356,7 +356,7 @@ K-Means does a fair job handling this data set as well.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/latent-dirichlet-allocation.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/latent-dirichlet-allocation.html b/docs/latest/algorithms/map-reduce/clustering/latent-dirichlet-allocation.html index 4706b6e..4b4afc1 100644 --- a/docs/latest/algorithms/map-reduce/clustering/latent-dirichlet-allocation.html +++ b/docs/latest/algorithms/map-reduce/clustering/latent-dirichlet-allocation.html @@ -322,7 +322,7 @@ post-processing step.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/llr---log-likelihood-ratio.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/llr---log-likelihood-ratio.html b/docs/latest/algorithms/map-reduce/clustering/llr---log-likelihood-ratio.html index eaf0b66..eb2295e 100644 --- a/docs/latest/algorithms/map-reduce/clustering/llr---log-likelihood-ratio.html +++ b/docs/latest/algorithms/map-reduce/clustering/llr---log-likelihood-ratio.html @@ -216,7 +216,7 @@ number of training examples and where features are highly interdependent.</p> <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/spectral-clustering.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/spectral-clustering.html b/docs/latest/algorithms/map-reduce/clustering/spectral-clustering.html index 697b563..78cb515 100644 --- a/docs/latest/algorithms/map-reduce/clustering/spectral-clustering.html +++ b/docs/latest/algorithms/map-reduce/clustering/spectral-clustering.html @@ -173,16 +173,16 @@ <ol> <li> - <p>Computing a similarity (or <em>affinity</em>) matrix <code class="highlighter-rouge">$$\mathbf{A}$$</code> from the data. This involves determining a pairwise distance function <code class="highlighter-rouge">$$f$$</code> that takes a pair of data points and returns a scalar.</p> + <p>Computing a similarity (or <em>affinity</em>) matrix <code class="highlighter-rouge">\(\mathbf{A}\)</code> from the data. This involves determining a pairwise distance function <code class="highlighter-rouge">\(f\)</code> that takes a pair of data points and returns a scalar.</p> </li> <li> - <p>Computing a graph Laplacian <code class="highlighter-rouge">$$\mathbf{L}$$</code> from the affinity matrix. There are several types of graph Laplacians; which is used will often depends on the situation.</p> + <p>Computing a graph Laplacian <code class="highlighter-rouge">\(\mathbf{L}\)</code> from the affinity matrix. There are several types of graph Laplacians; which is used will often depends on the situation.</p> </li> <li> - <p>Computing the eigenvectors and eigenvalues of <code class="highlighter-rouge">$$\mathbf{L}$$</code>. The degree of this decomposition is often modulated by <code class="highlighter-rouge">$$k$$</code>, or the number of clusters. Put another way, <code class="highlighter-rouge">$$k$$</code> eigenvectors and eigenvalues are computed.</p> + <p>Computing the eigenvectors and eigenvalues of <code class="highlighter-rouge">\(\mathbf{L}\)</code>. The degree of this decomposition is often modulated by <code class="highlighter-rouge">\(k\)</code>, or the number of clusters. Put another way, <code class="highlighter-rouge">\(k\)</code> eigenvectors and eigenvalues are computed.</p> </li> <li> - <p>The <code class="highlighter-rouge">$$k$$</code> eigenvectors are used as âproxyâ data for the original dataset, and fed into k-means clustering. The resulting cluster assignments are transparently passed back to the original data.</p> + <p>The <code class="highlighter-rouge">\(k\)</code> eigenvectors are used as âproxyâ data for the original dataset, and fed into k-means clustering. The resulting cluster assignments are transparently passed back to the original data.</p> </li> </ol> @@ -196,11 +196,11 @@ <h2 id="input">Input</h2> -<p>The input format for the algorithm currently takes the form of a Hadoop-backed affinity matrix in the form of text files. Each line of the text file specifies a single element of the affinity matrix: the row index <code class="highlighter-rouge">$$i$$</code>, the column index <code class="highlighter-rouge">$$j$$</code>, and the value:</p> +<p>The input format for the algorithm currently takes the form of a Hadoop-backed affinity matrix in the form of text files. Each line of the text file specifies a single element of the affinity matrix: the row index <code class="highlighter-rouge">\(i\)</code>, the column index <code class="highlighter-rouge">\(j\)</code>, and the value:</p> <p><code class="highlighter-rouge">i, j, value</code></p> -<p>The affinity matrix is symmetric, and any unspecified <code class="highlighter-rouge">$$i, j$$</code> pairs are assumed to be 0 for sparsity. The row and column indices are 0-indexed. Thus, only the non-zero entries of either the upper or lower triangular need be specified.</p> +<p>The affinity matrix is symmetric, and any unspecified <code class="highlighter-rouge">\(i, j\)</code> pairs are assumed to be 0 for sparsity. The row and column indices are 0-indexed. Thus, only the non-zero entries of either the upper or lower triangular need be specified.</p> <p>The matrix elements specified in the text files are collected into a Mahout <code class="highlighter-rouge">DistributedRowMatrix</code>.</p> @@ -266,7 +266,7 @@ <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/clustering/streaming-k-means.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/clustering/streaming-k-means.html b/docs/latest/algorithms/map-reduce/clustering/streaming-k-means.html index f41a349..ce276f1 100644 --- a/docs/latest/algorithms/map-reduce/clustering/streaming-k-means.html +++ b/docs/latest/algorithms/map-reduce/clustering/streaming-k-means.html @@ -361,7 +361,7 @@ The algorithm can be instructed to take multiple independent runs (using the <em <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/map-reduce/index.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/map-reduce/index.html b/docs/latest/algorithms/map-reduce/index.html index ee71478..663b6a2 100644 --- a/docs/latest/algorithms/map-reduce/index.html +++ b/docs/latest/algorithms/map-reduce/index.html @@ -211,7 +211,7 @@ <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/preprocessors/AsFactor.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/preprocessors/AsFactor.html b/docs/latest/algorithms/preprocessors/AsFactor.html index d5469bc..ca9b664 100644 --- a/docs/latest/algorithms/preprocessors/AsFactor.html +++ b/docs/latest/algorithms/preprocessors/AsFactor.html @@ -202,7 +202,7 @@ <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/preprocessors/MeanCenter.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/preprocessors/MeanCenter.html b/docs/latest/algorithms/preprocessors/MeanCenter.html index a39e939..4c4d57b 100644 --- a/docs/latest/algorithms/preprocessors/MeanCenter.html +++ b/docs/latest/algorithms/preprocessors/MeanCenter.html @@ -197,7 +197,7 @@ <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/preprocessors/StandardScaler.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/preprocessors/StandardScaler.html b/docs/latest/algorithms/preprocessors/StandardScaler.html index e422df9..bf63666 100644 --- a/docs/latest/algorithms/preprocessors/StandardScaler.html +++ b/docs/latest/algorithms/preprocessors/StandardScaler.html @@ -211,7 +211,7 @@ is trivial, however when testing the function on smaller datasets the practicion <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/preprocessors/index.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/preprocessors/index.html b/docs/latest/algorithms/preprocessors/index.html index 21b2749..6152d00 100644 --- a/docs/latest/algorithms/preprocessors/index.html +++ b/docs/latest/algorithms/preprocessors/index.html @@ -182,7 +182,7 @@ <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/reccomenders/cco.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/reccomenders/cco.html b/docs/latest/algorithms/reccomenders/cco.html index f0905bd..0e8d8c7 100644 --- a/docs/latest/algorithms/reccomenders/cco.html +++ b/docs/latest/algorithms/reccomenders/cco.html @@ -642,7 +642,7 @@ on the popularity field. If we use the ecom example but use the query to get â <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/reccomenders/d-als.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/reccomenders/d-als.html b/docs/latest/algorithms/reccomenders/d-als.html index 01ca25f..d6e32d4 100644 --- a/docs/latest/algorithms/reccomenders/d-als.html +++ b/docs/latest/algorithms/reccomenders/d-als.html @@ -174,7 +174,7 @@ TODO: Find the ALS Page</p> <h2 id="algorithm">Algorithm</h2> -<p>For the classic QR decomposition of the form <code class="highlighter-rouge">$$\mathbf{A}=\mathbf{QR},\mathbf{A}\in\mathbb{R}^{m\times n}$$</code> a distributed version is fairly easily achieved if <code class="highlighter-rouge">$$\mathbf{A}$$</code> is tall and thin such that <code class="highlighter-rouge">$$\mathbf{A}^{\top}\mathbf{A}$$</code> fits in memory, i.e. <em>m</em> is large but <em>n</em> < ~5000 Under such circumstances, only <code class="highlighter-rouge">$$\mathbf{A}$$</code> and <code class="highlighter-rouge">$$\mathbf{Q}$$</code> are distributed matricies and <code class="highlighter-rouge">$$\mathbf{A^{\top}A}$$</code> and <code class="highlighter-rouge">$$\mathbf{R}$$</code> are in-core products. We just compute the in-core version of the Cholesky decomposition in the form of <code class="highlighter-rouge">$$\mathbf{LL}^{\top}= \mathbf{A}^{\top}\mathbf{A}$$</code>. After that we take <code class="highlighter-rouge">$$\mathbf{R}= \mathbf{L}^{\top}$$</co de> and <code class="highlighter-rouge">$$\mathbf{Q}=\mathbf{A}\left(\mathbf{L}^{\top}\right)^{-1}$$</code>. The latter is easily achieved by multiplying each verticle block of <code class="highlighter-rouge">$$\mathbf{A}$$</code> by <code class="highlighter-rouge">$$\left(\mathbf{L}^{\top}\right)^{-1}$$</code>. (There is no actual matrix inversion happening).</p> +<p>For the classic QR decomposition of the form <code class="highlighter-rouge">\(\mathbf{A}=\mathbf{QR},\mathbf{A}\in\mathbb{R}^{m\times n}\)</code> a distributed version is fairly easily achieved if <code class="highlighter-rouge">\(\mathbf{A}\)</code> is tall and thin such that <code class="highlighter-rouge">\(\mathbf{A}^{\top}\mathbf{A}\)</code> fits in memory, i.e. <em>m</em> is large but <em>n</em> < ~5000 Under such circumstances, only <code class="highlighter-rouge">\(\mathbf{A}\)</code> and <code class="highlighter-rouge">\(\mathbf{Q}\)</code> are distributed matricies and <code class="highlighter-rouge">\(\mathbf{A^{\top}A}\)</code> and <code class="highlighter-rouge">\(\mathbf{R}\)</code> are in-core products. We just compute the in-core version of the Cholesky decomposition in the form of <code class="highlighter-rouge">\(\mathbf{LL}^{\top}= \mathbf{A}^{\top}\mathbf{A}\)</code>. After that we take <code class="highlighter-rouge">\(\mathbf{R}= \mathbf{L}^{\top}\)</co de> and <code class="highlighter-rouge">\(\mathbf{Q}=\mathbf{A}\left(\mathbf{L}^{\top}\right)^{-1}\)</code>. The latter is easily achieved by multiplying each verticle block of <code class="highlighter-rouge">\(\mathbf{A}\)</code> by <code class="highlighter-rouge">\(\left(\mathbf{L}^{\top}\right)^{-1}\)</code>. (There is no actual matrix inversion happening).</p> <h2 id="implementation">Implementation</h2> @@ -224,7 +224,7 @@ val(drmQ, inCoreR) = dqrThin(drma) <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer> http://git-wip-us.apache.org/repos/asf/mahout/blob/9b7db9c6/docs/latest/algorithms/reccomenders/index.html ---------------------------------------------------------------------- diff --git a/docs/latest/algorithms/reccomenders/index.html b/docs/latest/algorithms/reccomenders/index.html index 5d13d81..d6e44b0 100644 --- a/docs/latest/algorithms/reccomenders/index.html +++ b/docs/latest/algorithms/reccomenders/index.html @@ -203,7 +203,7 @@ and <a href="http://occamsmachete.com/ml/2014/09/09/mahout-on-spark-whats-new-i <footer class="footer bg-light"> <div class="container text-center small"> - Copyright © 2014-2017 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. + Copyright © 2014-2018 The Apache Software Foundation, Licensed under the Apache License, Version 2.0. </div> </footer>
