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<div class="section" id="torch-cholesky">
<h1>torch.cholesky<a class="headerlink" href="#torch-cholesky" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="torch.cholesky">
<code class="sig-prename descclassname">torch.</code><code class="sig-name descname">cholesky</code><span class="sig-paren">(</span><em class="sig-param">input</em>, <em class="sig-param">upper=False</em>, <em class="sig-param">out=None</em><span class="sig-paren">)</span> → Tensor<a class="headerlink" href="#torch.cholesky" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes the Cholesky decomposition of a symmetric positive-definite
matrix <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span></span></span></span>
</span> or for batches of symmetric positive-definite matrices.</p>
<p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">upper</span></code> is <code class="docutils literal notranslate"><span class="pre">True</span></code>, the returned matrix <code class="docutils literal notranslate"><span class="pre">U</span></code> is upper-triangular, and
the decomposition has the form:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><msup><mi>U</mi><mi>T</mi></msup><mi>U</mi></mrow><annotation encoding="application/x-tex">A = U^TU</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8913309999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span></span>
</div><p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">upper</span></code> is <code class="docutils literal notranslate"><span class="pre">False</span></code>, the returned matrix <code class="docutils literal notranslate"><span class="pre">L</span></code> is lower-triangular, and
the decomposition has the form:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><mi>L</mi><msup><mi>L</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">A = LL^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8913309999999999em;vertical-align:0em;"></span><span class="mord mathnormal">L</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></span>
</div><p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">upper</span></code> is <code class="docutils literal notranslate"><span class="pre">True</span></code>, and <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span></span></span></span>
</span> is a batch of symmetric positive-definite
matrices, then the returned tensor will be composed of upper-triangular Cholesky factors
of each of the individual matrices. Similarly, when <code class="xref py py-attr docutils literal notranslate"><span class="pre">upper</span></code> is <code class="docutils literal notranslate"><span class="pre">False</span></code>, the returned
tensor will be composed of lower-triangular Cholesky factors of each of the individual
matrices.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>input</strong> (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the input tensor <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span></span></span></span>
</span> of size <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>∗</mo><mo separator="true">,</mo><mi>n</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, n, n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">∗</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>
</span> where <cite>*</cite> is zero or more
batch dimensions consisting of symmetric positive-definite matrices.</p></li>
<li><p><strong>upper</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a><em>, </em><em>optional</em>) – flag that indicates whether to return a
upper or lower triangular matrix. Default: <code class="docutils literal notranslate"><span class="pre">False</span></code></p></li>
<li><p><strong>out</strong> (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the output matrix</p></li>
</ul>
</dd>
</dl>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">mm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">t</span><span class="p">())</span> <span class="c1"># make symmetric positive-definite</span>
<span class="gp">>>> </span><span class="n">l</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cholesky</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([[ 2.4112, -0.7486, 1.4551],</span>
<span class="go"> [-0.7486, 1.3544, 0.1294],</span>
<span class="go"> [ 1.4551, 0.1294, 1.6724]])</span>
<span class="gp">>>> </span><span class="n">l</span>
<span class="go">tensor([[ 1.5528, 0.0000, 0.0000],</span>
<span class="go"> [-0.4821, 1.0592, 0.0000],</span>
<span class="go"> [ 0.9371, 0.5487, 0.7023]])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">mm</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span><span class="o">.</span><span class="n">t</span><span class="p">())</span>
<span class="go">tensor([[ 2.4112, -0.7486, 1.4551],</span>
<span class="go"> [-0.7486, 1.3544, 0.1294],</span>
<span class="go"> [ 1.4551, 0.1294, 1.6724]])</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="mf">1e-03</span> <span class="c1"># make symmetric positive-definite</span>
<span class="gp">>>> </span><span class="n">l</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">cholesky</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">z</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">l</span><span class="p">,</span> <span class="n">l</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">z</span> <span class="o">-</span> <span class="n">a</span><span class="p">))</span> <span class="c1"># Max non-zero</span>
<span class="go">tensor(2.3842e-07)</span>
</pre></div>
</div>
</dd></dl>
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