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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="py function">
<dt id="torch.cholesky">
<code class="sig-prename descclassname"><span class="pre">torch.</span></code><code class="sig-name descname"><span class="pre">cholesky</span></code><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">input</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">upper</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">False</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">out</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span> → <a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a><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:0.6833em;"></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:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8913em;"></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:0.8913em;"><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:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8913em;"></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:0.8913em;"><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:0.6833em;"></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>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p><a class="reference internal" href="#torch.cholesky" title="torch.cholesky"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.cholesky()</span></code></a> is deprecated in favor of <a class="reference internal" href="torch.linalg.cholesky.html#torch.linalg.cholesky" title="torch.linalg.cholesky"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.cholesky()</span></code></a>
and will be removed in a future PyTorch release.</p>
<p><code class="docutils literal notranslate"><span class="pre">L</span> <span class="pre">=</span> <span class="pre">torch.cholesky(A)</span></code> should be replaced with</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">L</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">cholesky</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
</pre></div>
</div>
<p><code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">=</span> <span class="pre">torch.cholesky(A,</span> <span class="pre">upper=True)</span></code> should be replaced with</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">U</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</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="o">.</span><span class="n">mH</span>
</pre></div>
</div>
<p>This transform will produce equivalent results for all valid (symmetric positive definite) inputs.</p>
</div>
<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:0.6833em;"></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:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></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 v3.10)"><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>
</ul>
</dd>
<dt class="field-even">Keyword Arguments</dt>
<dd class="field-even"><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>
</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">a</span> <span class="o">@</span> <span class="n">a</span><span class="o">.</span><span class="n">mT</span> <span class="o">+</span> <span class="mf">1e-3</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">l</span> <span class="o">@</span> <span class="n">l</span><span class="o">.</span><span class="n">mT</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="c1"># Example for batched input</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">@</span> <span class="n">a</span><span class="o">.</span><span class="n">mT</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">l</span> <span class="o">@</span> <span class="n">l</span><span class="o">.</span><span class="n">mT</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">dist</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="go">tensor(2.3842e-07)</span>
</pre></div>
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