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<div class="section" id="torch-linalg-norm">
<h1>torch.linalg.norm<a class="headerlink" href="#torch-linalg-norm" title="Permalink to this headline">¶</a></h1>
<dl class="py function">
<dt id="torch.linalg.norm">
<code class="sig-prename descclassname"><span class="pre">torch.linalg.</span></code><code class="sig-name descname"><span class="pre">norm</span></code><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">A</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">ord</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">dim</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">keepdim</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>, <em class="sig-param"><span class="n"><span class="pre">dtype</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> → <span class="pre">Tensor</span><a class="headerlink" href="#torch.linalg.norm" title="Permalink to this definition">¶</a></dt>
<dd><p>Computes a vector or matrix norm.</p>
<p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code> is complex valued, it computes the norm of <code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code><cite>.abs()</cite></p>
<p>Supports input of float, double, cfloat and cdouble dtypes.</p>
<p>Whether this function computes a vector or matrix norm is determined as follows:</p>
<ul class="simple">
<li><p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> is an <cite>int</cite>, the vector norm will be computed.</p></li>
<li><p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> is a <cite>2</cite>-<cite>tuple</cite>, the matrix norm will be computed.</p></li>
<li><p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code><cite>= None</cite> and <code class="xref py py-attr docutils literal notranslate"><span class="pre">ord</span></code><cite>= None</cite>,
<code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code> will be flattened to 1D and the <cite>2</cite>-norm of the resulting vector will be computed.</p></li>
<li><p>If <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code><cite>= None</cite> and <code class="xref py py-attr docutils literal notranslate"><span class="pre">ord</span></code> <cite>!= None</cite>, <code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code> must be 1D or 2D.</p></li>
</ul>
<p><code class="xref py py-attr docutils literal notranslate"><span class="pre">ord</span></code> defines the norm that is computed. The following norms are supported:</p>
<table class="docutils colwidths-auto align-default">
<thead>
<tr class="row-odd"><th class="head"><p><code class="xref py py-attr docutils literal notranslate"><span class="pre">ord</span></code></p></th>
<th class="head"><p>norm for matrices</p></th>
<th class="head"><p>norm for vectors</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><cite>None</cite> (default)</p></td>
<td><p>Frobenius norm</p></td>
<td><p><cite>2</cite>-norm (see below)</p></td>
</tr>
<tr class="row-odd"><td><p><cite>‘fro’</cite></p></td>
<td><p>Frobenius norm</p></td>
<td><p>– not supported –</p></td>
</tr>
<tr class="row-even"><td><p><cite>‘nuc’</cite></p></td>
<td><p>nuclear norm</p></td>
<td><p>– not supported –</p></td>
</tr>
<tr class="row-odd"><td><p><cite>inf</cite></p></td>
<td><p><cite>max(sum(abs(x), dim=1))</cite></p></td>
<td><p><cite>max(abs(x))</cite></p></td>
</tr>
<tr class="row-even"><td><p><cite>-inf</cite></p></td>
<td><p><cite>min(sum(abs(x), dim=1))</cite></p></td>
<td><p><cite>min(abs(x))</cite></p></td>
</tr>
<tr class="row-odd"><td><p><cite>0</cite></p></td>
<td><p>– not supported –</p></td>
<td><p><cite>sum(x != 0)</cite></p></td>
</tr>
<tr class="row-even"><td><p><cite>1</cite></p></td>
<td><p><cite>max(sum(abs(x), dim=0))</cite></p></td>
<td><p>as below</p></td>
</tr>
<tr class="row-odd"><td><p><cite>-1</cite></p></td>
<td><p><cite>min(sum(abs(x), dim=0))</cite></p></td>
<td><p>as below</p></td>
</tr>
<tr class="row-even"><td><p><cite>2</cite></p></td>
<td><p>largest singular value</p></td>
<td><p>as below</p></td>
</tr>
<tr class="row-odd"><td><p><cite>-2</cite></p></td>
<td><p>smallest singular value</p></td>
<td><p>as below</p></td>
</tr>
<tr class="row-even"><td><p>other <cite>int</cite> or <cite>float</cite></p></td>
<td><p>– not supported –</p></td>
<td><p><cite>sum(abs(x)^{ord})^{(1 / ord)}</cite></p></td>
</tr>
</tbody>
</table>
<p>where <cite>inf</cite> refers to <cite>float(‘inf’)</cite>, NumPy’s <cite>inf</cite> object, or any equivalent object.</p>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="torch.linalg.vector_norm.html#torch.linalg.vector_norm" title="torch.linalg.vector_norm"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.vector_norm()</span></code></a> computes a vector norm.</p>
<p><a class="reference internal" href="torch.linalg.matrix_norm.html#torch.linalg.matrix_norm" title="torch.linalg.matrix_norm"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.matrix_norm()</span></code></a> computes a matrix norm.</p>
<p>The above functions are often clearer and more flexible than using <a class="reference internal" href="#torch.linalg.norm" title="torch.linalg.norm"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.norm()</span></code></a>.
For example, <cite>torch.linalg.norm(A, ord=1, dim=(0, 1))</cite> always
computes a matrix norm, but with <cite>torch.linalg.vector_norm(A, ord=1, dim=(0, 1))</cite> it is possible
to compute a vector norm over the two dimensions.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>A</strong> (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – tensor of shape <cite>(*, n)</cite> or <cite>(*, m, n)</cite> where <cite>*</cite> is zero or more batch dimensions</p></li>
<li><p><strong>ord</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.10)"><em>int</em></a><em>, </em><a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.10)"><em>float</em></a><em>, </em><em>inf</em><em>, </em><em>-inf</em><em>, </em><em>'fro'</em><em>, </em><em>'nuc'</em><em>, </em><em>optional</em>) – order of norm. Default: <cite>None</cite></p></li>
<li><p><strong>dim</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.10)"><em>int</em></a><em>, </em><em>Tuple</em><em>[</em><a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.10)"><em>int</em></a><em>]</em><em>, </em><em>optional</em>) – dimensions over which to compute
the vector or matrix norm. See above for the behavior when <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code><cite>= None</cite>.
Default: <cite>None</cite></p></li>
<li><p><strong>keepdim</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>) – If set to <cite>True</cite>, the reduced dimensions are retained
in the result as dimensions with size one. Default: <cite>False</cite></p></li>
</ul>
</dd>
<dt class="field-even">Keyword Arguments</dt>
<dd class="field-even"><ul class="simple">
<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>) – output tensor. Ignored if <cite>None</cite>. Default: <cite>None</cite>.</p></li>
<li><p><strong>dtype</strong> (<code class="xref py py-class docutils literal notranslate"><span class="pre">torch.dtype</span></code>, optional) – If specified, the input tensor is cast to
<code class="xref py py-attr docutils literal notranslate"><span class="pre">dtype</span></code> before performing the operation, and the returned tensor’s type
will be <code class="xref py py-attr docutils literal notranslate"><span class="pre">dtype</span></code>. Default: <cite>None</cite></p></li>
</ul>
</dd>
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>A real-valued tensor, even when <code class="xref py py-attr docutils literal notranslate"><span class="pre">A</span></code> is complex.</p>
</dd>
</dl>
<p>Examples:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">linalg</span> <span class="k">as</span> <span class="n">LA</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">arange</span><span class="p">(</span><span class="mi">9</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float</span><span class="p">)</span> <span class="o">-</span> <span class="mi">4</span>
<span class="gp">>>> </span><span class="n">a</span>
<span class="go">tensor([-4., -3., -2., -1., 0., 1., 2., 3., 4.])</span>
<span class="gp">>>> </span><span class="n">B</span> <span class="o">=</span> <span class="n">a</span><span class="o">.</span><span class="n">reshape</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">B</span>
<span class="go">tensor([[-4., -3., -2.],</span>
<span class="go"> [-1., 0., 1.],</span>
<span class="go"> [ 2., 3., 4.]])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">tensor(7.7460)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
<span class="go">tensor(7.7460)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="s1">'fro'</span><span class="p">)</span>
<span class="go">tensor(7.7460)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">))</span>
<span class="go">tensor(4.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">))</span>
<span class="go">tensor(9.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">))</span>
<span class="go">tensor(0.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="o">-</span><span class="nb">float</span><span class="p">(</span><span class="s1">'inf'</span><span class="p">))</span>
<span class="go">tensor(2.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">tensor(20.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="go">tensor(7.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">tensor(0.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="go">tensor(6.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">tensor(7.7460)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">B</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="go">tensor(7.3485)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="go">tensor(0.)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">B</span><span class="o">.</span><span class="n">double</span><span class="p">(),</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="go">tensor(1.8570e-16, dtype=torch.float64)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="go">tensor(5.8480)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">)</span>
<span class="go">tensor(0.)</span>
</pre></div>
</div>
<p>Using the <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> argument to compute vector norms:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">c</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mf">3.</span><span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">tensor([1.4142, 2.2361, 5.0000])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">tensor([3.7417, 4.2426])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="nb">ord</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="go">tensor([6., 6.])</span>
</pre></div>
</div>
<p>Using the <code class="xref py py-attr docutils literal notranslate"><span class="pre">dim</span></code> argument to compute matrix norms:</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">arange</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">torch</span><span class="o">.</span><span class="n">float</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</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="mi">2</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">dim</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
<span class="go">tensor([ 3.7417, 11.2250])</span>
<span class="gp">>>> </span><span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:]),</span> <span class="n">LA</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">A</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="p">:,</span> <span class="p">:])</span>
<span class="go">(tensor(3.7417), tensor(11.2250))</span>
</pre></div>
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