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<div class="section" id="complex-numbers">
<span id="complex-numbers-doc"></span><h1>Complex Numbers<a class="headerlink" href="#complex-numbers" title="Permalink to this heading">¶</a></h1>
<p>Complex numbers are numbers that can be expressed in the form <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>j</mi></mrow><annotation encoding="application/x-tex">a + bj</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">bj</span></span></span></span></span>, where a and b are real numbers,
and <em>j</em> is called the imaginary unit, which satisfies the equation <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>j</mi><mn>2</mn></msup><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">j^2 = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></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:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span></span>. Complex numbers frequently occur in mathematics and
engineering, especially in topics like signal processing. Traditionally many users and libraries (e.g., TorchAudio) have
handled complex numbers by representing the data in float tensors with shape <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><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(..., 2)</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">2</span><span class="mclose">)</span></span></span></span></span> where the last
dimension contains the real and imaginary values.</p>
<p>Tensors of complex dtypes provide a more natural user experience while working with complex numbers. Operations on
complex tensors (e.g., <a class="reference internal" href="generated/torch.mv.html#torch.mv" title="torch.mv"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.mv()</span></code></a>, <a class="reference internal" href="generated/torch.matmul.html#torch.matmul" title="torch.matmul"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.matmul()</span></code></a>) are likely to be faster and more memory efficient
than operations on float tensors mimicking them. Operations involving complex numbers in PyTorch are optimized
to use vectorized assembly instructions and specialized kernels (e.g. LAPACK, cuBlas).</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Spectral operations in the <a class="reference external" href="https://pytorch.org/docs/stable/fft.html#torch-fft">torch.fft module</a> support
native complex tensors.</p>
</div>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Complex tensors is a beta feature and subject to change.</p>
</div>
<div class="section" id="creating-complex-tensors">
<h2>Creating Complex Tensors<a class="headerlink" href="#creating-complex-tensors" title="Permalink to this heading">¶</a></h2>
<p>We support two complex dtypes: <cite>torch.cfloat</cite> and <cite>torch.cdouble</cite></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</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">2</span><span class="p">,</span><span class="mi">2</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">cfloat</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">x</span>
<span class="go">tensor([[-0.4621-0.0303j, -0.2438-0.5874j],</span>
<span class="go"> [ 0.7706+0.1421j, 1.2110+0.1918j]])</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The default dtype for complex tensors is determined by the default floating point dtype.
If the default floating point dtype is <cite>torch.float64</cite> then complex numbers are inferred to
have a dtype of <cite>torch.complex128</cite>, otherwise they are assumed to have a dtype of <cite>torch.complex64</cite>.</p>
</div>
<p>All factory functions apart from <a class="reference internal" href="generated/torch.linspace.html#torch.linspace" title="torch.linspace"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linspace()</span></code></a>, <a class="reference internal" href="generated/torch.logspace.html#torch.logspace" title="torch.logspace"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.logspace()</span></code></a>, and <a class="reference internal" href="generated/torch.arange.html#torch.arange" title="torch.arange"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.arange()</span></code></a> are
supported for complex tensors.</p>
</div>
<div class="section" id="transition-from-the-old-representation">
<h2>Transition from the old representation<a class="headerlink" href="#transition-from-the-old-representation" title="Permalink to this heading">¶</a></h2>
<p>Users who currently worked around the lack of complex tensors with real tensors of shape <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><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(..., 2)</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">2</span><span class="mclose">)</span></span></span></span></span>
can easily to switch using the complex tensors in their code using <a class="reference internal" href="generated/torch.view_as_complex.html#torch.view_as_complex" title="torch.view_as_complex"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.view_as_complex()</span></code></a>
and <a class="reference internal" href="generated/torch.view_as_real.html#torch.view_as_real" title="torch.view_as_real"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.view_as_real()</span></code></a>. Note that these functions don’t perform any copy and return a
view of the input tensor.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x</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="gp">>>> </span><span class="n">x</span>
<span class="go">tensor([[ 0.6125, -0.1681],</span>
<span class="go"> [-0.3773, 1.3487],</span>
<span class="go"> [-0.0861, -0.7981]])</span>
<span class="gp">>>> </span><span class="n">y</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">view_as_complex</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">y</span>
<span class="go">tensor([ 0.6125-0.1681j, -0.3773+1.3487j, -0.0861-0.7981j])</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">view_as_real</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="go">tensor([[ 0.6125, -0.1681],</span>
<span class="go"> [-0.3773, 1.3487],</span>
<span class="go"> [-0.0861, -0.7981]])</span>
</pre></div>
</div>
</div>
<div class="section" id="accessing-real-and-imag">
<h2>Accessing real and imag<a class="headerlink" href="#accessing-real-and-imag" title="Permalink to this heading">¶</a></h2>
<p>The real and imaginary values of a complex tensor can be accessed using the <code class="xref py py-attr docutils literal notranslate"><span class="pre">real</span></code> and
<code class="xref py py-attr docutils literal notranslate"><span class="pre">imag</span></code>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Accessing <cite>real</cite> and <cite>imag</cite> attributes doesn’t allocate any memory, and in-place updates on the
<cite>real</cite> and <cite>imag</cite> tensors will update the original complex tensor. Also, the
returned <cite>real</cite> and <cite>imag</cite> tensors are not contiguous.</p>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">real</span>
<span class="go">tensor([ 0.6125, -0.3773, -0.0861])</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">imag</span>
<span class="go">tensor([-0.1681, 1.3487, -0.7981])</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">real</span><span class="o">.</span><span class="n">mul_</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="go">tensor([ 1.2250, -0.7546, -0.1722])</span>
<span class="gp">>>> </span><span class="n">y</span>
<span class="go">tensor([ 1.2250-0.1681j, -0.7546+1.3487j, -0.1722-0.7981j])</span>
<span class="gp">>>> </span><span class="n">y</span><span class="o">.</span><span class="n">real</span><span class="o">.</span><span class="n">stride</span><span class="p">()</span>
<span class="go">(2,)</span>
</pre></div>
</div>
</div>
<div class="section" id="angle-and-abs">
<h2>Angle and abs<a class="headerlink" href="#angle-and-abs" title="Permalink to this heading">¶</a></h2>
<p>The angle and absolute values of a complex tensor can be computed using <a class="reference internal" href="generated/torch.angle.html#torch.angle" title="torch.angle"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.angle()</span></code></a> and
<a class="reference internal" href="generated/torch.abs.html#torch.abs" title="torch.abs"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.abs()</span></code></a>.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">x1</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="mi">3</span><span class="n">j</span><span class="p">,</span> <span class="mi">4</span><span class="o">+</span><span class="mi">4</span><span class="n">j</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">x1</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span>
<span class="go">tensor([3.0000, 5.6569])</span>
<span class="gp">>>> </span><span class="n">x1</span><span class="o">.</span><span class="n">angle</span><span class="p">()</span>
<span class="go">tensor([1.5708, 0.7854])</span>
</pre></div>
</div>
</div>
<div class="section" id="linear-algebra">
<h2>Linear Algebra<a class="headerlink" href="#linear-algebra" title="Permalink to this heading">¶</a></h2>
<p>Many linear algebra operations, like <a class="reference internal" href="generated/torch.matmul.html#torch.matmul" title="torch.matmul"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.matmul()</span></code></a>, <a class="reference internal" href="generated/torch.linalg.svd.html#torch.linalg.svd" title="torch.linalg.svd"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.svd()</span></code></a>, <a class="reference internal" href="generated/torch.linalg.solve.html#torch.linalg.solve" title="torch.linalg.solve"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.linalg.solve()</span></code></a> etc., support complex numbers.
If you’d like to request an operation we don’t currently support, please <a class="reference external" href="https://github.com/pytorch/pytorch/issues?q=is%3Aissue+is%3Aopen+complex">search</a>
if an issue has already been filed and if not, <a class="reference external" href="https://github.com/pytorch/pytorch/issues/new/choose">file one</a>.</p>
</div>
<div class="section" id="serialization">
<h2>Serialization<a class="headerlink" href="#serialization" title="Permalink to this heading">¶</a></h2>
<p>Complex tensors can be serialized, allowing data to be saved as complex values.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">save</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="s1">'complex_tensor.pt'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="s1">'complex_tensor.pt'</span><span class="p">)</span>
<span class="go">tensor([ 0.6125-0.1681j, -0.3773+1.3487j, -0.0861-0.7981j])</span>
</pre></div>
</div>
</div>
<div class="section" id="autograd">
<h2>Autograd<a class="headerlink" href="#autograd" title="Permalink to this heading">¶</a></h2>
<p>PyTorch supports autograd for complex tensors. The gradient computed is the Conjugate Wirtinger derivative,
the negative of which is precisely the direction of steepest descent used in Gradient Descent algorithm. Thus,
all the existing optimizers can be implemented to work out of the box with complex parameters. For more details,
check out the note <a class="reference internal" href="notes/autograd.html#complex-autograd-doc"><span class="std std-ref">Autograd for Complex Numbers</span></a>.</p>
</div>
<div class="section" id="optimizers">
<h2>Optimizers<a class="headerlink" href="#optimizers" title="Permalink to this heading">¶</a></h2>
<p>Semantically, we define stepping through a PyTorch optimizer with complex parameters as being equivalent to stepping
through the same optimizer on the <a class="reference internal" href="generated/torch.view_as_real.html#torch.view_as_real" title="torch.view_as_real"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.view_as_real()</span></code></a> equivalent of the complex params. More concretely:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</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">complex64</span><span class="p">)</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">)]</span>
<span class="gp">>>> </span><span class="n">real_params</span> <span class="o">=</span> <span class="p">[</span><span class="n">torch</span><span class="o">.</span><span class="n">view_as_real</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="n">params</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">complex_optim</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">AdamW</span><span class="p">(</span><span class="n">params</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">real_optim</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">optim</span><span class="o">.</span><span class="n">AdamW</span><span class="p">(</span><span class="n">real_params</span><span class="p">)</span>
</pre></div>
</div>
<p><cite>real_optim</cite> and <cite>complex_optim</cite> will compute the same updates on the parameters, though there may be slight numerical
discrepancies between the two optimizers, similar to numerical discrepancies between foreach vs forloop optimizers
and capturable vs default optimizers. For more details, see <a class="reference external" href="https://pytorch.org/docs/stable/notes/numerical_accuracy.html">https://pytorch.org/docs/stable/notes/numerical_accuracy.html</a>.</p>
<p>Specifically, while you can think of our optimizer’s handling of complex tensors as the same as optimizing over their
<cite>p.real</cite> and <cite>p.imag</cite> pieces separately, the implementation details are not precisely that. Note that the
<a class="reference internal" href="generated/torch.view_as_real.html#torch.view_as_real" title="torch.view_as_real"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.view_as_real()</span></code></a> equivalent will convert a complex tensor to a real tensor with shape <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><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(..., 2)</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">2</span><span class="mclose">)</span></span></span></span></span>,
whereas splitting a complex tensor into two tensors is 2 tensors 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><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(...)</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="mclose">)</span></span></span></span></span>. This distinction has no impact on
pointwise optimizers (like AdamW) but will cause slight discrepancy in optimizers that do global reductions (like LBFGS).
We currently do not have optimizers that do per-Tensor reductions and thus do not yet define this behavior. Open an issue
if you have a use case that requires precisely defining this behavior.</p>
<p>We do not fully support the following subsystems:</p>
<ul class="simple">
<li><p>Quantization</p></li>
<li><p>JIT</p></li>
<li><p>Sparse Tensors</p></li>
<li><p>Distributed</p></li>
</ul>
<p>If any of these would help your use case, please <a class="reference external" href="https://github.com/pytorch/pytorch/issues?q=is%3Aissue+is%3Aopen+complex">search</a>
if an issue has already been filed and if not, <a class="reference external" href="https://github.com/pytorch/pytorch/issues/new/choose">file one</a>.</p>
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