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<div class="section" id="torch-lobpcg">
<h1>torch.lobpcg<a class="headerlink" href="#torch-lobpcg" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="torch.lobpcg">
<code class="sig-prename descclassname">torch.</code><code class="sig-name descname">lobpcg</code><span class="sig-paren">(</span><em class="sig-param">A: torch.Tensor</em>, <em class="sig-param">k: Optional[int] = None</em>, <em class="sig-param">B: Optional[torch.Tensor] = None</em>, <em class="sig-param">X: Optional[torch.Tensor] = None</em>, <em class="sig-param">n: Optional[int] = None</em>, <em class="sig-param">iK: Optional[torch.Tensor] = None</em>, <em class="sig-param">niter: Optional[int] = None</em>, <em class="sig-param">tol: Optional[float] = None</em>, <em class="sig-param">largest: Optional[bool] = None</em>, <em class="sig-param">method: Optional[str] = None</em>, <em class="sig-param">tracker: None = None</em>, <em class="sig-param">ortho_iparams: Optional[Dict[str</em>, <em class="sig-param">int]] = None</em>, <em class="sig-param">ortho_fparams: Optional[Dict[str</em>, <em class="sig-param">float]] = None</em>, <em class="sig-param">ortho_bparams: Optional[Dict[str</em>, <em class="sig-param">bool]] = None</em><span class="sig-paren">)</span> → Tuple[torch.Tensor, torch.Tensor]<a class="reference internal" href="../_modules/torch/_lobpcg.html#lobpcg"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.lobpcg" title="Permalink to this definition">¶</a></dt>
<dd><p>Find the k largest (or smallest) eigenvalues and the corresponding
eigenvectors of a symmetric positive defined generalized
eigenvalue problem using matrix-free LOBPCG methods.</p>
<p>This function is a front-end to the following LOBPCG algorithms
selectable via <cite>method</cite> argument:</p>
<blockquote>
<div><p><cite>method=”basic”</cite> - the LOBPCG method introduced by Andrew
Knyazev, see [Knyazev2001]. A less robust method, may fail when
Cholesky is applied to singular input.</p>
<p><cite>method=”ortho”</cite> - the LOBPCG method with orthogonal basis
selection [StathopoulosEtal2002]. A robust method.</p>
</div></blockquote>
<p>Supported inputs are dense, sparse, and batches of dense matrices.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>In general, the basic method spends least time per
iteration. However, the robust methods converge much faster and
are more stable. So, the usage of the basic method is generally
not recommended but there exist cases where the usage of the
basic method may be preferred.</p>
</div>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>The backward method does not support sparse and complex inputs.
It works only when <cite>B</cite> is not provided (i.e. <cite>B == None</cite>).
We are actively working on extensions, and the details of
the algorithms are going to be published promptly.</p>
</div>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>While it is assumed that <cite>A</cite> is symmetric, <cite>A.grad</cite> is not.
To make sure that <cite>A.grad</cite> is symmetric, so that <cite>A - t * A.grad</cite> is symmetric
in first-order optimization routines, prior to running <cite>lobpcg</cite>
we do the following symmetrization map: <cite>A -> (A + A.t()) / 2</cite>.
The map is performed only when the <cite>A</cite> requires gradients.</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>) – the input tensor 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>m</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, m, m)</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">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span>
</span></p></li>
<li><p><strong>B</strong> (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>, </em><em>optional</em>) – the input tensor 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>m</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, m,
m)</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">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span>
</span>. When not specified, <cite>B</cite> is interpereted as
identity matrix.</p></li>
<li><p><strong>X</strong> (<em>tensor</em><em>, </em><em>optional</em>) – the input tensor 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>m</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, m, 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">m</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>k <= n <= m</cite>. When specified, it is used as
initial approximation of eigenvectors. X must be a
dense tensor.</p></li>
<li><p><strong>iK</strong> (<em>tensor</em><em>, </em><em>optional</em>) – the input tensor 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>m</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, m,
m)</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">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span>
</span>. When specified, it will be used as preconditioner.</p></li>
<li><p><strong>k</strong> (<em>integer</em><em>, </em><em>optional</em>) – the number of requested
eigenpairs. Default is the number of <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</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" style="margin-right:0.07847em;">X</span></span></span></span>
</span>
columns (when specified) or <cite>1</cite>.</p></li>
<li><p><strong>n</strong> (<em>integer</em><em>, </em><em>optional</em>) – if <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</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" style="margin-right:0.07847em;">X</span></span></span></span>
</span> is not specified then <cite>n</cite>
specifies the size of the generated random
approximation of eigenvectors. Default value for <cite>n</cite>
is <cite>k</cite>. If <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</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" style="margin-right:0.07847em;">X</span></span></span></span>
</span> is specified, the value of <cite>n</cite>
(when specified) must be the number of <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</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" style="margin-right:0.07847em;">X</span></span></span></span>
</span>
columns.</p></li>
<li><p><strong>tol</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.9)"><em>float</em></a><em>, </em><em>optional</em>) – residual tolerance for stopping
criterion. Default is <cite>feps ** 0.5</cite> where <cite>feps</cite> is
smallest non-zero floating-point number of the given
input tensor <cite>A</cite> data type.</p></li>
<li><p><strong>largest</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.9)"><em>bool</em></a><em>, </em><em>optional</em>) – when True, solve the eigenproblem for
the largest eigenvalues. Otherwise, solve the
eigenproblem for smallest eigenvalues. Default is
<cite>True</cite>.</p></li>
<li><p><strong>method</strong> (<a class="reference external" href="https://docs.python.org/3/library/stdtypes.html#str" title="(in Python v3.9)"><em>str</em></a><em>, </em><em>optional</em>) – select LOBPCG method. See the
description of the function above. Default is
“ortho”.</p></li>
<li><p><strong>niter</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.9)"><em>int</em></a><em>, </em><em>optional</em>) – maximum number of iterations. When
reached, the iteration process is hard-stopped and
the current approximation of eigenpairs is returned.
For infinite iteration but until convergence criteria
is met, use <cite>-1</cite>.</p></li>
<li><p><strong>tracker</strong> (<em>callable</em><em>, </em><em>optional</em>) – <p>a function for tracing the
iteration process. When specified, it is called at
each iteration step with LOBPCG instance as an
argument. The LOBPCG instance holds the full state of
the iteration process in the following attributes:</p>
<blockquote>
<div><p><cite>iparams</cite>, <cite>fparams</cite>, <cite>bparams</cite> - dictionaries of
integer, float, and boolean valued input
parameters, respectively</p>
<p><cite>ivars</cite>, <cite>fvars</cite>, <cite>bvars</cite>, <cite>tvars</cite> - dictionaries
of integer, float, boolean, and Tensor valued
iteration variables, respectively.</p>
<p><cite>A</cite>, <cite>B</cite>, <cite>iK</cite> - input Tensor arguments.</p>
<p><cite>E</cite>, <cite>X</cite>, <cite>S</cite>, <cite>R</cite> - iteration Tensor variables.</p>
</div></blockquote>
<p>For instance:</p>
<blockquote>
<div><p><cite>ivars[“istep”]</cite> - the current iteration step
<cite>X</cite> - the current approximation of eigenvectors
<cite>E</cite> - the current approximation of eigenvalues
<cite>R</cite> - the current residual
<cite>ivars[“converged_count”]</cite> - the current number of converged eigenpairs
<cite>tvars[“rerr”]</cite> - the current state of convergence criteria</p>
</div></blockquote>
<p>Note that when <cite>tracker</cite> stores Tensor objects from
the LOBPCG instance, it must make copies of these.</p>
<p>If <cite>tracker</cite> sets <cite>bvars[“force_stop”] = True</cite>, the
iteration process will be hard-stopped.</p>
</p></li>
<li><p><strong>ortho_fparams, ortho_bparams</strong> (<em>ortho_iparams</em><em>,</em>) – various parameters to LOBPCG algorithm when using
<cite>method=”ortho”</cite>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p><p>tensor of eigenvalues 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>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, k)</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" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>
</span></p>
<p>X (Tensor): tensor of eigenvectors 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>m</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*, m, k)</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">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>
</span></p>
</p>
</dd>
<dt class="field-odd">Return type</dt>
<dd class="field-odd"><p>E (<a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor">Tensor</a>)</p>
</dd>
</dl>
<p class="rubric">References</p>
<p>[Knyazev2001] Andrew V. Knyazev. (2001) Toward the Optimal
Preconditioned Eigensolver: Locally Optimal Block Preconditioned
Conjugate Gradient Method. SIAM J. Sci. Comput., 23(2),
517-541. (25 pages)
<a class="reference external" href="https://epubs.siam.org/doi/abs/10.1137/S1064827500366124">https://epubs.siam.org/doi/abs/10.1137/S1064827500366124</a></p>
<p>[StathopoulosEtal2002] Andreas Stathopoulos and Kesheng
Wu. (2002) A Block Orthogonalization Procedure with Constant
Synchronization Requirements. SIAM J. Sci. Comput., 23(6),
2165-2182. (18 pages)
<a class="reference external" href="https://epubs.siam.org/doi/10.1137/S1064827500370883">https://epubs.siam.org/doi/10.1137/S1064827500370883</a></p>
<p>[DuerschEtal2018] Jed A. Duersch, Meiyue Shao, Chao Yang, Ming
Gu. (2018) A Robust and Efficient Implementation of LOBPCG.
SIAM J. Sci. Comput., 40(5), C655-C676. (22 pages)
<a class="reference external" href="https://epubs.siam.org/doi/abs/10.1137/17M1129830">https://epubs.siam.org/doi/abs/10.1137/17M1129830</a></p>
</dd></dl>
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