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<h1>Source code for torch._lowrank</h1><div class="highlight"><pre>
<span></span><span class="sd">"""Implement various linear algebra algorithms for low rank matrices.</span>
<span class="sd">"""</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"svd_lowrank"</span><span class="p">,</span> <span class="s2">"pca_lowrank"</span><span class="p">]</span>
<span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Optional</span><span class="p">,</span> <span class="n">Tuple</span>
<span class="kn">import</span> <span class="nn">torch</span>
<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">Tensor</span>
<span class="kn">from</span> <span class="nn">.</span> <span class="kn">import</span> <span class="n">_linalg_utils</span> <span class="k">as</span> <span class="n">_utils</span>
<span class="kn">from</span> <span class="nn">.overrides</span> <span class="kn">import</span> <span class="n">handle_torch_function</span><span class="p">,</span> <span class="n">has_torch_function</span>
<span class="k">def</span> <span class="nf">get_approximate_basis</span><span class="p">(</span>
<span class="n">A</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">q</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">niter</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">M</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span>
<span class="p">)</span> <span class="o">-></span> <span class="n">Tensor</span><span class="p">:</span>
<span class="w"> </span><span class="sd">"""Return tensor :math:`Q` with :math:`q` orthonormal columns such</span>
<span class="sd"> that :math:`Q Q^H A` approximates :math:`A`. If :math:`M` is</span>
<span class="sd"> specified, then :math:`Q` is such that :math:`Q Q^H (A - M)`</span>
<span class="sd"> approximates :math:`A - M`.</span>
<span class="sd"> .. note:: The implementation is based on the Algorithm 4.4 from</span>
<span class="sd"> Halko et al, 2009.</span>
<span class="sd"> .. note:: For an adequate approximation of a k-rank matrix</span>
<span class="sd"> :math:`A`, where k is not known in advance but could be</span>
<span class="sd"> estimated, the number of :math:`Q` columns, q, can be</span>
<span class="sd"> choosen according to the following criteria: in general,</span>
<span class="sd"> :math:`k <= q <= min(2*k, m, n)`. For large low-rank</span>
<span class="sd"> matrices, take :math:`q = k + 5..10`. If k is</span>
<span class="sd"> relatively small compared to :math:`min(m, n)`, choosing</span>
<span class="sd"> :math:`q = k + 0..2` may be sufficient.</span>
<span class="sd"> .. note:: To obtain repeatable results, reset the seed for the</span>
<span class="sd"> pseudorandom number generator</span>
<span class="sd"> Args::</span>
<span class="sd"> A (Tensor): the input tensor of size :math:`(*, m, n)`</span>
<span class="sd"> q (int): the dimension of subspace spanned by :math:`Q`</span>
<span class="sd"> columns.</span>
<span class="sd"> niter (int, optional): the number of subspace iterations to</span>
<span class="sd"> conduct; ``niter`` must be a</span>
<span class="sd"> nonnegative integer. In most cases, the</span>
<span class="sd"> default value 2 is more than enough.</span>
<span class="sd"> M (Tensor, optional): the input tensor's mean of size</span>
<span class="sd"> :math:`(*, 1, n)`.</span>
<span class="sd"> References::</span>
<span class="sd"> - Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding</span>
<span class="sd"> structure with randomness: probabilistic algorithms for</span>
<span class="sd"> constructing approximate matrix decompositions,</span>
<span class="sd"> arXiv:0909.4061 [math.NA; math.PR], 2009 (available at</span>
<span class="sd"> `arXiv <http://arxiv.org/abs/0909.4061>`_).</span>
<span class="sd"> """</span>
<span class="n">niter</span> <span class="o">=</span> <span class="mi">2</span> <span class="k">if</span> <span class="n">niter</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">niter</span>
<span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span>
<span class="n">dtype</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">get_floating_dtype</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="n">matmul</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">matmul</span>
<span class="n">R</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="n">n</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">A</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
<span class="c1"># The following code could be made faster using torch.geqrf + torch.ormqr</span>
<span class="c1"># but geqrf is not differentiable</span>
<span class="n">A_H</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transjugate</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">Q</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">qr</span><span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">R</span><span class="p">))</span><span class="o">.</span><span class="n">Q</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">niter</span><span class="p">):</span>
<span class="n">Q</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">qr</span><span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A_H</span><span class="p">,</span> <span class="n">Q</span><span class="p">))</span><span class="o">.</span><span class="n">Q</span>
<span class="n">Q</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">qr</span><span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Q</span><span class="p">))</span><span class="o">.</span><span class="n">Q</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">M_H</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transjugate</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
<span class="n">Q</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">qr</span><span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">R</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">R</span><span class="p">))</span><span class="o">.</span><span class="n">Q</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">niter</span><span class="p">):</span>
<span class="n">Q</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">qr</span><span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A_H</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M_H</span><span class="p">,</span> <span class="n">Q</span><span class="p">))</span><span class="o">.</span><span class="n">Q</span>
<span class="n">Q</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">qr</span><span class="p">(</span><span class="n">matmul</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">Q</span><span class="p">))</span><span class="o">.</span><span class="n">Q</span>
<span class="k">return</span> <span class="n">Q</span>
<div class="viewcode-block" id="svd_lowrank"><a class="viewcode-back" href="../../generated/torch.svd_lowrank.html#torch.svd_lowrank">[docs]</a><span class="k">def</span> <span class="nf">svd_lowrank</span><span class="p">(</span>
<span class="n">A</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span>
<span class="n">q</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="mi">6</span><span class="p">,</span>
<span class="n">niter</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span>
<span class="n">M</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
<span class="p">)</span> <span class="o">-></span> <span class="n">Tuple</span><span class="p">[</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">Tensor</span><span class="p">]:</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Return the singular value decomposition ``(U, S, V)`` of a matrix,</span>
<span class="sd"> batches of matrices, or a sparse matrix :math:`A` such that</span>
<span class="sd"> :math:`A \approx U diag(S) V^T`. In case :math:`M` is given, then</span>
<span class="sd"> SVD is computed for the matrix :math:`A - M`.</span>
<span class="sd"> .. note:: The implementation is based on the Algorithm 5.1 from</span>
<span class="sd"> Halko et al, 2009.</span>
<span class="sd"> .. note:: To obtain repeatable results, reset the seed for the</span>
<span class="sd"> pseudorandom number generator</span>
<span class="sd"> .. note:: The input is assumed to be a low-rank matrix.</span>
<span class="sd"> .. note:: In general, use the full-rank SVD implementation</span>
<span class="sd"> :func:`torch.linalg.svd` for dense matrices due to its 10-fold</span>
<span class="sd"> higher performance characteristics. The low-rank SVD</span>
<span class="sd"> will be useful for huge sparse matrices that</span>
<span class="sd"> :func:`torch.linalg.svd` cannot handle.</span>
<span class="sd"> Args::</span>
<span class="sd"> A (Tensor): the input tensor of size :math:`(*, m, n)`</span>
<span class="sd"> q (int, optional): a slightly overestimated rank of A.</span>
<span class="sd"> niter (int, optional): the number of subspace iterations to</span>
<span class="sd"> conduct; niter must be a nonnegative</span>
<span class="sd"> integer, and defaults to 2</span>
<span class="sd"> M (Tensor, optional): the input tensor's mean of size</span>
<span class="sd"> :math:`(*, 1, n)`.</span>
<span class="sd"> References::</span>
<span class="sd"> - Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding</span>
<span class="sd"> structure with randomness: probabilistic algorithms for</span>
<span class="sd"> constructing approximate matrix decompositions,</span>
<span class="sd"> arXiv:0909.4061 [math.NA; math.PR], 2009 (available at</span>
<span class="sd"> `arXiv <https://arxiv.org/abs/0909.4061>`_).</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">jit</span><span class="o">.</span><span class="n">is_scripting</span><span class="p">():</span>
<span class="n">tensor_ops</span> <span class="o">=</span> <span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">M</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">set</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="nb">type</span><span class="p">,</span> <span class="n">tensor_ops</span><span class="p">))</span><span class="o">.</span><span class="n">issubset</span><span class="p">(</span>
<span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">,</span> <span class="nb">type</span><span class="p">(</span><span class="kc">None</span><span class="p">))</span>
<span class="p">)</span> <span class="ow">and</span> <span class="n">has_torch_function</span><span class="p">(</span><span class="n">tensor_ops</span><span class="p">):</span>
<span class="k">return</span> <span class="n">handle_torch_function</span><span class="p">(</span>
<span class="n">svd_lowrank</span><span class="p">,</span> <span class="n">tensor_ops</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M</span>
<span class="p">)</span>
<span class="k">return</span> <span class="n">_svd_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M</span><span class="p">)</span></div>
<span class="k">def</span> <span class="nf">_svd_lowrank</span><span class="p">(</span>
<span class="n">A</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span>
<span class="n">q</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="mi">6</span><span class="p">,</span>
<span class="n">niter</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span>
<span class="n">M</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="n">Tensor</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span>
<span class="p">)</span> <span class="o">-></span> <span class="n">Tuple</span><span class="p">[</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">Tensor</span><span class="p">]:</span>
<span class="n">q</span> <span class="o">=</span> <span class="mi">6</span> <span class="k">if</span> <span class="n">q</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">q</span>
<span class="n">m</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span>
<span class="n">matmul</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">matmul</span>
<span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">M_t</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">M_t</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
<span class="n">A_t</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="c1"># Algorithm 5.1 in Halko et al 2009, slightly modified to reduce</span>
<span class="c1"># the number conjugate and transpose operations</span>
<span class="k">if</span> <span class="n">m</span> <span class="o"><</span> <span class="n">n</span> <span class="ow">or</span> <span class="n">n</span> <span class="o">></span> <span class="n">q</span><span class="p">:</span>
<span class="c1"># computing the SVD approximation of a transpose in</span>
<span class="c1"># order to keep B shape minimal (the m < n case) or the V</span>
<span class="c1"># shape small (the n > q case)</span>
<span class="n">Q</span> <span class="o">=</span> <span class="n">get_approximate_basis</span><span class="p">(</span><span class="n">A_t</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M_t</span><span class="p">)</span>
<span class="n">Q_c</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
<span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">B_t</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">Q_c</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">B_t</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">Q_c</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">m</span><span class="p">,</span> <span class="p">(</span><span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">m</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">q</span><span class="p">,</span> <span class="p">(</span><span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o"><=</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span>
<span class="n">U</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Vh</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">svd</span><span class="p">(</span><span class="n">B_t</span><span class="p">,</span> <span class="n">full_matrices</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">V</span> <span class="o">=</span> <span class="n">Vh</span><span class="o">.</span><span class="n">mH</span>
<span class="n">V</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">V</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">Q</span> <span class="o">=</span> <span class="n">get_approximate_basis</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M</span><span class="p">)</span>
<span class="n">Q_c</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">conjugate</span><span class="p">(</span><span class="n">Q</span><span class="p">)</span>
<span class="k">if</span> <span class="n">M</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A_t</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">matmul</span><span class="p">(</span><span class="n">A_t</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span> <span class="o">-</span> <span class="n">matmul</span><span class="p">(</span><span class="n">M_t</span><span class="p">,</span> <span class="n">Q_c</span><span class="p">)</span>
<span class="n">B_t</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">==</span> <span class="n">q</span><span class="p">,</span> <span class="p">(</span><span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o">==</span> <span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="o"><=</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span> <span class="n">B_t</span><span class="o">.</span><span class="n">shape</span>
<span class="n">U</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">Vh</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">svd</span><span class="p">(</span><span class="n">B_t</span><span class="p">,</span> <span class="n">full_matrices</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">V</span> <span class="o">=</span> <span class="n">Vh</span><span class="o">.</span><span class="n">mH</span>
<span class="n">U</span> <span class="o">=</span> <span class="n">Q</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">U</span><span class="p">)</span>
<span class="k">return</span> <span class="n">U</span><span class="p">,</span> <span class="n">S</span><span class="p">,</span> <span class="n">V</span>
<div class="viewcode-block" id="pca_lowrank"><a class="viewcode-back" href="../../generated/torch.pca_lowrank.html#torch.pca_lowrank">[docs]</a><span class="k">def</span> <span class="nf">pca_lowrank</span><span class="p">(</span>
<span class="n">A</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">q</span><span class="p">:</span> <span class="n">Optional</span><span class="p">[</span><span class="nb">int</span><span class="p">]</span> <span class="o">=</span> <span class="kc">None</span><span class="p">,</span> <span class="n">center</span><span class="p">:</span> <span class="nb">bool</span> <span class="o">=</span> <span class="kc">True</span><span class="p">,</span> <span class="n">niter</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">2</span>
<span class="p">)</span> <span class="o">-></span> <span class="n">Tuple</span><span class="p">[</span><span class="n">Tensor</span><span class="p">,</span> <span class="n">Tensor</span><span class="p">,</span> <span class="n">Tensor</span><span class="p">]:</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Performs linear Principal Component Analysis (PCA) on a low-rank</span>
<span class="sd"> matrix, batches of such matrices, or sparse matrix.</span>
<span class="sd"> This function returns a namedtuple ``(U, S, V)`` which is the</span>
<span class="sd"> nearly optimal approximation of a singular value decomposition of</span>
<span class="sd"> a centered matrix :math:`A` such that :math:`A = U diag(S) V^T`.</span>
<span class="sd"> .. note:: The relation of ``(U, S, V)`` to PCA is as follows:</span>
<span class="sd"> - :math:`A` is a data matrix with ``m`` samples and</span>
<span class="sd"> ``n`` features</span>
<span class="sd"> - the :math:`V` columns represent the principal directions</span>
<span class="sd"> - :math:`S ** 2 / (m - 1)` contains the eigenvalues of</span>
<span class="sd"> :math:`A^T A / (m - 1)` which is the covariance of</span>
<span class="sd"> ``A`` when ``center=True`` is provided.</span>
<span class="sd"> - ``matmul(A, V[:, :k])`` projects data to the first k</span>
<span class="sd"> principal components</span>
<span class="sd"> .. note:: Different from the standard SVD, the size of returned</span>
<span class="sd"> matrices depend on the specified rank and q</span>
<span class="sd"> values as follows:</span>
<span class="sd"> - :math:`U` is m x q matrix</span>
<span class="sd"> - :math:`S` is q-vector</span>
<span class="sd"> - :math:`V` is n x q matrix</span>
<span class="sd"> .. note:: To obtain repeatable results, reset the seed for the</span>
<span class="sd"> pseudorandom number generator</span>
<span class="sd"> Args:</span>
<span class="sd"> A (Tensor): the input tensor of size :math:`(*, m, n)`</span>
<span class="sd"> q (int, optional): a slightly overestimated rank of</span>
<span class="sd"> :math:`A`. By default, ``q = min(6, m,</span>
<span class="sd"> n)``.</span>
<span class="sd"> center (bool, optional): if True, center the input tensor,</span>
<span class="sd"> otherwise, assume that the input is</span>
<span class="sd"> centered.</span>
<span class="sd"> niter (int, optional): the number of subspace iterations to</span>
<span class="sd"> conduct; niter must be a nonnegative</span>
<span class="sd"> integer, and defaults to 2.</span>
<span class="sd"> References::</span>
<span class="sd"> - Nathan Halko, Per-Gunnar Martinsson, and Joel Tropp, Finding</span>
<span class="sd"> structure with randomness: probabilistic algorithms for</span>
<span class="sd"> constructing approximate matrix decompositions,</span>
<span class="sd"> arXiv:0909.4061 [math.NA; math.PR], 2009 (available at</span>
<span class="sd"> `arXiv <http://arxiv.org/abs/0909.4061>`_).</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">jit</span><span class="o">.</span><span class="n">is_scripting</span><span class="p">():</span>
<span class="k">if</span> <span class="nb">type</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span> <span class="ow">and</span> <span class="n">has_torch_function</span><span class="p">((</span><span class="n">A</span><span class="p">,)):</span>
<span class="k">return</span> <span class="n">handle_torch_function</span><span class="p">(</span>
<span class="n">pca_lowrank</span><span class="p">,</span> <span class="p">(</span><span class="n">A</span><span class="p">,),</span> <span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="o">=</span><span class="n">q</span><span class="p">,</span> <span class="n">center</span><span class="o">=</span><span class="n">center</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span>
<span class="p">)</span>
<span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="o">-</span><span class="mi">2</span><span class="p">:]</span>
<span class="k">if</span> <span class="n">q</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">q</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="k">elif</span> <span class="ow">not</span> <span class="p">(</span><span class="n">q</span> <span class="o">>=</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">q</span> <span class="o"><=</span> <span class="nb">min</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)):</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span>
<span class="sa">f</span><span class="s2">"q(=</span><span class="si">{</span><span class="n">q</span><span class="si">}</span><span class="s2">) must be non-negative integer and not greater than min(m, n)=</span><span class="si">{</span><span class="nb">min</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span>
<span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="p">(</span><span class="n">niter</span> <span class="o">>=</span> <span class="mi">0</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="sa">f</span><span class="s2">"niter(=</span><span class="si">{</span><span class="n">niter</span><span class="si">}</span><span class="s2">) must be non-negative integer"</span><span class="p">)</span>
<span class="n">dtype</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">get_floating_dtype</span><span class="p">(</span><span class="n">A</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">center</span><span class="p">:</span>
<span class="k">return</span> <span class="n">_svd_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
<span class="k">if</span> <span class="n">_utils</span><span class="o">.</span><span class="n">is_sparse</span><span class="p">(</span><span class="n">A</span><span class="p">):</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">!=</span> <span class="mi">2</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"pca_lowrank input is expected to be 2-dimensional tensor"</span><span class="p">)</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">sum</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="o">-</span><span class="mi">2</span><span class="p">,))</span> <span class="o">/</span> <span class="n">m</span>
<span class="c1"># reshape c</span>
<span class="n">column_indices</span> <span class="o">=</span> <span class="n">c</span><span class="o">.</span><span class="n">indices</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">indices</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span>
<span class="mi">2</span><span class="p">,</span>
<span class="nb">len</span><span class="p">(</span><span class="n">column_indices</span><span class="p">),</span>
<span class="n">dtype</span><span class="o">=</span><span class="n">column_indices</span><span class="o">.</span><span class="n">dtype</span><span class="p">,</span>
<span class="n">device</span><span class="o">=</span><span class="n">column_indices</span><span class="o">.</span><span class="n">device</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">indices</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">column_indices</span>
<span class="n">C_t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">sparse_coo_tensor</span><span class="p">(</span>
<span class="n">indices</span><span class="p">,</span> <span class="n">c</span><span class="o">.</span><span class="n">values</span><span class="p">(),</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">A</span><span class="o">.</span><span class="n">device</span>
<span class="p">)</span>
<span class="n">ones_m1_t</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">dtype</span><span class="p">,</span> <span class="n">device</span><span class="o">=</span><span class="n">A</span><span class="o">.</span><span class="n">device</span><span class="p">)</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">_utils</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">sparse</span><span class="o">.</span><span class="n">mm</span><span class="p">(</span><span class="n">C_t</span><span class="p">,</span> <span class="n">ones_m1_t</span><span class="p">))</span>
<span class="k">return</span> <span class="n">_svd_lowrank</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="n">M</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">C</span> <span class="o">=</span> <span class="n">A</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">dim</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,),</span> <span class="n">keepdim</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="k">return</span> <span class="n">_svd_lowrank</span><span class="p">(</span><span class="n">A</span> <span class="o">-</span> <span class="n">C</span><span class="p">,</span> <span class="n">q</span><span class="p">,</span> <span class="n">niter</span><span class="o">=</span><span class="n">niter</span><span class="p">,</span> <span class="n">M</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span></div>
</pre></div>
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