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<span class="target" id="module-torch.sparse"></span><section id="torch-sparse">
<span id="sparse-docs"></span><h1>torch.sparse<a class="headerlink" href="#torch-sparse" title="Permalink to this heading">¶</a></h1>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>The PyTorch API of sparse tensors is in beta and may change in the near future.
We highly welcome feature requests, bug reports and general suggestions as Github issues.</p>
</div>
<section id="why-and-when-to-use-sparsity">
<h2>Why and when to use sparsity<a class="headerlink" href="#why-and-when-to-use-sparsity" title="Permalink to this heading">¶</a></h2>
<p>By default PyTorch stores <a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><code class="xref py py-class docutils literal notranslate"><span class="pre">torch.Tensor</span></code></a> stores elements contiguously
physical memory. This leads to efficient implementations of various array
processing algorithms that require fast access to elements.</p>
<p>Now, some users might decide to represent data such as graph adjacency
matrices, pruned weights or points clouds by Tensors whose <em>elements are
mostly zero valued</em>. We recognize these are important applications and aim
to provide performance optimizations for these use cases via sparse storage formats.</p>
<p>Various sparse storage formats such as COO, CSR/CSC, LIL, etc. have been
developed over the years. While they differ in exact layouts, they all
compress data through efficient representation of zero valued elements.
We call the uncompressed values <em>specified</em> in contrast to <em>unspecified</em>,
compressed elements.</p>
<p>By compressing repeat zeros sparse storage formats aim to save memory
and computational resources on various CPUs and GPUs. Especially for high
degrees of sparsity or highly structured sparsity this can have significant
performance implications. As such sparse storage formats can be seen as a
performance optimization.</p>
<p>Like many other performance optimization sparse storage formats are not
always advantageous. When trying sparse formats for your use case
you might find your execution time to decrease rather than increase.</p>
<p>Please feel encouraged to open a Github issue if you analytically
expected to see a stark increase in performance but measured a
degradation instead. This helps us prioritize the implementation
of efficient kernels and wider performance optimizations.</p>
<p>We make it easy to try different sparsity layouts, and convert between them,
without being opinionated on what’s best for your particular application.</p>
</section>
<section id="functionality-overview">
<h2>Functionality overview<a class="headerlink" href="#functionality-overview" title="Permalink to this heading">¶</a></h2>
<p>We want it to be straightforward to construct a sparse Tensor from a
given dense Tensor by providing conversion routines for each layout.</p>
<p>In the next example we convert a 2D Tensor with default dense (strided)
layout to a 2D Tensor backed by the COO memory layout. Only values and
indices of non-zero elements are stored in this case.</p>
<div class="doctest 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">tensor</span><span class="p">([[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">2.</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">a</span><span class="o">.</span><span class="n">to_sparse</span><span class="p">()</span>
<span class="go">tensor(indices=tensor([[0, 1],</span>
<span class="go"> [1, 0]]),</span>
<span class="go"> values=tensor([2., 3.]),</span>
<span class="go"> size=(2, 2), nnz=2, layout=torch.sparse_coo)</span>
</pre></div>
</div>
<p>PyTorch currently supports <a class="reference internal" href="#sparse-coo-docs"><span class="std std-ref">COO</span></a>, <a class="reference internal" href="#sparse-csr-docs"><span class="std std-ref">CSR</span></a>,
<a class="reference internal" href="#sparse-csc-docs"><span class="std std-ref">CSC</span></a>, <a class="reference internal" href="#sparse-bsr-docs"><span class="std std-ref">BSR</span></a>, and <a class="reference internal" href="#sparse-bsc-docs"><span class="std std-ref">BSC</span></a>.
Please see the references for more details.</p>
<p>Note that we provide slight generalizations of these formats.</p>
<p>Batching: Devices such as GPUs require batching for optimal performance and
thus we support batch dimensions.</p>
<p>We currently offer a very simple version of batching where each component of a sparse format
itself is batched. This also requires the same number of specified elements per batch entry.
In this example we construct a 3D (batched) CSR Tensor from a 3D dense Tensor.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">t</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="mi">0</span><span class="p">],</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="p">[[</span><span class="mf">4.</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mf">5.</span><span class="p">,</span> <span class="mf">6.</span><span class="p">]]])</span>
<span class="gp">>>> </span><span class="n">t</span><span class="o">.</span><span class="n">dim</span><span class="p">()</span>
<span class="go">3</span>
<span class="gp">>>> </span><span class="n">t</span><span class="o">.</span><span class="n">to_sparse_csr</span><span class="p">()</span>
<span class="go">tensor(crow_indices=tensor([[0, 1, 3],</span>
<span class="go"> [0, 1, 3]]),</span>
<span class="go"> col_indices=tensor([[0, 0, 1],</span>
<span class="go"> [0, 0, 1]]),</span>
<span class="go"> values=tensor([[1., 2., 3.],</span>
<span class="go"> [4., 5., 6.]]), size=(2, 2, 2), nnz=3,</span>
<span class="go"> layout=torch.sparse_csr)</span>
</pre></div>
</div>
<p>Dense dimensions: On the other hand, some data such as Graph embeddings might be
better viewed as sparse collections of vectors instead of scalars.</p>
<p>In this example we create a 3D Hybrid COO Tensor with 2 sparse and 1 dense dimension
from a 3D strided Tensor. If an entire row in the 3D strided Tensor is zero, it is
not stored. If however any of the values in the row are non-zero, they are stored
entirely. This reduces the number of indices since we need one index one per row instead
of one per element. But it also increases the amount of storage for the values. Since
only rows that are <em>entirely</em> zero can be emitted and the presence of any non-zero
valued elements cause the entire row to be stored.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">t</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">0.</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</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="p">[[</span><span class="mf">0.</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">4.</span><span class="p">]]])</span>
<span class="gp">>>> </span><span class="n">t</span><span class="o">.</span><span class="n">to_sparse</span><span class="p">(</span><span class="n">sparse_dim</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="go">tensor(indices=tensor([[0, 1],</span>
<span class="go"> [1, 1]]),</span>
<span class="go"> values=tensor([[1., 2.],</span>
<span class="go"> [3., 4.]]),</span>
<span class="go"> size=(2, 2, 2), nnz=2, layout=torch.sparse_coo)</span>
</pre></div>
</div>
</section>
<section id="operator-overview">
<h2>Operator overview<a class="headerlink" href="#operator-overview" title="Permalink to this heading">¶</a></h2>
<p>Fundamentally, operations on Tensor with sparse storage formats behave the same as
operations on Tensor with strided (or other) storage formats. The particularities of
storage, that is the physical layout of the data, influences the performance of
an operation but shhould not influence the semantics.</p>
<p>We are actively increasing operator coverage for sparse tensors. Users should not
expect support same level of support as for dense Tensors yet.
See our <a class="reference internal" href="#sparse-ops-docs"><span class="std std-ref">operator</span></a> documentation for a list.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">b</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">0</span><span class="p">,</span> <span class="mi">0</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="mi">3</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">b_s</span> <span class="o">=</span> <span class="n">b</span><span class="o">.</span><span class="n">to_sparse_csr</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">b_s</span><span class="o">.</span><span class="n">cos</span><span class="p">()</span>
<span class="gt">Traceback (most recent call last):</span>
File <span class="nb">"<stdin>"</span>, line <span class="m">1</span>, in <span class="n"><module></span>
<span class="gr">RuntimeError</span>: <span class="n">unsupported tensor layout: SparseCsr</span>
<span class="gp">>>> </span><span class="n">b_s</span><span class="o">.</span><span class="n">sin</span><span class="p">()</span>
<span class="go">tensor(crow_indices=tensor([0, 3, 6]),</span>
<span class="go"> col_indices=tensor([2, 3, 4, 0, 1, 3]),</span>
<span class="go"> values=tensor([ 0.8415, 0.9093, 0.1411, -0.7568, -0.9589, -0.2794]),</span>
<span class="go"> size=(2, 6), nnz=6, layout=torch.sparse_csr)</span>
</pre></div>
</div>
<p>As shown in the example above, we don’t support non-zero preserving unary
operators such as cos. The output of a non-zero preserving unary operation
will not be able to take advantage of sparse storage formats to the same
extent as the input and potentially result in a catastrophic increase in memory.
We instead rely on the user to explicitly convert to a dense Tensor first and
then run the operation.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">b_s</span><span class="o">.</span><span class="n">to_dense</span><span class="p">()</span><span class="o">.</span><span class="n">cos</span><span class="p">()</span>
<span class="go">tensor([[ 1.0000, -0.4161],</span>
<span class="go"> [-0.9900, 1.0000]])</span>
</pre></div>
</div>
<p>We are aware that some users want to ignore compressed zeros for operations such
as <cite>cos</cite> instead of preserving the exact semantics of the operation. For this we
can point to torch.masked and its MaskedTensor, which is in turn also backed and
powered by sparse storage formats and kernels.</p>
<p>Also note that, for now, the user doesn’t have a choice of the output layout. For example,
adding a sparse Tensor to a regular strided Tensor results in a strided Tensor. Some
users might prefer for this to stay a sparse layout, because they know the result will
still be sufficiently sparse.</p>
<div class="doctest 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">b</span><span class="o">.</span><span class="n">to_sparse</span><span class="p">()</span>
<span class="go">tensor([[0., 3.],</span>
<span class="go"> [3., 0.]])</span>
</pre></div>
</div>
<p>We acknowledge that access to kernels that can efficiently produce different output
layouts can be very useful. A subsequent operation might significantly benefit from
receiving a particular layout. We are working on an API to control the result layout
and recognize it is an important feature to plan a more optimal path of execution for
any given model.</p>
</section>
<section id="sparse-coo-tensors">
<span id="sparse-coo-docs"></span><h2>Sparse COO tensors<a class="headerlink" href="#sparse-coo-tensors" title="Permalink to this heading">¶</a></h2>
<p>PyTorch implements the so-called Coordinate format, or COO
format, as one of the storage formats for implementing sparse
tensors. In COO format, the specified elements are stored as tuples
of element indices and the corresponding values. In particular,</p>
<blockquote>
<div><ul class="simple">
<li><p>the indices of specified elements are collected in <code class="docutils literal notranslate"><span class="pre">indices</span></code>
tensor of size <code class="docutils literal notranslate"><span class="pre">(ndim,</span> <span class="pre">nse)</span></code> and with element type
<code class="docutils literal notranslate"><span class="pre">torch.int64</span></code>,</p></li>
<li><p>the corresponding values are collected in <code class="docutils literal notranslate"><span class="pre">values</span></code> tensor of
size <code class="docutils literal notranslate"><span class="pre">(nse,)</span></code> and with an arbitrary integer or floating point
number element type,</p></li>
</ul>
</div></blockquote>
<p>where <code class="docutils literal notranslate"><span class="pre">ndim</span></code> is the dimensionality of the tensor and <code class="docutils literal notranslate"><span class="pre">nse</span></code> is the
number of specified elements.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The memory consumption of a sparse COO tensor is at least <code class="docutils literal notranslate"><span class="pre">(ndim</span> <span class="pre">*</span>
<span class="pre">8</span> <span class="pre">+</span> <span class="pre"><size</span> <span class="pre">of</span> <span class="pre">element</span> <span class="pre">type</span> <span class="pre">in</span> <span class="pre">bytes>)</span> <span class="pre">*</span> <span class="pre">nse</span></code> bytes (plus a constant
overhead from storing other tensor data).</p>
<p>The memory consumption of a strided tensor is at least
<code class="docutils literal notranslate"><span class="pre">product(<tensor</span> <span class="pre">shape>)</span> <span class="pre">*</span> <span class="pre"><size</span> <span class="pre">of</span> <span class="pre">element</span> <span class="pre">type</span> <span class="pre">in</span> <span class="pre">bytes></span></code>.</p>
<p>For example, the memory consumption of a 10 000 x 10 000 tensor
with 100 000 non-zero 32-bit floating point numbers is at least
<code class="docutils literal notranslate"><span class="pre">(2</span> <span class="pre">*</span> <span class="pre">8</span> <span class="pre">+</span> <span class="pre">4)</span> <span class="pre">*</span> <span class="pre">100</span> <span class="pre">000</span> <span class="pre">=</span> <span class="pre">2</span> <span class="pre">000</span> <span class="pre">000</span></code> bytes when using COO tensor
layout and <code class="docutils literal notranslate"><span class="pre">10</span> <span class="pre">000</span> <span class="pre">*</span> <span class="pre">10</span> <span class="pre">000</span> <span class="pre">*</span> <span class="pre">4</span> <span class="pre">=</span> <span class="pre">400</span> <span class="pre">000</span> <span class="pre">000</span></code> bytes when using
the default strided tensor layout. Notice the 200 fold memory
saving from using the COO storage format.</p>
</div>
<section id="construction">
<h3>Construction<a class="headerlink" href="#construction" title="Permalink to this heading">¶</a></h3>
<p>A sparse COO tensor can be constructed by providing the two tensors of
indices and values, as well as the size of the sparse tensor (when it
cannot be inferred from the indices and values tensors) to a function
<a class="reference internal" href="generated/torch.sparse_coo_tensor.html#torch.sparse_coo_tensor" title="torch.sparse_coo_tensor"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.sparse_coo_tensor()</span></code></a>.</p>
<p>Suppose we want to define a sparse tensor with the entry 3 at location
(0, 2), entry 4 at location (1, 0), and entry 5 at location (1, 2).
Unspecified elements are assumed to have the same value, fill value,
which is zero by default. We would then write:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">i</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="go"> [2, 0, 2]]</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">s</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">i</span><span class="p">,</span> <span class="n">v</span><span class="p">,</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="gp">>>> </span><span class="n">s</span>
<span class="go">tensor(indices=tensor([[0, 1, 1],</span>
<span class="go"> [2, 0, 2]]),</span>
<span class="go"> values=tensor([3, 4, 5]),</span>
<span class="go"> size=(2, 3), nnz=3, layout=torch.sparse_coo)</span>
<span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">to_dense</span><span class="p">()</span>
<span class="go">tensor([[0, 0, 3],</span>
<span class="go"> [4, 0, 5]])</span>
</pre></div>
</div>
<p>Note that the input <code class="docutils literal notranslate"><span class="pre">i</span></code> is NOT a list of index tuples. If you want
to write your indices this way, you should transpose before passing them to
the sparse constructor:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">i</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</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="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span> <span class="p">]</span>
<span class="gp">>>> </span><span class="n">s</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="nb">list</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="o">*</span><span class="n">i</span><span class="p">)),</span> <span class="n">v</span><span class="p">,</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="gp">>>> </span><span class="c1"># Or another equivalent formulation to get s</span>
<span class="gp">>>> </span><span class="n">s</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">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">(</span><span class="n">i</span><span class="p">)</span><span class="o">.</span><span class="n">t</span><span class="p">(),</span> <span class="n">v</span><span class="p">,</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="gp">>>> </span><span class="n">torch</span><span class="o">.</span><span class="n">sparse_coo_tensor</span><span class="p">(</span><span class="n">i</span><span class="o">.</span><span class="n">t</span><span class="p">(),</span> <span class="n">v</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">Size</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="o">.</span><span class="n">to_dense</span><span class="p">()</span>
<span class="go">tensor([[0, 0, 3],</span>
<span class="go"> [4, 0, 5]])</span>
</pre></div>
</div>
<p>An empty sparse COO tensor can be constructed by specifying its size
only:</p>
<div class="doctest 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">sparse_coo_tensor</span><span class="p">(</span><span class="n">size</span><span class="o">=</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="go">tensor(indices=tensor([], size=(2, 0)),</span>
<span class="go"> values=tensor([], size=(0,)),</span>
<span class="go"> size=(2, 3), nnz=0, layout=torch.sparse_coo)</span>
</pre></div>
</div>
</section>
<section id="sparse-hybrid-coo-tensors">
<span id="sparse-hybrid-coo-docs"></span><h3>Sparse hybrid COO tensors<a class="headerlink" href="#sparse-hybrid-coo-tensors" title="Permalink to this heading">¶</a></h3>
<p>Pytorch implements an extension of sparse tensors with scalar values
to sparse tensors with (contiguous) tensor values. Such tensors are
called hybrid tensors.</p>
<p>PyTorch hybrid COO tensor extends the sparse COO tensor by allowing
the <code class="docutils literal notranslate"><span class="pre">values</span></code> tensor to be a multi-dimensional tensor so that we
have:</p>
<blockquote>
<div><ul class="simple">
<li><p>the indices of specified elements are collected in <code class="docutils literal notranslate"><span class="pre">indices</span></code>
tensor of size <code class="docutils literal notranslate"><span class="pre">(sparse_dims,</span> <span class="pre">nse)</span></code> and with element type
<code class="docutils literal notranslate"><span class="pre">torch.int64</span></code>,</p></li>
<li><p>the corresponding (tensor) values are collected in <code class="docutils literal notranslate"><span class="pre">values</span></code>
tensor of size <code class="docutils literal notranslate"><span class="pre">(nse,</span> <span class="pre">dense_dims)</span></code> and with an arbitrary integer
or floating point number element type.</p></li>
</ul>
</div></blockquote>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>We use (M + K)-dimensional tensor to denote a N-dimensional sparse
hybrid tensor, where M and K are the numbers of sparse and dense
dimensions, respectively, such that M + K == N holds.</p>
</div>
<p>Suppose we want to create a (2 + 1)-dimensional tensor with the entry
[3, 4] at location (0, 2), entry [5, 6] at location (1, 0), and entry
[7, 8] at location (1, 2). We would write</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">i</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="go"> [2, 0, 2]]</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">s</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">i</span><span class="p">,</span> <span class="n">v</span><span class="p">,</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="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">s</span>
<span class="go">tensor(indices=tensor([[0, 1, 1],</span>
<span class="go"> [2, 0, 2]]),</span>
<span class="go"> values=tensor([[3, 4],</span>
<span class="go"> [5, 6],</span>
<span class="go"> [7, 8]]),</span>
<span class="go"> size=(2, 3, 2), nnz=3, layout=torch.sparse_coo)</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">to_dense</span><span class="p">()</span>
<span class="go">tensor([[[0, 0],</span>
<span class="go"> [0, 0],</span>
<span class="go"> [3, 4]],</span>
<span class="go"> [[5, 6],</span>
<span class="go"> [0, 0],</span>
<span class="go"> [7, 8]]])</span>
</pre></div>
</div>
<p>In general, if <code class="docutils literal notranslate"><span class="pre">s</span></code> is a sparse COO tensor and <code class="docutils literal notranslate"><span class="pre">M</span> <span class="pre">=</span>
<span class="pre">s.sparse_dim()</span></code>, <code class="docutils literal notranslate"><span class="pre">K</span> <span class="pre">=</span> <span class="pre">s.dense_dim()</span></code>, then we have the following
invariants:</p>
<blockquote>
<div><ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">M</span> <span class="pre">+</span> <span class="pre">K</span> <span class="pre">==</span> <span class="pre">len(s.shape)</span> <span class="pre">==</span> <span class="pre">s.ndim</span></code> - dimensionality of a tensor
is the sum of the number of sparse and dense dimensions,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">s.indices().shape</span> <span class="pre">==</span> <span class="pre">(M,</span> <span class="pre">nse)</span></code> - sparse indices are stored
explicitly,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">s.values().shape</span> <span class="pre">==</span> <span class="pre">(nse,)</span> <span class="pre">+</span> <span class="pre">s.shape[M</span> <span class="pre">:</span> <span class="pre">M</span> <span class="pre">+</span> <span class="pre">K]</span></code> - the values
of a hybrid tensor are K-dimensional tensors,</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">s.values().layout</span> <span class="pre">==</span> <span class="pre">torch.strided</span></code> - values are stored as
strided tensors.</p></li>
</ul>
</div></blockquote>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Dense dimensions always follow sparse dimensions, that is, mixing
of dense and sparse dimensions is not supported.</p>
</div>
</section>
<section id="uncoalesced-sparse-coo-tensors">
<span id="sparse-uncoalesced-coo-docs"></span><h3>Uncoalesced sparse COO tensors<a class="headerlink" href="#uncoalesced-sparse-coo-tensors" title="Permalink to this heading">¶</a></h3>
<p>PyTorch sparse COO tensor format permits sparse <em>uncoalesced</em> tensors,
where there may be duplicate coordinates in the indices; in this case,
the interpretation is that the value at that index is the sum of all
duplicate value entries. For example, one can specify multiple values,
<code class="docutils literal notranslate"><span class="pre">3</span></code> and <code class="docutils literal notranslate"><span class="pre">4</span></code>, for the same index <code class="docutils literal notranslate"><span class="pre">1</span></code>, that leads to an 1-D
uncoalesced tensor:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">i</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">s</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">i</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,))</span>
<span class="gp">>>> </span><span class="n">s</span>
<span class="go">tensor(indices=tensor([[1, 1]]),</span>
<span class="go"> values=tensor( [3, 4]),</span>
<span class="go"> size=(3,), nnz=2, layout=torch.sparse_coo)</span>
</pre></div>
</div>
<p>while the coalescing process will accumulate the multi-valued elements
into a single value using summation:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">coalesce</span><span class="p">()</span>
<span class="go">tensor(indices=tensor([[1]]),</span>
<span class="go"> values=tensor([7]),</span>
<span class="go"> size=(3,), nnz=1, layout=torch.sparse_coo)</span>
</pre></div>
</div>
<p>In general, the output of <a class="reference internal" href="generated/torch.Tensor.coalesce.html#torch.Tensor.coalesce" title="torch.Tensor.coalesce"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.coalesce()</span></code></a> method is a
sparse tensor with the following properties:</p>
<ul class="simple">
<li><p>the indices of specified tensor elements are unique,</p></li>
<li><p>the indices are sorted in lexicographical order,</p></li>
<li><p><a class="reference internal" href="generated/torch.Tensor.is_coalesced.html#torch.Tensor.is_coalesced" title="torch.Tensor.is_coalesced"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.is_coalesced()</span></code></a> returns <code class="docutils literal notranslate"><span class="pre">True</span></code>.</p></li>
</ul>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>For the most part, you shouldn’t have to care whether or not a
sparse tensor is coalesced or not, as most operations will work
identically given a sparse coalesced or uncoalesced tensor.</p>
<p>However, some operations can be implemented more efficiently on
uncoalesced tensors, and some on coalesced tensors.</p>
<p>For instance, addition of sparse COO tensors is implemented by
simply concatenating the indices and values tensors:</p>
<div class="doctest 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">sparse_coo_tensor</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">(</span><span class="mi">2</span><span class="p">,))</span>
<span class="gp">>>> </span><span class="n">b</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="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">],</span> <span class="p">(</span><span class="mi">2</span><span class="p">,))</span>
<span class="gp">>>> </span><span class="n">a</span> <span class="o">+</span> <span class="n">b</span>
<span class="go">tensor(indices=tensor([[0, 0, 1, 1]]),</span>
<span class="go"> values=tensor([7, 8, 5, 6]),</span>
<span class="go"> size=(2,), nnz=4, layout=torch.sparse_coo)</span>
</pre></div>
</div>
<p>If you repeatedly perform an operation that can produce duplicate
entries (e.g., <a class="reference internal" href="generated/torch.Tensor.add.html#torch.Tensor.add" title="torch.Tensor.add"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.Tensor.add()</span></code></a>), you should occasionally
coalesce your sparse tensors to prevent them from growing too large.</p>
<p>On the other hand, the lexicographical ordering of indices can be
advantageous for implementing algorithms that involve many element
selection operations, such as slicing or matrix products.</p>
</div>
</section>
<section id="working-with-sparse-coo-tensors">
<h3>Working with sparse COO tensors<a class="headerlink" href="#working-with-sparse-coo-tensors" title="Permalink to this heading">¶</a></h3>
<p>Let’s consider the following example:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">i</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="go"> [2, 0, 2]]</span>
<span class="gp">>>> </span><span class="n">v</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">],</span> <span class="p">[</span><span class="mi">5</span><span class="p">,</span> <span class="mi">6</span><span class="p">],</span> <span class="p">[</span><span class="mi">7</span><span class="p">,</span> <span class="mi">8</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">s</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">i</span><span class="p">,</span> <span class="n">v</span><span class="p">,</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="mi">2</span><span class="p">))</span>
</pre></div>
</div>
<p>As mentioned above, a sparse COO tensor is a <a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><code class="xref py py-class docutils literal notranslate"><span class="pre">torch.Tensor</span></code></a>
instance and to distinguish it from the <cite>Tensor</cite> instances that use
some other layout, on can use <a class="reference internal" href="generated/torch.Tensor.is_sparse.html#torch.Tensor.is_sparse" title="torch.Tensor.is_sparse"><code class="xref py py-attr docutils literal notranslate"><span class="pre">torch.Tensor.is_sparse</span></code></a> or
<code class="xref py py-attr docutils literal notranslate"><span class="pre">torch.Tensor.layout</span></code> properties:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="nb">isinstance</span><span class="p">(</span><span class="n">s</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="go">True</span>
<span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">is_sparse</span>
<span class="go">True</span>
<span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">layout</span> <span class="o">==</span> <span class="n">torch</span><span class="o">.</span><span class="n">sparse_coo</span>
<span class="go">True</span>
</pre></div>
</div>
<p>The number of sparse and dense dimensions can be acquired using
methods <a class="reference internal" href="generated/torch.Tensor.sparse_dim.html#torch.Tensor.sparse_dim" title="torch.Tensor.sparse_dim"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.sparse_dim()</span></code></a> and
<a class="reference internal" href="generated/torch.Tensor.dense_dim.html#torch.Tensor.dense_dim" title="torch.Tensor.dense_dim"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.dense_dim()</span></code></a>, respectively. For instance:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">sparse_dim</span><span class="p">(),</span> <span class="n">s</span><span class="o">.</span><span class="n">dense_dim</span><span class="p">()</span>
<span class="go">(2, 1)</span>
</pre></div>
</div>
<p>If <code class="docutils literal notranslate"><span class="pre">s</span></code> is a sparse COO tensor then its COO format data can be
acquired using methods <a class="reference internal" href="generated/torch.Tensor.indices.html#torch.Tensor.indices" title="torch.Tensor.indices"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.indices()</span></code></a> and
<a class="reference internal" href="generated/torch.Tensor.values.html#torch.Tensor.values" title="torch.Tensor.values"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.values()</span></code></a>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Currently, one can acquire the COO format data only when the tensor
instance is coalesced:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">indices</span><span class="p">()</span>
<span class="go">RuntimeError: Cannot get indices on an uncoalesced tensor, please call .coalesce() first</span>
</pre></div>
</div>
<p>For acquiring the COO format data of an uncoalesced tensor, use
<code class="xref py py-func docutils literal notranslate"><span class="pre">torch.Tensor._values()</span></code> and <code class="xref py py-func docutils literal notranslate"><span class="pre">torch.Tensor._indices()</span></code>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">_indices</span><span class="p">()</span>
<span class="go">tensor([[0, 1, 1],</span>
<span class="go"> [2, 0, 2]])</span>
</pre></div>
</div>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Calling <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor._values()</span></code> will return a <em>detached</em> tensor.
To track gradients, <code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.coalesce().values()</span></code> must be
used instead.</p>
</div>
</div>
<p>Constructing a new sparse COO tensor results a tensor that is not
coalesced:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span><span class="o">.</span><span class="n">is_coalesced</span><span class="p">()</span>
<span class="go">False</span>
</pre></div>
</div>
<p>but one can construct a coalesced copy of a sparse COO tensor using
the <a class="reference internal" href="generated/torch.Tensor.coalesce.html#torch.Tensor.coalesce" title="torch.Tensor.coalesce"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.coalesce()</span></code></a> method:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s2</span> <span class="o">=</span> <span class="n">s</span><span class="o">.</span><span class="n">coalesce</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">s2</span><span class="o">.</span><span class="n">indices</span><span class="p">()</span>
<span class="go">tensor([[0, 1, 1],</span>
<span class="go"> [2, 0, 2]])</span>
</pre></div>
</div>
<p>When working with uncoalesced sparse COO tensors, one must take into
an account the additive nature of uncoalesced data: the values of the
same indices are the terms of a sum that evaluation gives the value of
the corresponding tensor element. For example, the scalar
multiplication on a sparse uncoalesced tensor could be implemented by
multiplying all the uncoalesced values with the scalar because <code class="docutils literal notranslate"><span class="pre">c</span> <span class="pre">*</span>
<span class="pre">(a</span> <span class="pre">+</span> <span class="pre">b)</span> <span class="pre">==</span> <span class="pre">c</span> <span class="pre">*</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">c</span> <span class="pre">*</span> <span class="pre">b</span></code> holds. However, any nonlinear operation,
say, a square root, cannot be implemented by applying the operation to
uncoalesced data because <code class="docutils literal notranslate"><span class="pre">sqrt(a</span> <span class="pre">+</span> <span class="pre">b)</span> <span class="pre">==</span> <span class="pre">sqrt(a)</span> <span class="pre">+</span> <span class="pre">sqrt(b)</span></code> does not
hold in general.</p>
<p>Slicing (with positive step) of a sparse COO tensor is supported only
for dense dimensions. Indexing is supported for both sparse and dense
dimensions:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span>
<span class="go">tensor(indices=tensor([[0, 2]]),</span>
<span class="go"> values=tensor([[5, 6],</span>
<span class="go"> [7, 8]]),</span>
<span class="go"> size=(3, 2), nnz=2, layout=torch.sparse_coo)</span>
<span class="gp">>>> </span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="go">tensor(6)</span>
<span class="gp">>>> </span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">:]</span>
<span class="go">tensor([6])</span>
</pre></div>
</div>
<p>In PyTorch, the fill value of a sparse tensor cannot be specified
explicitly and is assumed to be zero in general. However, there exists
operations that may interpret the fill value differently. For
instance, <a class="reference internal" href="generated/torch.sparse.softmax.html#torch.sparse.softmax" title="torch.sparse.softmax"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.sparse.softmax()</span></code></a> computes the softmax with the
assumption that the fill value is negative infinity.</p>
</section>
</section>
<section id="sparse-compressed-tensors">
<span id="sparse-compressed-docs"></span><h2>Sparse Compressed Tensors<a class="headerlink" href="#sparse-compressed-tensors" title="Permalink to this heading">¶</a></h2>
<p>Sparse Compressed Tensors represents a class of sparse tensors that
have a common feature of compressing the indices of a certain dimension
using an encoding that enables certain optimizations on linear algebra
kernels of sparse compressed tensors. This encoding is based on the
<a class="reference external" href="https://en.wikipedia.org/wiki/Sparse_matrix#Compressed_sparse_row_(CSR,_CRS_or_Yale_format)">Compressed Sparse Row (CSR)</a> format that PyTorch sparse compressed
tensors extend with the support of sparse tensor batches, allowing
multi-dimensional tensor values, and storing sparse tensor values in
dense blocks.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>We use (B + M + K)-dimensional tensor to denote a N-dimensional
sparse compressed hybrid tensor, where B, M, and K are the numbers
of batch, sparse, and dense dimensions, respectively, such that
<code class="docutils literal notranslate"><span class="pre">B</span> <span class="pre">+</span> <span class="pre">M</span> <span class="pre">+</span> <span class="pre">K</span> <span class="pre">==</span> <span class="pre">N</span></code> holds. The number of sparse dimensions for
sparse compressed tensors is always two, <code class="docutils literal notranslate"><span class="pre">M</span> <span class="pre">==</span> <span class="pre">2</span></code>.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>We say that an indices tensor <code class="docutils literal notranslate"><span class="pre">compressed_indices</span></code> uses CSR
encoding if the following invariants are satisfied:</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">compressed_indices</span></code> is a contiguous strided 32 or 64 bit
integer tensor</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">compressed_indices</span></code> shape is <code class="docutils literal notranslate"><span class="pre">(*batchsize,</span>
<span class="pre">compressed_dim_size</span> <span class="pre">+</span> <span class="pre">1)</span></code> where <code class="docutils literal notranslate"><span class="pre">compressed_dim_size</span></code> is the
number of compressed dimensions (e.g. rows or columns)</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">compressed_indices[...,</span> <span class="pre">0]</span> <span class="pre">==</span> <span class="pre">0</span></code> where <code class="docutils literal notranslate"><span class="pre">...</span></code> denotes batch
indices</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">compressed_indices[...,</span> <span class="pre">compressed_dim_size]</span> <span class="pre">==</span> <span class="pre">nse</span></code> where
<code class="docutils literal notranslate"><span class="pre">nse</span></code> is the number of specified elements</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre"><=</span> <span class="pre">compressed_indices[...,</span> <span class="pre">i]</span> <span class="pre">-</span> <span class="pre">compressed_indices[...,</span> <span class="pre">i</span> <span class="pre">-</span>
<span class="pre">1]</span> <span class="pre"><=</span> <span class="pre">plain_dim_size</span></code> for <code class="docutils literal notranslate"><span class="pre">i=1,</span> <span class="pre">...,</span> <span class="pre">compressed_dim_size</span></code>,
where <code class="docutils literal notranslate"><span class="pre">plain_dim_size</span></code> is the number of plain dimensions
(orthogonal to compressed dimensions, e.g. columns or rows).</p></li>
</ul>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The generalization of sparse compressed layouts to N-dimensional
tensors can lead to some confusion regarding the count of specified
elements. When a sparse compressed tensor contains batch dimensions
the number of specified elements will correspond to the number of such
elements per-batch. When a sparse compressed tensor has dense dimensions
the element considered is now the K-dimensional array. Also for block
sparse compressed layouts the 2-D block is considered as the element
being specified. Take as an example a 3-dimensional block sparse
tensor, with one batch dimension of length <code class="docutils literal notranslate"><span class="pre">b</span></code>, and a block
shape of <code class="docutils literal notranslate"><span class="pre">p,</span> <span class="pre">q</span></code>. If this tensor has <code class="docutils literal notranslate"><span class="pre">n</span></code> specified elements, then
in fact we have <code class="docutils literal notranslate"><span class="pre">n</span></code> blocks specified per batch. This tensor would
have <code class="docutils literal notranslate"><span class="pre">values</span></code> with shape <code class="docutils literal notranslate"><span class="pre">(b,</span> <span class="pre">n,</span> <span class="pre">p,</span> <span class="pre">q)</span></code>. This interpretation of the
number of specified elements comes from all sparse compressed layouts
being derived from the compression of a 2-dimensional matrix. Batch
dimensions are treated as stacking of sparse matrices, dense dimensions
change the meaning of the element from a simple scalar value to an
array with its own dimensions.</p>
</div>
<section id="sparse-csr-tensor">
<span id="sparse-csr-docs"></span><h3>Sparse CSR Tensor<a class="headerlink" href="#sparse-csr-tensor" title="Permalink to this heading">¶</a></h3>
<p>The primary advantage of the CSR format over the COO format is better
use of storage and much faster computation operations such as sparse
matrix-vector multiplication using MKL and MAGMA backends.</p>
<p>In the simplest case, a (0 + 2 + 0)-dimensional sparse CSR tensor
consists of three 1-D tensors: <code class="docutils literal notranslate"><span class="pre">crow_indices</span></code>, <code class="docutils literal notranslate"><span class="pre">col_indices</span></code> and
<code class="docutils literal notranslate"><span class="pre">values</span></code>:</p>
<blockquote>
<div><ul class="simple">
<li><p>The <code class="docutils literal notranslate"><span class="pre">crow_indices</span></code> tensor consists of compressed row
indices. This is a 1-D tensor of size <code class="docutils literal notranslate"><span class="pre">nrows</span> <span class="pre">+</span> <span class="pre">1</span></code> (the number of
rows plus 1). The last element of <code class="docutils literal notranslate"><span class="pre">crow_indices</span></code> is the number
of specified elements, <code class="docutils literal notranslate"><span class="pre">nse</span></code>. This tensor encodes the index in
<code class="docutils literal notranslate"><span class="pre">values</span></code> and <code class="docutils literal notranslate"><span class="pre">col_indices</span></code> depending on where the given row
starts. Each successive number in the tensor subtracted by the
number before it denotes the number of elements in a given row.</p></li>
<li><p>The <code class="docutils literal notranslate"><span class="pre">col_indices</span></code> tensor contains the column indices of each
element. This is a 1-D tensor of size <code class="docutils literal notranslate"><span class="pre">nse</span></code>.</p></li>
<li><p>The <code class="docutils literal notranslate"><span class="pre">values</span></code> tensor contains the values of the CSR tensor
elements. This is a 1-D tensor of size <code class="docutils literal notranslate"><span class="pre">nse</span></code>.</p></li>
</ul>
</div></blockquote>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The index tensors <code class="docutils literal notranslate"><span class="pre">crow_indices</span></code> and <code class="docutils literal notranslate"><span class="pre">col_indices</span></code> should have
element type either <code class="docutils literal notranslate"><span class="pre">torch.int64</span></code> (default) or
<code class="docutils literal notranslate"><span class="pre">torch.int32</span></code>. If you want to use MKL-enabled matrix operations,
use <code class="docutils literal notranslate"><span class="pre">torch.int32</span></code>. This is as a result of the default linking of
pytorch being with MKL LP64, which uses 32 bit integer indexing.</p>
</div>
<p>In the general case, the (B + 2 + K)-dimensional sparse CSR tensor
consists of two (B + 1)-dimensional index tensors <code class="docutils literal notranslate"><span class="pre">crow_indices</span></code> and
<code class="docutils literal notranslate"><span class="pre">col_indices</span></code>, and of (1 + K)-dimensional <code class="docutils literal notranslate"><span class="pre">values</span></code> tensor such
that</p>
<blockquote>
<div><ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">crow_indices.shape</span> <span class="pre">==</span> <span class="pre">(*batchsize,</span> <span class="pre">nrows</span> <span class="pre">+</span> <span class="pre">1)</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">col_indices.shape</span> <span class="pre">==</span> <span class="pre">(*batchsize,</span> <span class="pre">nse)</span></code></p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">values.shape</span> <span class="pre">==</span> <span class="pre">(nse,</span> <span class="pre">*densesize)</span></code></p></li>
</ul>
</div></blockquote>
<p>while the shape of the sparse CSR tensor is <code class="docutils literal notranslate"><span class="pre">(*batchsize,</span> <span class="pre">nrows,</span>
<span class="pre">ncols,</span> <span class="pre">*densesize)</span></code> where <code class="docutils literal notranslate"><span class="pre">len(batchsize)</span> <span class="pre">==</span> <span class="pre">B</span></code> and
<code class="docutils literal notranslate"><span class="pre">len(densesize)</span> <span class="pre">==</span> <span class="pre">K</span></code>.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The batches of sparse CSR tensors are dependent: the number of
specified elements in all batches must be the same. This somewhat
artificial constraint allows efficient storage of the indices of
different CSR batches.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>The number of sparse and dense dimensions can be acquired using
<a class="reference internal" href="generated/torch.Tensor.sparse_dim.html#torch.Tensor.sparse_dim" title="torch.Tensor.sparse_dim"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.sparse_dim()</span></code></a> and <a class="reference internal" href="generated/torch.Tensor.dense_dim.html#torch.Tensor.dense_dim" title="torch.Tensor.dense_dim"><code class="xref py py-meth docutils literal notranslate"><span class="pre">torch.Tensor.dense_dim()</span></code></a>
methods. The batch dimensions can be computed from the tensor
shape: <code class="docutils literal notranslate"><span class="pre">batchsize</span> <span class="pre">=</span> <span class="pre">tensor.shape[:-tensor.sparse_dim()</span> <span class="pre">-</span>
<span class="pre">tensor.dense_dim()]</span></code>.</p>
</div>