# torch.sparse¶

Warning

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.

## Why and when to use sparsity¶

By default PyTorch stores torch.Tensor stores elements contiguously physical memory. This leads to efficient implementations of various array processing algorithms that require fast access to elements.

Now, some users might decide to represent data such as graph adjacency matrices, pruned weights or points clouds by Tensors whose elements are mostly zero valued. We recognize these are important applications and aim to provide performance optimizations for these use cases via sparse storage formats.

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 specified in contrast to unspecified, compressed elements.

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.

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.

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.

We make it easy to try different sparsity layouts, and convert between them, without being opinionated on what’s best for your particular application.

## Functionality overview¶

We want it to be straightforward to construct a sparse Tensor from a given dense Tensor by providing conversion routines for each layout.

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.

>>> a = torch.tensor([[0, 2.], [3, 0]])
>>> a.to_sparse()
tensor(indices=tensor([[0, 1],
[1, 0]]),
values=tensor([2., 3.]),
size=(2, 2), nnz=2, layout=torch.sparse_coo)


PyTorch currently supports COO, CSR, CSC, BSR, and BSC. Please see the references for more details.

Note that we provide slight generalizations of these formats.

Batching: Devices such as GPUs require batching for optimal performance and thus we support batch dimensions.

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.

>>> t = torch.tensor([[[1., 0], [2., 3.]], [[4., 0], [5., 6.]]])
>>> t.dim()
3
>>> t.to_sparse_csr()
tensor(crow_indices=tensor([[0, 1, 3],
[0, 1, 3]]),
col_indices=tensor([[0, 0, 1],
[0, 0, 1]]),
values=tensor([[1., 2., 3.],
[4., 5., 6.]]), size=(2, 2, 2), nnz=3,
layout=torch.sparse_csr)


Dense dimensions: On the other hand, some data such as Graph embeddings might be better viewed as sparse collections of vectors instead of scalars.

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 entirely zero can be emitted and the presence of any non-zero valued elements cause the entire row to be stored.

>>> t = torch.tensor([[[0., 0], [1., 2.]], [[0., 0], [3., 4.]]])
>>> t.to_sparse(sparse_dim=2)
tensor(indices=tensor([[0, 1],
[1, 1]]),
values=tensor([[1., 2.],
[3., 4.]]),
size=(2, 2, 2), nnz=2, layout=torch.sparse_coo)


## Operator overview¶

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.

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 operator documentation for a list.

>>> b = torch.tensor([[0, 0, 1, 2, 3, 0], [4, 5, 0, 6, 0, 0]])
>>> b_s = b.to_sparse_csr()
>>> b_s.cos()
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
RuntimeError: unsupported tensor layout: SparseCsr
>>> b_s.sin()
tensor(crow_indices=tensor([0, 3, 6]),
col_indices=tensor([2, 3, 4, 0, 1, 3]),
values=tensor([ 0.8415,  0.9093,  0.1411, -0.7568, -0.9589, -0.2794]),
size=(2, 6), nnz=6, layout=torch.sparse_csr)


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.

>>> b_s.to_dense().cos()
tensor([[ 1.0000, -0.4161],
[-0.9900,  1.0000]])


We are aware that some users want to ignore compressed zeros for operations such as cos 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.

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.

>>> a + b.to_sparse()
tensor([[0., 3.],
[3., 0.]])


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.

## Sparse COO tensors¶

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,

• the indices of specified elements are collected in indices tensor of size (ndim, nse) and with element type torch.int64,

• the corresponding values are collected in values tensor of size (nse,) and with an arbitrary integer or floating point number element type,

where ndim is the dimensionality of the tensor and nse is the number of specified elements.

Note

The memory consumption of a sparse COO tensor is at least (ndim * 8 + <size of element type in bytes>) * nse bytes (plus a constant overhead from storing other tensor data).

The memory consumption of a strided tensor is at least product(<tensor shape>) * <size of element type in bytes>.

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 (2 * 8 + 4) * 100 000 = 2 000 000 bytes when using COO tensor layout and 10 000 * 10 000 * 4 = 400 000 000 bytes when using the default strided tensor layout. Notice the 200 fold memory saving from using the COO storage format.

### Construction¶

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 torch.sparse_coo_tensor().

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:

>>> i = [[0, 1, 1],
[2, 0, 2]]
>>> v =  [3, 4, 5]
>>> s = torch.sparse_coo_tensor(i, v, (2, 3))
>>> s
tensor(indices=tensor([[0, 1, 1],
[2, 0, 2]]),
values=tensor([3, 4, 5]),
size=(2, 3), nnz=3, layout=torch.sparse_coo)
>>> s.to_dense()
tensor([[0, 0, 3],
[4, 0, 5]])


Note that the input i 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:

>>> i = [[0, 2], [1, 0], [1, 2]]
>>> v =  [3,      4,      5    ]
>>> s = torch.sparse_coo_tensor(list(zip(*i)), v, (2, 3))
>>> # Or another equivalent formulation to get s
>>> s = torch.sparse_coo_tensor(torch.tensor(i).t(), v, (2, 3))
>>> torch.sparse_coo_tensor(i.t(), v, torch.Size([2,3])).to_dense()
tensor([[0, 0, 3],
[4, 0, 5]])


An empty sparse COO tensor can be constructed by specifying its size only:

>>> torch.sparse_coo_tensor(size=(2, 3))
tensor(indices=tensor([], size=(2, 0)),
values=tensor([], size=(0,)),
size=(2, 3), nnz=0, layout=torch.sparse_coo)


### Sparse hybrid COO tensors¶

Pytorch implements an extension of sparse tensors with scalar values to sparse tensors with (contiguous) tensor values. Such tensors are called hybrid tensors.

PyTorch hybrid COO tensor extends the sparse COO tensor by allowing the values tensor to be a multi-dimensional tensor so that we have:

• the indices of specified elements are collected in indices tensor of size (sparse_dims, nse) and with element type torch.int64,

• the corresponding (tensor) values are collected in values tensor of size (nse, dense_dims) and with an arbitrary integer or floating point number element type.

Note

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.

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

>>> i = [[0, 1, 1],
[2, 0, 2]]
>>> v =  [[3, 4], [5, 6], [7, 8]]
>>> s = torch.sparse_coo_tensor(i, v, (2, 3, 2))
>>> s
tensor(indices=tensor([[0, 1, 1],
[2, 0, 2]]),
values=tensor([[3, 4],
[5, 6],
[7, 8]]),
size=(2, 3, 2), nnz=3, layout=torch.sparse_coo)

>>> s.to_dense()
tensor([[[0, 0],
[0, 0],
[3, 4]],
[[5, 6],
[0, 0],
[7, 8]]])


In general, if s is a sparse COO tensor and M = s.sparse_dim(), K = s.dense_dim(), then we have the following invariants:

• M + K == len(s.shape) == s.ndim - dimensionality of a tensor is the sum of the number of sparse and dense dimensions,

• s.indices().shape == (M, nse) - sparse indices are stored explicitly,

• s.values().shape == (nse,) + s.shape[M : M + K] - the values of a hybrid tensor are K-dimensional tensors,

• s.values().layout == torch.strided - values are stored as strided tensors.

Note

Dense dimensions always follow sparse dimensions, that is, mixing of dense and sparse dimensions is not supported.

### Uncoalesced sparse COO tensors¶

PyTorch sparse COO tensor format permits sparse uncoalesced 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, 3 and 4, for the same index 1, that leads to an 1-D uncoalesced tensor:

>>> i = [[1, 1]]
>>> v =  [3, 4]
>>> s=torch.sparse_coo_tensor(i, v, (3,))
>>> s
tensor(indices=tensor([[1, 1]]),
values=tensor(  [3, 4]),
size=(3,), nnz=2, layout=torch.sparse_coo)


while the coalescing process will accumulate the multi-valued elements into a single value using summation:

>>> s.coalesce()
tensor(indices=tensor([[1]]),
values=tensor([7]),
size=(3,), nnz=1, layout=torch.sparse_coo)


In general, the output of torch.Tensor.coalesce() method is a sparse tensor with the following properties:

Note

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.

However, some operations can be implemented more efficiently on uncoalesced tensors, and some on coalesced tensors.

For instance, addition of sparse COO tensors is implemented by simply concatenating the indices and values tensors:

>>> a = torch.sparse_coo_tensor([[1, 1]], [5, 6], (2,))
>>> b = torch.sparse_coo_tensor([[0, 0]], [7, 8], (2,))
>>> a + b
tensor(indices=tensor([[0, 0, 1, 1]]),
values=tensor([7, 8, 5, 6]),
size=(2,), nnz=4, layout=torch.sparse_coo)


If you repeatedly perform an operation that can produce duplicate entries (e.g., torch.Tensor.add()), you should occasionally coalesce your sparse tensors to prevent them from growing too large.

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.

### Working with sparse COO tensors¶

Let’s consider the following example:

>>> i = [[0, 1, 1],
[2, 0, 2]]
>>> v =  [[3, 4], [5, 6], [7, 8]]
>>> s = torch.sparse_coo_tensor(i, v, (2, 3, 2))


As mentioned above, a sparse COO tensor is a torch.Tensor instance and to distinguish it from the Tensor instances that use some other layout, on can use torch.Tensor.is_sparse or torch.Tensor.layout properties:

>>> isinstance(s, torch.Tensor)
True
>>> s.is_sparse
True
>>> s.layout == torch.sparse_coo
True


The number of sparse and dense dimensions can be acquired using methods torch.Tensor.sparse_dim() and torch.Tensor.dense_dim(), respectively. For instance:

>>> s.sparse_dim(), s.dense_dim()
(2, 1)


If s is a sparse COO tensor then its COO format data can be acquired using methods torch.Tensor.indices() and torch.Tensor.values().

Note

Currently, one can acquire the COO format data only when the tensor instance is coalesced:

>>> s.indices()
RuntimeError: Cannot get indices on an uncoalesced tensor, please call .coalesce() first


For acquiring the COO format data of an uncoalesced tensor, use torch.Tensor._values() and torch.Tensor._indices():

>>> s._indices()
tensor([[0, 1, 1],
[2, 0, 2]])


Warning

Calling torch.Tensor._values() will return a detached tensor. To track gradients, torch.Tensor.coalesce().values() must be used instead.

Constructing a new sparse COO tensor results a tensor that is not coalesced:

>>> s.is_coalesced()
False


but one can construct a coalesced copy of a sparse COO tensor using the torch.Tensor.coalesce() method:

>>> s2 = s.coalesce()
>>> s2.indices()
tensor([[0, 1, 1],
[2, 0, 2]])


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 c * (a + b) == c * a + c * b holds. However, any nonlinear operation, say, a square root, cannot be implemented by applying the operation to uncoalesced data because sqrt(a + b) == sqrt(a) + sqrt(b) does not hold in general.

Slicing (with positive step) of a sparse COO tensor is supported only for dense dimensions. Indexing is supported for both sparse and dense dimensions:

>>> s[1]
tensor(indices=tensor([[0, 2]]),
values=tensor([[5, 6],
[7, 8]]),
size=(3, 2), nnz=2, layout=torch.sparse_coo)
>>> s[1, 0, 1]
tensor(6)
>>> s[1, 0, 1:]
tensor([6])


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, torch.sparse.softmax() computes the softmax with the assumption that the fill value is negative infinity.

## Sparse Compressed Tensors¶

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 Compressed Sparse Row (CSR) 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.

Note

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 B + M + K == N holds. The number of sparse dimensions for sparse compressed tensors is always two, M == 2.

Note

We say that an indices tensor compressed_indices uses CSR encoding if the following invariants are satisfied:

• compressed_indices is a contiguous strided 32 or 64 bit integer tensor

• compressed_indices shape is (*batchsize, compressed_dim_size + 1) where compressed_dim_size is the number of compressed dimensions (e.g. rows or columns)

• compressed_indices[..., 0] == 0 where ... denotes batch indices

• compressed_indices[..., compressed_dim_size] == nse where nse is the number of specified elements

• 0 <= compressed_indices[..., i] - compressed_indices[..., i - 1] <= plain_dim_size for i=1, ..., compressed_dim_size, where plain_dim_size is the number of plain dimensions (orthogonal to compressed dimensions, e.g. columns or rows).

Note

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 b, and a block shape of p, q. If this tensor has n specified elements, then in fact we have n blocks specified per batch. This tensor would have values with shape (b, n, p, q). 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.

### Sparse CSR Tensor¶

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.

In the simplest case, a (0 + 2 + 0)-dimensional sparse CSR tensor consists of three 1-D tensors: crow_indices, col_indices and values:

• The crow_indices tensor consists of compressed row indices. This is a 1-D tensor of size nrows + 1 (the number of rows plus 1). The last element of crow_indices is the number of specified elements, nse. This tensor encodes the index in values and col_indices 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.

• The col_indices tensor contains the column indices of each element. This is a 1-D tensor of size nse.

• The values tensor contains the values of the CSR tensor elements. This is a 1-D tensor of size nse.

Note

The index tensors crow_indices and col_indices should have element type either torch.int64 (default) or torch.int32. If you want to use MKL-enabled matrix operations, use torch.int32. This is as a result of the default linking of pytorch being with MKL LP64, which uses 32 bit integer indexing.

In the general case, the (B + 2 + K)-dimensional sparse CSR tensor consists of two (B + 1)-dimensional index tensors crow_indices and col_indices, and of (1 + K)-dimensional values tensor such that

• crow_indices.shape == (*batchsize, nrows + 1)

• col_indices.shape == (*batchsize, nse)

• values.shape == (nse, *densesize)

while the shape of the sparse CSR tensor is (*batchsize, nrows, ncols, *densesize) where len(batchsize) == B and len(densesize) == K.

Note

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.

Note

The number of sparse and dense dimensions can be acquired using torch.Tensor.sparse_dim() and torch.Tensor.dense_dim() methods. The batch dimensions can be computed from the tensor shape: batchsize = tensor.shape[:-tensor.sparse_dim() - tensor.dense_dim()].

Note

The memory consumption of a sparse CSR tensor is at least (nrows * 8 + (8 + <size of element type in bytes> * prod(densesize)) * nse) * prod(batchsize) bytes (plus a constant overhead from storing other tensor data).

With the same example data of the note in sparse COO format introduction, the memory consumption of a 10 000 x 10 000 tensor with 100 000 non-zero 32-bit floating point numbers is at least (10000 * 8 + (8 + 4 * 1) * 100 000) * 1 = 1 280 000 bytes when using CSR tensor layout. Notice the 1.6 and 310 fold savings from using CSR storage format compared to using the COO and strided formats, respectively.

#### Construction of CSR tensors¶

Sparse CSR tensors can be directly constructed by using the torch.sparse_csr_tensor() function. The user must supply the row and column indices and values tensors separately where the row indices must be specified using the CSR compression encoding. The size argument is optional and will be deduced from the crow_indices and col_indices if it is not present.

>>> crow_indices = torch.tensor([0, 2, 4])
>>> col_indices = torch.tensor([0, 1, 0, 1])
>>> values = torch.tensor([1, 2, 3, 4])
>>> csr = torch.sparse_csr_tensor(crow_indices, col_indices, values, dtype=torch.float64)
>>> csr
tensor(crow_indices=tensor([0, 2, 4]),
col_indices=tensor([0, 1, 0, 1]),
values=tensor([1., 2., 3., 4.]), size=(2, 2), nnz=4,
dtype=torch.float64)
>>> csr.to_dense()
tensor([[1., 2.],
[3., 4.]], dtype=torch.float64)


Note

The values of sparse dimensions in deduced size is computed from the size of crow_indices and the maximal index value in col_indices. If the number of columns needs to be larger than in the deduced size then the size argument must be specified explicitly.

The simplest way of constructing a 2-D sparse CSR tensor from a strided or sparse COO tensor is to use torch.Tensor.to_sparse_csr() method. Any zeros in the (strided) tensor will be interpreted as missing values in the sparse tensor:

>>> a = torch.tensor([[0, 0, 1, 0], [1, 2, 0, 0], [0, 0, 0, 0]], dtype=torch.float64)
>>> sp = a.to_sparse_csr()
>>> sp
tensor(crow_indices=tensor([0, 1, 3, 3]),
col_indices=tensor([2, 0, 1]),
values=tensor([1., 1., 2.]), size=(3, 4), nnz=3, dtype=torch.float64)


#### CSR Tensor Operations¶

The sparse matrix-vector multiplication can be performed with the tensor.matmul() method. This is currently the only math operation supported on CSR tensors.

>>> vec = torch.randn(4, 1, dtype=torch.float64)
>>> sp.matmul(vec)
tensor([[0.9078],
[1.3180],
[0.0000]], dtype=torch.float64)


### Sparse CSC Tensor¶

The sparse CSC (Compressed Sparse Column) tensor format implements the CSC format for storage of 2 dimensional tensors with an extension to supporting batches of sparse CSC tensors and values being multi-dimensional tensors.

Note

Sparse CSC tensor is essentially a transpose of the sparse CSR tensor when the transposition is about swapping the sparse dimensions.

Similarly to sparse CSR tensors, a sparse CSC tensor consists of three tensors: ccol_indices, row_indices and values:

• The ccol_indices tensor consists of compressed column indices. This is a (B + 1)-D tensor of shape (*batchsize, ncols + 1). The last element is the number of specified elements, nse. This tensor encodes the index in values and row_indices depending on where the given column starts. Each successive number in the tensor subtracted by the number before it denotes the number of elements in a given column.

• The row_indices tensor contains the row indices of each element. This is a (B + 1)-D tensor of shape (*batchsize, nse).

• The values tensor contains the values of the CSC tensor elements. This is a (1 + K)-D tensor of shape (nse, *densesize).

#### Construction of CSC tensors¶

Sparse CSC tensors can be directly constructed by using the torch.sparse_csc_tensor() function. The user must supply the row and column indices and values tensors separately where the column indices must be specified using the CSR compression encoding. The size argument is optional and will be deduced from the row_indices and ccol_indices tensors if it is not present.

>>> ccol_indices = torch.tensor([0, 2, 4])
>>> row_indices = torch.tensor([0, 1, 0, 1])
>>> values = torch.tensor([1, 2, 3, 4])
>>> csc = torch.sparse_csc_tensor(ccol_indices, row_indices, values, dtype=torch.float64)
>>> csc
tensor(ccol_indices=tensor([0, 2, 4]),
row_indices=tensor([0, 1, 0, 1]),
values=tensor([1., 2., 3., 4.]), size=(2, 2), nnz=4,
dtype=torch.float64, layout=torch.sparse_csc)
>>> csc.to_dense()
tensor([[1., 3.],
[2., 4.]], dtype=torch.float64)


Note

The sparse CSC tensor constructor function has the compressed column indices argument before the row indices argument.

The (0 + 2 + 0)-dimensional sparse CSC tensors can be constructed from any two-dimensional tensor using torch.Tensor.to_sparse_csc() method. Any zeros in the (strided) tensor will be interpreted as missing values in the sparse tensor:

>>> a = torch.tensor([[0, 0, 1, 0], [1, 2, 0, 0], [0, 0, 0, 0]], dtype=torch.float64)
>>> sp = a.to_sparse_csc()
>>> sp
tensor(ccol_indices=tensor([0, 1, 2, 3, 3]),
row_indices=tensor([1, 1, 0]),
values=tensor([1., 2., 1.]), size=(3, 4), nnz=3, dtype=torch.float64,
layout=torch.sparse_csc)


### Sparse BSR Tensor¶

The sparse BSR (Block compressed Sparse Row) tensor format implements the BSR format for storage of two-dimensional tensors with an extension to supporting batches of sparse BSR tensors and values being blocks of multi-dimensional tensors.

A sparse BSR tensor consists of three tensors: crow_indices, col_indices and values:

• The crow_indices tensor consists of compressed row indices. This is a (B + 1)-D tensor of shape (*batchsize, nrowblocks + 1). The last element is the number of specified blocks, nse. This tensor encodes the index in values and col_indices depending on where the given column block starts. Each successive number in the tensor subtracted by the number before it denotes the number of blocks in a given row.

• The col_indices tensor contains the column block indices of each element. This is a (B + 1)-D tensor of shape (*batchsize, nse).

• The values tensor contains the values of the sparse BSR tensor elements collected into two-dimensional blocks. This is a (1 + 2 + K)-D tensor of shape (nse, nrowblocks, ncolblocks, *densesize).

#### Construction of BSR tensors¶

Sparse BSR tensors can be directly constructed by using the torch.sparse_bsr_tensor() function. The user must supply the row and column block indices and values tensors separately where the row block indices must be specified using the CSR compression encoding. The size argument is optional and will be deduced from the crow_indices and col_indices tensors if it is not present.

>>> crow_indices = torch.tensor([0, 2, 4])
>>> col_indices = torch.tensor([0, 1, 0, 1])
>>> values = torch.tensor([[[0, 1, 2], [6, 7, 8]],
...                        [[3, 4, 5], [9, 10, 11]],
...                        [[12, 13, 14], [18, 19, 20]],
...                        [[15, 16, 17], [21, 22, 23]]])
>>> bsr = torch.sparse_bsr_tensor(crow_indices, col_indices, values, dtype=torch.float64)
>>> bsr
tensor(crow_indices=tensor([0, 2, 4]),
col_indices=tensor([0, 1, 0, 1]),
values=tensor([[[ 0.,  1.,  2.],
[ 6.,  7.,  8.]],
[[ 3.,  4.,  5.],
[ 9., 10., 11.]],
[[12., 13., 14.],
[18., 19., 20.]],
[[15., 16., 17.],
[21., 22., 23.]]]),
size=(4, 6), nnz=4, dtype=torch.float64, layout=torch.sparse_bsr)
>>> bsr.to_dense()
tensor([[ 0.,  1.,  2.,  3.,  4.,  5.],
[ 6.,  7.,  8.,  9., 10., 11.],
[12., 13., 14., 15., 16., 17.],
[18., 19., 20., 21., 22., 23.]], dtype=torch.float64)


The (0 + 2 + 0)-dimensional sparse BSR tensors can be constructed from any two-dimensional tensor using torch.Tensor.to_sparse_bsr() method that also requires the specification of the values block size:

>>> dense = torch.tensor([[0, 1, 2, 3, 4, 5],
...                       [6, 7, 8, 9, 10, 11],
...                       [12, 13, 14, 15, 16, 17],
...                       [18, 19, 20, 21, 22, 23]])
>>> bsr = dense.to_sparse_bsr(blocksize=(2, 3))
>>> bsr
tensor(crow_indices=tensor([0, 2, 4]),
col_indices=tensor([0, 1, 0, 1]),
values=tensor([[[ 0,  1,  2],
[ 6,  7,  8]],
[[ 3,  4,  5],
[ 9, 10, 11]],
[[12, 13, 14],
[18, 19, 20]],
[[15, 16, 17],
[21, 22, 23]]]), size=(4, 6), nnz=4,
layout=torch.sparse_bsr)


### Sparse BSC Tensor¶

The sparse BSC (Block compressed Sparse Column) tensor format implements the BSC format for storage of two-dimensional tensors with an extension to supporting batches of sparse BSC tensors and values being blocks of multi-dimensional tensors.

A sparse BSC tensor consists of three tensors: ccol_indices, row_indices and values:

• The ccol_indices tensor consists of compressed column indices. This is a (B + 1)-D tensor of shape (*batchsize, ncolblocks + 1). The last element is the number of specified blocks, nse. This tensor encodes the index in values and row_indices depending on where the given row block starts. Each successive number in the tensor subtracted by the number before it denotes the number of blocks in a given column.

• The row_indices tensor contains the row block indices of each element. This is a (B + 1)-D tensor of shape (*batchsize, nse).

• The values tensor contains the values of the sparse BSC tensor elements collected into two-dimensional blocks. This is a (1 + 2 + K)-D tensor of shape (nse, nrowblocks, ncolblocks, *densesize).

#### Construction of BSC tensors¶

Sparse BSC tensors can be directly constructed by using the torch.sparse_bsc_tensor() function. The user must supply the row and column block indices and values tensors separately where the column block indices must be specified using the CSR compression encoding. The size argument is optional and will be deduced from the ccol_indices and row_indices tensors if it is not present.

>>> ccol_indices = torch.tensor([0, 2, 4])
>>> row_indices = torch.tensor([0, 1, 0, 1])
>>> values = torch.tensor([[[0, 1, 2], [6, 7, 8]],
...                        [[3, 4, 5], [9, 10, 11]],
...                        [[12, 13, 14], [18, 19, 20]],
...                        [[15, 16, 17], [21, 22, 23]]])
>>> bsc = torch.sparse_bsc_tensor(ccol_indices, row_indices, values, dtype=torch.float64)
>>> bsc
tensor(ccol_indices=tensor([0, 2, 4]),
row_indices=tensor([0, 1, 0, 1]),
values=tensor([[[ 0.,  1.,  2.],
[ 6.,  7.,  8.]],
[[ 3.,  4.,  5.],
[ 9., 10., 11.]],
[[12., 13., 14.],
[18., 19., 20.]],
[[15., 16., 17.],
[21., 22., 23.]]]), size=(4, 6), nnz=4,
dtype=torch.float64, layout=torch.sparse_bsc)


### Tools for working with sparse compressed tensors¶

All sparse compressed tensors — CSR, CSC, BSR, and BSC tensors — are conceptionally very similar in that their indices data is split into two parts: so-called compressed indices that use the CSR encoding, and so-called plain indices that are orthogonal to the compressed indices. This allows various tools on these tensors to share the same implementations that are parameterized by tensor layout.

#### Construction of sparse compressed tensors¶

Sparse CSR, CSC, BSR, and CSC tensors can be constructed by using torch.sparse_compressed_tensor() function that have the same interface as the above discussed constructor functions torch.sparse_csr_tensor(), torch.sparse_csc_tensor(), torch.sparse_bsr_tensor(), and torch.sparse_bsc_tensor(), respectively, but with an extra required layout argument. The following example illustrates a method of constructing CSR and CSC tensors using the same input data by specifying the corresponding layout parameter to the torch.sparse_compressed_tensor() function:

>>> compressed_indices = torch.tensor([0, 2, 4])
>>> plain_indices = torch.tensor([0, 1, 0, 1])
>>> values = torch.tensor([1, 2, 3, 4])
>>> csr = torch.sparse_compressed_tensor(compressed_indices, plain_indices, values, layout=torch.sparse_csr)
>>> csr
tensor(crow_indices=tensor([0, 2, 4]),
col_indices=tensor([0, 1, 0, 1]),
values=tensor([1, 2, 3, 4]), size=(2, 2), nnz=4,
layout=torch.sparse_csr)
>>> csc = torch.sparse_compressed_tensor(compressed_indices, plain_indices, values, layout=torch.sparse_csc)
>>> csc
tensor(ccol_indices=tensor([0, 2, 4]),
row_indices=tensor([0, 1, 0, 1]),
values=tensor([1, 2, 3, 4]), size=(2, 2), nnz=4,
layout=torch.sparse_csc)
>>> (csr.transpose(0, 1).to_dense() == csc.to_dense()).all()
tensor(True)


## Supported operations¶

### Linear Algebra operations¶

The following table summarizes supported Linear Algebra operations on sparse matrices where the operands layouts may vary. Here T[layout] denotes a tensor with a given layout. Similarly, M[layout] denotes a matrix (2-D PyTorch tensor), and V[layout] denotes a vector (1-D PyTorch tensor). In addition, f denotes a scalar (float or 0-D PyTorch tensor), * is element-wise multiplication, and @ is matrix multiplication.

PyTorch operation

Layout signature

torch.mv()

no

M[sparse_coo] @ V[strided] -> V[strided]

torch.mv()

no

M[sparse_csr] @ V[strided] -> V[strided]

torch.matmul()

no

M[sparse_coo] @ M[strided] -> M[strided]

torch.matmul()

no

M[sparse_csr] @ M[strided] -> M[strided]

torch.mm()

no

M[sparse_coo] @ M[strided] -> M[strided]

torch.sparse.mm()

yes

M[sparse_coo] @ M[strided] -> M[strided]

torch.smm()

no

M[sparse_coo] @ M[strided] -> M[sparse_coo]

torch.hspmm()

no

M[sparse_coo] @ M[strided] -> M[hybrid sparse_coo]

torch.bmm()

no

T[sparse_coo] @ T[strided] -> T[strided]

torch.addmm()

no

f * M[strided] + f * (M[sparse_coo] @ M[strided]) -> M[strided]

torch.sparse.addmm()

yes

f * M[strided] + f * (M[sparse_coo] @ M[strided]) -> M[strided]

torch.sspaddmm()

no

f * M[sparse_coo] + f * (M[sparse_coo] @ M[strided]) -> M[sparse_coo]

torch.lobpcg()

no

GENEIG(M[sparse_coo]) -> M[strided], M[strided]

torch.pca_lowrank()

yes

PCA(M[sparse_coo]) -> M[strided], M[strided], M[strided]

torch.svd_lowrank()

yes

SVD(M[sparse_coo]) -> M[strided], M[strided], M[strided]

where “Sparse grad?” column indicates if the PyTorch operation supports backward with respect to sparse matrix argument. All PyTorch operations, except torch.smm(), support backward with respect to strided matrix arguments.

Note

Currently, PyTorch does not support matrix multiplication with the layout signature M[strided] @ M[sparse_coo]. However, applications can still compute this using the matrix relation D @ S == (S.t() @ D.t()).t().

### Tensor methods and sparse¶

The following Tensor methods are related to sparse tensors:

 Tensor.is_sparse Is True if the Tensor uses sparse storage layout, False otherwise. Tensor.is_sparse_csr Is True if the Tensor uses sparse CSR storage layout, False otherwise. Tensor.dense_dim Return the number of dense dimensions in a sparse tensor self. Tensor.sparse_dim Return the number of sparse dimensions in a sparse tensor self. Tensor.sparse_mask Returns a new sparse tensor with values from a strided tensor self filtered by the indices of the sparse tensor mask. Tensor.to_sparse Returns a sparse copy of the tensor. Tensor.to_sparse_coo Convert a tensor to coordinate format. Tensor.to_sparse_csr Convert a tensor to compressed row storage format (CSR). Tensor.to_sparse_csc Convert a tensor to compressed column storage (CSC) format. Tensor.to_sparse_bsr Convert a CSR tensor to a block sparse row (BSR) storage format of given blocksize. Tensor.to_sparse_bsc Convert a CSR tensor to a block sparse column (BSC) storage format of given blocksize. Tensor.to_dense Creates a strided copy of self if self is not a strided tensor, otherwise returns self. Tensor.values Return the values tensor of a sparse COO tensor.

The following Tensor methods are specific to sparse COO tensors:

 Tensor.coalesce Returns a coalesced copy of self if self is an uncoalesced tensor. Tensor.sparse_resize_ Resizes self sparse tensor to the desired size and the number of sparse and dense dimensions. Tensor.sparse_resize_and_clear_ Removes all specified elements from a sparse tensor self and resizes self to the desired size and the number of sparse and dense dimensions. Tensor.is_coalesced Returns True if self is a sparse COO tensor that is coalesced, False otherwise. Tensor.indices Return the indices tensor of a sparse COO tensor.

The following methods are specific to sparse CSR tensors and sparse BSR tensors:

 Tensor.crow_indices Returns the tensor containing the compressed row indices of the self tensor when self is a sparse CSR tensor of layout sparse_csr. Tensor.col_indices Returns the tensor containing the column indices of the self tensor when self is a sparse CSR tensor of layout sparse_csr.

The following methods are specific to sparse CSC tensors and sparse BSC tensors:

The following Tensor methods support sparse COO tensors:

### Torch functions specific to sparse Tensors¶

 sparse_coo_tensor Constructs a sparse tensor in COO(rdinate) format with specified values at the given indices. sparse_csr_tensor Constructs a sparse tensor in CSR (Compressed Sparse Row) with specified values at the given crow_indices and col_indices. sparse_csc_tensor Constructs a sparse tensor in CSC (Compressed Sparse Column) with specified values at the given ccol_indices and row_indices. sparse_bsr_tensor Constructs a sparse tensor in BSR (Block Compressed Sparse Row)) with specified 2-dimensional blocks at the given crow_indices and col_indices. sparse_bsc_tensor Constructs a sparse tensor in BSC (Block Compressed Sparse Column)) with specified 2-dimensional blocks at the given ccol_indices and row_indices. sparse_compressed_tensor Constructs a sparse tensor in Compressed Sparse format - CSR, CSC, BSR, or BSC - with specified values at the given compressed_indices and plain_indices. sparse.sum Returns the sum of each row of the sparse tensor input in the given dimensions dim. sparse.addmm This function does exact same thing as torch.addmm() in the forward, except that it supports backward for sparse COO matrix mat1. sparse.sampled_addmm Performs a matrix multiplication of the dense matrices mat1 and mat2 at the locations specified by the sparsity pattern of input. sparse.mm Performs a matrix multiplication of the sparse matrix mat1 sspaddmm Matrix multiplies a sparse tensor mat1 with a dense tensor mat2, then adds the sparse tensor input to the result. hspmm Performs a matrix multiplication of a sparse COO matrix mat1 and a strided matrix mat2. smm Performs a matrix multiplication of the sparse matrix input with the dense matrix mat. sparse.softmax Applies a softmax function. sparse.log_softmax Applies a softmax function followed by logarithm. sparse.spdiags Creates a sparse 2D tensor by placing the values from rows of diagonals along specified diagonals of the output

### Other functions¶

The following torch functions support sparse tensors:

### Unary functions¶

We aim to support all zero-preserving unary functions.

If you find that we are missing a zero-preserving unary function that you need, please feel encouraged to open an issue for a feature request. As always please kindly try the search function first before opening an issue.

The following operators currently support sparse COO/CSR/CSC/BSR/CSR tensor inputs.