# torch.sparse¶

## Introduction¶

PyTorch provides torch.Tensor to represent a multi-dimensional array containing elements of a single data type. By default, array elements are stored contiguously in memory leading to efficient implementations of various array processing algorithms that relay on the fast access to array elements. However, there exists an important class of multi-dimensional arrays, so-called sparse arrays, where the contiguous memory storage of array elements turns out to be suboptimal. Sparse arrays have a property of having a vast portion of elements being equal to zero which means that a lot of memory as well as processor resources can be spared if only the non-zero elements are stored or/and processed. Various sparse storage formats (such as COO, CSR/CSC, LIL, etc.) have been developed that are optimized for a particular structure of non-zero elements in sparse arrays as well as for specific operations on the arrays.

Note

When talking about storing only non-zero elements of a sparse array, the usage of adjective “non-zero” is not strict: one is allowed to store also zeros in the sparse array data structure. Hence, in the following, we use “specified elements” for those array elements that are actually stored. In addition, the unspecified elements are typically assumed to have zero value, but not only, hence we use the term “fill value” to denote such elements.

Note

Using a sparse storage format for storing sparse arrays can be advantageous only when the size and sparsity levels of arrays are high. Otherwise, for small-sized or low-sparsity arrays using the contiguous memory storage format is likely the most efficient approach.

Warning

The PyTorch API of sparse tensors is in beta and may change in the near future.

## Sparse COO tensors¶

Currently, PyTorch implements the so-called Coordinate format, or COO format, as the default sparse storage format for storing 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)


### Hybrid sparse 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 hybrid sparse 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 uncoalesced sparse 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 coalesced or uncoalesced sparse 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]])


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 an uncoalesced sparse 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.

## Supported 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.matmul()

no

M[sparse_coo] @ 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().

class torch.Tensor

The following methods are specific to sparse tensors:

is_sparse

Is True if the Tensor uses sparse storage layout, False otherwise.

dense_dim() → int

Return the number of dense dimensions in a sparse tensor self.

Warning

Throws an error if self is not a sparse tensor.

See also Tensor.sparse_dim() and hybrid tensors.

sparse_dim() → int

Return the number of sparse dimensions in a sparse tensor self.

Warning

Throws an error if self is not a sparse tensor.

See also Tensor.dense_dim() and hybrid tensors.

sparse_mask(mask) → Tensor

Returns a new sparse tensor with values from a strided tensor self filtered by the indices of the sparse tensor mask. The values of mask sparse tensor are ignored. self and mask tensors must have the same shape.

Note

The returned sparse tensor has the same indices as the sparse tensor mask, even when the corresponding values in self are zeros.

Parameters

mask (Tensor) – a sparse tensor whose indices are used as a filter

Example:

>>> nse = 5
>>> dims = (5, 5, 2, 2)
>>> I = torch.cat([torch.randint(0, dims[0], size=(nse,)),
...                torch.randint(0, dims[1], size=(nse,))], 0).reshape(2, nse)
>>> V = torch.randn(nse, dims[2], dims[3])
>>> S = torch.sparse_coo_tensor(I, V, dims).coalesce()
>>> D = torch.randn(dims)
tensor(indices=tensor([[0, 0, 0, 2],
[0, 1, 4, 3]]),
values=tensor([[[ 1.6550,  0.2397],
[-0.1611, -0.0779]],

[[ 0.2326, -1.0558],
[ 1.4711,  1.9678]],

[[-0.5138, -0.0411],
[ 1.9417,  0.5158]],

[[ 0.0793,  0.0036],
[-0.2569, -0.1055]]]),
size=(5, 5, 2, 2), nnz=4, layout=torch.sparse_coo)

sparse_resize_(size, sparse_dim, dense_dim) → Tensor

Resizes self sparse tensor to the desired size and the number of sparse and dense dimensions.

Note

If the number of specified elements in self is zero, then size, sparse_dim, and dense_dim can be any size and positive integers such that len(size) == sparse_dim + dense_dim.

If self specifies one or more elements, however, then each dimension in size must not be smaller than the corresponding dimension of self, sparse_dim must equal the number of sparse dimensions in self, and dense_dim must equal the number of dense dimensions in self.

Warning

Throws an error if self is not a sparse tensor.

Parameters
• size (torch.Size) – the desired size. If self is non-empty sparse tensor, the desired size cannot be smaller than the original size.

• sparse_dim (int) – the number of sparse dimensions

• dense_dim (int) – the number of dense dimensions

sparse_resize_and_clear_(size, sparse_dim, dense_dim) → Tensor

Removes all specified elements from a sparse tensor self and resizes self to the desired size and the number of sparse and dense dimensions.

Parameters
• size (torch.Size) – the desired size.

• sparse_dim (int) – the number of sparse dimensions

• dense_dim (int) – the number of dense dimensions

to_dense() → Tensor

Creates a strided copy of self.

Warning

Throws an error if self is a strided tensor.

Example:

>>> s = torch.sparse_coo_tensor(
...        torch.tensor([[1, 1],
...                      [0, 2]]),
...        torch.tensor([9, 10]),
...        size=(3, 3))
>>> s.to_dense()
tensor([[ 0,  0,  0],
[ 9,  0, 10],
[ 0,  0,  0]])

to_sparse(sparseDims) → Tensor

Returns a sparse copy of the tensor. PyTorch supports sparse tensors in coordinate format.

Parameters

sparseDims (int, optional) – the number of sparse dimensions to include in the new sparse tensor

Example:

>>> d = torch.tensor([[0, 0, 0], [9, 0, 10], [0, 0, 0]])
>>> d
tensor([[ 0,  0,  0],
[ 9,  0, 10],
[ 0,  0,  0]])
>>> d.to_sparse()
tensor(indices=tensor([[1, 1],
[0, 2]]),
values=tensor([ 9, 10]),
size=(3, 3), nnz=2, layout=torch.sparse_coo)
>>> d.to_sparse(1)
tensor(indices=tensor([[1]]),
values=tensor([[ 9,  0, 10]]),
size=(3, 3), nnz=1, layout=torch.sparse_coo)

coalesce() → Tensor

Returns a coalesced copy of self if self is an uncoalesced tensor.

Returns self if self is a coalesced tensor.

Warning

Throws an error if self is not a sparse COO tensor.

is_coalesced() → bool

Returns True if self is a sparse COO tensor that is coalesced, False otherwise.

Warning

Throws an error if self is not a sparse COO tensor.

indices() → Tensor

Return the indices tensor of a sparse COO tensor.

Warning

Throws an error if self is not a sparse COO tensor.

See also Tensor.values().

Note

This method can only be called on a coalesced sparse tensor. See Tensor.coalesce() for details.

values() → Tensor

Return the values tensor of a sparse COO tensor.

Warning

Throws an error if self is not a sparse COO tensor.

See also Tensor.indices().

Note

This method can only be called on a coalesced sparse tensor. See Tensor.coalesce() for details.

The following torch.Tensor methods support sparse COO tensors:

## Sparse tensor functions¶

torch.sparse_coo_tensor(indices, values, size=None, *, dtype=None, device=None, requires_grad=False) → Tensor

Constructs a sparse tensor in COO(rdinate) format with specified values at the given indices.

Note

This function returns an uncoalesced tensor.

Parameters
• indices (array_like) – Initial data for the tensor. Can be a list, tuple, NumPy ndarray, scalar, and other types. Will be cast to a torch.LongTensor internally. The indices are the coordinates of the non-zero values in the matrix, and thus should be two-dimensional where the first dimension is the number of tensor dimensions and the second dimension is the number of non-zero values.

• values (array_like) – Initial values for the tensor. Can be a list, tuple, NumPy ndarray, scalar, and other types.

• size (list, tuple, or torch.Size, optional) – Size of the sparse tensor. If not provided the size will be inferred as the minimum size big enough to hold all non-zero elements.

Keyword Arguments
• dtype (torch.dtype, optional) – the desired data type of returned tensor. Default: if None, infers data type from values.

• device (torch.device, optional) – the desired device of returned tensor. Default: if None, uses the current device for the default tensor type (see torch.set_default_tensor_type()). device will be the CPU for CPU tensor types and the current CUDA device for CUDA tensor types.

• requires_grad (bool, optional) – If autograd should record operations on the returned tensor. Default: False.

Example:

>>> i = torch.tensor([[0, 1, 1],
...                   [2, 0, 2]])
>>> v = torch.tensor([3, 4, 5], dtype=torch.float32)
>>> torch.sparse_coo_tensor(i, v, [2, 4])
tensor(indices=tensor([[0, 1, 1],
[2, 0, 2]]),
values=tensor([3., 4., 5.]),
size=(2, 4), nnz=3, layout=torch.sparse_coo)

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

>>> torch.sparse_coo_tensor(i, v, [2, 4],
...                         dtype=torch.float64,
...                         device=torch.device('cuda:0'))
tensor(indices=tensor([[0, 1, 1],
[2, 0, 2]]),
values=tensor([3., 4., 5.]),
device='cuda:0', size=(2, 4), nnz=3, dtype=torch.float64,
layout=torch.sparse_coo)

# Create an empty sparse tensor with the following invariants:
#   1. sparse_dim + dense_dim = len(SparseTensor.shape)
#   2. SparseTensor._indices().shape = (sparse_dim, nnz)
#   3. SparseTensor._values().shape = (nnz, SparseTensor.shape[sparse_dim:])
#
# For instance, to create an empty sparse tensor with nnz = 0, dense_dim = 0 and
# sparse_dim = 1 (hence indices is a 2D tensor of shape = (1, 0))
>>> S = torch.sparse_coo_tensor(torch.empty([1, 0]), [], [1])
tensor(indices=tensor([], size=(1, 0)),
values=tensor([], size=(0,)),
size=(1,), nnz=0, layout=torch.sparse_coo)

# and to create an empty sparse tensor with nnz = 0, dense_dim = 1 and
# sparse_dim = 1
>>> S = torch.sparse_coo_tensor(torch.empty([1, 0]), torch.empty([0, 2]), [1, 2])
tensor(indices=tensor([], size=(1, 0)),
values=tensor([], size=(0, 2)),
size=(1, 2), nnz=0, layout=torch.sparse_coo)

torch.sparse.sum(input, dim=None, dtype=None)[source]

Returns the sum of each row of the sparse tensor input in the given dimensions dim. If dim is a list of dimensions, reduce over all of them. When sum over all sparse_dim, this method returns a dense tensor instead of a sparse tensor.

All summed dim are squeezed (see torch.squeeze()), resulting an output tensor having dim fewer dimensions than input.

During backward, only gradients at nnz locations of input will propagate back. Note that the gradients of input is coalesced.

Parameters
• input (Tensor) – the input sparse tensor

• dim (int or tuple of python:ints) – a dimension or a list of dimensions to reduce. Default: reduce over all dims.

• dtype (torch.dtype, optional) – the desired data type of returned Tensor. Default: dtype of input.

Example:

>>> nnz = 3
>>> dims = [5, 5, 2, 3]
>>> I = torch.cat([torch.randint(0, dims[0], size=(nnz,)),
torch.randint(0, dims[1], size=(nnz,))], 0).reshape(2, nnz)
>>> V = torch.randn(nnz, dims[2], dims[3])
>>> size = torch.Size(dims)
>>> S = torch.sparse_coo_tensor(I, V, size)
>>> S
tensor(indices=tensor([[2, 0, 3],
[2, 4, 1]]),
values=tensor([[[-0.6438, -1.6467,  1.4004],
[ 0.3411,  0.0918, -0.2312]],

[[ 0.5348,  0.0634, -2.0494],
[-0.7125, -1.0646,  2.1844]],

[[ 0.1276,  0.1874, -0.6334],
[-1.9682, -0.5340,  0.7483]]]),
size=(5, 5, 2, 3), nnz=3, layout=torch.sparse_coo)

# when sum over only part of sparse_dims, return a sparse tensor
>>> torch.sparse.sum(S, [1, 3])
tensor(indices=tensor([[0, 2, 3]]),
values=tensor([[-1.4512,  0.4073],
[-0.8901,  0.2017],
[-0.3183, -1.7539]]),
size=(5, 2), nnz=3, layout=torch.sparse_coo)

# when sum over all sparse dim, return a dense tensor
# with summed dims squeezed
>>> torch.sparse.sum(S, [0, 1, 3])
tensor([-2.6596, -1.1450])

torch.sparse.addmm(mat, mat1, mat2, beta=1.0, alpha=1.0)[source]

This function does exact same thing as torch.addmm() in the forward, except that it supports backward for sparse matrix mat1. mat1 need to have sparse_dim = 2. Note that the gradients of mat1 is a coalesced sparse tensor.

Parameters
• mat (Tensor) – a dense matrix to be added

• mat1 (Tensor) – a sparse matrix to be multiplied

• mat2 (Tensor) – a dense matrix to be multiplied

• beta (Number, optional) – multiplier for mat ($\beta$ )

• alpha (Number, optional) – multiplier for $mat1 @ mat2$ ($\alpha$ )

torch.sparse.mm(mat1, mat2)[source]

Performs a matrix multiplication of the sparse matrix mat1 and the (sparse or strided) matrix mat2. Similar to torch.mm(), If mat1 is a $(n \times m)$ tensor, mat2 is a $(m \times p)$ tensor, out will be a $(n \times p)$ tensor. mat1 need to have sparse_dim = 2. This function also supports backward for both matrices. Note that the gradients of mat1 is a coalesced sparse tensor.

Parameters
• mat1 (SparseTensor) – the first sparse matrix to be multiplied

• mat2 (Tensor) – the second matrix to be multiplied, which could be sparse or dense

Shape:

The format of the output tensor of this function follows: - sparse x sparse -> sparse - sparse x dense -> dense

Example:

>>> a = torch.randn(2, 3).to_sparse().requires_grad_(True)
>>> a
tensor(indices=tensor([[0, 0, 0, 1, 1, 1],
[0, 1, 2, 0, 1, 2]]),
values=tensor([ 1.5901,  0.0183, -0.6146,  1.8061, -0.0112,  0.6302]),
size=(2, 3), nnz=6, layout=torch.sparse_coo, requires_grad=True)

>>> b = torch.randn(3, 2, requires_grad=True)
>>> b
tensor([[-0.6479,  0.7874],
[-1.2056,  0.5641],

>>> y = torch.sparse.mm(a, b)
>>> y
tensor([[-0.3323,  1.8723],
>>> y.sum().backward()
tensor(indices=tensor([[0, 0, 0, 1, 1, 1],
[0, 1, 2, 0, 1, 2]]),
values=tensor([ 0.1394, -0.6415, -2.1639,  0.1394, -0.6415, -2.1639]),
size=(2, 3), nnz=6, layout=torch.sparse_coo)

torch.sspaddmm(input, mat1, mat2, *, beta=1, alpha=1, out=None) → Tensor

Matrix multiplies a sparse tensor mat1 with a dense tensor mat2, then adds the sparse tensor input to the result.

Note: This function is equivalent to torch.addmm(), except input and mat1 are sparse.

Parameters
• input (Tensor) – a sparse matrix to be added

• mat1 (Tensor) – a sparse matrix to be matrix multiplied

• mat2 (Tensor) – a dense matrix to be matrix multiplied

Keyword Arguments
• beta (Number, optional) – multiplier for mat ($\beta$ )

• alpha (Number, optional) – multiplier for $mat1 @ mat2$ ($\alpha$ )

• out (Tensor, optional) – the output tensor.

torch.hspmm(mat1, mat2, *, out=None) → Tensor

Performs a matrix multiplication of a sparse COO matrix mat1 and a strided matrix mat2. The result is a (1 + 1)-dimensional hybrid COO matrix.

Parameters
• mat1 (Tensor) – the first sparse matrix to be matrix multiplied

• mat2 (Tensor) – the second strided matrix to be matrix multiplied

Keyword Arguments

{out}

torch.smm(input, mat) → Tensor

Performs a matrix multiplication of the sparse matrix input with the dense matrix mat.

Parameters
• input (Tensor) – a sparse matrix to be matrix multiplied

• mat (Tensor) – a dense matrix to be matrix multiplied

torch.sparse.softmax(input, dim, dtype=None)[source]

Applies a softmax function.

Softmax is defined as:

$\text{Softmax}(x_{i}) = \frac{exp(x_i)}{\sum_j exp(x_j)}$

where $i, j$ run over sparse tensor indices and unspecified entries are ignores. This is equivalent to defining unspecified entries as negative infinity so that $exp(x_k) = 0$ when the entry with index $k$ has not specified.

It is applied to all slices along dim, and will re-scale them so that the elements lie in the range [0, 1] and sum to 1.

Parameters
• input (Tensor) – input

• dim (int) – A dimension along which softmax will be computed.

• dtype (torch.dtype, optional) – the desired data type of returned tensor. If specified, the input tensor is casted to dtype before the operation is performed. This is useful for preventing data type overflows. Default: None

torch.sparse.log_softmax(input, dim, dtype=None)[source]

Applies a softmax function followed by logarithm.

See softmax for more details.

Parameters
• input (Tensor) – input

• dim (int) – A dimension along which softmax will be computed.

• dtype (torch.dtype, optional) – the desired data type of returned tensor. If specified, the input tensor is casted to dtype before the operation is performed. This is useful for preventing data type overflows. Default: None

## Other functions¶

The following torch functions support sparse COO tensors: