torch.sparse¶
Introduction¶
PyTorch provides torch.Tensor
to represent a
multidimensional 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 multidimensional arrays, socalled 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 nonzero 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 nonzero elements in sparse arrays as well as
for specific operations on the arrays.
Note
When talking about storing only nonzero elements of a sparse array, the usage of adjective “nonzero” 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 smallsized or lowsparsity 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¶
PyTorch implements the socalled 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 typetorch.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 nonzero 32bit 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 multidimensional tensor so that we
have:
the indices of specified elements are collected in
indices
tensor of size(sparse_dims, nse)
and with element typetorch.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 Ndimensional 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 Kdimensional 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 1D
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 multivalued 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:
the indices of specified tensor elements are unique,
the indices are sorted in lexicographical order,
torch.Tensor.is_coalesced()
returnsTrue
.
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]])
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 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.
Sparse CSR Tensor¶
The CSR (Compressed Sparse Row) sparse tensor format implements the CSR format for storage of 2 dimensional tensors. Although there is no support for Ndimensional tensors, the primary advantage over the COO format is better use of storage and much faster computation operations such as sparse matrixvector multiplication using MKL and MAGMA backends. CUDA support does not exist as of now.
A CSR sparse tensor consists of three 1D tensors: crow_indices
, col_indices
and values
:
The
crow_indices
tensor consists of compressed row indices. This is a 1D tensor of sizesize[0] + 1
. The last element is the number of nonzeros. This tensor encodes the index invalues
andcol_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 value. This is a 1D tensor of sizennz
.The
values
tensor contains the values of the CSR tensor. This is a 1D tensor of sizennz
.
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 MKLenabled 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.
Construction of CSR tensors¶
Sparse CSR matrices can be directly constructed by using the torch.sparse_csr_tensor()
method. The user must supply the row and column indices and values tensors separately.
The size
argument is optional and will be deduced from the 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.double)
>>> 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)
CSR Tensor Operations¶
The simplest way of constructing a sparse CSR tensor from a strided or sparse COO
tensor is to use tensor.to_sparse_csr()
. 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)
The sparse matrixvector 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)
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 (2D PyTorch tensor), and V[layout]
denotes a vector (1D PyTorch tensor). In addition, f
denotes a
scalar (float or 0D PyTorch tensor), *
is elementwise
multiplication, and @
is matrix multiplication.
PyTorch operation 
Sparse grad? 
Layout signature 

no 


no 


no 


no 


no 


yes 


no 


no 


no 


no 


yes 


no 


no 


yes 


yes 

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:
Is 

Return the number of dense dimensions in a sparse tensor 

Return the number of sparse dimensions in a sparse tensor 

Returns a new sparse tensor with values from a strided tensor 

Returns a sparse copy of the tensor. 


Convert a tensor to coordinate format. 

Convert a tensor to compressed row storage format. 
Return the indices tensor of a sparse COO tensor. 

Return the values tensor of a sparse COO tensor. 
The following Tensor methods are specific to sparse COO tensors:
Returns a coalesced copy of 

Resizes 

Removes all specified elements from a sparse tensor 

Returns 

Creates a strided copy of 
The following methods are specific to sparse CSR tensors:

Returns the tensor containing the compressed row indices of the 

Returns the tensor containing the column indices of the 
The following Tensor methods support sparse COO tensors:
add()
add_()
addmm()
addmm_()
any()
asin()
asin_()
arcsin()
arcsin_()
bmm()
clone()
deg2rad()
deg2rad_()
detach()
detach_()
dim()
div()
div_()
floor_divide()
floor_divide_()
get_device()
index_select()
isnan()
log1p()
log1p_()
mm()
mul()
mul_()
mv()
narrow_copy()
neg()
neg_()
negative()
negative_()
numel()
rad2deg()
rad2deg_()
resize_as_()
size()
pow()
sqrt()
square()
smm()
sspaddmm()
sub()
sub_()
t()
t_()
transpose()
transpose_()
zero_()
Torch functions specific to sparse Tensors¶
Constructs a sparse tensor in COO(rdinate) format with specified values at the given 

Constructs a sparse tensor in CSR (Compressed Sparse Row) with specified values at the given 

Returns the sum of each row of the sparse tensor 

This function does exact same thing as 

Performs a matrix multiplication of the dense matrices 

Performs a matrix multiplication of the sparse matrix 

Matrix multiplies a sparse tensor 

Performs a matrix multiplication of a sparse COO matrix 

Performs a matrix multiplication of the sparse matrix 

Applies a softmax function. 

Applies a softmax function followed by logarithm. 
Other functions¶
The following torch
functions support sparse tensors:
cat()
dstack()
empty()
empty_like()
hstack()
index_select()
is_complex()
is_floating_point()
is_nonzero()
is_same_size()
is_signed()
is_tensor()
lobpcg()
mm()
native_norm()
pca_lowrank()
select()
stack()
svd_lowrank()
unsqueeze()
vstack()
zeros()
zeros_like()