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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.


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.


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.


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,

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


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.


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:


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:


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:


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().


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 Sparse grad? 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.


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().

The following methods are specific to sparse tensors <sparse-docs>:











The following torch.Tensor methods support sparse COO tensors <sparse-coo-docs>:

~torch.Tensor.add ~torch.Tensor.add_ ~torch.Tensor.addmm ~torch.Tensor.addmm_ ~torch.Tensor.any ~torch.Tensor.asin ~torch.Tensor.asin_ ~torch.Tensor.arcsin ~torch.Tensor.arcsin_ ~torch.Tensor.bmm ~torch.Tensor.clone ~torch.Tensor.deg2rad ~torch.Tensor.deg2rad_ ~torch.Tensor.detach ~torch.Tensor.detach_ ~torch.Tensor.dim ~torch.Tensor.div ~torch.Tensor.div_ ~torch.Tensor.floor_divide ~torch.Tensor.floor_divide_ ~torch.Tensor.get_device ~torch.Tensor.index_select ~torch.Tensor.isnan ~torch.Tensor.log1p ~torch.Tensor.log1p_ ~torch.Tensor.mm ~torch.Tensor.mul ~torch.Tensor.mul_ ~torch.Tensor.mv ~torch.Tensor.narrow_copy ~torch.Tensor.neg ~torch.Tensor.neg_ ~torch.Tensor.negative ~torch.Tensor.negative_ ~torch.Tensor.numel ~torch.Tensor.rad2deg ~torch.Tensor.rad2deg_ ~torch.Tensor.resize_as_ ~torch.Tensor.size ~torch.Tensor.pow ~torch.Tensor.sqrt ~torch.Tensor.square ~torch.Tensor.smm ~torch.Tensor.sspaddmm ~torch.Tensor.sub ~torch.Tensor.sub_ ~torch.Tensor.t ~torch.Tensor.t_ ~torch.Tensor.transpose ~torch.Tensor.transpose_ ~torch.Tensor.zero_

Sparse tensor functions










Other functions

The following torch functions support sparse COO tensors <sparse-coo-docs>:

~torch.cat ~torch.dstack ~torch.empty ~torch.empty_like ~torch.hstack ~torch.index_select ~torch.is_complex ~torch.is_floating_point ~torch.is_nonzero ~torch.is_same_size ~torch.is_signed ~torch.is_tensor ~torch.lobpcg ~torch.mm ~torch.native_norm ~torch.pca_lowrank ~torch.select ~torch.stack ~torch.svd_lowrank ~torch.unsqueeze ~torch.vstack ~torch.zeros ~torch.zeros_like