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

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.


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:

  • 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.


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.


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()
       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() returns True.


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)
>>> s.is_sparse
>>> s.layout == torch.sparse_coo

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()

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]
>>> s[1, 0, 1:]

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 N-dimensional tensors, the primary advantage 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. CUDA support does not exist as of now.

A CSR sparse 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 size[0] + 1. The last element is the number of non-zeros. 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 value. This is a 1-D tensor of size nnz.

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


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.

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,
>>> 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 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)
        [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 (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


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


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



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



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


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


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



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



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



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



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



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



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



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



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



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().

Tensor methods and sparse

The following Tensor methods are related to sparse tensors:


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


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


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


Returns a new sparse tensor with values from a strided tensor self filtered by the indices of the sparse tensor mask.


Returns a sparse copy of the tensor.


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 self if self is an uncoalesced tensor.


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


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


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


Creates a strided copy of self.

The following methods are specific to sparse CSR tensors:


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


Returns the tensor containing the column indices of the self tensor when self is a sparse CSR tensor of layout sparse_csr.

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 indices.


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


Returns the sum of each row of the sparse tensor input in the given dimensions dim.


This function does exact same thing as torch.addmm() in the forward, except that it supports backward for sparse matrix mat1.

Performs a matrix multiplication of the sparse matrix mat1 and the (sparse or strided) matrix mat2.


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


Performs a matrix multiplication of a sparse COO matrix mat1 and a strided matrix mat2.


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


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()


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