In general, sparse matrices provide the same functionality as regular matrices. The difference lies in the way the elements of sparse matrices are represented and stored in memory. Only the non-zero elements of the latter are stored. This has some potential advantages: first, this may obviously lead to reduced memory usage and, second, clever storage methods may lead to reduced computation time through the use of sparse specific algorithms. We usually refer to the generically stored matrices as dense matrices.
Aesara’s sparse package provides efficient algorithms, but its use is not recommended in all cases or for all matrices. As an obvious example, consider the case where the sparsity proportion is very low. The sparsity proportion refers to the ratio of the number of zero elements to the number of all elements in a matrix. A low sparsity proportion may result in the use of more space in memory since not only the actual data is stored, but also the position of nearly every element of the matrix. This would also require more computation time whereas a dense matrix representation along with regular optimized algorithms might do a better job. Other examples may be found at the nexus of the specific purpose and structure of the matrices. More documentation may be found in the SciPy Sparse Reference.
Since sparse matrices are not stored in contiguous arrays, there are several
ways to represent them in memory. This is usually designated by the so-called
of the matrix. Since Aesara’s sparse matrix package is based on the SciPy
sparse package, complete information about sparse matrices can be found
in the SciPy documentation. Like SciPy, Aesara does not implement sparse formats for
arrays with a number of dimensions different from two.
So far, Aesara implements two
formats of sparse matrix:
Those are almost identical except that
csc is based on the columns of the
csr is based on its rows. They both have the same purpose:
to provide for the use of efficient algorithms performing linear algebra operations.
A disadvantage is that they fail to give an efficient way to modify the sparsity structure
of the underlying matrix, i.e. adding new elements. This means that if you are
planning to add new elements in a sparse matrix very often in your computational graph,
perhaps a tensor variable could be a better choice.
More documentation may be found in the Sparse Library Reference.
Before going further, here are the
import statements that are assumed for the rest of the
>>> import aesara >>> import numpy as np >>> import scipy.sparse as sp >>> from aesara import sparse
Compressed Sparse Format¶
Aesara supports two compressed sparse formats:
csr, respectively based on columns
and rows. They have both the same attributes:
dataattribute is a one-dimensional
ndarraywhich contains all the non-zero elements of the sparse matrix.
indptrattributes are used to store the position of the data in the sparse matrix.
shapeattribute is exactly the same as the
shapeattribute of a dense (i.e. generic) matrix. It can be explicitly specified at the creation of a sparse matrix if it cannot be inferred from the first three attributes.
Which format should I use?¶
At the end, the format does not affect the length of the
indices attributes. They are both
completely fixed by the number of elements you want to store. The only thing that changes with the format
csc format, the matrix is compressed along columns so a lower number of columns will
result in less memory use. On the other hand, with the
csr format, the matrix is compressed along
the rows and with a matrix that have a lower number of rows,
csr format is a better choice. So here is the rule:
If shape > shape, use
csc format. Otherwise, use
Sometimes, since the sparse module is young, ops does not exist for both format. So here is what may be the most relevant rule:
Use the format compatible with the ops in your computation graph.
The documentation about the ops and their supported format may be found in the Sparse Library Reference.
Handling Sparse in Aesara¶
Most of the ops in Aesara depend on the
format of the sparse matrix.
That is why there are two kinds of constructors of sparse variables:
csr_matrix. These can be called with the usual
dtype parameters, but no
broadcastable flags are
allowed. This is forbidden since the sparse package, as the SciPy sparse module,
does not provide any way to handle a number of dimensions different from two.
The set of all accepted
dtype for the sparse matrices can be found in
>>> sparse.all_dtypes set(['int8', 'int16', 'int32', 'int64', 'uint8', 'uint16', 'uint32', 'uint64', 'float32', 'float64', 'complex64', 'complex128'])
To and Fro¶
To move back and forth from a dense matrix to a sparse matrix representation, Aesara
csc_from_dense functions. No additional detail must be provided. Here is
an example that performs a full cycle from sparse to sparse:
>>> x = sparse.csc_matrix(name='x', dtype='float32') >>> y = sparse.dense_from_sparse(x) >>> z = sparse.csc_from_dense(y)
Properties and Construction¶
Although sparse variables do not allow direct access to their properties,
this can be accomplished using the
csm_properties function. This will return
a tuple of one-dimensional
tensor variables that represents the internal characteristics
of the sparse matrix.
In order to reconstruct a sparse matrix from some properties, the functions
CSR can be used. This will create the sparse matrix in the desired
format. As an example, the following code reconstructs a
csc matrix into
>>> x = sparse.csc_matrix(name='x', dtype='int64') >>> data, indices, indptr, shape = sparse.csm_properties(x) >>> y = sparse.CSR(data, indices, indptr, shape) >>> f = aesara.function([x], y) >>> a = sp.csc_matrix(np.asarray([[0, 1, 1], [0, 0, 0], [1, 0, 0]])) >>> print(a.toarray()) [[0 1 1] [0 0 0] [1 0 0]] >>> print(f(a).toarray()) [[0 0 1] [1 0 0] [1 0 0]]
The last example shows that one format can be obtained from transposition of
the other. Indeed, when calling the
the sparse characteristics of the resulting matrix cannot be the same as the one
provided as input.
Several ops are set to make use of the very peculiar structure of the sparse matrices. These ops are said to be structured and simply do not perform any computations on the zero elements of the sparse matrix. They can be thought as being applied only to the data attribute of the latter. Note that these structured ops provide a structured gradient. More explication below.
>>> x = sparse.csc_matrix(name='x', dtype='float32') >>> y = sparse.structured_add(x, 2) >>> f = aesara.function([x], y) >>> a = sp.csc_matrix(np.asarray([[0, 0, -1], [0, -2, 1], [3, 0, 0]], dtype='float32')) >>> print(a.toarray()) [[ 0. 0. -1.] [ 0. -2. 1.] [ 3. 0. 0.]] >>> print(f(a).toarray()) [[ 0. 0. 1.] [ 0. 0. 3.] [ 5. 0. 0.]]
The gradients of the ops in the sparse module can also be structured. Some ops provide a flag to indicate if the gradient is to be structured or not. The documentation can be used to determine if the gradient of an op is regular or structured or if its implementation can be modified. Similarly to structured ops, when a structured gradient is calculated, the computation is done only for the non-zero elements of the sparse matrix.
More documentation regarding the gradients of specific ops can be found in the Sparse Library Reference.