sparse – Symbolic Sparse Matrices

In the tutorial section, you can find a sparse tutorial.

The sparse submodule is not loaded when we import PyTensor. You must import pytensor.sparse to enable it.

The sparse module provides the same functionality as the tensor module. The difference lies under the covers because sparse matrices do not store data in a contiguous array. The sparse module has been used in:

• NLP: Dense linear transformations of sparse vectors.

• Audio: Filterbank in the Fourier domain.

Compressed Sparse Format

This section tries to explain how information is stored for the two sparse formats of SciPy supported by PyTensor.

PyTensor supports two compressed sparse formats: csc and csr, respectively based on columns and rows. They have both the same attributes: data, indices, indptr and shape.

• The data attribute is a one-dimensional ndarray which contains all the non-zero elements of the sparse matrix.

• The indices and indptr attributes are used to store the position of the data in the sparse matrix.

• The shape attribute is exactly the same as the shape attribute 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.

CSC Matrix

In the Compressed Sparse Column format, indices stands for indexes inside the column vectors of the matrix and indptr tells where the column starts in the data and in the indices attributes. indptr can be thought of as giving the slice which must be applied to the other attribute in order to get each column of the matrix. In other words, slice(indptr[i], indptr[i+1]) corresponds to the slice needed to find the i-th column of the matrix in the data and indices fields.

The following example builds a matrix and returns its columns. It prints the i-th column, i.e. a list of indices in the column and their corresponding value in the second list.

>>> import numpy as np
>>> import scipy.sparse as sp
>>> data = np.asarray([7, 8, 9])
>>> indices = np.asarray([0, 1, 2])
>>> indptr = np.asarray([0, 2, 3, 3])
>>> m = sp.csc_matrix((data, indices, indptr), shape=(3, 3))
>>> m.toarray()
array([[7, 0, 0],
       [8, 0, 0],
       [0, 9, 0]])
>>> i = 0
>>> m.indices[m.indptr[i]:m.indptr[i+1]], m.data[m.indptr[i]:m.indptr[i+1]]
(array([0, 1], dtype=int32), array([7, 8]))
>>> i = 1
>>> m.indices[m.indptr[i]:m.indptr[i+1]], m.data[m.indptr[i]:m.indptr[i+1]]
(array([2], dtype=int32), array([9]))
>>> i = 2
>>> m.indices[m.indptr[i]:m.indptr[i+1]], m.data[m.indptr[i]:m.indptr[i+1]]
(array([], dtype=int32), array([], dtype=int64))

CSR Matrix

In the Compressed Sparse Row format, indices stands for indexes inside the row vectors of the matrix and indptr tells where the row starts in the data and in the indices attributes. indptr can be thought of as giving the slice which must be applied to the other attribute in order to get each row of the matrix. In other words, slice(indptr[i], indptr[i+1]) corresponds to the slice needed to find the i-th row of the matrix in the data and indices fields.

The following example builds a matrix and returns its rows. It prints the i-th row, i.e. a list of indices in the row and their corresponding value in the second list.

>>> import numpy as np
>>> import scipy.sparse as sp
>>> data = np.asarray([7, 8, 9])
>>> indices = np.asarray([0, 1, 2])
>>> indptr = np.asarray([0, 2, 3, 3])
>>> m = sp.csr_matrix((data, indices, indptr), shape=(3, 3))
>>> m.toarray()
array([[7, 8, 0],
       [0, 0, 9],
       [0, 0, 0]])
>>> i = 0
>>> m.indices[m.indptr[i]:m.indptr[i+1]], m.data[m.indptr[i]:m.indptr[i+1]]
(array([0, 1], dtype=int32), array([7, 8]))
>>> i = 1
>>> m.indices[m.indptr[i]:m.indptr[i+1]], m.data[m.indptr[i]:m.indptr[i+1]]
(array([2], dtype=int32), array([9]))
>>> i = 2
>>> m.indices[m.indptr[i]:m.indptr[i+1]], m.data[m.indptr[i]:m.indptr[i+1]]
(array([], dtype=int32), array([], dtype=int64))

List of Implemented Operations

• Moving from and to sparse

  • dense_from_sparse. Both grads are implemented. Structured by default.

  • csr_from_dense, csc_from_dense. The grad implemented is structured.

  • PyTensor SparseVariable objects have a method toarray() that is the same as dense_from_sparse.

• Construction of Sparses and their Properties

  • CSM and CSC, CSR to construct a matrix. The grad implemented is regular.

  • csm_properties. to get the properties of a sparse matrix. The grad implemented is regular.

  • csm_indices(x), csm_indptr(x), csm_data(x) and csm_shape(x) or x.shape.

  • sp_ones_like. The grad implemented is regular.

  • sp_zeros_like. The grad implemented is regular.

  • square_diagonal. The grad implemented is regular.

  • construct_sparse_from_list. The grad implemented is regular.

• Cast

  • cast with bcast, wcast, icast, lcast, fcast, dcast, ccast, and zcast. The grad implemented is regular.

• Transpose

  • transpose. The grad implemented is regular.

• Basic Arithmetic

  • neg. The grad implemented is regular.

  • eq.

  • neq.

  • gt.

  • ge.

  • lt.

  • le.

  • add. The grad implemented is regular.

  • sub. The grad implemented is regular.

  • mul. The grad implemented is regular.

  • col_scale to multiply by a vector along the columns. The grad implemented is structured.

  • row_scale to multiply by a vector along the rows. The grad implemented is structured.

• Monoid (Element-wise operation with only one sparse input).

  They all have a structured grad.

  • structured_sigmoid

  • structured_exp

  • structured_log

  • structured_pow

  • structured_minimum

  • structured_maximum

  • structured_add

  • sin

  • arcsin

  • tan

  • arctan

  • sinh

  • arcsinh

  • tanh

  • arctanh

  • rad2deg

  • deg2rad

  • rint

  • ceil

  • floor

  • trunc

  • sign

  • log1p

  • expm1

  • sqr

  • sqrt

• Dot Product

  • dot.

    • One of the inputs must be sparse, the other sparse or dense.

    • The grad implemented is regular.

    • No C code for perform and no C code for grad.

    • Returns a dense for perform and a dense for grad.

  • structured_dot.

    • The first input is sparse, the second can be sparse or dense.

    • The grad implemented is structured.

    • C code for perform and grad.

    • It returns a sparse output if both inputs are sparse and dense one if one of the inputs is dense.

    • Returns a sparse grad for sparse inputs and dense grad for dense inputs.

  • true_dot.

    • The first input is sparse, the second can be sparse or dense.

    • The grad implemented is regular.

    • No C code for perform and no C code for grad.

    • Returns a Sparse.

    • The gradient returns a Sparse for sparse inputs and by default a dense for dense inputs. The parameter grad_preserves_dense can be set to False to return a sparse grad for dense inputs.

  • sampling_dot.

    • Both inputs must be dense.

    • The grad implemented is structured for p.

    • Sample of the dot and sample of the gradient.

    • C code for perform but not for grad.

    • Returns sparse for perform and grad.

  • usmm.

    • You shouldn’t insert this op yourself!

      • There is a rewrite that transforms a dot to Usmm when possible.

    • This Op is the equivalent of gemm for sparse dot.

    • There is no grad implemented for this Op.

    • One of the inputs must be sparse, the other sparse or dense.

    • Returns a dense from perform.

• Slice Operations

  • sparse_variable[N, N], returns a tensor scalar. There is no grad implemented for this operation.

  • sparse_variable[M:N, O:P], returns a sparse matrix There is no grad implemented for this operation.

  • Sparse variables don’t support [M, N:O] and [M:N, O] as we don’t support sparse vectors and returning a sparse matrix would break the numpy interface. Use [M:M+1, N:O] and [M:N, O:O+1] instead.

  • diag. The grad implemented is regular.

• Concatenation

  • hstack. The grad implemented is regular.

  • vstack. The grad implemented is regular.

• Probability

  There is no grad implemented for these operations.

  • Poisson and poisson

  • Binomial and csc_fbinomial, csc_dbinomial csr_fbinomial, csr_dbinomial

  • Multinomial and multinomial

• Internal Representation

  They all have a regular grad implemented.

  • ensure_sorted_indices.

  • remove0.

  • clean to resort indices and remove zeros

• To help testing

  • tests.sparse.test_basic.sparse_random_inputs()

sparse – Sparse Op

Classes for handling sparse matrices.

To read about different sparse formats, see http://www-users.cs.umn.edu/~saad/software/SPARSKIT/paper.ps

TODO: Automatic methods for determining best sparse format?

class pytensor.sparse.basic.CSM(format, kmap=None)

Construct a CSM matrix from constituent parts.

Notes

The grad method returns a dense vector, so it provides a regular grad.

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(data, indices, indptr, shape)

Parameters:

• data – One dimensional tensor representing the data of the sparse matrix to construct.

• indices – One dimensional tensor of integers representing the indices of the sparse matrix to construct.

• indptr – One dimensional tensor of integers representing the indice pointer for the sparse matrix to construct.

• shape – One dimensional tensor of integers representing the shape of the sparse matrix to construct.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.CSMGrad(kmap=None)

Compute the gradient of a CSM.

Note

CSM creates a matrix from data, indices, and indptr vectors; it’s gradient is the gradient of the data vector only. There are two complexities to calculate this gradient:

1. The gradient may be sparser than the input matrix defined by (data, indices, indptr). In this case, the data vector of the gradient will have less elements than the data vector of the input because sparse formats remove 0s. Since we are only returning the gradient of the data vector, the relevant 0s need to be added back. 2. The elements in the sparse dimension are not guaranteed to be sorted. Therefore, the input data vector may have a different order than the gradient data vector.

make_node(x_data, x_indices, x_indptr, x_shape, g_data, g_indices, g_indptr, g_shape)

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:

node – The constructed Apply node.

Return type:

Apply

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.CSMProperties(kmap=None)

Create arrays containing all the properties of a given sparse matrix.

More specifically, this Op extracts the .data, .indices, .indptr and .shape fields.

For specific field, csm_data, csm_indices, csm_indptr and csm_shape are provided.

Notes

The grad implemented is regular, i.e. not structured. infer_shape method is not available for this Op.

We won’t implement infer_shape for this op now. This will ask that we implement an GetNNZ op, and this op will keep the dependence on the input of this op. So this won’t help to remove computations in the graph. To remove computation, we will need to make an infer_sparse_pattern feature to remove computations. Doing this is trickier then the infer_shape feature. For example, how do we handle the case when some op create some 0 values? So there is dependence on the values themselves. We could write an infer_shape for the last output that is the shape, but I dough this will get used.

We don’t return a view of the shape, we create a new ndarray from the shape tuple.

grad(inputs, g)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(csm)

The output vectors correspond to the tuple (data, indices, indptr, shape), i.e. the properties of a csm array.

Parameters:

csm – Sparse matrix in CSR or CSC format.

perform(node, inputs, out)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

view_map = {0: [0], 1: [0], 2: [0]}

A dict that maps output indices to the input indices of which they are a view.

Examples

view_map = {0: [1]} # first output is a view of second input
view_map = {1: [0]} # second output is a view of first input

class pytensor.sparse.basic.Cast(out_type)

grad(inputs, outputs_gradients)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x)

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:

node – The constructed Apply node.

Return type:

Apply

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.ColScaleCSC

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x, s)

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:

node – The constructed Apply node.

Return type:

Apply

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.ConstructSparseFromList

Constructs a sparse matrix out of a list of 2-D matrix rows.

Notes

The grad implemented is regular, i.e. not structured.

R_op(inputs, eval_points)

Construct a graph for the R-operator.

This method is primarily used by Rop.

Parameters:

• inputs – The Op inputs.

• eval_points – A Variable or list of Variables with the same length as inputs. Each element of eval_points specifies the value of the corresponding input at the point where the R-operator is to be evaluated.

Return type:

rval[i] should be Rop(f=f_i(inputs), wrt=inputs, eval_points=eval_points).

grad(inputs, grads)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x, values, ilist)

This creates a sparse matrix with the same shape as x. Its values are the rows of values moved. It operates similar to the following pseudo-code:

output = csc_matrix.zeros_like(x, dtype=values.dtype)
for in_idx, out_idx in enumerate(ilist):
    output[out_idx] = values[in_idx]

Parameters:

• x – A dense matrix that specifies the output shape.

• values – A dense matrix with the values to use for output.

• ilist – A dense vector with the same length as the number of rows of values. It specifies where in the output to put the corresponding rows.

perform(node, inp, out_)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.DenseFromSparse(structured=True)

Convert a sparse matrix to a dense one.

Notes

The grad implementation can be controlled through the constructor via the structured parameter. True will provide a structured grad while False will provide a regular grad. By default, the grad is structured.

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x)

Parameters:

x – A sparse matrix.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.Diag

Extract the diagonal of a square sparse matrix as a dense vector.

Notes

The grad implemented is regular, i.e. not structured, since the output is a dense vector.

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x)

Parameters:

x – A square sparse matrix in csc format.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.EnsureSortedIndices(inplace)

Re-sort indices of a sparse matrix.

CSR column indices are not necessarily sorted. Likewise for CSC row indices. Use ensure_sorted_indices when sorted indices are required (e.g. when passing data to other libraries).

Notes

The grad implemented is regular, i.e. not structured.

grad(inputs, output_grad)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x)

Parameters:

x – A sparse matrix.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.GetItem2Lists

Select elements of sparse matrix, returning them in a vector.

grad(inputs, g_outputs)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x, ind1, ind2)

Parameters:

• x – Sparse matrix.

• index – List of two lists, first list indicating the row of each element and second list indicating its column.

perform(node, inp, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.GetItem2ListsGrad

make_node(x, ind1, ind2, gz)

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:

node – The constructed Apply node.

Return type:

Apply

perform(node, inp, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.GetItem2d

Implement a subtensor of sparse variable, returning a sparse matrix.

If you want to take only one element of a sparse matrix see GetItemScalar that returns a tensor scalar.

Notes

Subtensor selection always returns a matrix, so indexing with [a:b, c:d] is forced. If one index is a scalar, for instance, x[a:b, c] or x[a, b:c], an error will be raised. Use instead x[a:b, c:c+1] or x[a:a+1, b:c].

The above indexing methods are not supported because the return value would be a sparse matrix rather than a sparse vector, which is a deviation from numpy indexing rule. This decision is made largely to preserve consistency between numpy and pytensor. This may be revised when sparse vectors are supported.

The grad is not implemented for this op.

make_node(x, index)

Parameters:

• x – Sparse matrix.

• index – Tuple of slice object.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.GetItemList

Select row of sparse matrix, returning them as a new sparse matrix.

grad(inputs, g_outputs)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x, index)

Parameters:

• x – Sparse matrix.

• index – List of rows.

perform(node, inp, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.GetItemListGrad

make_node(x, index, gz)

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:

node – The constructed Apply node.

Return type:

Apply

perform(node, inp, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.GetItemScalar

Subtensor of a sparse variable that takes two scalars as index and returns a scalar.

If you want to take a slice of a sparse matrix see GetItem2d that returns a sparse matrix.

Notes

The grad is not implemented for this op.

make_node(x, index)

Parameters:

• x – Sparse matrix.

• index – Tuple of scalars.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.HStack(format=None, dtype=None)

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

perform(node, block, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.Neg

Negative of the sparse matrix (i.e. multiply by -1).

Notes

The grad is regular, i.e. not structured.

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x)

Parameters:

x – Sparse matrix.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.Remove0(inplace=False)

Remove explicit zeros from a sparse matrix.

Notes

The grad implemented is regular, i.e. not structured.

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x)

Parameters:

x – Sparse matrix.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.RowScaleCSC

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x, s)

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:

node – The constructed Apply node.

Return type:

Apply

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

view_map = {0: [0]}

A dict that maps output indices to the input indices of which they are a view.

Examples

view_map = {0: [1]} # first output is a view of second input
view_map = {1: [0]} # second output is a view of first input

class pytensor.sparse.basic.SparseFromDense(format)

Convert a dense matrix to a sparse matrix.

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x)

Parameters:

x – A dense matrix.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.SquareDiagonal

Produce a square sparse (csc) matrix with a diagonal given by a dense vector.

Notes

The grad implemented is regular, i.e. not structured.

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(diag)

Parameters:

x – Dense vector for the diagonal.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

class pytensor.sparse.basic.Stack(format=None, dtype=None)

make_node(*mat)

Construct an Apply node that represent the application of this operation to the given inputs.

This must be implemented by sub-classes.

Returns:

node – The constructed Apply node.

Return type:

Apply

class pytensor.sparse.basic.Transpose

Transpose of a sparse matrix.

Notes

The returned matrix will not be in the same format. csc matrix will be changed in csr matrix and csr matrix in csc matrix.

The grad is regular, i.e. not structured.

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

make_node(x)

Parameters:

x – Sparse matrix.

perform(node, inputs, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

view_map = {0: [0]}

A dict that maps output indices to the input indices of which they are a view.

Examples

view_map = {0: [1]} # first output is a view of second input
view_map = {1: [0]} # second output is a view of first input

class pytensor.sparse.basic.VStack(format=None, dtype=None)

grad(inputs, gout)

Construct a graph for the gradient with respect to each input variable.

Each returned Variable represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output. If the output is not differentiable with respect to an input, then this method should return an instance of type NullType for that input.

Using the reverse-mode AD characterization given in [1]_, for a \(C = f(A, B)\) representing the function implemented by the Op and its two arguments \(A\) and \(B\), given by the Variables in inputs, the values returned by Op.grad represent the quantities \(\bar{A} \equiv \frac{\partial S_O}{A}\) and \(\bar{B}\), for some scalar output term \(S_O\) of \(C\) in

\[\operatorname{Tr}\left(\bar{C}^\top dC\right) = \operatorname{Tr}\left(\bar{A}^\top dA\right) + \operatorname{Tr}\left(\bar{B}^\top dB\right)\]

Parameters:

• inputs – The input variables.

• output_grads – The gradients of the output variables.

Returns:

The gradients with respect to each Variable in inputs.

Return type:

grads

References

[1]

Giles, Mike. 2008. “An Extended Collection of Matrix Derivative Results for Forward and Reverse Mode Automatic Differentiation.”

perform(node, block, outputs)

Calculate the function on the inputs and put the variables in the output storage.

Parameters:

• node – The symbolic Apply node that represents this computation.

• inputs – Immutable sequence of non-symbolic/numeric inputs. These are the values of each Variable in node.inputs.

• output_storage – List of mutable single-element lists (do not change the length of these lists). Each sub-list corresponds to value of each Variable in node.outputs. The primary purpose of this method is to set the values of these sub-lists.

Notes

The output_storage list might contain data. If an element of output_storage is not None, it has to be of the right type, for instance, for a TensorVariable, it has to be a NumPy ndarray with the right number of dimensions and the correct dtype. Its shape and stride pattern can be arbitrary. It is not guaranteed that such pre-set values were produced by a previous call to this Op.perform(); they could’ve been allocated by another Op’s perform method. An Op is free to reuse output_storage as it sees fit, or to discard it and allocate new memory.

pytensor.sparse.basic.as_sparse(x, name=None, ndim=None, **kwargs)

Wrapper around SparseVariable constructor to construct a Variable with a sparse matrix with the same dtype and format.

Parameters:

x – A sparse matrix.

Returns:

SparseVariable version of x.

Return type:

object

pytensor.sparse.basic.as_sparse_variable(x, name=None, ndim=None, **kwargs)

Wrapper around SparseVariable constructor to construct a Variable with a sparse matrix with the same dtype and format.

Parameters:

x – A sparse matrix.

Returns:

SparseVariable version of x.

Return type:

object

pytensor.sparse.basic.cast(variable, dtype)

Cast sparse variable to the desired dtype.

Parameters:

• variable – Sparse matrix.

• dtype – The dtype wanted.

Return type:

Same as x but having dtype as dtype.

Notes

The grad implemented is regular, i.e. not structured.

pytensor.sparse.basic.clean(x)

Remove explicit zeros from a sparse matrix, and re-sort indices.

CSR column indices are not necessarily sorted. Likewise for CSC row indices. Use clean when sorted indices are required (e.g. when passing data to other libraries) and to ensure there are no zeros in the data.

Parameters:

x – A sparse matrix.

Returns:

The same as x with indices sorted and zeros removed.

Return type:

A sparse matrix

Notes

The grad implemented is regular, i.e. not structured.

pytensor.sparse.basic.col_scale(x, s)

Scale each columns of a sparse matrix by the corresponding element of a dense vector.

Parameters:

• x – A sparse matrix.

• s – A dense vector with length equal to the number of columns of x.

Returns:

• A sparse matrix in the same format as x which each column had been

• multiply by the corresponding element of s.

Notes

The grad implemented is structured.

pytensor.sparse.basic.csm_data(csm)

Return the data field of the sparse variable.

pytensor.sparse.basic.csm_grad

alias of CSMGrad

pytensor.sparse.basic.csm_indices(csm)

Return the indices field of the sparse variable.

pytensor.sparse.basic.csm_indptr(csm)

Return the indptr field of the sparse variable.

pytensor.sparse.basic.csm_shape(csm)

Return the shape field of the sparse variable.

pytensor.sparse.basic.hstack(blocks, format=None, dtype=None)

Stack sparse matrices horizontally (column wise).

This wrap the method hstack from scipy.

Parameters:

• blocks – List of sparse array of compatible shape.

• format – String representing the output format. Default is csc.

• dtype – Output dtype.

Returns:

The concatenation of the sparse array column wise.

Return type:

array

Notes

The number of line of the sparse matrix must agree.

The grad implemented is regular, i.e. not structured.

pytensor.sparse.basic.row_scale(x, s)

Scale each row of a sparse matrix by the corresponding element of a dense vector.

Parameters:

• x – A sparse matrix.

• s – A dense vector with length equal to the number of rows of x.

Returns:

A sparse matrix in the same format as x whose each row has been multiplied by the corresponding element of s.

Return type:

A sparse matrix

Notes

The grad implemented is structured.

pytensor.sparse.basic.sp_ones_like(x)

Construct a sparse matrix of ones with the same sparsity pattern.

Parameters:

x – Sparse matrix to take the sparsity pattern.

Returns:

The same as x with data changed for ones.

Return type:

A sparse matrix

pytensor.sparse.basic.sp_zeros_like(x)

Construct a sparse matrix of zeros.

Parameters:

x – Sparse matrix to take the shape.

Returns:

The same as x with zero entries for all element.

Return type:

A sparse matrix

pytensor.sparse.basic.sparse_formats = ['csc', 'csr']

Types of sparse matrices to use for testing.

pytensor.sparse.basic.vstack(blocks, format=None, dtype=None)

Stack sparse matrices vertically (row wise).

This wrap the method vstack from scipy.

Parameters:

• blocks – List of sparse array of compatible shape.

• format – String representing the output format. Default is csc.

• dtype – Output dtype.

Returns:

The concatenation of the sparse array row wise.

Return type:

array

Notes

The number of column of the sparse matrix must agree.

The grad implemented is regular, i.e. not structured.

pytensor.sparse.sparse_grad(var)

This function return a new variable whose gradient will be stored in a sparse format instead of dense.

Currently only variable created by AdvancedSubtensor1 is supported. i.e. a_tensor_var[an_int_vector].

New in version 0.6rc4.