tensor.slinalg – Linear Algebra Ops Using Scipy

Note

This module is not imported by default. You need to import it to use it.

API

class theano.tensor.slinalg.Cholesky(lower=True, on_error='raise')

Return a triangular matrix square root of positive semi-definite x.

L = cholesky(X, lower=True) implies dot(L, L.T) == X.

Parameters

• lower (bool, default=True) – Whether to return the lower or upper cholesky factor

• on_error (['raise', 'nan']) – If on_error is set to ‘raise’, this Op will raise a scipy.linalg.LinAlgError if the matrix is not positive definite. If on_error is set to ‘nan’, it will return a matrix containing nans instead.

L_op(inputs, outputs, gradients)

Cholesky decomposition reverse-mode gradient update.

Symbolic expression for reverse-mode Cholesky gradient taken from 1

References

1

I. Murray, “Differentiation of the Cholesky decomposition”, http://arxiv.org/abs/1602.07527

make_node(x)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters

• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these 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 not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.slinalg.CholeskyGrad(lower=True)

make_node(x, l, dz)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)

Implements the “reverse-mode” gradient 2 for the Cholesky factorization of a positive-definite matrix.

References

2

S. P. Smith. “Differentiation of the Cholesky Algorithm”. Journal of Computational and Graphical Statistics, Vol. 4, No. 2 (Jun.,1995), pp. 134-147 http://www.jstor.org/stable/1390762

class theano.tensor.slinalg.Eigvalsh(lower=True)

Generalized eigenvalues of a Hermitian positive definite eigensystem.

make_node(a, b)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters

• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these 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 not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.slinalg.EigvalshGrad(lower=True)

Gradient of generalized eigenvalues of a Hermitian positive definite eigensystem.

make_node(a, b, gw)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters

• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these 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 not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.slinalg.Expm

Compute the matrix exponential of a square array.

make_node(A)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters

• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these 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 not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.slinalg.ExpmGrad

Gradient of the matrix exponential of a square array.

make_node(A, gw)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters

• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these 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 not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

class theano.tensor.slinalg.Solve(A_structure='general', lower=False, overwrite_A=False, overwrite_b=False)

Solve a system of linear equations.

For on CPU and GPU.

L_op(inputs, outputs, output_gradients)

Reverse-mode gradient updates for matrix solve operation c = A \ b.

Symbolic expression for updates taken from 3.

References

3

M. B. Giles, “An extended collection of matrix derivative results for forward and reverse mode automatic differentiation”, http://eprints.maths.ox.ac.uk/1079/

make_node(A, b)

Create a “apply” nodes for the inputs in that order.

perform(node, inputs, output_storage)

Required: Calculate the function on the inputs and put the variables in the output storage. Return None.

Parameters

• node (Apply instance) – Contains the symbolic inputs and outputs.

• inputs (list) – Sequence of inputs (immutable).

• output_storage (list) – List of mutable 1-element lists (do not change the length of these 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 not is guaranteed that it was produced by a previous call to impl. It could be allocated by another Op impl is free to reuse it as it sees fit, or to discard it and allocate new memory.

Raises

MethodNotDefined – The subclass does not override this method.

theano.tensor.slinalg.kron(a, b)

Kronecker product.

Same as scipy.linalg.kron(a, b).

Parameters

• a (array_like) –

• b (array_like) –

Returns

Return type

array_like with a.ndim + b.ndim - 2 dimensions

Notes

numpy.kron(a, b) != scipy.linalg.kron(a, b)! They don’t have the same shape and order when a.ndim != b.ndim != 2.

theano.tensor.slinalg.solve_symmetric = Solve{('symmetric', False, False, False)}

Optimized implementation of theano.tensor.slinalg.solve() when A is symmetric.

theano.tensor.slinalg.solve(a, b)

Solves the equation a x = b for x, where a is a matrix and b can be either a vector or a matrix.

Note

Parameters

• a ((M, M) symbolix matrix) – A square matrix

• b ((M,) or (M, N) symbolic vector or matrix) – Right hand side matrix in a x = b

Returns

x – x will have the same shape as b

Return type

(M, ) or (M, N) symbolic vector or matrix

theano.tensor.slinalg.solve_lower_triangular(a, b)

Optimized implementation of theano.tensor.slinalg.solve() when A is lower triangular.

theano.tensor.slinalg.solve_upper_triangular(a, b)

Optimized implementation of theano.tensor.slinalg.solve() when A is upper triangular.