`tensor.nlinalg` – Linear Algebra Ops Using Numpy¶

Note

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

API¶

class theano.tensor.nlinalg.AllocDiag[source]¶

Allocates a square matrix with the given vector as its diagonal.

make_node(_x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.Det[source]¶

Matrix determinant. Input should be a square matrix.

make_node(x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.Eig[source]¶

Compute the eigenvalues and right eigenvectors of a square array.

make_node(x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.Eigh(UPLO='L')[source]¶

Return the eigenvalues and eigenvectors of a Hermitian or symmetric matrix.

grad(inputs, g_outputs)[source]¶

The gradient function should return

$\sum_n\left(W_n\frac{\partial\,w_n} {\partial a_{ij}} + \sum_k V_{nk}\frac{\partial\,v_{nk}} {\partial a_{ij}}\right),$

where [ $W$ , $V$ ] corresponds to g_outputs, $a$ to inputs, and $(w, v)=\mbox{eig}(a)$ .

Analytic formulae for eigensystem gradients are well-known in perturbation theory:

$\frac{\partial\,w_n} {\partial a_{ij}} = v_{in}\,v_{jn}$

$\frac{\partial\,v_{kn}} {\partial a_{ij}} = \sum_{m\ne n}\frac{v_{km}v_{jn}}{w_n-w_m}$

make_node(x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.EighGrad(UPLO='L')[source]¶

Gradient of an eigensystem of a Hermitian matrix.

make_node(x, w, v, gw, gv)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶: Implements the “reverse-mode” gradient for the eigensystem of a square matrix.

class theano.tensor.nlinalg.MatrixInverse[source]¶

Computes the inverse of a matrix $A$ .

Given a square matrix $A$ , matrix_inverse returns a square matrix $A_{inv}$ such that the dot product $A \cdot A_{inv}$ and $A_{inv} \cdot A$ equals the identity matrix $I$ .

Notes

When possible, the call to this op will be optimized to the call of solve.

R_op(inputs, eval_points)[source]¶

The gradient function should return

$\frac{\partial X^{-1}}{\partial X}V,$

where $V$ corresponds to g_outputs and $X$ to inputs. Using the matrix cookbook, one can deduce that the relation corresponds to

$X^{-1} \cdot V \cdot X^{-1}.$

grad(inputs, g_outputs)[source]¶

The gradient function should return

$V\frac{\partial X^{-1}}{\partial X},$

where $V$ corresponds to g_outputs and $X$ to inputs. Using the matrix cookbook, one can deduce that the relation corresponds to

$(X^{-1} \cdot V^{T} \cdot X^{-1})^T.$

make_node(x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.MatrixPinv[source]¶

Computes the pseudo-inverse of a matrix $A$ .

The pseudo-inverse of a matrix $A$ , denoted $A^+$ , is defined as: “the matrix that ‘solves’ [the least-squares problem] $Ax = b$ ,” i.e., if $\bar{x}$ is said solution, then $A^+$ is that matrix such that $\bar{x} = A^+b$ .

Note that $Ax=AA^+b$ , so $AA^+$ is close to the identity matrix. This method is not faster than matrix_inverse. Its strength comes from that it works for non-square matrices. If you have a square matrix though, matrix_inverse can be both more exact and faster to compute. Also this op does not get optimized into a solve op.

L_op(inputs, outputs, g_outputs)[source]¶

The gradient function should return

$V\frac{\partial X^+}{\partial X},$

where $V$ corresponds to g_outputs and $X$ to inputs. According to Wikipedia, this corresponds to

$(-X^+ V^T X^+ + X^+ X^{+T} V (I - X X^+) + (I - X^+ X) V X^{+T} X^+)^T.$

make_node(x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.QRFull(mode)[source]¶

Full QR Decomposition.

Computes the QR decomposition of a matrix. Factor the matrix a as qr, where q is orthonormal and r is upper-triangular.

make_node(x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.QRIncomplete(mode)[source]¶

Incomplete QR Decomposition.

Computes the QR decomposition of a matrix. Factor the matrix a as qr and return a single matrix R.

make_node(x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.SVD(full_matrices=True, compute_uv=True)[source]¶

Parameters

full_matrices (bool, optional) – If True (default), u and v have the shapes (M, M) and (N, N), respectively. Otherwise, the shapes are (M, K) and (K, N), respectively, where K = min(M, N).
compute_uv (bool, optional) – Whether or not to compute u and v in addition to s. True by default.

make_node(x)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.TensorInv(ind=2)[source]¶

Class wrapper for tensorinv() function; Theano utilization of numpy.linalg.tensorinv;

make_node(a)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.TensorSolve(axes=None)[source]¶

Theano utilization of numpy.linalg.tensorsolve Class wrapper for tensorsolve function.

make_node(a, b)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.diag(x)[source]¶

Numpy-compatibility method If x is a matrix, return its diagonal. If x is a vector return a matrix with it as its diagonal.

This method does not support the k argument that numpy supports.

class theano.tensor.nlinalg.lstsq[source]¶

make_node(x, y, rcond)[source]¶: Create a “apply” nodes for the inputs in that order.

perform(node, inputs, outputs)[source]¶

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.nlinalg.matrix_dot(*args)[source]¶

Shorthand for product between several dots.

Given $N$ matrices $A_0, A_1, .., A_N$ , matrix_dot will generate the matrix product between all in the given order, namely $A_0 \cdot A_1 \cdot A_2 \cdot .. \cdot A_N$ .

theano.tensor.nlinalg.matrix_power(M, n)[source]¶

Raise a square matrix to the (integer) power n.

Parameters

M (Tensor variable) –
n (Python int) –

theano.tensor.nlinalg.qr(a, mode='reduced')[source]¶

Computes the QR decomposition of a matrix. Factor the matrix a as qr, where q is orthonormal and r is upper-triangular.

Parameters

a (array_like, shape (M, N)) – Matrix to be factored.
mode ({'reduced', 'complete', 'r', 'raw'}, optional) –
If K = min(M, N), then

’reduced’
returns q, r with dimensions (M, K), (K, N)

’complete’
returns q, r with dimensions (M, M), (M, N)

’r’
returns r only with dimensions (K, N)

’raw’
returns h, tau with dimensions (N, M), (K,)

Note that array h returned in ‘raw’ mode is transposed for calling Fortran.

Default mode is ‘reduced’

Returns

q (matrix of float or complex, optional) – A matrix with orthonormal columns. When mode = ‘complete’ the result is an orthogonal/unitary matrix depending on whether or not a is real/complex. The determinant may be either +/- 1 in that case.
r (matrix of float or complex, optional) – The upper-triangular matrix.

theano.tensor.nlinalg.svd(a, full_matrices=1, compute_uv=1)[source]¶

This function performs the SVD on CPU.

Parameters

full_matrices (bool, optional) – If True (default), u and v have the shapes (M, M) and (N, N), respectively. Otherwise, the shapes are (M, K) and (K, N), respectively, where K = min(M, N).
compute_uv (bool, optional) – Whether or not to compute u and v in addition to s. True by default.

Returns

U, V, D

Return type

matrices

theano.tensor.nlinalg.tensorinv(a, ind=2)[source]¶

Does not run on GPU; Theano utilization of numpy.linalg.tensorinv;

Compute the ‘inverse’ of an N-dimensional array. The result is an inverse for a relative to the tensordot operation tensordot(a, b, ind), i. e., up to floating-point accuracy, tensordot(tensorinv(a), a, ind) is the “identity” tensor for the tensordot operation.

Parameters

a (array_like) – Tensor to ‘invert’. Its shape must be ‘square’, i. e., prod(a.shape[:ind]) == prod(a.shape[ind:]).
ind (int, optional) – Number of first indices that are involved in the inverse sum. Must be a positive integer, default is 2.

Returns

b – a’s tensordot inverse, shape a.shape[ind:] + a.shape[:ind].

Return type

ndarray

Raises

LinAlgError – If a is singular or not ‘square’ (in the above sense).

theano.tensor.nlinalg.tensorsolve(a, b, axes=None)[source]¶

Theano utilization of numpy.linalg.tensorsolve. Does not run on GPU!

Solve the tensor equation a x = b for x. It is assumed that all indices of x are summed over in the product, together with the rightmost indices of a, as is done in, for example, tensordot(a, x, axes=len(b.shape)).

Parameters

a (array_like) – Coefficient tensor, of shape b.shape + Q. Q, a tuple, equals the shape of that sub-tensor of a consisting of the appropriate number of its rightmost indices, and must be such that prod(Q) == prod(b.shape) (in which sense a is said to be ‘square’).
b (array_like) – Right-hand tensor, which can be of any shape.
axes (tuple of ints, optional) – Axes in a to reorder to the right, before inversion. If None (default), no reordering is done.

Returns

x

Return type

ndarray, shape Q

Raises

LinAlgError – If a is singular or not ‘square’ (in the above sense).

theano.tensor.nlinalg.trace(X)[source]¶

Returns the sum of diagonal elements of matrix X.

Notes

Works on GPU since 0.6rc4.

tensor.nlinalg – Linear Algebra Ops Using Numpy¶

API¶

`tensor.nlinalg` – Linear Algebra Ops Using Numpy¶