tensor.nlinalg – Linear Algebra Ops Using Numpy

Note

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

API

pytensor.tensor.nlinalg.kron(a, b)

Kronecker product.

Same as np.kron(a, b)

Parameters:

• a (array_like) –

• b (array_like) –

Return type:

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

pytensor.tensor.nlinalg.matrix_dot(*args)

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 \(A_0 \cdot A_1 \cdot A_2 \cdot .. \cdot A_N\).

pytensor.tensor.nlinalg.matrix_power(M, n)

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

This implementation uses exponentiation by squaring which is significantly faster than the naive implementation. The time complexity for exponentiation by squaring is :math: mathcal{O}((n log M)^k)

Parameters:

• M (TensorVariable) –

• n (int) –

pytensor.tensor.nlinalg.norm(x, ord=None, axis=None, keepdims=False)

Matrix or vector norm.

Parameters:

• x (TensorVariable) – Tensor to take norm of.

• ord (float, str or int, optional) –

  Order of norm. If ord is a str, it must be one of the following:

  • ’fro’ or ‘f’ : Frobenius norm

  • ’nuc’ : nuclear norm

  • ’inf’ : Infinity norm

  • ’-inf’ : Negative infinity norm

  If an integer, order can be one of -2, -1, 0, 1, or 2. Otherwise ord must be a float.

  Default is the Frobenius (L2) norm.

• axis (tuple of int, optional) – Axes over which to compute the norm. If None, norm of entire matrix (or vector) is computed. Row or column norms can be computed by passing a single integer; this will treat a matrix like a batch of vectors.

• keepdims (bool) – If True, dummy axes will be inserted into the output so that norm.dnim == x.dnim. Default is False.

Returns:

Norm of x along axes specified by axis.

Return type:

TensorVariable

Notes

Batched dimensions are supported to the left of the core dimensions. For example, if x is a 3D tensor with shape (2, 3, 4), then norm(x) will compute the norm of each 3x4 matrix in the batch.

If the input is a 2D tensor and should be treated as a batch of vectors, the axis argument must be specified.

pytensor.tensor.nlinalg.pinv(x, hermitian=False)

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.

pytensor.tensor.nlinalg.qr(a, mode='reduced')

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.

pytensor.tensor.nlinalg.slogdet(x)

Compute the sign and (natural) logarithm of the determinant of an array.

Returns a naive graph which is optimized later using rewrites with the det operation.

Parameters:

x ((..., M, M) tensor or tensor_like) – Input tensor, has to be square.

Returns:

• A tuple with the following attributes

• sign ((…) tensor_like) – A number representing the sign of the determinant. For a real matrix, this is 1, 0, or -1.

• logabsdet ((…) tensor_like) – The natural log of the absolute value of the determinant.

• If the determinant is zero, then sign will be 0 and logabsdet

• will be -inf. In all cases, the determinant is equal to

• sign * exp(logabsdet).

pytensor.tensor.nlinalg.svd(a, full_matrices=True, compute_uv=True)

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

pytensor.tensor.nlinalg.tensorinv(a, ind=2)

PyTensor 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).

pytensor.tensor.nlinalg.tensorsolve(a, b, axes=None)

PyTensor utilization of numpy.linalg.tensorsolve.

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).

pytensor.tensor.nlinalg.trace(X)

Returns the sum of diagonal elements of matrix X.