lab.linear_algebra module

lab.linear_algebra.T

Shorthand for transpose.

lab.linear_algebra.chol

Shorthand for cholesky.

lab.linear_algebra.cholesky

Compute the Cholesky decomposition.

Parameters

a (tensor) – Matrix to decompose.

Returns

Cholesky decomposition.

Return type

tensor

lab.linear_algebra.cholesky_solve

Solve the linear system a x = b given the Cholesky factorisation of a.

Parameters

• a (tensor) – Cholesky factorisation of a.

• b (tensor) – RHS b.

Returns

Solution x.

Return type

tensor

lab.linear_algebra.cholsolve

Shorthand for cholesky_solve.

lab.linear_algebra.det

Compute the determinant of a.

Parameters

a (tensor) – Matrix to compute determinant of.

Returns

Determinant of a

Return type

scalar

lab.linear_algebra.dot

Shorthand for matmul.

lab.linear_algebra.epsilon = 1e-12

Magnitude of diagonal to regularise matrices with.

lab.linear_algebra.ew_dists

Compute the Euclidean norm of the element-wise differences between two matrices where rows correspond to elements and columns to features.

Parameters

• a (matrix) – First matrix.

• b (matrix, optional) – Second matrix. Defaults to a.

Returns

Euclidean norm of the element-wise differences between the

elements of a and b.

Return type

matrix

lab.linear_algebra.ew_dists2

Compute the square the Euclidean norm of the element-wise differences between two matrices where rows correspond to elements and columns to features.

Parameters

• a (matrix) – First matrix.

• b (matrix, optional) – Second matrix. Defaults to a.

Returns

Square of the Euclidean norm of the element-wise differences

between the elements of a and b.

Return type

matrix

lab.linear_algebra.ew_sums

Compute the Euclidean norm of the element-wise sums between two matrices where rows correspond to elements and columns to features.

Parameters

• a (matrix) – First matrix.

• b (matrix, optional) – Second matrix. Defaults to a.

Returns

Euclidean norm of the element-wise sums between the

elements of a and b.

Return type

matrix

lab.linear_algebra.ew_sums2

Compute the square the Euclidean norm of the element-wise sums between two matrices where rows correspond to elements and columns to features.

Parameters

• a (matrix) – First matrix.

• b (matrix, optional) – Second matrix. Defaults to a.

Returns

Square of the Euclidean norm of the element-wise sums

between the elements of a and b.

Return type

matrix

lab.linear_algebra.expm

Compute the matrix exponential of a.

Parameters

a (tensor) – Matrix to matrix exponential of.

Returns

Matrix exponential of a

Return type

scalar

lab.linear_algebra.inv

Compute the inverse of a.

Parameters

a (tensor) – Matrix to compute inverse of.

Returns

Inverse of a.

Return type

tensor

lab.linear_algebra.kron

Kronecker product.

Parameters

• a (tensor) – First matrix.

• b (tensor) – Second matrix.

Returns

Kronecker product of a and b.

Return type

tensor

lab.linear_algebra.logdet

Compute the log-determinant of a.

Parameters

a (tensor) – Matrix to compute log-determinant of.

Returns

Log-determinant of a

Return type

scalar

lab.linear_algebra.logm

Compute the matrix logarithm of a.

Parameters

a (tensor) – Matrix to matrix logarithm of.

Returns

Matrix logarithm of a

Return type

scalar

lab.linear_algebra.matmul

Matrix multiplication.

Parameters

• a (tensor) – First matrix.

• b (tensor) – Second matrix.

• tr_a (bool, optional) – Transpose first matrix. Defaults to False.

• tr_b (bool, optional) – Transpose second matrix. Defaults to False.

Returns

Matrix product of a and b.

Return type

tensor

lab.linear_algebra.mm

Shorthand for matmul.

lab.linear_algebra.outer

Compute the outer product between two vectors or matrices.

Parameters

• a (tensor) – First tensor.

• b (tensor) – Second tensor.

Returns

Outer product of a and b.

Return type

tensor

lab.linear_algebra.pw_dists

Compute the Euclidean norm of the pairwise differences between two matrices where rows correspond to elements and columns to features.

Parameters

• a (matrix) – First matrix.

• b (matrix, optional) – Second matrix. Defaults to a.

Returns

Euclidean norm of the pairwise differences between the

elements of a and b.

Return type

matrix

lab.linear_algebra.pw_dists2

Compute the square the Euclidean norm of the pairwise differences between two matrices where rows correspond to elements and columns to features.

Parameters

• a (matrix) – First matrix.

• b (matrix, optional) – Second matrix. Defaults to a.

Returns

Square of the Euclidean norm of the pairwise differences

between the elements of a and b.

Return type

matrix

lab.linear_algebra.pw_sums

Compute the Euclidean norm of the pairwise sums between two matrices where rows correspond to elements and columns to features.

Parameters

• a (matrix) – First matrix.

• b (matrix, optional) – Second matrix. Defaults to a.

Returns

Euclidean norm of the pairwise sums between the

elements of a and b.

Return type

matrix

lab.linear_algebra.pw_sums2

Compute the square the Euclidean norm of the pairwise sums between two matrices where rows correspond to elements and columns to features.

Parameters

• a (matrix) – First matrix.

• b (matrix, optional) – Second matrix. Defaults to a.

Returns

Square of the Euclidean norm of the pairwise sums

between the elements of a and b.

Return type

matrix

lab.linear_algebra.reg

Add a diagonal to a matrix.

Parameters

• a (matrix) – Matrix to add a diagonal to.

• diag (float, optional) – Magnitude of the diagonal to add. Defaults to .linear_algebra.epsilon.

• clip (bool, optional) – Let diag be at least .linear_algebra.epsilon. Defaults to True.

Returns

Regularised version of a.

Return type

matrix

lab.linear_algebra.solve

Solve the linear system a x = b.

Parameters

• a (tensor) – LHS a.

• b (tensor) – RHS b.

Returns

Solution x.

Return type

tensor

lab.linear_algebra.svd

Compute the singular value decomposition.

Note

PyTorch does not allow batch computation.

Parameters

• a (tensor) – Matrix to decompose.

• compute_uv (bool, optional) – Also compute U and V. Defaults to True.

Returns

(U, S, V) is compute_uv is True and just S otherwise.

Return type

tuple

lab.linear_algebra.t

Shorthand for transpose.

lab.linear_algebra.toeplitz_solve

Solve the linear system toep(a, b) x = c where toep(a, b) is a Toeplitz matrix.

Parameters

• a (tensor) – First column of the Toeplitz matrix.

• b (tensor, optional) – Except for the first element, first row of the Toeplitz matrix. Defaults to a[1:].

• c (tensor) – RHS c.

Returns

Solution x.

Return type

tensor

lab.linear_algebra.toepsolve

Shorthand for toeplitz_solve.

lab.linear_algebra.trace

Compute the trace of a tensor.

Parameters

• a (tensor) – Tensor to compute trace of.

• axis1 (int, optional) – First dimension to compute trace over. Defaults to 0.

• axis2 (int, optional) – Second dimension to compute trace over. Defaults to 1.

Returns

Trace.

Return type

tensor

lab.linear_algebra.transpose

Transpose a matrix.

Parameters

• a (tensor) – Matrix to transpose.

• perm (list[int] or tuple[int], optional) – Permutation. Defaults to switching the last two axes.

Returns

Transposition of a.

Return type

tensor

lab.linear_algebra.triangular_solve

Solve the linear system a x = b where a is triangular.

Parameters

• a (tensor) – Triangular matrix a.

• b (tensor) – RHS b.

• lower_a (bool, optional) – Indicate that a is lower triangular instead of upper triangular. Defaults to True.

Returns

Solution x.

Return type

tensor

lab.linear_algebra.trisolve

Shorthand for triangular_solve.