covmats.CovViaCholesky

class covmats.CovViaCholesky(*args, **kwargs)

Representation of a covariance via a Cholesky factorization.

Parameters:

cholesky (array_like) – The lower triangular Cholesky factor of the covariance matrix.

Notes

Let the covariance matrix be \(A\) and \(L\) be the lower Cholesky factor such that \(L L^T = A\). Whitening of a data point \(x\) is performed by computing \(L^{-1} x\). \(\log\det{A}\) is calculated as \(2tr(\log{L})\), where the \(\log\) operation is performed element-wise.

This Covariance class does not support singular covariance matrices because the Cholesky decomposition does not exist for a singular covariance matrix.

Examples

Prepare a symmetric positive definite covariance matrix A and a data point x.

>>> import numpy as np
>>> import covmats
>>> rng = np.random.default_rng()
>>> n = 5
>>> A = rng.random(size=(n, n))
>>> A = A @ A.T  # make the covariance symmetric positive definite
>>> x = rng.random(size=n)

Perform the Cholesky decomposition of A and create the Covariance object.

>>> L = np.linalg.cholesky(A)
>>> cov = covmats.CovViaCholesky(L)

Compare the functionality of the Covariance object against reference implementation.

>>> from scipy.linalg import solve_triangular
>>> res = cov.whiten(x)
>>> ref = solve_triangular(L, x, lower=True)
>>> np.allclose(res, ref)
True
>>> res = cov.log_pdet
>>> ref = np.linalg.slogdet(A)[-1]
>>> np.allclose(res, ref)
True

__init__(L: _Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | complex | bytes | str | _NestedSequence[complex | bytes | str], covariance: _Buffer | _SupportsArray[dtype[Any]] | _NestedSequence[_SupportsArray[dtype[Any]]] | complex | bytes | str | _NestedSequence[complex | bytes | str] | None = None) → None

Initialize the instance.

Parameters:

• L (NDArrayFloat) – Lower triangle of the covariance matrix Cholesky factorization.

• covariance (Optional[NDArrayFloat], optional) – Dense covariance matrix, by default None

Properties

H	Hermitian adjoint.
L	Lower triangle of the covariance matrix Cholesky factorization.
T	Transpose this linear operator.
covariance	Explicit dense representation of the covariance matrix.
log_pdet	Log of the pseudo-determinant of the covariance matrix.
n_pts	Number of points in the domain (n).
ndim
precision	Explicit dense representation of the precision matrix.
rank	Rank of the covariance matrix.
shape	Shape of the covariance matrix (n, n).
subspace_size	Subspace size of the covariance matrix.

Methods

adjoint	Hermitian adjoint.
colorize	Perform a colorizing transformation on data.
dot	Matrix-matrix or matrix-vector multiplication.
from_cholesky	Representation of a covariance provided via choleksy factorization.
from_diagonal	Representation of a covariance provided via diagonal.
from_eigendecomposition	Representation of a covariance provided via eigendecomposition.
from_precision	Return a representation of a covariance from its precision matrix.
get_diagonal	Return the diagonal entries of the matrix (variances).
get_trace	Return the trace of the covariance matrix (sum of diagonal elements).
matmat	Matrix-matrix multiplication.
matvec	Matrix-vector multiplication.
rmatmat	Adjoint matrix-matrix multiplication.
rmatvec	Adjoint matrix-vector multiplication.
sample_mvnormal	Draw samples from the multivariate normal N(0, A).
solve	Solve Ax = b, with A, the current covariance matrix instance.
todense	Explicit dense representation of the covariance matrix with shape (n, n).
transpose	Transpose this linear operator.
whiten	Perform a whitening transformation on data.