covmats.SparseCholeskyFactor.colorize_inv_adjoint

SparseCholeskyFactor.colorize_inv_adjoint(y: ndarray[tuple[Any, ...], dtype[floating]]) → ndarray[tuple[Any, ...], dtype[floating]]

Perform the adjoint (transpose) of SparseCholeskyFactor.colorize_inv().

Parameters:

y (array_like) – An array of points in data space, with the last dimension equal to n_pts.

Returns:

y_ – The transformed array, same shape as y.

Return type:

array_like

Note

Writing colorize_inv() as the linear map \(\mathrm{colorize\_inv}(x) = \mathbf{P}^{T} \mathbf{L}^{-T} \mathbf{D}^{-1/2} x\) (colorize_inv() solves \(\mathbf{L}^{T} z = \mathbf{D}^{-1/2} x\) then applies \(\mathbf{P}^{T}\)), its adjoint is \(\mathrm{colorize\_inv}^{T}(y) = \mathbf{D}^{-1/2} \mathbf{L}^{-1} \mathbf{P} y\), obtained here by forward-solving \(\mathbf{L} z = \mathbf{P} y\) (lower-triangular, unlike the backward solve used by colorize_inv() itself) and then applying sqrtinvD.

Examples

>>> import numpy as np
>>> import scipy as sp
>>> import covmats
>>> rng = np.random.default_rng(0)
>>> n = 4
>>> B = rng.random(size=(n, n))
>>> cov_array = B @ B.T + n * np.eye(n)
>>> L, D, P = sp.linalg.ldl(cov_array)
>>> scf = covmats.SparseCholeskyFactor(
...     sp.sparse.csc_array(L), sp.sparse.csc_array(D), P
... )
>>> x = rng.standard_normal(n)
>>> y = rng.standard_normal(n)
>>> np.allclose(
...     scf.colorize_inv(x[None, :])[0] @ y,
...     x @ scf.colorize_inv_adjoint(y[None, :])[0],
... )
True