def approximate_cholesky(self, epsilon=1e-6):
r""" Compute low-rank approximation to the Cholesky decomposition of target matrix.
The decomposition will be conducted while ensuring that the spectrum of `A_k^{-1}` is positive.
Parameters
----------
epsilon : float, optional, default 1e-6
Cutoff for eigenvalue norms. If negative eigenvalues occur, with norms larger than epsilon, the largest
negative eigenvalue norm will be used instead of epsilon, i.e. a band including all negative eigenvalues
will be cut off.
Returns
-------
L : ndarray((n,m), dtype=float)
Cholesky matrix such that `A \approx L L^{\top}`. Number of columns :math:`m` is most at the number of columns
used in the Nystroem approximation, but may be smaller depending on epsilon.
"""
# compute the Eigenvalues of C0 using Schur factorization
Wk = self._C_k[self._columns, :]
L0 = spd_inv_split(Wk, epsilon=epsilon)
L = np.dot(self._C_k, L0)
return L
def _diagonalize(self):
"""Performs SVD on covariance matrices and save left, right singular vectors and values in the model.
Parameters
----------
scaling : None or string, default=None
Scaling to be applied to the VAMP modes upon transformation
* None: no scaling will be applied, variance of the singular
functions is 1
* 'kinetic map' or 'km': singular functions are scaled by
singular value. Note that only the left singular functions
induce a kinetic map.
"""
L0 = spd_inv_split(self.C00, epsilon=self.epsilon)
self._rank0 = L0.shape[1] if L0.ndim == 2 else 1
Lt = spd_inv_split(self.Ctt, epsilon=self.epsilon)
self._rankt = Lt.shape[1] if Lt.ndim == 2 else 1
W = np.dot(L0.T, self.C0t).dot(Lt)
from scipy.linalg import svd
A, s, BT = svd(W, compute_uv=True, lapack_driver='gesvd')
self._singular_values = s
# don't pass any values in the argument list that call _diagonalize again!!!
m = VAMPModel._dimension(self._rank0, self._rankt, self.dim,
self._singular_values)
U = np.dot(L0, A[:, :m])
V = np.dot(Lt, BT[:m, :].T)
# scale vectors
if self.scaling is not None:
U *= s[np.newaxis,
0:m] # scaled left singular functions induce a kinetic map
V *= s[
np.newaxis, 0:
m] # scaled right singular functions induce a kinetic map wrt. backward propagator
self._U = U
self._V = V
self._svd_performed = True
def _finish_estimation(self):
R = spd_inv_split(self._covar.cov, epsilon=self.epsilon, canonical_signs=True)
# Set the new correlation matrix:
M = R.shape[1]
K = np.dot(R.T, np.dot((self._covar.cov_tau), R))
K = np.vstack((K, np.dot((self._covar.mean_tau - self._covar.mean), R)))
ex1 = np.zeros((M + 1, 1))
ex1[M, 0] = 1.0
self._K = np.hstack((K, ex1))
self._R = R
self._estimation_finished = True
self._estimated = True