"""Random symmetric positive definite matrices."""
from typing import Optional, Sequence
import numpy as np
import scipy.stats
import probnum.utils as _utils
from probnum.type import IntArgType, RandomStateArgType
[docs]def random_spd_matrix(
dim: IntArgType,
spectrum: Sequence = None,
random_state: Optional[RandomStateArgType] = None,
) -> np.ndarray:
"""Random symmetric positive definite matrix.
Constructs a random symmetric positive definite matrix from a given spectrum. An
orthogonal matrix :math:`Q` with :math:`\\operatorname{det}(Q)` (a rotation) is
sampled with respect to the Haar measure and the diagonal matrix
containing the eigenvalues is rotated accordingly resulting in :math:`A=Q
\\operatorname{diag}(\\lambda_1, \\dots, \\lambda_n)Q^\\top`. If no spectrum is
provided, one is randomly drawn from a Gamma distribution.
Parameters
----------
dim
Matrix dimension.
spectrum
Eigenvalues of the matrix.
random_state
Random state of the random variable. If None (or np.random), the global
:mod:`numpy.random` state is used. If integer, it is used to seed the local
:class:`~numpy.random.RandomState` instance.
See Also
--------
random_sparse_spd_matrix : Generate a random sparse symmetric positive definite matrix.
Examples
--------
>>> from probnum.problems.zoo.linalg import random_spd_matrix
>>> mat = random_spd_matrix(dim=5, random_state=0)
>>> mat
array([[10.49868572, -0.80840778, 0.79781892, 1.9229059 , 0.73413367],
[-0.80840778, 15.79117417, 0.52641887, -1.8727916 , -0.9309482 ],
[ 0.79781892, 0.52641887, 15.56457452, 1.26004438, -1.44969733],
[ 1.9229059 , -1.8727916 , 1.26004438, 8.59057287, -0.44955394],
[ 0.73413367, -0.9309482 , -1.44969733, -0.44955394, 9.77198568]])
Check for symmetry and positive definiteness.
>>> np.all(mat == mat.T)
True
>>> np.linalg.eigvals(mat)
array([ 6.93542496, 10.96494454, 9.34928449, 16.25401501, 16.71332395])
"""
# Initialization
random_state = _utils.as_random_state(random_state)
if spectrum is None:
# Create a custom ordered spectrum if none is given.
spectrum_shape: float = 10.0
spectrum_scale: float = 1.0
spectrum_offset: float = 0.0
spectrum = scipy.stats.gamma.rvs(
spectrum_shape,
loc=spectrum_offset,
scale=spectrum_scale,
size=dim,
random_state=random_state,
)
spectrum = np.sort(spectrum)[::-1]
else:
spectrum = np.asarray(spectrum)
if not np.all(spectrum > 0):
raise ValueError(f"Eigenvalues must be positive, but are {spectrum}.")
# Early exit for d=1 -- special_ortho_group does not like this case.
if dim == 1:
return spectrum.reshape((1, 1))
# Draw orthogonal matrix with respect to the Haar measure
orth_mat = scipy.stats.special_ortho_group.rvs(dim, random_state=random_state)
spd_mat = orth_mat @ np.diag(spectrum) @ orth_mat.T
# Symmetrize to avoid numerically not symmetric matrix
# Since A commutes with itself (AA' = A'A = AA) the eigenvalues do not change.
return 0.5 * (spd_mat + spd_mat.T)
[docs]def random_sparse_spd_matrix(
dim: IntArgType,
density: float,
chol_entry_min: float = 0.1,
chol_entry_max: float = 1.0,
random_state: Optional[RandomStateArgType] = None,
) -> np.ndarray:
"""Random sparse symmetric positive definite matrix.
Constructs a random sparse symmetric positive definite matrix for a given degree
of sparsity. The matrix is constructed from its Cholesky factor :math:`L`. Its
diagonal is set to one and all other entries of the lower triangle are sampled
from a uniform distribution with bounds :code:`[chol_entry_min, chol_entry_max]`.
The resulting sparse matrix is then given by :math:`A=LL^\\top`.
Parameters
----------
dim
Matrix dimension.
density
Degree of sparsity of the off-diagonal entries of the Cholesky factor.
Between 0 and 1 where 1 represents a dense matrix.
chol_entry_min
Lower bound on the entries of the Cholesky factor.
chol_entry_max
Upper bound on the entries of the Cholesky factor.
random_state
Random state of the random variable. If None (or np.random), the global
:mod:`numpy.random` state is used. If integer, it is used to seed the local
:class:`~numpy.random.RandomState` instance.
See Also
--------
random_spd_matrix : Generate a random symmetric positive definite matrix.
Examples
--------
>>> from probnum.problems.zoo.linalg import random_sparse_spd_matrix
>>> sparsemat = random_sparse_spd_matrix(dim=5, density=0.1, random_state=42)
>>> sparsemat
array([[1. , 0. , 0. , 0. , 0. ],
[0. , 1. , 0. , 0. , 0. ],
[0. , 0. , 1. , 0. , 0.24039507],
[0. , 0. , 0. , 1. , 0. ],
[0. , 0. , 0.24039507, 0. , 1.05778979]])
"""
# Initialization
random_state = _utils.as_random_state(random_state)
if not 0 <= density <= 1:
raise ValueError(f"Density must be between 0 and 1, but is {density}.")
chol = np.eye(dim)
num_off_diag_cholesky = int(0.5 * dim * (dim - 1))
num_nonzero_entries = int(num_off_diag_cholesky * density)
if num_nonzero_entries > 0:
# Draw entries of lower triangle (below diagonal) according to sparsity level
entry_ids = np.mask_indices(n=dim, mask_func=np.tril, k=-1)
idx_samples = random_state.choice(
a=num_off_diag_cholesky, size=num_nonzero_entries, replace=False
)
nonzero_entry_ids = (entry_ids[0][idx_samples], entry_ids[1][idx_samples])
# Fill Cholesky factor
chol[nonzero_entry_ids] = random_state.uniform(
low=chol_entry_min, high=chol_entry_max, size=num_nonzero_entries
)
return chol @ chol.T