# Source code for probnum.problems.zoo.linalg._random_spd_matrix

"""Random symmetric positive definite matrices."""

from typing import Sequence

import numpy as np
import scipy.stats

from probnum.typing import IntArgType

[docs]def random_spd_matrix(
rng: np.random.Generator,
dim: IntArgType,
spectrum: Sequence = None,
) -> np.ndarray:
r"""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
----------
rng
Random number generator.
dim
Matrix dimension.
spectrum
Eigenvalues of the matrix.

--------
random_sparse_spd_matrix : Generate a random sparse symmetric positive definite matrix.

Examples
--------
>>> import numpy as np
>>> from probnum.problems.zoo.linalg import random_spd_matrix
>>> rng = np.random.default_rng(1)
>>> mat = random_spd_matrix(rng, dim=5)
>>> mat
array([[10.24394619,  0.05484236,  0.39575826, -0.70032495, -0.75482692],
[ 0.05484236, 11.31516868,  0.6968935 , -0.13877394,  0.52783063],
[ 0.39575826,  0.6968935 , 11.5728974 ,  0.21214568,  1.07692458],
[-0.70032495, -0.13877394,  0.21214568,  9.88674751, -1.09750511],
[-0.75482692,  0.52783063,  1.07692458, -1.09750511, 10.193655  ]])

Check for symmetry and positive definiteness.

>>> np.all(mat == mat.T)
True
>>> np.linalg.eigvals(mat)
array([ 8.09147328, 12.7635956 , 10.84504988, 10.73086331, 10.78143272])
"""

# Initialization
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=rng,
)
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=rng)
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(
rng: np.random.Generator,
dim: IntArgType,
density: float,
chol_entry_min: float = 0.1,
chol_entry_max: float = 1.0,
format="csr",  # pylint: disable="redefined-builtin"
) -> scipy.sparse.spmatrix:
r"""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 nonzero 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
----------
rng
Random number generator.
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.
format
Sparse matrix format.

--------
random_spd_matrix : Generate a random symmetric positive definite matrix.

Examples
--------
>>> import numpy as np
>>> from probnum.problems.zoo.linalg import random_sparse_spd_matrix
>>> rng = np.random.default_rng(42)
>>> sparsemat = random_sparse_spd_matrix(rng, dim=5, density=0.1)
>>> sparsemat
<5x5 sparse matrix of type '<class 'numpy.float64'>'
with 9 stored elements in Compressed Sparse Row format>
>>> sparsemat.todense()
matrix([[1.        , 0.        , 0.87273813, 0.        , 0.        ],
[0.        , 1.        , 0.        , 0.        , 0.        ],
[0.87273813, 0.        , 1.76167184, 0.        , 0.        ],
[0.        , 0.        , 0.        , 1.        , 0.72763123],
[0.        , 0.        , 0.        , 0.72763123, 1.5294472 ]])
"""

# Initialization
if not 0 <= density <= 1:
raise ValueError(f"Density must be between 0 and 1, but is {density}.")
chol = scipy.sparse.eye(dim, format="csr")
num_off_diag_cholesky = int(0.5 * dim * (dim - 1))
num_nonzero_entries = int(num_off_diag_cholesky * density)

if num_nonzero_entries > 0:
sparse_matrix = scipy.sparse.rand(
m=dim,
n=dim,
format="csr",
density=density,
random_state=rng,
)

# Rescale entries
sparse_matrix.data *= chol_entry_max - chol_entry_min
sparse_matrix.data += chol_entry_min

# Extract lower triangle
chol += scipy.sparse.tril(A=sparse_matrix, k=-1, format=format)

return (chol @ chol.T).asformat(format=format)