def closest_rotation_matrix(mat):
"""
Compute the closest rotation matrix of a given matrix mat,
in terms of the Frobenius norm.
"""
mat = gs.to_ndarray(mat, to_ndim=3)
n_mats, mat_dim_1, mat_dim_2 = mat.shape
assert mat_dim_1 == mat_dim_2
if mat_dim_1 == 3:
mat_unitary_u, diag_s, mat_unitary_v = gs.linalg.svd(mat)
rot_mat = gs.matmul(mat_unitary_u, mat_unitary_v)
mask = gs.where(gs.linalg.det(rot_mat) < 0)
new_mat_diag_s = gs.tile(gs.diag([1, 1, -1]), len(mask))
rot_mat[mask] = gs.matmul(
gs.matmul(mat_unitary_u[mask], new_mat_diag_s),
mat_unitary_v[mask])
else:
aux_mat = gs.matmul(gs.transpose(mat, axes=(0, 2, 1)), mat)
inv_sqrt_mat = gs.zeros_like(mat)
for i in range(n_mats):
sym_mat = aux_mat[i]
assert spd_matrices_space.is_symmetric(sym_mat)
inv_sqrt_mat[i] = gs.linalg.inv(spd_matrices_space.sqrtm(sym_mat))
rot_mat = gs.matmul(mat, inv_sqrt_mat)
assert rot_mat.ndim == 3
return rot_mat
def projection(self, mat):
"""
Project a matrix on SO(n), using the Frobenius norm.
"""
# TODO(nina): projection when the point_type is not 'matrix'?
mat = gs.to_ndarray(mat, to_ndim=3)
n_mats, mat_dim_1, mat_dim_2 = mat.shape
assert mat_dim_1 == mat_dim_2 == self.n
if self.n == 3:
mat_unitary_u, diag_s, mat_unitary_v = gs.linalg.svd(mat)
rot_mat = gs.matmul(mat_unitary_u, mat_unitary_v)
mask = gs.nonzero(gs.linalg.det(rot_mat) < 0)
diag = gs.array([1, 1, -1])
new_mat_diag_s = gs.tile(gs.diag(diag), len(mask))
rot_mat[mask] = gs.matmul(
gs.matmul(mat_unitary_u[mask], new_mat_diag_s),
mat_unitary_v[mask])
else:
aux_mat = gs.matmul(gs.transpose(mat, axes=(0, 2, 1)), mat)
inv_sqrt_mat = gs.zeros_like(mat)
for i in range(n_mats):
sym_mat = aux_mat[i]
assert spd_matrices_space.is_symmetric(sym_mat)
inv_sqrt_mat[i] = gs.linalg.inv(
spd_matrices_space.sqrtm(sym_mat))
rot_mat = gs.matmul(mat, inv_sqrt_mat)
assert gs.ndim(rot_mat) == 3
return rot_mat
def test_make_symmetric_and_is_symmetric_vectorization(self):
n_samples = self.n_samples
mats = gs.random.rand(n_samples, 5, 5)
results = spd_matrices_space.make_symmetric(mats)
self.assertTrue(gs.all(spd_matrices_space.is_symmetric(results)))
def test_is_symmetric_vectorization(self):
n_samples = self.n_samples
points = self.space.random_uniform(n_samples=n_samples)
self.assertTrue(gs.all(spd_matrices_space.is_symmetric(points)))
def test_is_symmetric(self):
sym_mat = gs.array([[1, 2], [2, 1]])
self.assertTrue(spd_matrices_space.is_symmetric(sym_mat))
not_a_sym_mat = gs.array([[1., 0.6, -3.], [6., -7., 0.], [0., 7., 8.]])
self.assertFalse(spd_matrices_space.is_symmetric(not_a_sym_mat))