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 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 left_exp_from_identity(self, tangent_vec):
"""
Compute the *left* Riemannian exponential from the identity of the
Lie group of tangent vector tangent_vec.
The left Riemannian exponential has a special role since the
left Riemannian exponential of the canonical metric parameterizes
the points.
Note: In the case where the method is called by a right-invariant
metric, it used the left-invariant metric associated to the same
inner-product at the identity.
"""
import geomstats.spd_matrices_space as spd_matrices_space
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2)
tangent_vec = self.group.regularize_tangent_vec_at_identity(
tangent_vec=tangent_vec, metric=self)
sqrt_inner_product_mat = spd_matrices_space.sqrtm(
self.inner_product_mat_at_identity)
mat = gs.transpose(sqrt_inner_product_mat, axes=(0, 2, 1))
exp = gs.matmul(tangent_vec, mat)
exp = gs.squeeze(exp, axis=0)
exp = self.group.regularize(exp)
return exp
def test_sqrtm(self):
n_samples = self.n_samples
points = self.space.random_uniform(n_samples=n_samples)
result = spd_matrices_space.sqrtm(points)
expected = gs.zeros((n_samples, self.n, self.n))
for i in range(n_samples):
expected[i] = scipy.linalg.sqrtm(points[i])
self.assertTrue(gs.allclose(result, expected))
def left_exp_from_identity(self, tangent_vec):
"""
Riemannian exponential of a tangent vector wrt the identity associated
to the left-invariant metric.
If the method is called by a right-invariant metric, it uses the
left-invariant metric associated to the same inner-product matrix
at the identity.
"""
import geomstats.spd_matrices_space as spd_matrices_space
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2)
tangent_vec = self.group.regularize_tangent_vec_at_identity(
tangent_vec=tangent_vec, metric=self)
sqrt_inner_product_mat = spd_matrices_space.sqrtm(
self.inner_product_mat_at_identity)
mat = gs.transpose(sqrt_inner_product_mat, axes=(0, 2, 1))
exp = gs.einsum('ni,nij->nj', tangent_vec, mat)
exp = self.group.regularize(exp)
return exp
def left_log_from_identity(self, point):
"""
Riemannian logarithm of a point wrt the identity associated
to the left-invariant metric.
If the method is called by a right-invariant metric, it uses the
left-invariant metric associated to the same inner-product matrix
at the identity.
"""
import geomstats.spd_matrices_space as spd_matrices_space
point = self.group.regularize(point)
inner_prod_mat = self.inner_product_mat_at_identity
inv_inner_prod_mat = gs.linalg.inv(inner_prod_mat)
sqrt_inv_inner_prod_mat = spd_matrices_space.sqrtm(inv_inner_prod_mat)
assert sqrt_inv_inner_prod_mat.shape == ((1, ) +
(self.group.dimension, ) * 2)
aux = gs.squeeze(sqrt_inv_inner_prod_mat, axis=0)
log = gs.matmul(point, aux)
log = self.group.regularize_tangent_vec_at_identity(tangent_vec=log,
metric=self)
assert log.ndim == 2
return log
def left_log_from_identity(self, point):
"""
Compute the *left* Riemannian logarithm from the identity of the
Lie group of tangent vector tangent_vec.
The left Riemannian logarithm has a special role since the
left Riemannian logarithm of the canonical metric parameterizes
the points.
"""
import geomstats.spd_matrices_space as spd_matrices_space
point = self.group.regularize(point)
inner_prod_mat = self.inner_product_mat_at_identity
inv_inner_prod_mat = gs.linalg.inv(inner_prod_mat)
sqrt_inv_inner_prod_mat = spd_matrices_space.sqrtm(inv_inner_prod_mat)
assert sqrt_inv_inner_prod_mat.shape == ((1, ) +
(self.group.dimension, ) * 2)
aux = gs.squeeze(sqrt_inv_inner_prod_mat, axis=0)
log = gs.matmul(point, aux)
log = self.group.regularize_tangent_vec_at_identity(tangent_vec=log,
metric=self)
assert log.ndim == 2
return log