def rotation_vector_from_quaternion(self, quaternion):
"""Convert a unit quaternion into a rotation vector.
Parameters
----------
quaternion : array-like, shape=[..., 4]
Returns
-------
rot_vec : array-like, shape=[..., 3]
"""
cos_half_angle = quaternion[:, 0]
cos_half_angle = gs.clip(cos_half_angle, -1, 1)
half_angle = gs.arccos(cos_half_angle)
half_angle = gs.to_ndarray(half_angle, to_ndim=2, axis=1)
mask_0 = gs.isclose(half_angle, 0.)
mask_not_0 = ~mask_0
rotation_axis = gs.divide(
quaternion[:, 1:],
gs.sin(half_angle) * gs.cast(mask_not_0, gs.float32) +
gs.cast(mask_0, gs.float32))
rot_vec = gs.array(2 * half_angle * rotation_axis *
gs.cast(mask_not_0, gs.float32))
rot_vec = self.regularize(rot_vec)
return rot_vec
def __init__(self,
group,
inner_product_mat_at_identity=None,
left_or_right='left'):
if inner_product_mat_at_identity is None:
inner_product_mat_at_identity = gs.eye(self.group.dimension)
inner_product_mat_at_identity = gs.to_ndarray(
inner_product_mat_at_identity, to_ndim=3)
mat_shape = inner_product_mat_at_identity.shape
assert mat_shape == (1, ) + (group.dimension, ) * 2, mat_shape
assert left_or_right in ('left', 'right')
eigenvalues = gs.linalg.eigvalsh(inner_product_mat_at_identity)
mask_pos_eigval = gs.greater(eigenvalues, 0.)
n_pos_eigval = gs.sum(gs.cast(mask_pos_eigval, gs.int32))
mask_neg_eigval = gs.less(eigenvalues, 0.)
n_neg_eigval = gs.sum(gs.cast(mask_neg_eigval, gs.int32))
mask_null_eigval = gs.isclose(eigenvalues, 0.)
n_null_eigval = gs.sum(gs.cast(mask_null_eigval, gs.int32))
self.group = group
if inner_product_mat_at_identity is None:
inner_product_mat_at_identity = gs.eye(self.group.dimension)
self.inner_product_mat_at_identity = inner_product_mat_at_identity
self.left_or_right = left_or_right
self.signature = (n_pos_eigval, n_null_eigval, n_neg_eigval)
def __init__(self,
group,
algebra=None,
metric_mat_at_identity=None,
left_or_right='left',
**kwargs):
super(InvariantMetric, self).__init__(dim=group.dim, **kwargs)
self.group = group
self.lie_algebra = algebra
if metric_mat_at_identity is None:
metric_mat_at_identity = gs.eye(self.group.dim)
geomstats.errors.check_parameter_accepted_values(
left_or_right, 'left_or_right', ['left', 'right'])
eigenvalues = gs.linalg.eigvalsh(metric_mat_at_identity)
mask_pos_eigval = gs.greater(eigenvalues, 0.)
n_pos_eigval = gs.sum(gs.cast(mask_pos_eigval, gs.int32))
mask_neg_eigval = gs.less(eigenvalues, 0.)
n_neg_eigval = gs.sum(gs.cast(mask_neg_eigval, gs.int32))
mask_null_eigval = gs.isclose(eigenvalues, 0.)
n_null_eigval = gs.sum(gs.cast(mask_null_eigval, gs.int32))
self.metric_mat_at_identity = metric_mat_at_identity
self.left_or_right = left_or_right
self.signature = (n_pos_eigval, n_null_eigval, n_neg_eigval)
def test_random_von_mises_kappa(self):
# check concentration parameter for dispersed distribution
kappa = 1.0
n_points = 100000
for dim in [2, 9]:
sphere = Hypersphere(dim)
points = sphere.random_von_mises_fisher(kappa=kappa,
n_samples=n_points)
sum_points = gs.sum(points, axis=0)
mean_norm = gs.linalg.norm(sum_points) / n_points
kappa_estimate = (mean_norm * (dim + 1.0 - mean_norm**2) /
(1.0 - mean_norm**2))
kappa_estimate = gs.cast(kappa_estimate, gs.float64)
p = dim + 1
n_steps = 100
for _ in range(n_steps):
bessel_func_1 = scipy.special.iv(p / 2.0, kappa_estimate)
bessel_func_2 = scipy.special.iv(p / 2.0 - 1.0, kappa_estimate)
ratio = bessel_func_1 / bessel_func_2
denominator = 1.0 - ratio**2 - (p -
1.0) * ratio / kappa_estimate
mean_norm = gs.cast(mean_norm, gs.float64)
kappa_estimate = kappa_estimate - (ratio -
mean_norm) / denominator
result = kappa_estimate
expected = kappa
self.assertAllClose(result, expected, atol=KAPPA_ESTIMATION_TOL)
def _log_translation_transform(self, rot_vec):
rot_vec = gs.to_ndarray(rot_vec, to_ndim=2, axis=1)
exp_transform = self._exp_translation_transform(rot_vec)
if rot_vec.dtype == gs.float32:
mask_close_0 = gs.isclose(rot_vec, 0., atol=1e-6)
else:
mask_close_0 = gs.isclose(rot_vec, 0.)
mask_else = ~mask_close_0
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
inv_determinant = gs.zeros_like(rot_vec)
inv_determinant += 0.5 * mask_close_0_float / (
TAYLOR_COEFFS_1_AT_0[0]
+ TAYLOR_COEFFS_1_AT_0[1] * rot_vec ** 2
+ TAYLOR_COEFFS_1_AT_0[2] * rot_vec ** 6)
rot_vec = rot_vec + 1e-3 * mask_close_0_float
inv_determinant += mask_else_float * (
rot_vec ** 2 / (2 * (1 - gs.cos(rot_vec))))
transform = gs.einsum(
'il, ijk -> ijk', inv_determinant,
gs.transpose(exp_transform, axes=[0, 2, 1]))
return transform
def log(self, point, base_point):
"""
Riemannian logarithm of a point wrt a base point.
"""
point = gs.to_ndarray(point, to_ndim=2)
base_point = gs.to_ndarray(base_point, to_ndim=2)
norm_base_point = self.embedding_metric.norm(base_point)
norm_point = self.embedding_metric.norm(point)
inner_prod = self.embedding_metric.inner_product(base_point, point)
cos_angle = inner_prod / (norm_base_point * norm_point)
cos_angle = gs.clip(cos_angle, -1., 1.)
angle = gs.arccos(cos_angle)
angle = gs.to_ndarray(angle, to_ndim=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
mask_0 = gs.isclose(angle, 0.)
mask_else = gs.equal(mask_0, gs.array(False))
mask_0_float = gs.cast(mask_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1 += mask_0_float * (1. + INV_SIN_TAYLOR_COEFFS[1] * angle**2 +
INV_SIN_TAYLOR_COEFFS[3] * angle**4 +
INV_SIN_TAYLOR_COEFFS[5] * angle**6 +
INV_SIN_TAYLOR_COEFFS[7] * angle**8)
coef_2 += mask_0_float * (1. + INV_TAN_TAYLOR_COEFFS[1] * angle**2 +
INV_TAN_TAYLOR_COEFFS[3] * angle**4 +
INV_TAN_TAYLOR_COEFFS[5] * angle**6 +
INV_TAN_TAYLOR_COEFFS[7] * angle**8)
# This avoids division by 0.
angle += mask_0_float * 1.
coef_1 += mask_else_float * angle / gs.sin(angle)
coef_2 += mask_else_float * angle / gs.tan(angle)
log = (gs.einsum('ni,nj->nj', coef_1, point) -
gs.einsum('ni,nj->nj', coef_2, base_point))
mask_same_values = gs.isclose(point, base_point)
mask_else = gs.equal(mask_same_values, gs.array(False))
mask_else_float = gs.cast(mask_else, gs.float32)
mask_else_float = gs.to_ndarray(mask_else_float, to_ndim=1)
mask_else_float = gs.to_ndarray(mask_else_float, to_ndim=2)
mask_not_same_points = gs.sum(mask_else_float, axis=1)
mask_same_points = gs.isclose(mask_not_same_points, 0.)
mask_same_points = gs.cast(mask_same_points, gs.float32)
mask_same_points = gs.to_ndarray(mask_same_points, to_ndim=2, axis=1)
mask_same_points_float = gs.cast(mask_same_points, gs.float32)
log -= mask_same_points_float * log
return log
def quaternion_from_rotation_vector(self, rot_vec):
"""Convert a rotation vector into a unit quaternion.
Parameters
----------
rot_vec : array-like, shape=[..., 3]
Returns
-------
quaternion : array-like, shape=[..., 4]
"""
rot_vec = self.regularize(rot_vec)
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
mask_0 = gs.isclose(angle, 0.)
mask_not_0 = ~mask_0
rotation_axis = gs.divide(
rot_vec,
angle * gs.cast(mask_not_0, gs.float32) +
gs.cast(mask_0, gs.float32))
quaternion = gs.concatenate(
(gs.cos(angle / 2), gs.sin(angle / 2) * rotation_axis[:]), axis=1)
return quaternion
def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""
Inner product defined by the Riemannian metric at point base_point
between tangent vectors tangent_vec_a and tangent_vec_b.
"""
tangent_vec_a = gs.to_ndarray(tangent_vec_a, to_ndim=2)
tangent_vec_b = gs.to_ndarray(tangent_vec_b, to_ndim=2)
n_tangent_vec_a, _ = tangent_vec_a.shape
n_tangent_vec_b, _ = tangent_vec_b.shape
inner_prod_mat = self.inner_product_matrix(base_point)
inner_prod_mat = gs.to_ndarray(inner_prod_mat, to_ndim=3)
n_mats, _, _ = inner_prod_mat.shape
n_inner_prod = gs.maximum(n_tangent_vec_a, n_tangent_vec_b)
n_inner_prod = gs.maximum(n_inner_prod, n_mats)
n_tiles_a = gs.divide(n_inner_prod, n_tangent_vec_a)
n_tiles_a = gs.cast(n_tiles_a, gs.int32)
tangent_vec_a = gs.tile(tangent_vec_a, [n_tiles_a, 1])
n_tiles_b = gs.divide(n_inner_prod, n_tangent_vec_b)
n_tiles_b = gs.cast(n_tiles_b, gs.int32)
tangent_vec_b = gs.tile(tangent_vec_b, [n_tiles_b, 1])
n_tiles_mat = gs.divide(n_inner_prod, n_mats)
n_tiles_mat = gs.cast(n_tiles_mat, gs.int32)
inner_prod_mat = gs.tile(inner_prod_mat, [n_tiles_mat, 1, 1])
aux = gs.einsum('nj,njk->nk', tangent_vec_a, inner_prod_mat)
inner_prod = gs.einsum('nk,nk->n', aux, tangent_vec_b)
inner_prod = gs.to_ndarray(inner_prod, to_ndim=2, axis=1)
assert gs.ndim(inner_prod) == 2, inner_prod.shape
return inner_prod
def _fit_extrinsic(self, X, y, weights=None, compute_training_score=False):
"""Estimate the parameters using the extrinsic gradient descent.
Estimate the intercept and the coefficient defining the
geodesic regression model, using the extrinsic gradient.
Parameters
----------
X : {array-like, sparse matrix}, shape=[...,}]
Training input samples.
y : array-like, shape=[..., {dim, [n,n]}]
Training target values.
weights : array-like, shape=[...,]
Weights associated to the points.
Optional, default: None.
compute_training_score : bool
Whether to compute R^2.
Optional, default: False.
Returns
-------
self : object
Returns self.
"""
shape = (
y.shape[-1:] if self.space.default_point_type == "vector" else y.shape[-2:]
)
intercept_init, coef_init = self.initialize_parameters(y)
intercept_hat = self.space.projection(intercept_init)
coef_hat = self.space.to_tangent(coef_init, intercept_hat)
initial_guess = gs.vstack([gs.flatten(intercept_hat), gs.flatten(coef_hat)])
objective_with_grad = gs.autodiff.value_and_grad(
lambda param: self._loss(X, y, param, shape, weights), to_numpy=True
)
res = minimize(
objective_with_grad,
initial_guess,
method="CG",
jac=True,
options={"disp": self.verbose, "maxiter": self.max_iter},
tol=self.tol,
)
intercept_hat, coef_hat = gs.split(gs.array(res.x), 2)
intercept_hat = gs.reshape(intercept_hat, shape)
intercept_hat = gs.cast(intercept_hat, dtype=y.dtype)
coef_hat = gs.reshape(coef_hat, shape)
coef_hat = gs.cast(coef_hat, dtype=y.dtype)
self.intercept_ = self.space.projection(intercept_hat)
self.coef_ = self.space.to_tangent(coef_hat, self.intercept_)
if compute_training_score:
variance = gs.sum(self.metric.squared_dist(y, self.intercept_))
self.training_score_ = 1 - 2 * res.fun / variance
return self
def test_log(self, n, point, base_point, expected):
group = self.space(n)
expected = gs.cast(gs.array(expected), gs.float64)
point = gs.cast(gs.array(point), gs.float64)
base_point = (None if base_point is None else gs.cast(
gs.array(base_point), gs.float64))
self.assertAllClose(group.log(point, base_point), expected)
def test_exp_vectorization(self):
point = gs.array(
[
[[2.0, 0.0, 0.0], [0.0, 3.0, 0.0], [0.0, 0.0, 4.0]],
[[1.0, 0.0, 0.0], [0.0, 5.0, 0.0], [0.0, 0.0, 6.0]],
]
)
expected = gs.array(
[
[
[7.38905609, 0.0, 0.0],
[0.0, 20.0855369, 0.0],
[0.0, 0.0, 54.5981500],
],
[
[2.718281828, 0.0, 0.0],
[0.0, 148.413159, 0.0],
[0.0, 0.0, 403.42879349],
],
]
)
expected = gs.cast(expected, gs.float64)
point = gs.cast(point, gs.float64)
result = self.group.exp(point)
self.assertAllClose(result, expected)
def projection(self, point, atol=gs.atol):
"""Project a point in ambient space to the parameter set.
The parameter is floored to `gs.atol` if it is negative
and to '1-gs.atol' if it is greater than 1.
Parameters
----------
point : array-like, shape=[...,]
Point in ambient space.
atol : float
Tolerance to evaluate positivity.
Returns
-------
projected : array-like, shape=[...,]
Projected point.
"""
point = gs.cast(gs.array(point), dtype=gs.float32)
projected = gs.where(
gs.logical_or(point < atol, point > 1 - atol),
(1 - atol) * gs.cast(
(point > 1 - atol), gs.float32) + atol * gs.cast(
(point < atol), gs.float32),
point,
)
return gs.squeeze(projected)
def _exp_translation_transform(self, rot_vec):
n_samples = rot_vec.shape[0]
base_1 = gs.array([gs.eye(2)] * n_samples)
base_2 = -gs.array(
[self.rotations.skew_matrix_from_vector(gs.ones(1))] * n_samples)
mask_close_0 = gs.isclose(rot_vec, 0.)
mask_else = ~mask_close_0
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
cos_coef = gs.zeros_like(rot_vec)
sin_coef = gs.zeros_like(rot_vec)
cos_coef += mask_close_0_float * (
TAYLOR_COEFFS_1_AT_0[0] * rot_vec
+ TAYLOR_COEFFS_1_AT_0[1] * rot_vec ** 3
+ TAYLOR_COEFFS_1_AT_0[2] * rot_vec ** 5)
sin_coef += mask_close_0_float * (
1
- TAYLOR_COEFFS_2_AT_0[0] * rot_vec ** 2
- TAYLOR_COEFFS_2_AT_0[2] * rot_vec ** 4)
rot_vec = rot_vec + mask_close_0_float * 1e-6
cos_coef += mask_else_float * ((1. - gs.cos(rot_vec)) / rot_vec)
sin_coef += mask_else_float * (gs.sin(rot_vec) / rot_vec)
sin_term = gs.einsum('...i,...jk->...jk', sin_coef, base_1)
cos_term = gs.einsum('...i,...jk->...jk', cos_coef, base_2)
transform = sin_term + cos_term
return transform
def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""
Inner product between two tangent vectors at a base point.
"""
tangent_vec_a = gs.to_ndarray(tangent_vec_a, to_ndim=2)
tangent_vec_b = gs.to_ndarray(tangent_vec_b, to_ndim=2)
n_tangent_vec_a = gs.shape(tangent_vec_a)[0]
n_tangent_vec_b = gs.shape(tangent_vec_b)[0]
inner_prod_mat = self.inner_product_matrix(base_point)
inner_prod_mat = gs.to_ndarray(inner_prod_mat, to_ndim=3)
n_mats = gs.shape(inner_prod_mat)[0]
n_inner_prod = gs.maximum(n_tangent_vec_a, n_tangent_vec_b)
n_inner_prod = gs.maximum(n_inner_prod, n_mats)
n_tiles_a = gs.divide(n_inner_prod, n_tangent_vec_a)
n_tiles_a = gs.cast(n_tiles_a, gs.int32)
tangent_vec_a = gs.tile(tangent_vec_a, [n_tiles_a, 1])
n_tiles_b = gs.divide(n_inner_prod, n_tangent_vec_b)
n_tiles_b = gs.cast(n_tiles_b, gs.int32)
tangent_vec_b = gs.tile(tangent_vec_b, [n_tiles_b, 1])
n_tiles_mat = gs.divide(n_inner_prod, n_mats)
n_tiles_mat = gs.cast(n_tiles_mat, gs.int32)
inner_prod_mat = gs.tile(inner_prod_mat, [n_tiles_mat, 1, 1])
aux = gs.einsum('nj,njk->nk', tangent_vec_a, inner_prod_mat)
inner_prod = gs.einsum('nk,nk->n', aux, tangent_vec_b)
inner_prod = gs.to_ndarray(inner_prod, to_ndim=2, axis=1)
assert gs.ndim(inner_prod) == 2, inner_prod.shape
return inner_prod
def test_exp(self, n, tangent_vec, base_point, expected):
group = self.space(n)
expected = gs.cast(gs.array(expected), gs.float64)
tangent_vec = gs.cast(gs.array(tangent_vec), gs.float64)
base_point = (None if base_point is None else gs.cast(
gs.array(base_point), gs.float64))
self.assertAllClose(group.exp(tangent_vec, base_point),
gs.array(expected))
def _log_translation_transform(self, rot_vec):
"""Compute matrix associated to rot_vec for the translation part in log.
Parameters
----------
rot_vec : array-like, shape=[..., 3]
Returns
-------
transform : array-like, shape=[..., 3, 3]
Matrix to be applied to the translation part in log
"""
n_samples = rot_vec.shape[0]
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_mat = self.rotations.skew_matrix_from_vector(rot_vec)
sq_skew_mat = gs.matmul(skew_mat, skew_mat)
mask_close_0 = gs.isclose(angle, 0.0)
mask_close_pi = gs.isclose(angle, gs.pi)
mask_else = ~mask_close_0 & ~mask_close_pi
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_close_pi_float = gs.cast(mask_close_pi, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
mask_0 = gs.isclose(angle, 0.0, atol=1e-7)
mask_0_float = gs.cast(mask_0, gs.float32)
angle += mask_0_float * gs.ones_like(angle)
coef_1 = -0.5 * gs.ones_like(angle)
coef_2 = gs.zeros_like(angle)
coef_2 += mask_close_0_float * (1.0 / 12.0 + angle**2 / 720.0 +
angle**4 / 30240.0 +
angle**6 / 1209600.0)
delta_angle = angle - gs.pi
coef_2 += mask_close_pi_float * (
1.0 / PI2 + (PI2 - 8.0) * delta_angle / (4.0 * PI3) -
((PI2 - 12.0) * delta_angle**2 / (4.0 * PI4)) +
((-192.0 + 12.0 * PI2 + PI4) * delta_angle**3 / (48.0 * PI5)) -
((-240.0 + 12.0 * PI2 + PI4) * delta_angle**4 / (48.0 * PI6)) +
((-2880.0 + 120.0 * PI2 + 10.0 * PI4 + PI6) * delta_angle**5 /
(480.0 * PI7)) -
((-3360 + 120.0 * PI2 + 10.0 * PI4 + PI6) * delta_angle**6 /
(480.0 * PI8)))
psi = 0.5 * angle * gs.sin(angle) / (1 - gs.cos(angle))
coef_2 += mask_else_float * (1 - psi) / (angle**2)
term_1 = gs.einsum("...i,...ij->...ij", coef_1, skew_mat)
term_2 = gs.einsum("...i,...ij->...ij", coef_2, sq_skew_mat)
term_id = gs.array([gs.eye(3)] * n_samples)
transform = term_id + term_1 + term_2
return transform
def regularize_tangent_vec_at_identity(
self, tangent_vec, metric=None):
"""Regularize a tangent vector at the identify.
In 3D, regularize a tangent_vector by getting its norm at the identity,
determined by the metric, to be less than pi.
Parameters
----------
tangent_vec : array-like, shape=[..., 3]]
metric : RiemannianMetric, optional
default: self.left_canonical_metric
Returns
-------
regularized_vec : array-like, shape=[..., 3]]
"""
if metric is None:
metric = self.left_canonical_metric
tangent_vec_metric_norm = metric.norm(tangent_vec)
tangent_vec_canonical_norm = gs.linalg.norm(tangent_vec, axis=-1)
mask_norm_0 = gs.isclose(tangent_vec_metric_norm, 0.)
mask_canonical_norm_0 = gs.isclose(tangent_vec_canonical_norm, 0.)
mask_0 = mask_norm_0 | mask_canonical_norm_0
mask_else = ~mask_0
# This avoids division by 0.
mask_0_float = gs.cast(mask_0, gs.float32) + self.epsilon
mask_else_float = gs.cast(mask_else, gs.float32) + self.epsilon
regularized_vec = gs.zeros_like(tangent_vec)
regularized_vec += gs.einsum(
'...,...i->...i', mask_0_float, tangent_vec)
tangent_vec_canonical_norm += mask_0_float
coef = gs.zeros_like(tangent_vec_metric_norm)
coef += mask_else_float * (
tangent_vec_metric_norm
/ tangent_vec_canonical_norm)
coef_tangent_vec = gs.einsum(
'...,...i->...i', coef, tangent_vec)
regularized_vec += gs.einsum(
'...,...i->...i',
mask_else_float,
self.regularize(coef_tangent_vec))
coef += mask_0_float
regularized_vec = gs.einsum(
'...,...i->...i', 1. / coef, regularized_vec)
regularized_vec = gs.einsum(
'...,...i->...i', mask_else_float, regularized_vec)
return regularized_vec
def _exp_translation_transform(self, rot_vec):
"""Compute matrix associated to rot_vec for the translation part in exp.
Parameters
----------
rot_vec : array-like, shape=[..., 3]
Returns
-------
transform : array-like, shape=[..., 3, 3]
Matrix to be applied to the translation part in exp.
"""
n_samples = rot_vec.shape[0]
angle = gs.linalg.norm(rot_vec, axis=-1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_mat = self.rotations.skew_matrix_from_vector(rot_vec)
sq_skew_mat = gs.matmul(skew_mat, skew_mat)
mask_0 = gs.equal(angle, 0.)
mask_close_0 = gs.isclose(angle, 0.) & ~mask_0
mask_else = ~mask_0 & ~mask_close_0
mask_0_float = gs.cast(mask_0, gs.float32)
mask_close_0_float = gs.cast(mask_close_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1 += mask_0_float * 1. / 2. * gs.ones_like(angle)
coef_2 += mask_0_float * 1. / 6. * gs.ones_like(angle)
coef_1 += mask_close_0_float * (
TAYLOR_COEFFS_1_AT_0[0]
+ TAYLOR_COEFFS_1_AT_0[2] * angle ** 2
+ TAYLOR_COEFFS_1_AT_0[4] * angle ** 4
+ TAYLOR_COEFFS_1_AT_0[6] * angle ** 6)
coef_2 += mask_close_0_float * (
TAYLOR_COEFFS_2_AT_0[0]
+ TAYLOR_COEFFS_2_AT_0[2] * angle ** 2
+ TAYLOR_COEFFS_2_AT_0[4] * angle ** 4
+ TAYLOR_COEFFS_2_AT_0[6] * angle ** 6)
angle += mask_0_float * 1.
coef_1 += mask_else_float * ((1. - gs.cos(angle)) / angle ** 2)
coef_2 += mask_else_float * ((angle - gs.sin(angle)) / angle ** 3)
term_1 = gs.einsum('...i,...ij->...ij', coef_1, skew_mat)
term_2 = gs.einsum('...i,...ij->...ij', coef_2, sq_skew_mat)
term_id = gs.array([gs.eye(3)] * n_samples)
transform = term_id + term_1 + term_2
return transform
def exp(self, tangent_vec, base_point):
"""
Riemannian exponential of a tangent vector wrt to a base point.
Parameters
----------
tangent_vec : array-like, shape=[n_samples, dimension + 1]
or shape=[1, dimension + 1]
base_point : array-like, shape=[n_samples, dimension + 1]
or shape=[1, dimension + 1]
Returns
-------
exp : array-like, shape=[n_samples, dimension + 1]
or shape=[1, dimension + 1]
"""
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2)
base_point = gs.to_ndarray(base_point, to_ndim=2)
sq_norm_tangent_vec = self.embedding_metric.squared_norm(
tangent_vec)
norm_tangent_vec = gs.sqrt(sq_norm_tangent_vec)
mask_0 = gs.isclose(sq_norm_tangent_vec, 0.)
mask_0 = gs.to_ndarray(mask_0, to_ndim=1)
mask_else = ~mask_0
mask_else = gs.to_ndarray(mask_else, to_ndim=1)
mask_0_float = gs.cast(mask_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
coef_1 = gs.zeros_like(norm_tangent_vec)
coef_2 = gs.zeros_like(norm_tangent_vec)
coef_1 += mask_0_float * (
1. + COSH_TAYLOR_COEFFS[2] * norm_tangent_vec ** 2
+ COSH_TAYLOR_COEFFS[4] * norm_tangent_vec ** 4
+ COSH_TAYLOR_COEFFS[6] * norm_tangent_vec ** 6
+ COSH_TAYLOR_COEFFS[8] * norm_tangent_vec ** 8)
coef_2 += mask_0_float * (
1. + SINH_TAYLOR_COEFFS[3] * norm_tangent_vec ** 2
+ SINH_TAYLOR_COEFFS[5] * norm_tangent_vec ** 4
+ SINH_TAYLOR_COEFFS[7] * norm_tangent_vec ** 6
+ SINH_TAYLOR_COEFFS[9] * norm_tangent_vec ** 8)
# This avoids dividing by 0.
norm_tangent_vec += mask_0_float * 1.0
coef_1 += mask_else_float * (gs.cosh(norm_tangent_vec))
coef_2 += mask_else_float * (
(gs.sinh(norm_tangent_vec) / (norm_tangent_vec)))
exp = (gs.einsum('ni,nj->nj', coef_1, base_point)
+ gs.einsum('ni,nj->nj', coef_2, tangent_vec))
hyperbolic_space = HyperbolicSpace(dimension=self.dimension)
exp = hyperbolic_space.regularize(exp)
return exp
def exp(self, tangent_vec, base_point):
"""Compute the Riemannian exponential of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[..., dim + 1]
Tangent vector at a base point.
base_point : array-like, shape=[..., dim + 1]
Point in hyperbolic space.
Returns
-------
exp : array-like, shape=[..., dim + 1]
Point in hyperbolic space equal to the Riemannian exponential
of tangent_vec at the base point.
"""
sq_norm_tangent_vec = self.embedding_metric.squared_norm(
tangent_vec)
sq_norm_tangent_vec = gs.clip(sq_norm_tangent_vec, 0, math.inf)
norm_tangent_vec = gs.sqrt(sq_norm_tangent_vec)
mask_0 = gs.isclose(sq_norm_tangent_vec, 0.)
mask_0 = gs.to_ndarray(mask_0, to_ndim=1)
mask_else = ~mask_0
mask_else = gs.to_ndarray(mask_else, to_ndim=1)
mask_0_float = gs.cast(mask_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
coef_1 = gs.zeros_like(norm_tangent_vec)
coef_2 = gs.zeros_like(norm_tangent_vec)
coef_1 += mask_0_float * (
1. + COSH_TAYLOR_COEFFS[2] * norm_tangent_vec ** 2
+ COSH_TAYLOR_COEFFS[4] * norm_tangent_vec ** 4
+ COSH_TAYLOR_COEFFS[6] * norm_tangent_vec ** 6
+ COSH_TAYLOR_COEFFS[8] * norm_tangent_vec ** 8)
coef_2 += mask_0_float * (
1. + SINH_TAYLOR_COEFFS[3] * norm_tangent_vec ** 2
+ SINH_TAYLOR_COEFFS[5] * norm_tangent_vec ** 4
+ SINH_TAYLOR_COEFFS[7] * norm_tangent_vec ** 6
+ SINH_TAYLOR_COEFFS[9] * norm_tangent_vec ** 8)
# This avoids dividing by 0.
norm_tangent_vec += mask_0_float * 1.0
coef_1 += mask_else_float * (gs.cosh(norm_tangent_vec))
coef_2 += mask_else_float * (
(gs.sinh(norm_tangent_vec) / (norm_tangent_vec)))
exp = (
gs.einsum('...,...j->...j', coef_1, base_point)
+ gs.einsum('...,...j->...j', coef_2, tangent_vec))
hyperbolic_space = Hyperboloid(dim=self.dim)
exp = hyperbolic_space.regularize(exp)
return exp
def test_powerm(self):
"""Test of powerm method."""
sym_n = SymmetricMatrices(self.n)
expected = gs.array(
[[[1, 1. / 4., 0.], [1. / 4, 2., 0.], [0., 0., 1.]]])
expected = gs.cast(expected, gs.float64)
power = gs.array(1. / 2)
power = gs.cast(power, gs.float64)
result = sym_n.powerm(expected, power)
result = gs.matmul(result, gs.transpose(result, (0, 2, 1)))
self.assertAllClose(result, expected)
def log(self, point, base_point):
"""
Riemannian logarithm of a point wrt a base point.
Parameters
----------
point : array-like, shape=[n_samples, dimension + 1]
or shape=[1, dimension + 1]
base_point : array-like, shape=[n_samples, dimension + 1]
or shape=[1, dimension + 1]
Returns
-------
log : array-like, shape=[n_samples, dimension + 1]
or shape=[1, dimension + 1]
"""
point = gs.to_ndarray(point, to_ndim=2)
base_point = gs.to_ndarray(base_point, to_ndim=2)
angle = self.dist(base_point, point)
angle = gs.to_ndarray(angle, to_ndim=1)
angle = gs.to_ndarray(angle, to_ndim=2)
mask_0 = gs.isclose(angle, 0.)
mask_else = ~mask_0
mask_0_float = gs.cast(mask_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1 += mask_0_float * (
1. + INV_SINH_TAYLOR_COEFFS[1] * angle ** 2
+ INV_SINH_TAYLOR_COEFFS[3] * angle ** 4
+ INV_SINH_TAYLOR_COEFFS[5] * angle ** 6
+ INV_SINH_TAYLOR_COEFFS[7] * angle ** 8)
coef_2 += mask_0_float * (
1. + INV_TANH_TAYLOR_COEFFS[1] * angle ** 2
+ INV_TANH_TAYLOR_COEFFS[3] * angle ** 4
+ INV_TANH_TAYLOR_COEFFS[5] * angle ** 6
+ INV_TANH_TAYLOR_COEFFS[7] * angle ** 8)
# This avoids dividing by 0.
angle += mask_0_float * 1.
coef_1 += mask_else_float * (angle / gs.sinh(angle))
coef_2 += mask_else_float * (angle / gs.tanh(angle))
log = (gs.einsum('ni,nj->nj', coef_1, point)
- gs.einsum('ni,nj->nj', coef_2, base_point))
return log
def log(self, point, base_point):
"""Compute Riemannian logarithm of a point wrt a base point.
If point_type = 'poincare' then base_point belongs
to the Poincare ball and point is a vector in the Euclidean
space of the same dimension as the ball.
Parameters
----------
point : array-like, shape=[..., dim + 1]
Point in hyperbolic space.
base_point : array-like, shape=[..., dim + 1]
Point in hyperbolic space.
Returns
-------
log : array-like, shape=[..., dim + 1]
Tangent vector at the base point equal to the Riemannian logarithm
of point at the base point.
"""
angle = self.dist(base_point, point) / self.scale
angle = gs.to_ndarray(angle, to_ndim=1)
mask_0 = gs.isclose(angle, 0.)
mask_else = ~mask_0
mask_0_float = gs.cast(mask_0, gs.float32)
mask_else_float = gs.cast(mask_else, gs.float32)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1 += mask_0_float * (
1. + INV_SINH_TAYLOR_COEFFS[1] * angle ** 2
+ INV_SINH_TAYLOR_COEFFS[3] * angle ** 4
+ INV_SINH_TAYLOR_COEFFS[5] * angle ** 6
+ INV_SINH_TAYLOR_COEFFS[7] * angle ** 8)
coef_2 += mask_0_float * (
1. + INV_TANH_TAYLOR_COEFFS[1] * angle ** 2
+ INV_TANH_TAYLOR_COEFFS[3] * angle ** 4
+ INV_TANH_TAYLOR_COEFFS[5] * angle ** 6
+ INV_TANH_TAYLOR_COEFFS[7] * angle ** 8)
# This avoids dividing by 0.
angle += mask_0_float * 1.
coef_1 += mask_else_float * (angle / gs.sinh(angle))
coef_2 += mask_else_float * (angle / gs.tanh(angle))
log_term_1 = gs.einsum('...,...j->...j', coef_1, point)
log_term_2 = - gs.einsum('...,...j->...j', coef_2, base_point)
log = log_term_1 + log_term_2
return log
def matrix_from_rotation_vector(self, rot_vec):
"""Convert rotation vector to rotation matrix.
Parameters
----------
rot_vec: array-like, shape=[..., 3]
Returns
-------
rot_mat: array-like, shape=[..., 3]
"""
rot_vec = self.regularize(rot_vec)
angle = gs.linalg.norm(rot_vec, axis=1)
angle = gs.to_ndarray(angle, to_ndim=2, axis=1)
skew_rot_vec = self.skew_matrix_from_vector(rot_vec)
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
# This avoids dividing by 0.
mask_0 = gs.isclose(angle, 0.)
mask_0_float = gs.cast(mask_0, gs.float32) + self.epsilon
coef_1 += mask_0_float * (1. - (angle ** 2) / 6.)
coef_2 += mask_0_float * (1. / 2. - angle ** 2)
# This avoids dividing by 0.
mask_else = ~mask_0
mask_else_float = gs.cast(mask_else, gs.float32) + self.epsilon
angle += mask_0_float
coef_1 += mask_else_float * (gs.sin(angle) / angle)
coef_2 += mask_else_float * (
(1. - gs.cos(angle)) / (angle ** 2))
coef_1 = gs.squeeze(coef_1, axis=1)
coef_2 = gs.squeeze(coef_2, axis=1)
term_1 = (gs.eye(self.dim)
+ gs.einsum('n,njk->njk', coef_1, skew_rot_vec))
squared_skew_rot_vec = gs.einsum(
'nij,njk->nik', skew_rot_vec, skew_rot_vec)
term_2 = gs.einsum('n,njk->njk', coef_2, squared_skew_rot_vec)
return term_1 + term_2
def test_log_and_exp_general_case_general_dim(self):
"""
Test that the Riemannian exponential
and the Riemannian logarithm are inverse.
Expect their composition to give the identity function.
"""
# Riemannian Log then Riemannian Exp
dim = 5
n_samples = self.n_samples
h5 = Hyperboloid(dim=dim)
h5_metric = h5.metric
base_point = h5.random_point()
point = h5.random_point()
point = gs.cast(point, gs.float64)
base_point = gs.cast(base_point, gs.float64)
one_log = h5_metric.log(point=point, base_point=base_point)
result = h5_metric.exp(tangent_vec=one_log, base_point=base_point)
expected = point
self.assertAllClose(result, expected)
# Test vectorization of log
base_point = gs.stack([base_point] * n_samples, axis=0)
point = gs.stack([point] * n_samples, axis=0)
expected = gs.stack([one_log] * n_samples, axis=0)
log = h5_metric.log(point=point, base_point=base_point)
result = log
self.assertAllClose(gs.shape(result), (n_samples, dim + 1))
self.assertAllClose(result, expected)
result = h5_metric.exp(tangent_vec=log, base_point=base_point)
expected = point
self.assertAllClose(gs.shape(result), (n_samples, dim + 1))
self.assertAllClose(result, expected)
# Test vectorization of exp
tangent_vec = gs.stack([one_log] * n_samples, axis=0)
exp = h5_metric.exp(tangent_vec=tangent_vec, base_point=base_point)
result = exp
expected = point
self.assertAllClose(gs.shape(result), (n_samples, dim + 1))
self.assertAllClose(result, expected)
def test_squared_dist_bureswasserstein_vectorization(self):
"""Test of SPDMetricBuresWasserstein.squared_dist method."""
point_a = self.space.random_point(2)
point_b = gs.array([[9., 0., 0.], [0., 5., 0.], [0., 0., 1.]])
point_a = gs.cast(point_a, gs.float64)
point_b = gs.cast(point_b, gs.float64)
metric = self.metric_bureswasserstein
result = metric.squared_dist(point_a, point_b)
log = metric.log(point=point_b, base_point=point_a)
expected = metric.squared_norm(vector=log, base_point=point_a)
self.assertAllClose(result, expected)
def regularize(self, point):
"""Regularize a point to be in accordance with convention.
In 3D, regularize the norm of the rotation vector,
to be between 0 and pi, following the axis-angle
representation's convention.
If the angle angle is between pi and 2pi,
the function computes its complementary in 2pi and
inverts the direction of the rotation axis.
Parameters
----------
point : array-like, shape=[...,3]
Returns
-------
regularized_point : array-like, shape=[..., 3]
"""
regularized_point = point
angle = gs.linalg.norm(regularized_point, axis=-1)
mask_0 = gs.isclose(angle, 0.)
mask_not_0 = ~mask_0
mask_pi = gs.isclose(angle, gs.pi)
# This avoids division by 0.
mask_0_float = gs.cast(mask_0, gs.float32) + self.epsilon
mask_not_0_float = (
gs.cast(mask_not_0, gs.float32)
+ self.epsilon)
mask_pi_float = gs.cast(mask_pi, gs.float32) + self.epsilon
k = gs.floor(angle / (2 * gs.pi) + .5)
angle += mask_0_float
norms_ratio = gs.zeros_like(angle)
norms_ratio += mask_not_0_float * (
1. - 2. * gs.pi * k / angle)
norms_ratio += mask_0_float
norms_ratio += mask_pi_float * (
gs.pi / angle
- (1. - 2. * gs.pi * k / angle))
regularized_point = gs.einsum(
'...,...i->...i', norms_ratio, regularized_point)
return regularized_point
def square_root_velocity(self, curve):
"""Compute the square root velocity representation of a curve.
The velocity is computed using the log map. The case of several curves
is handled through vectorization. In that case, an index selection
procedure allows to get rid of the log between the end point of
curve[k, :, :] and the starting point of curve[k + 1, :, :].
Parameters
----------
curve :
Returns
-------
srv :
"""
curve = gs.to_ndarray(curve, to_ndim=3)
n_curves, n_sampling_points, n_coords = curve.shape
srv_shape = (n_curves, n_sampling_points - 1, n_coords)
curve = gs.reshape(curve, (n_curves * n_sampling_points, n_coords))
coef = gs.cast(gs.array(n_sampling_points - 1), gs.float32)
velocity = coef * self.ambient_metric.log(point=curve[1:, :],
base_point=curve[:-1, :])
velocity_norm = self.ambient_metric.norm(velocity, curve[:-1, :])
srv = velocity / gs.sqrt(velocity_norm)
index = gs.arange(n_curves * n_sampling_points - 1)
mask = ~gs.equal((index + 1) % n_sampling_points, 0)
index_select = gs.gather(index, gs.squeeze(gs.where(mask)))
srv = gs.reshape(gs.gather(srv, index_select), srv_shape)
return srv
def apply_func_to_eigvals(x, function, check_positive=False):
"""
Apply function to eigenvalues and reconstruct the matrix.
Parameters
----------
x : array_like, shape=[..., n, n]
Symmetric matrix.
function : callable
Function to apply to eigenvalues.
Returns
-------
x : array_like, shape=[..., n, n]
Symmetric matrix.
"""
eigvals, eigvecs = gs.linalg.eigh(x)
if check_positive:
if gs.any(gs.cast(eigvals, gs.float32) < 0.):
logging.warning('Negative eigenvalue encountered in'
' {}'.format(function.__name__))
eigvals = function(eigvals)
eigvals = algebra_utils.from_vector_to_diagonal_matrix(eigvals)
transp_eigvecs = Matrices.transpose(eigvecs)
reconstuction = gs.matmul(eigvecs, eigvals)
reconstuction = gs.matmul(reconstuction, transp_eigvecs)
return reconstuction
def flip_determinant(matrix, det):
"""Change sign of the determinant if it is negative.
For a batch of matrices, multiply the matrices which have negative
determinant by a diagonal matrix :math: `diag(1,...,1,-1) from the right.
This changes the sign of the last column of the matrix.
Parameters
----------
matrix : array-like, shape=[...,n ,m]
Matrix to transform.
det : array-like, shape=[...]
Determinant of matrix, or any other scalar to use as threshold to
determine whether to change the sign of the last column of matrix.
Returns
-------
matrix_flipped : array-like, shape=[..., n, m]
Matrix with the sign of last column changed if det < 0.
"""
if gs.any(det < 0):
ones = gs.ones(matrix.shape[-1])
reflection_vec = gs.concatenate([ones[:-1], gs.array([-1.0])], axis=0)
mask = gs.cast(det < 0, matrix.dtype)
sign = mask[..., None] * reflection_vec + (1.0 - mask)[..., None] * ones
return gs.einsum("...ij,...j->...ij", matrix, sign)
return matrix