def parallel_transport(self, tangent_vec_a, tangent_vec_b, base_point):
"""Compute the parallel transport of a tangent vector.
Closed-form solution for the parallel transport of a tangent vector a
along the geodesic defined by exp_(base_point)(tangent_vec_b).
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim + 1]
Tangent vector at base point to be transported.
tangent_vec_b : array-like, shape=[..., dim + 1]
Tangent vector at base point, along which the parallel transport
is computed.
base_point : array-like, shape=[..., dim + 1]
Point on the hypersphere.
Returns
-------
transported_tangent_vec: array-like, shape=[..., dim + 1]
Transported tangent vector at exp_(base_point)(tangent_vec_b).
"""
theta = self.embedding_metric.norm(tangent_vec_b)
normalized_b = gs.einsum('...,...i->...i', 1 / theta, tangent_vec_b)
pb = self.embedding_metric.inner_product(tangent_vec_a, normalized_b)
p_orth = tangent_vec_a - gs.einsum('...,...i->...i', pb, normalized_b)
transported = \
gs.einsum('...,...i->...i', gs.sinh(theta) * pb, base_point)\
+ gs.einsum('...,...i->...i', gs.cosh(theta) * pb, normalized_b)\
+ p_orth
return transported
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 parallel_transport(self,
tangent_vec,
base_point,
direction=None,
end_point=None):
r"""Compute the parallel transport of a tangent vector.
Closed-form solution for the parallel transport of a tangent vector
along the geodesic between two points `base_point` and `end_point`
or alternatively defined by :math:`t\mapsto exp_(base_point)(
t*direction)`.
Parameters
----------
tangent_vec : array-like, shape=[..., dim + 1]
Tangent vector at base point to be transported.
base_point : array-like, shape=[..., dim + 1]
Point on the hyperboloid.
direction : array-like, shape=[..., dim + 1]
Tangent vector at base point, along which the parallel transport
is computed.
Optional, default : None.
end_point : array-like, shape=[..., dim + 1]
Point on the hyperboloid. Point to transport to. Unused if `tangent_vec_b`
is given.
Optional, default : None.
Returns
-------
transported_tangent_vec: array-like, shape=[..., dim + 1]
Transported tangent vector at exp_(base_point)(tangent_vec_b).
"""
if direction is None:
if end_point is not None:
direction = self.log(end_point, base_point)
else:
raise ValueError(
"Either an end_point or a tangent_vec_b must be given to define the"
" geodesic along which to transport.")
theta = self.embedding_metric.norm(direction)
eps = gs.where(theta == 0.0, 1.0, theta)
normalized_b = gs.einsum("...,...i->...i", 1 / eps, direction)
pb = self.embedding_metric.inner_product(tangent_vec, normalized_b)
p_orth = tangent_vec - gs.einsum("...,...i->...i", pb, normalized_b)
transported = (gs.einsum("...,...i->...i",
gs.sinh(theta) * pb, base_point) +
gs.einsum("...,...i->...i",
gs.cosh(theta) * pb, normalized_b) + p_orth)
return transported
def empirical_frechet_var_bubble(n_samples, theta, dim, n_expectation=1000):
"""Variance of the empirical Fréchet mean for a bubble distribution.
Draw n_sampless from a bubble distribution, computes its empirical
Fréchet mean and the square distance to the asymptotic mean. This
is repeated n_expectation times to compute an approximation of its
expectation (i.e. its variance) by sampling.
The bubble distribution is an isotropic distributions on a Riemannian
hyper sub-sphere of radius 0 < theta = around the north pole of the
hyperbolic space of dimension dim.
Parameters
----------
n_samples: number of samples to draw
theta: radius of the bubble distribution
dim: dimension of the hyperbolic space (embedded in R^{1,dim})
n_expectation: number of computations for approximating the expectation
Returns
-------
tuple (variance, std-dev on the computed variance)
"""
assert dim > 1, "Dim > 1 needed to draw a uniform sample on sub-sphere"
var = []
hyperbole = Hyperbolic(dimension=dim)
bubble = Hypersphere(dimension=dim - 1)
origin = gs.zeros(dim + 1)
origin[0] = 1.0
for k in range(n_expectation):
# Sample n points from the uniform distribution on a sub-sphere
# of radius theta (i.e cos(theta) in ambient space)
data = gs.zeros((n_samples, dim + 1), dtype=gs.float64)
directions = bubble.random_uniform(n_samples)
for i in range(n_samples):
for j in range(dim):
data[i, j + 1] = gs.sinh(theta) * directions[i, j]
data[i, 0] = gs.cosh(theta)
current_mean = _adaptive_gradient_descent(data,
metric=hyperbole.metric,
n_max_iterations=64,
init_points=[origin])
var.append(hyperbole.metric.squared_dist(origin, current_mean))
return np.mean(var), 2 * np.std(var) / np.sqrt(n_expectation)
def exp(self, tangent_vec, base_point):
"""
Compute the Riemannian exponential at point base_point
of tangent vector tangent_vec wrt the metric obtained by
embedding of the hyperbolic space in the Minkowski space.
This gives a point on the hyperbolic space.
:param base_point: a point on the hyperbolic space
:param vector: vector
:returns riem_exp: a point on the hyperbolic space
"""
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)
coef_1 = gs.zeros_like(norm_tangent_vec)
coef_2 = gs.zeros_like(norm_tangent_vec)
coef_1[mask_0] = (1. +
COSH_TAYLOR_COEFFS[2] * norm_tangent_vec[mask_0]**2 +
COSH_TAYLOR_COEFFS[4] * norm_tangent_vec[mask_0]**4 +
COSH_TAYLOR_COEFFS[6] * norm_tangent_vec[mask_0]**6 +
COSH_TAYLOR_COEFFS[8] * norm_tangent_vec[mask_0]**8)
coef_2[mask_0] = (1. +
SINH_TAYLOR_COEFFS[3] * norm_tangent_vec[mask_0]**2 +
SINH_TAYLOR_COEFFS[5] * norm_tangent_vec[mask_0]**4 +
SINH_TAYLOR_COEFFS[7] * norm_tangent_vec[mask_0]**6 +
SINH_TAYLOR_COEFFS[9] * norm_tangent_vec[mask_0]**8)
coef_1[mask_else] = gs.cosh(norm_tangent_vec[mask_else])
coef_2[mask_else] = (gs.sinh(norm_tangent_vec[mask_else]) /
norm_tangent_vec[mask_else])
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):
"""
Riemannian exponential of a tangent vector wrt to a base point.
"""
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)
coef_1 = gs.zeros_like(norm_tangent_vec)
coef_2 = gs.zeros_like(norm_tangent_vec)
coef_1[mask_0] = (1. +
COSH_TAYLOR_COEFFS[2] * norm_tangent_vec[mask_0]**2 +
COSH_TAYLOR_COEFFS[4] * norm_tangent_vec[mask_0]**4 +
COSH_TAYLOR_COEFFS[6] * norm_tangent_vec[mask_0]**6 +
COSH_TAYLOR_COEFFS[8] * norm_tangent_vec[mask_0]**8)
coef_2[mask_0] = (1. +
SINH_TAYLOR_COEFFS[3] * norm_tangent_vec[mask_0]**2 +
SINH_TAYLOR_COEFFS[5] * norm_tangent_vec[mask_0]**4 +
SINH_TAYLOR_COEFFS[7] * norm_tangent_vec[mask_0]**6 +
SINH_TAYLOR_COEFFS[9] * norm_tangent_vec[mask_0]**8)
coef_1[mask_else] = gs.cosh(norm_tangent_vec[mask_else])
coef_2[mask_else] = (gs.sinh(norm_tangent_vec[mask_else]) /
norm_tangent_vec[mask_else])
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=[n_samples, dimension + 1]
Tangent vector at a base point.
base_point : array-like, shape=[n_samples, dimension + 1]
Point in hyperbolic space.
Returns
-------
exp : array-like, shape=[n_samples, dimension + 1]
Point in hyperbolic space equal to the Riemannian exponential
of tangent_vec at the base point.
"""
if self.point_type == 'extrinsic':
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)
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('ni,nj->nj', coef_1, base_point) +
gs.einsum('ni,nj->nj', coef_2, tangent_vec))
hyperbolic_space = Hyperbolic(dimension=self.dimension)
exp = hyperbolic_space.regularize(exp)
return exp
elif self.point_type == 'ball':
norm_base_point = gs.to_ndarray(gs.linalg.norm(base_point, -1), 2,
-1)
norm_base_point = gs.repeat(norm_base_point, base_point.shape[-1],
-1)
den = 1 - norm_base_point**2
norm_tan = gs.to_ndarray(gs.linalg.norm(tangent_vec, axis=-1), 2,
-1)
norm_tan = gs.repeat(norm_tan, base_point.shape[-1], -1)
lambda_base_point = 1 / den
direction = tangent_vec / norm_tan
factor = gs.tanh(lambda_base_point * norm_tan)
exp = self.mobius_add(base_point, direction * factor)
return exp
else:
raise NotImplementedError(
'exp is only implemented for ball and extrinsic')
