class HyperboloidMetric(HyperbolicMetric):
"""Class that defines operations using a hyperbolic metric.
Parameters
----------
dim : int
Dimension of the hyperbolic space.
point_type : str, {'extrinsic', 'intrinsic', etc}
Default coordinates to represent points in hyperbolic space.
Optional, default: 'extrinsic'.
scale : int
Scale of the hyperbolic space, defined as the set of points
in Minkowski space whose squared norm is equal to -scale.
Optional, default: 1.
"""
default_point_type = "vector"
default_coords_type = "extrinsic"
def __init__(self, dim, coords_type="extrinsic", scale=1):
super(HyperboloidMetric, self).__init__(dim=dim, scale=scale)
self.embedding_metric = MinkowskiMetric(dim + 1)
self.coords_type = coords_type
self.point_type = HyperbolicMetric.default_point_type
self.scale = scale
def metric_matrix(self, base_point=None):
"""Compute the inner product matrix.
Parameters
----------
base_point: array-like, shape=[..., dim + 1]
Base point.
Optional, default: None.
Returns
-------
inner_prod_mat: array-like, shape=[..., dim+1, dim + 1]
Inner-product matrix.
"""
self.embedding_metric.metric_matrix(base_point)
def _inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""Compute the inner-product of two tangent vectors at a base point.
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim + 1]
First tangent vector at base point.
tangent_vec_b : array-like, shape=[..., dim + 1]
Second tangent vector at base point.
base_point : array-like, shape=[..., dim + 1], optional
Point in hyperbolic space.
Returns
-------
inner_prod : array-like, shape=[...,]
Inner-product of the two tangent vectors.
"""
inner_prod = self.embedding_metric.inner_product(
tangent_vec_a, tangent_vec_b, base_point)
return inner_prod
def _squared_norm(self, vector, base_point=None):
"""Compute the squared norm of a vector.
Squared norm of a vector associated with the inner-product
at the tangent space at a base point.
Parameters
----------
vector : array-like, shape=[..., dim + 1]
Vector on the tangent space of the hyperbolic space at base point.
base_point : array-like, shape=[..., dim + 1], optional
Point in hyperbolic space in extrinsic coordinates.
Returns
-------
sq_norm : array-like, shape=[...,]
Squared norm of the vector.
"""
sq_norm = self.embedding_metric.squared_norm(vector)
return sq_norm
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)
coef_1 = utils.taylor_exp_even_func(sq_norm_tangent_vec,
utils.cosh_close_0,
order=5)
coef_2 = utils.taylor_exp_even_func(sq_norm_tangent_vec,
utils.sinch_close_0,
order=5)
exp = gs.einsum("...,...j->...j", coef_1, base_point) + gs.einsum(
"...,...j->...j", coef_2, tangent_vec)
exp = Hyperboloid(dim=self.dim).regularize(exp)
return exp
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
coef_1_ = utils.taylor_exp_even_func(angle**2,
utils.inv_sinch_close_0,
order=4)
coef_2_ = utils.taylor_exp_even_func(angle**2,
utils.inv_tanh_close_0,
order=4)
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 dist(self, point_a, point_b):
"""Compute the geodesic distance between two points.
Parameters
----------
point_a : array-like, shape=[..., dim + 1]
First point in hyperbolic space.
point_b : array-like, shape=[..., dim + 1]
Second point in hyperbolic space.
Returns
-------
dist : array-like, shape=[...,]
Geodesic distance between the two points.
"""
sq_norm_a = self.embedding_metric.squared_norm(point_a)
sq_norm_b = self.embedding_metric.squared_norm(point_b)
inner_prod = self.embedding_metric.inner_product(point_a, point_b)
cosh_angle = -inner_prod / gs.sqrt(sq_norm_a * sq_norm_b)
cosh_angle = gs.clip(cosh_angle, 1.0, 1e24)
dist = gs.arccosh(cosh_angle)
dist *= self.scale
return dist
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
class HyperboloidMetric(HyperbolicMetric):
"""Class that defines operations using a hyperbolic metric.
Parameters
----------
dim : int
Dimension of the hyperbolic space.
point_type : str, {'extrinsic', 'intrinsic', etc}
Default coordinates to represent points in hyperbolic space.
Optional, default: 'extrinsic'.
scale : int
Scale of the hyperbolic space, defined as the set of points
in Minkowski space whose squared norm is equal to -scale.
Optional, default: 1.
"""
default_point_type = 'vector'
default_coords_type = 'extrinsic'
def __init__(self, dim, coords_type='extrinsic', scale=1):
super(HyperboloidMetric, self).__init__(
dim=dim,
scale=scale)
self.embedding_metric = MinkowskiMetric(dim + 1)
self.coords_type = coords_type
self.point_type = HyperbolicMetric.default_point_type
self.scale = scale
def metric_matrix(self, base_point=None):
"""Compute the inner product matrix.
Parameters
----------
base_point: array-like, shape=[..., dim + 1]
Base point.
Optional, default: None.
Returns
-------
inner_prod_mat: array-like, shape=[..., dim+1, dim + 1]
Inner-product matrix.
"""
self.embedding_metric.metric_matrix(base_point)
def _inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""Compute the inner-product of two tangent vectors at a base point.
Parameters
----------
tangent_vec_a : array-like, shape=[..., dim + 1]
First tangent vector at base point.
tangent_vec_b : array-like, shape=[..., dim + 1]
Second tangent vector at base point.
base_point : array-like, shape=[..., dim + 1], optional
Point in hyperbolic space.
Returns
-------
inner_prod : array-like, shape=[...,]
Inner-product of the two tangent vectors.
"""
inner_prod = self.embedding_metric.inner_product(
tangent_vec_a, tangent_vec_b, base_point)
return inner_prod
def _squared_norm(self, vector, base_point=None):
"""Compute the squared norm of a vector.
Squared norm of a vector associated with the inner-product
at the tangent space at a base point.
Parameters
----------
vector : array-like, shape=[..., dim + 1]
Vector on the tangent space of the hyperbolic space at base point.
base_point : array-like, shape=[..., dim + 1], optional
Point in hyperbolic space in extrinsic coordinates.
Returns
-------
sq_norm : array-like, shape=[...,]
Squared norm of the vector.
"""
sq_norm = self.embedding_metric.squared_norm(vector)
return sq_norm
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)
coef_1 = utils.taylor_exp_even_func(
sq_norm_tangent_vec, utils.cosh_close_0, order=5)
coef_2 = utils.taylor_exp_even_func(
sq_norm_tangent_vec, utils.sinch_close_0, order=5)
exp = (
gs.einsum('...,...j->...j', coef_1, base_point)
+ gs.einsum('...,...j->...j', coef_2, tangent_vec))
exp = Hyperboloid(dim=self.dim).regularize(exp)
return exp
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
coef_1_ = utils.taylor_exp_even_func(
angle ** 2, utils.inv_sinch_close_0, order=4)
coef_2_ = utils.taylor_exp_even_func(
angle ** 2, utils.inv_tanh_close_0, order=4)
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 dist(self, point_a, point_b):
"""Compute the geodesic distance between two points.
Parameters
----------
point_a : array-like, shape=[..., dim + 1]
First point in hyperbolic space.
point_b : array-like, shape=[..., dim + 1]
Second point in hyperbolic space.
Returns
-------
dist : array-like, shape=[...,]
Geodesic distance between the two points.
"""
sq_norm_a = self.embedding_metric.squared_norm(point_a)
sq_norm_b = self.embedding_metric.squared_norm(point_b)
inner_prod = self.embedding_metric.inner_product(point_a, point_b)
cosh_angle = - inner_prod / gs.sqrt(sq_norm_a * sq_norm_b)
cosh_angle = gs.clip(cosh_angle, 1.0, 1e24)
dist = gs.arccosh(cosh_angle)
dist *= self.scale
return dist