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 inner_product_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.inner_product_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
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}, optional
Default coordinates to represent points in hyperbolic space.
scale : int, optional
Scale of the hyperbolic space, defined as the set of points
in Minkowski space whose squared norm is equal to -scale.
"""
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 inner_product_matrix(self, base_point=None):
"""Compute the inner product matrix.
Parameters
----------
base_point: array-like, shape=[..., dim+1]
Returns
-------
inner_prod_mat: array-like, shape=[..., dim+1, dim+1]
"""
self.embedding_metric.inner_product_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=[..., 1]
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=[..., 1]
Squared norm of the vector.
"""
sq_norm = self.embedding_metric.squared_norm(vector)
return sq_norm
@geomstats.vectorization.decorator(['else', 'vector', 'vector'])
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
@geomstats.vectorization.decorator(['else', 'vector', 'vector'])
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 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=[..., 1]
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