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)
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
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1[mask_0] = (1. + INV_SINH_TAYLOR_COEFFS[1] * angle[mask_0]**2 +
INV_SINH_TAYLOR_COEFFS[3] * angle[mask_0]**4 +
INV_SINH_TAYLOR_COEFFS[5] * angle[mask_0]**6 +
INV_SINH_TAYLOR_COEFFS[7] * angle[mask_0]**8)
coef_2[mask_0] = (1. + INV_TANH_TAYLOR_COEFFS[1] * angle[mask_0]**2 +
INV_TANH_TAYLOR_COEFFS[3] * angle[mask_0]**4 +
INV_TANH_TAYLOR_COEFFS[5] * angle[mask_0]**6 +
INV_TANH_TAYLOR_COEFFS[7] * angle[mask_0]**8)
coef_1[mask_else] = angle[mask_else] / gs.sinh(angle[mask_else])
coef_2[mask_else] = angle[mask_else] / gs.tanh(angle[mask_else])
log = (gs.einsum('ni,nj->nj', coef_1, point) -
gs.einsum('ni,nj->nj', coef_2, base_point))
return log
def exp(self, tangent_vec, base_point, **kwargs):
"""Compute the Riemannian exponential of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[..., dim]
Tangent vector at a base point.
base_point : array-like, shape=[..., dim]
Point in the Poincare ball.
Returns
-------
exp : array-like, shape=[..., dim]
Point in the Poincare ball equal to the Riemannian exponential
of tangent_vec at the base point.
"""
squared_norm_bp = gs.sum(base_point**2, axis=-1)
norm_tan = gs.linalg.norm(tangent_vec, axis=-1)
lambda_base_point = 1 / (1 - squared_norm_bp)
# This avoids dividing by 0
norm_tan_eps = gs.where(gs.isclose(norm_tan, 0.0), EPSILON, norm_tan)
direction = gs.einsum("...i,...->...i", tangent_vec, 1 / norm_tan_eps)
factor = gs.tanh(lambda_base_point * norm_tan)
exp = self.mobius_add(base_point,
gs.einsum("...,...i->...i", factor, direction))
return exp
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 exp(self, tangent_vec, base_point):
"""Compute the Riemannian exponential of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[n_samples, dim]
Tangent vector at a base point.
base_point : array-like, shape=[n_samples, dim]
Point in hyperbolic space.
Returns
-------
exp : array-like, shape=[n_samples, dim]
Point in hyperbolic space equal to the Riemannian exponential
of tangent_vec at the base point.
"""
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=2)
base_point = gs.to_ndarray(base_point, to_ndim=2)
norm_base_point =\
gs.expand_dims(gs.linalg.norm(base_point, axis=-1),
axis=-1)
den = 1 - norm_base_point**2
norm_tan =\
gs.expand_dims(gs.linalg.norm(tangent_vec, axis=-1),
axis=-1)
lambda_base_point = 1 / den
zero_tan =\
gs.isclose(gs.sum(tangent_vec * tangent_vec, axis=-1), 0.)
if gs.any(zero_tan):
if norm_tan[zero_tan].shape[0] != 0:
norm_tan[zero_tan] = EPSILON
direction = gs.einsum('...i,...k->...i', tangent_vec, 1 / norm_tan)
factor = gs.tanh(lambda_base_point * norm_tan)
exp = self.mobius_add(base_point, direction * factor)
zero_tan =\
gs.isclose(gs.sum(tangent_vec * tangent_vec, axis=-1), 0.)
if gs.any(zero_tan):
if exp[zero_tan].shape[0] != 0:
exp[zero_tan] = base_point[zero_tan]
return exp
def asymptotic_modulation(dim, theta):
"""Compute the asymptotic modulation factor.
Parameters
----------
dim: dimension of the sphere (embedded in R^{dim+1})
theta: radius of the bubble distribution
Returns
-------
tuple (modulation factor, std-dev on the modulation factor)
"""
gamma = 1.0 / dim + (1.0 - 1.0 / dim) * theta / gs.tanh(theta)
return (1.0 / gamma)**2
def exp(self, tangent_vec, base_point):
"""Compute the Riemannian exponential of a tangent vector.
Parameters
----------
tangent_vec : array-like, shape=[..., dim]
Tangent vector at a base point.
base_point : array-like, shape=[..., dim]
Point in hyperbolic space.
Returns
-------
exp : array-like, shape=[..., dim]
Point in hyperbolic space equal to the Riemannian exponential
of tangent_vec at the base point.
"""
norm_base_point = gs.linalg.norm(base_point, axis=-1)
norm_tan = gs.linalg.norm(tangent_vec, axis=-1)
den = 1 - norm_base_point ** 2
lambda_base_point = 1 / den
zero_tan = gs.isclose(gs.sum(tangent_vec ** 2, axis=-1), 0.)
if gs.any(zero_tan):
norm_tan = gs.assignment(norm_tan, EPSILON, zero_tan)
direction = gs.einsum('...i,...->...i', tangent_vec, 1 / norm_tan)
factor = gs.tanh(
gs.einsum('...,...->...', lambda_base_point, norm_tan))
exp = self.mobius_add(
base_point,
gs.einsum('...i,...->...i', direction, factor))
if gs.any(zero_tan):
exp = gs.assignment(
exp, base_point[zero_tan], zero_tan)
return exp
def log(self, point, base_point):
"""
Compute the Riemannian logarithm at point base_point,
of point wrt the metric obtained by
embedding of the hyperbolic space in the Minkowski space.
This gives a tangent vector at point base_point.
:param base_point: point on the hyperbolic space
:param point: point on the hyperbolic space
:returns riem_log: tangent vector at base_point
"""
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
coef_1 = gs.zeros_like(angle)
coef_2 = gs.zeros_like(angle)
coef_1[mask_0] = (1. + INV_SINH_TAYLOR_COEFFS[1] * angle[mask_0]**2 +
INV_SINH_TAYLOR_COEFFS[3] * angle[mask_0]**4 +
INV_SINH_TAYLOR_COEFFS[5] * angle[mask_0]**6 +
INV_SINH_TAYLOR_COEFFS[7] * angle[mask_0]**8)
coef_2[mask_0] = (1. + INV_TANH_TAYLOR_COEFFS[1] * angle[mask_0]**2 +
INV_TANH_TAYLOR_COEFFS[3] * angle[mask_0]**4 +
INV_TANH_TAYLOR_COEFFS[5] * angle[mask_0]**6 +
INV_TANH_TAYLOR_COEFFS[7] * angle[mask_0]**8)
coef_1[mask_else] = angle[mask_else] / gs.sinh(angle[mask_else])
coef_2[mask_else] = angle[mask_else] / gs.tanh(angle[mask_else])
log = (gs.einsum('ni,nj->nj', coef_1, point) -
gs.einsum('ni,nj->nj', coef_2, base_point))
return log
}
var_inv_tanc_close_0 = {
"function": lambda x: (1 - (x / gs.tan(x))) / x ** 2,
"coefficients": VAR_INV_TAN_TAYLOR_COEFFS,
}
sinch_close_0 = {
"function": lambda x: gs.sinh(x) / x,
"coefficients": SINHC_TAYLOR_COEFFS,
}
cosh_close_0 = {"function": gs.cosh, "coefficients": COSH_TAYLOR_COEFFS}
inv_sinch_close_0 = {
"function": lambda x: x / gs.sinh(x),
"coefficients": INV_SINHC_TAYLOR_COEFFS,
}
inv_tanh_close_0 = {
"function": lambda x: x / gs.tanh(x),
"coefficients": INV_TANH_TAYLOR_COEFFS,
}
arctanh_card_close_0 = {
"function": lambda x: gs.arctanh(x) / x,
"coefficients": ARCTANH_CARD_TAYLOR_COEFFS,
}
def from_vector_to_diagonal_matrix(vector, num_diag=0):
"""Create diagonal matrices from rows of a matrix.
Parameters
----------
vector : array-like, shape=[m, n]
num_diag : int
}
var_inv_tanc_close_0 = {
'function': lambda x: (1 - (x / gs.tan(x))) / x**2,
'coefficients': VAR_INV_TAN_TAYLOR_COEFFS
}
sinch_close_0 = {
'function': lambda x: gs.sinh(x) / x,
'coefficients': SINHC_TAYLOR_COEFFS
}
cosh_close_0 = {'function': gs.cosh, 'coefficients': COSH_TAYLOR_COEFFS}
inv_sinch_close_0 = {
'function': lambda x: x / gs.sinh(x),
'coefficients': INV_SINHC_TAYLOR_COEFFS
}
inv_tanh_close_0 = {
'function': lambda x: x / gs.tanh(x),
'coefficients': INV_TANH_TAYLOR_COEFFS
}
arctanh_card_close_0 = {
'function': lambda x: gs.arctanh(x) / x,
'coefficients': ARCTANH_CARD_TAYLOR_COEFFS
}
def from_vector_to_diagonal_matrix(vector):
"""Create diagonal matrices from rows of a matrix.
Parameters
----------
vector : array-like, shape=[m, n]
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=[n_samples, dimension + 1]
Point in hyperbolic space.
base_point : array-like, shape=[n_samples, dimension + 1]
Point in hyperbolic space.
Returns
-------
log : array-like, shape=[n_samples, dimension + 1]
Tangent vector at the base point equal to the Riemannian logarithm
of point at the base point.
"""
if self.point_type == 'extrinsic':
point = gs.to_ndarray(point, to_ndim=2)
base_point = gs.to_ndarray(base_point, to_ndim=2)
angle = self.dist(base_point, point) / self.scale
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
elif self.point_type == 'ball':
add_base_point = self.mobius_add(-base_point, point)
norm_add = gs.to_ndarray(gs.linalg.norm(add_base_point, axis=-1),
2, -1)
norm_add = gs.repeat(norm_add, base_point.shape[-1], -1)
norm_base_point = gs.to_ndarray(
gs.linalg.norm(base_point, axis=-1), 2, -1)
norm_base_point = gs.repeat(norm_base_point, base_point.shape[-1],
-1)
log = (1 - norm_base_point**2) * gs.arctanh(norm_add)\
* (add_base_point / norm_add)
mask_0 = gs.all(gs.isclose(norm_add, 0.))
log[mask_0] = 0
return log
else:
raise NotImplementedError(
'log is only implemented for ball and extrinsic')
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')
