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]
Point in hyperbolic space.
base_point : array-like, shape=[..., dim]
Point in hyperbolic space.
Returns
-------
log : array-like, shape=[..., dim]
Tangent vector at the base point equal to the Riemannian logarithm
of point at the base point.
"""
add_base_point = self.mobius_add(-base_point, point)
norm_add =\
gs.expand_dims(gs.linalg.norm(
add_base_point, axis=-1), axis=-1)
norm_base_point =\
gs.expand_dims(gs.linalg.norm(
base_point, axis=-1), axis=-1)
log = (1 - norm_base_point**2) * gs.arctanh(norm_add)
mask_0 = gs.isclose(gs.squeeze(norm_add, axis=-1), 0.)
mask_non0 = ~mask_0
add_base_point = gs.assignment(
add_base_point,
gs.zeros_like(add_base_point[mask_0]),
mask_0)
add_base_point = gs.assignment(
add_base_point,
add_base_point[mask_non0] / norm_add[mask_non0],
mask_non0)
log = gs.einsum(
'...i,...j->...j', log, add_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=[n_samples, dim]
Point in hyperbolic space.
base_point : array-like, shape=[n_samples, dim]
Point in hyperbolic space.
Returns
-------
log : array-like, shape=[n_samples, dim]
Tangent vector at the base point equal to the Riemannian logarithm
of point at the base point.
"""
base_point = gs.to_ndarray(base_point, to_ndim=2)
point = gs.to_ndarray(point, to_ndim=2)
add_base_point = self.mobius_add(-base_point, point)
norm_add =\
gs.expand_dims(gs.linalg.norm(
add_base_point, axis=-1), axis=-1)
norm_base_point =\
gs.expand_dims(gs.linalg.norm(
base_point, axis=-1), axis=-1)
log = (1 - norm_base_point**2) * gs.arctanh(norm_add)
log = gs.einsum('...i,...j->...j', log, (add_base_point / norm_add))
mask_0 = gs.isclose(gs.squeeze(norm_add, axis=-1), 0.)
if gs.any(mask_0):
log[mask_0] = 0
return log
}
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
number of diagonal in result matrix. If 0, the result matrix is a
diagonal matrix; if positive, the result matrix has an upper-right
non-zero diagonal; if negative, the result matrix has a lower-left
non-zero diagonal.
}
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]
Returns
-------
diagonals : array-like, shape=[m, n, n]
3-dimensional array where the `i`-th n-by-n array `diagonals[i, :, :]`
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')