class HyperbolicSpace(Manifold):
"""
Hyperbolic space embedded in Minkowski space.
Note: points are parameterized by the extrinsic
coordinates by defaults.
"""
def __init__(self, dimension):
self.dimension = dimension
self.metric = HyperbolicMetric(self.dimension)
self.embedding_metric = MinkowskiMetric(self.dimension + 1)
def belongs(self, point, tolerance=TOLERANCE):
"""
By definition, a point on the Hyperbolic space
has Minkowski squared norm -1.
Note: point must be given in extrinsic coordinates.
"""
point = vectorization.expand_dims(point, to_ndim=2)
_, point_dim = point.shape
if point_dim is not self.dimension + 1:
if point_dim is self.dimension:
logging.warning('Use the extrinsic coordinates to '
'represent points on the hypersphere.')
return False
sq_norm = self.embedding_metric.squared_norm(point)
diff = np.abs(sq_norm + 1)
return diff < tolerance
def intrinsic_to_extrinsic_coords(self, point_intrinsic):
"""
From the intrinsic coordinates in the hyperbolic space,
to the extrinsic coordinates in Minkowski space.
"""
point_intrinsic = vectorization.expand_dims(point_intrinsic, to_ndim=2)
n_points, _ = point_intrinsic.shape
dimension = self.dimension
point_extrinsic = np.zeros((n_points, dimension + 1))
point_extrinsic[:, 1:dimension + 1] = point_intrinsic[:, 0:dimension]
point_extrinsic[:, 0] = np.sqrt(
1. + np.linalg.norm(point_intrinsic, axis=1)**2)
assert np.all(self.belongs(point_extrinsic))
assert point_extrinsic.ndim == 2
return point_extrinsic
def extrinsic_to_intrinsic_coords(self, point_extrinsic):
"""
From the extrinsic coordinates in Minkowski space,
to the extrinsic coordinates in Hyperbolic space.
"""
point_extrinsic = vectorization.expand_dims(point_extrinsic, to_ndim=2)
assert np.all(self.belongs(point_extrinsic))
point_intrinsic = point_extrinsic[:, 1:]
assert point_intrinsic.ndim == 2
return point_intrinsic
def projection_to_tangent_space(self, vector, base_point):
"""
Project the vector vector onto the tangent space at base_point
T_{base_point}H
= { w s.t. embedding_inner_product(base_point, w) = 0 }
"""
assert self.belongs(base_point)
inner_prod = self.embedding_metric.inner_product(base_point, vector)
sq_norm_base_point = self.embedding_metric.squared_norm(base_point)
tangent_vec = vector - inner_prod * base_point / sq_norm_base_point
return tangent_vec
def random_uniform(self, n_samples=1, max_norm=1):
"""
Generate random elements on the hyperbolic space.
"""
point = (np.random.rand(n_samples, self.dimension) - .5) * max_norm
point = self.intrinsic_to_extrinsic_coords(point)
assert np.all(self.belongs(point))
assert point.ndim == 2
return point
class HyperbolicMetric(RiemannianMetric):
def __init__(self, dimension):
self.dimension = dimension
self.signature = (dimension, 0, 0)
self.embedding_metric = MinkowskiMetric(dimension + 1)
def squared_norm(self, vector, base_point=None):
"""
Squared norm associated to the Hyperbolic Metric.
"""
sq_norm = self.embedding_metric.squared_norm(vector)
return sq_norm
def exp_basis(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
"""
sq_norm_tangent_vec = self.embedding_metric.squared_norm(tangent_vec)
# TODO(nina): Fix, value error on this squared norm
norm_tangent_vec = np.sqrt(sq_norm_tangent_vec)
if np.isclose(norm_tangent_vec, 0):
coef_1 = (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 = (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)
else:
coef_1 = np.cosh(norm_tangent_vec)
coef_2 = np.sinh(norm_tangent_vec) / norm_tangent_vec
riem_exp = coef_1 * base_point + coef_2 * tangent_vec
return riem_exp
def log_basis(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
"""
angle = self.dist(base_point, point)
if np.isclose(angle, 0):
coef_1 = (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 = (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)
else:
coef_1 = angle / np.sinh(angle)
coef_2 = angle / np.tanh(angle)
return coef_1 * point - coef_2 * base_point
def dist(self, point_a, point_b):
"""
Compute the distance induced on the hyperbolic
space, from its embedding in the Minkowski space.
"""
if np.all(point_a == point_b):
return 0.
point_a = vectorization.expand_dims(point_a, to_ndim=2)
point_b = vectorization.expand_dims(point_b, to_ndim=2)
n_points_a, _ = point_a.shape
n_points_b, _ = point_b.shape
assert (n_points_a == n_points_b or n_points_a == 1 or n_points_b == 1)
n_dists = np.maximum(n_points_a, n_points_b)
dist = np.zeros((n_dists, 1))
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 / np.sqrt(sq_norm_a * sq_norm_b)
mask_cosh_less_1 = np.less_equal(cosh_angle, 1.)
mask_cosh_greater_1 = ~mask_cosh_less_1
dist[mask_cosh_less_1] = 0.
dist[mask_cosh_greater_1] = np.arccosh(cosh_angle[mask_cosh_greater_1])
return dist
class HyperbolicMetric(RiemannianMetric):
def __init__(self, dimension):
self.dimension = dimension
self.signature = (dimension, 0, 0)
self.embedding_metric = MinkowskiMetric(dimension + 1)
def squared_norm(self, vector, base_point=None):
"""
Squared norm associated to the Hyperbolic Metric.
"""
sq_norm = self.embedding_metric.squared_norm(vector)
return sq_norm
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 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
def dist(self, point_a, point_b):
"""
Compute the distance induced on the hyperbolic
space, from its embedding in the Minkowski space.
"""
if gs.all(gs.equal(point_a, point_b)):
return 0.
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, None)
dist = gs.arccosh(cosh_angle)
return dist
class HyperbolicMetric(RiemannianMetric):
def __init__(self, dimension):
super(HyperbolicMetric, self).__init__(dimension=dimension,
signature=(dimension, 0, 0))
self.embedding_metric = MinkowskiMetric(dimension + 1)
def inner_product(self, tangent_vec_a, tangent_vec_b, base_point=None):
"""
Inner product.
"""
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):
"""
Squared norm of a vector associated with the inner product
at the tangent space at a base point.
"""
sq_norm = self.embedding_metric.squared_norm(vector)
return sq_norm
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)
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 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
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 dist(self, point_a, point_b):
"""
Geodesic distance between 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)
return dist
class HyperbolicMetric(RiemannianMetric):
def __init__(self, dimension):
self.dimension = dimension
self.signature = (dimension, 0, 0)
self.embedding_metric = MinkowskiMetric(dimension + 1)
def squared_norm(self, vector, base_point=None):
"""
Squared norm of a vector associated with the inner product
at the tangent space at a base point.
"""
sq_norm = self.embedding_metric.squared_norm(vector)
return sq_norm
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 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 dist(self, point_a, point_b):
"""
Geodesic distance between two points.
"""
if gs.all(gs.equal(point_a, point_b)):
return 0.
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, None)
dist = gs.arccosh(cosh_angle)
return dist