def empirical_frechet_var_bubble(n_samples, theta, dim, n_expectation=1000):
"""Variance of the empirical Fréchet mean for a bubble distribution.
Draw n_sampless from a bubble distribution, computes its empirical
Fréchet mean and the square distance to the asymptotic mean. This
is repeated n_expectation times to compute an approximation of its
expectation (i.e. its variance) by sampling.
The bubble distribution is an isotropic distributions on a Riemannian
hyper sub-sphere of radius 0 < theta < Pi around the north pole of the
sphere of dimension dim.
Parameters
----------
n_samples : int
Number of samples to draw.
theta: float
Radius of the bubble distribution.
dim : int
Dimension of the sphere (embedded in R^{dim+1}).
n_expectation: int, optional (defaults to 1000)
Number of computations for approximating the expectation.
Returns
-------
tuple (variance, std-dev on the computed variance)
"""
if dim <= 1:
raise ValueError(
'Dim > 1 needed to draw a uniform sample on sub-sphere.')
var = []
sphere = Hypersphere(dim=dim)
bubble = Hypersphere(dim=dim - 1)
north_pole = gs.zeros(dim + 1)
north_pole[dim] = 1.0
for _ in range(n_expectation):
# Sample n points from the uniform distribution on a sub-sphere
# of radius theta (i.e cos(theta) in ambient space)
# TODO (nina): Add this code as a method of hypersphere
data = gs.zeros((n_samples, dim + 1), dtype=gs.float64)
directions = bubble.random_uniform(n_samples)
directions = gs.to_ndarray(directions, to_ndim=2)
for i in range(n_samples):
for j in range(dim):
data[i, j] = gs.sin(theta) * directions[i, j]
data[i, dim] = gs.cos(theta)
# TODO (nina): Use FrechetMean here
current_mean = _adaptive_gradient_descent(data,
metric=sphere.metric,
max_iter=32,
init_point=north_pole)
var.append(sphere.metric.squared_dist(north_pole, current_mean))
return gs.mean(var), 2 * gs.std(var) / gs.sqrt(n_expectation)
def empirical_frechet_mean_random_init_s2(data, n_init=1, init_points=[]):
"""Fréchet mean on S2 by gradient descent from multiple starting points"
Parameters
----------
data: empirical distribution on S2
n_init: number of initial points drawn uniformly at random on S2
init_points: list of initial points for the first gradient descent
Returns
-------
frechet mean list
"""
assert n_init >= 1, "Gradient descent needs at least one starting point"
dim = len(data[0]) - 1
sphere = Hypersphere(dimension=dim)
if len(init_points) == 0:
init_points = [sphere.random_uniform()]
# for a noncompact manifold, we need to revise this to a ball
# with a maximal radius
mean = _adaptive_gradient_descent(data,
metric=sphere.metric,
n_max_iterations=64,
init_points=init_points)
sigma_mean = mean_sq_dist_s2(mean, data)
# print ("variance {0} for FM {1}".format(sigFM,FM))
for i in range(n_init - 1):
init_points = sphere.random_uniform()
new_mean = _adaptive_gradient_descent(data,
metric=sphere.metric,
n_max_iterations=64,
init_points=init_points)
sigma_new_mean = mean_sq_dist_s2(new_mean, data)
if sigma_new_mean < sigma_mean:
mean = new_mean
sigma_mean = sigma_new_mean
# print ("new variance {0} for FM {1}".format(sigFM,FM))
return mean
def test_adaptive_gradient_descent_sphere(self):
n_tests = 100
result = gs.zeros(n_tests)
expected = gs.zeros(n_tests)
for i in range(n_tests):
# take 2 random points, compute their mean, and verify that
# log of each at the mean is opposite
points = self.sphere.random_uniform(n_samples=2)
mean = _adaptive_gradient_descent(points=points,
metric=self.sphere.metric)
logs = self.sphere.metric.log(point=points, base_point=mean)
result[i] = gs.linalg.norm(logs[1, :] + logs[0, :])
self.assertAllClose(expected, result, rtol=1e-10, atol=1e-10)
def empirical_frechet_var_bubble(n_samples, theta, dim, n_expectation=1000):
"""Variance of the empirical Fréchet mean for a bubble distribution.
Draw n_sampless from a bubble distribution, computes its empirical
Fréchet mean and the square distance to the asymptotic mean. This
is repeated n_expectation times to compute an approximation of its
expectation (i.e. its variance) by sampling.
The bubble distribution is an isotropic distributions on a Riemannian
hyper sub-sphere of radius 0 < theta = around the north pole of the
hyperbolic space of dimension dim.
Parameters
----------
n_samples: number of samples to draw
theta: radius of the bubble distribution
dim: dimension of the hyperbolic space (embedded in R^{1,dim})
n_expectation: number of computations for approximating the expectation
Returns
-------
tuple (variance, std-dev on the computed variance)
"""
assert dim > 1, "Dim > 1 needed to draw a uniform sample on sub-sphere"
var = []
hyperbole = Hyperbolic(dimension=dim)
bubble = Hypersphere(dimension=dim - 1)
origin = gs.zeros(dim + 1)
origin[0] = 1.0
for k in range(n_expectation):
# Sample n points from the uniform distribution on a sub-sphere
# of radius theta (i.e cos(theta) in ambient space)
data = gs.zeros((n_samples, dim + 1), dtype=gs.float64)
directions = bubble.random_uniform(n_samples)
for i in range(n_samples):
for j in range(dim):
data[i, j + 1] = gs.sinh(theta) * directions[i, j]
data[i, 0] = gs.cosh(theta)
current_mean = _adaptive_gradient_descent(data,
metric=hyperbole.metric,
n_max_iterations=64,
init_points=[origin])
var.append(hyperbole.metric.squared_dist(origin, current_mean))
return np.mean(var), 2 * np.std(var) / np.sqrt(n_expectation)