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_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
last_col = gs.cos(theta) * gs.ones(n_samples)
last_col = last_col[:, None] if (n_samples > 1) else last_col
directions = bubble.random_uniform(n_samples)
rest_col = gs.sin(theta) * directions
data = gs.concatenate([rest_col, last_col], axis=-1)
estimator = FrechetMean(sphere.metric,
max_iter=32,
method="adaptive",
init_point=north_pole)
estimator.fit(data)
current_mean = estimator.estimate_
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_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: number of samples to draw
theta: radius of the bubble distribution
dim: dimension of the sphere (embedded in R^{dim+1})
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 = []
sphere = Hypersphere(dimension=dim)
bubble = Hypersphere(dimension=dim - 1)
north_pole = gs.zeros(dim + 1)
north_pole[dim] = 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)
# 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)
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)
current_mean = sphere.metric.adaptive_gradientdescent_mean(
data, n_max_iterations=64, init_points=[north_pole])
var.append(sphere.metric.squared_dist(north_pole, current_mean))
return gs.mean(var), 2 * gs.std(var) / gs.sqrt(n_expectation)