"""Frechet mean.
Lead authors: Nicolas Guigui and Nina Miolane.
"""
import logging
import math
from sklearn.base import BaseEstimator
import geomstats.backend as gs
import geomstats.errors as error
import geomstats.vectorization
from geomstats.geometry.discrete_curves import SRVMetric
from geomstats.geometry.hypersphere import Hypersphere
EPSILON = 1e-4
[docs]def variance(points, base_point, metric, weights=None, point_type="vector"):
"""Variance of (weighted) points wrt a base point.
Parameters
----------
points : array-like, shape=[..., dim]
Points.
weights : array-like, shape=[...,]
Weights associated to the points.
Optional, default: None.
Returns
-------
var : float
Weighted variance of the points.
"""
n_points = geomstats.vectorization.get_n_points(points, point_type)
if weights is None:
weights = gs.ones((n_points,))
sum_weights = gs.sum(weights)
sq_dists = metric.squared_dist(base_point, points)
var = weights * sq_dists
var = gs.sum(var)
var /= sum_weights
return var
[docs]def linear_mean(points, weights=None, point_type="vector"):
"""Compute the weighted linear mean.
The linear mean is the Frechet mean when points:
- lie in a Euclidean space with Euclidean metric,
- lie in a Minkowski space with Minkowski metric.
Parameters
----------
points : array-like, shape=[..., dim]
Points to be averaged.
weights : array-like, shape=[...,]
Weights associated to the points.
Optional, default: None.
Returns
-------
mean : array-like, shape=[dim,]
Weighted linear mean of the points.
"""
if isinstance(points, list):
points = gs.stack(points, axis=0)
if isinstance(weights, list):
weights = gs.array(weights)
n_points = geomstats.vectorization.get_n_points(points, point_type)
if weights is None:
weights = gs.ones((n_points,))
sum_weights = gs.sum(weights)
einsum_str = "...,...j->...j"
if point_type == "matrix":
einsum_str = "...,...jk->...jk"
weighted_points = gs.einsum(einsum_str, weights, points)
mean = gs.sum(weighted_points, axis=0) / sum_weights
return mean
def _default_gradient_descent(
points,
metric,
weights,
max_iter,
point_type,
epsilon,
init_step_size,
verbose,
init_point=None,
):
"""Perform default gradient descent."""
if point_type == "vector":
points = gs.to_ndarray(points, to_ndim=2)
einsum_str = "n,nj->j"
else:
points = gs.to_ndarray(points, to_ndim=3)
einsum_str = "n,nij->ij"
n_points = gs.shape(points)[0]
if weights is None:
weights = gs.ones((n_points,))
mean = points[0] if init_point is None else init_point
if n_points == 1:
return mean
sum_weights = gs.sum(weights)
sq_dists_between_iterates = []
iteration = 0
sq_dist = 0.0
var = 0.0
norm_old = gs.linalg.norm(points)
step = init_step_size
while iteration < max_iter:
logs = metric.log(point=points, base_point=mean)
var = gs.sum(metric.squared_norm(logs, mean) * weights) / gs.sum(weights)
tangent_mean = gs.einsum(einsum_str, weights, logs)
tangent_mean /= sum_weights
norm = gs.linalg.norm(tangent_mean)
sq_dist = metric.squared_norm(tangent_mean, mean)
sq_dists_between_iterates.append(sq_dist)
var_is_0 = gs.isclose(var, 0.0)
metric_dim = metric.dim
if isinstance(metric, SRVMetric):
metric_dim = tangent_mean.shape[-2] * tangent_mean.shape[-1]
sq_dist_is_small = gs.less_equal(sq_dist, epsilon * metric_dim)
condition = ~gs.logical_or(var_is_0, sq_dist_is_small)
if not (condition or iteration == 0):
break
estimate_next = metric.exp(step * tangent_mean, mean)
mean = estimate_next
iteration += 1
if norm < norm_old:
norm_old = norm
elif norm > norm_old:
step = step / 2.0
if iteration == max_iter:
logging.warning(
"Maximum number of iterations {} reached. "
"The mean may be inaccurate".format(max_iter)
)
if verbose:
logging.info(
"n_iter: {}, final variance: {}, final dist: {}".format(
iteration, var, sq_dist
)
)
return mean
def _batch_gradient_descent(
points,
metric,
weights=None,
max_iter=32,
init_step_size=1e-3,
epsilon=5e-3,
point_type="vector",
verbose=False,
init_point=None,
):
"""Perform batch gradient descent."""
if point_type == "vector":
if points.ndim < 3:
return _default_gradient_descent(
points,
metric,
weights,
max_iter,
point_type,
epsilon,
init_step_size,
verbose,
)
einsum_str = "ni,nij->ij"
ndim = 1
else:
if points.ndim < 4:
return _default_gradient_descent(
points,
metric,
weights,
max_iter,
point_type,
epsilon,
init_step_size,
verbose,
)
einsum_str = "nk,nkij->kij"
ndim = 2
shape = points.shape
n_points = shape[0]
n_batch = shape[1]
if n_points == 1:
return points[0]
if weights is None:
weights = gs.ones((n_points, n_batch))
flat_shape = (n_batch * n_points,) + shape[-ndim:]
estimates = points[0] if init_point is None else init_point
points_flattened = gs.reshape(points, (n_points * n_batch,) + shape[-ndim:])
convergence = math.inf
iteration = 0
convergence_old = convergence
while convergence > epsilon and max_iter > iteration:
iteration += 1
estimates_broadcast, _ = gs.broadcast_arrays(estimates, points)
estimates_flattened = gs.reshape(estimates_broadcast, flat_shape)
tangent_grad = metric.log(points_flattened, estimates_flattened)
tangent_grad = gs.reshape(tangent_grad, shape)
tangent_mean = gs.einsum(einsum_str, weights, tangent_grad) / n_points
next_estimates = metric.exp(init_step_size * tangent_mean, estimates)
convergence = gs.sum(metric.squared_norm(tangent_mean, estimates))
estimates = next_estimates
if convergence < convergence_old:
convergence_old = convergence
elif convergence > convergence_old:
init_step_size = init_step_size / 2.0
if iteration == max_iter:
logging.warning(
"Maximum number of iterations {} reached. The "
"mean may be inaccurate".format(max_iter)
)
if verbose:
logging.info(
"n_iter: {}, final dist: {},"
"final step size: {}".format(iteration, convergence, init_step_size)
)
return estimates
def _adaptive_gradient_descent(
points,
metric,
weights=None,
max_iter=32,
epsilon=1e-12,
init_step_size=1.0,
init_point=None,
point_type="vector",
verbose=False,
):
"""Perform adaptive gradient descent.
Frechet mean of (weighted) points using adaptive time-steps
The loss function optimized is :math:`||M_1(x)||_x`
(where :math:`M_1(x)` is the tangent mean at x) rather than
the mean-square-distance (MSD) because this simplifies computations.
Adaptivity is done in a Levenberg-Marquardt style weighting variable tau
between the first order and the second order Gauss-Newton gradient descent.
Parameters
----------
points : array-like, shape=[..., dim]
Points to be averaged.
weights : array-like, shape=[..., 1], optional
Weights associated to the points.
max_iter : int, optional
Maximum number of iterations for the gradient descent.
init_point : array-like, shape=[{dim, [n, n]}]
Initial point.
Optional, default : None. In this case the first sample of the input data is
used.
epsilon : float, optional
Tolerance for stopping the gradient descent.
Returns
-------
current_mean: array-like, shape=[..., dim]
Weighted Frechet mean of the points.
"""
if point_type == "vector":
points = gs.to_ndarray(points, to_ndim=2)
einsum_str = "n,nj->j"
else:
points = gs.to_ndarray(points, to_ndim=3)
einsum_str = "n,nij->ij"
n_points = gs.shape(points)[0]
tau_max = 1e6
tau_mul_up = 1.6511111
tau_min = 1e-6
tau_mul_down = 0.1
if n_points == 1:
return points[0]
current_mean = points[0] if init_point is None else init_point
if weights is None:
weights = gs.ones((n_points,))
sum_weights = gs.sum(weights)
tau = init_step_size
iteration = 0
logs = metric.log(point=points, base_point=current_mean)
var = gs.sum(metric.squared_norm(logs, current_mean) * weights) / gs.sum(weights)
current_tangent_mean = gs.einsum(einsum_str, weights, logs)
current_tangent_mean /= sum_weights
sq_norm_current_tangent_mean = metric.squared_norm(
current_tangent_mean, base_point=current_mean
)
while sq_norm_current_tangent_mean > epsilon**2 and iteration < max_iter:
iteration += 1
shooting_vector = tau * current_tangent_mean
next_mean = metric.exp(tangent_vec=shooting_vector, base_point=current_mean)
logs = metric.log(point=points, base_point=next_mean)
var = gs.sum(metric.squared_norm(logs, current_mean) * weights) / gs.sum(
weights
)
next_tangent_mean = gs.einsum(einsum_str, weights, logs)
next_tangent_mean /= sum_weights
sq_norm_next_tangent_mean = metric.squared_norm(
next_tangent_mean, base_point=next_mean
)
if sq_norm_next_tangent_mean < sq_norm_current_tangent_mean:
current_mean = next_mean
current_tangent_mean = next_tangent_mean
sq_norm_current_tangent_mean = sq_norm_next_tangent_mean
tau = min(tau_max, tau_mul_up * tau)
else:
tau = max(tau_min, tau_mul_down * tau)
if iteration == max_iter:
logging.warning(
"Maximum number of iterations {} reached. "
"The mean may be inaccurate".format(max_iter)
)
if verbose:
logging.info(
"n_iter: {}, final variance: {}, final dist: {},"
" final_step_size: {}".format(
iteration, var, sq_norm_current_tangent_mean, tau
)
)
return current_mean
def _circle_mean(points):
"""Determine the mean on a circle.
Data are expected in radians in the range [-pi, pi). The mean is returned
in the same range. If the mean is unique, this algorithm is guaranteed to
find it. It is not vulnerable to local minima of the Frechet function. If
the mean is not unique, the algorithm only returns one of the means. Which
mean is returned depends on numerical rounding errors.
Reference
---------
..[HH15] Hotz, T. and S. F. Huckemann (2015), "Intrinsic means on the circle:
Uniqueness, locus and asymptotics", Annals of the Institute of
Statistical Mathematics 67 (1), 177–193.
https://arxiv.org/abs/1108.2141
"""
if points.ndim > 1:
points_ = Hypersphere.extrinsic_to_angle(points)
else:
points_ = gs.copy(points)
sample_size = points_.shape[0]
mean0 = gs.mean(points_)
var0 = gs.sum((points_ - mean0) ** 2)
sorted_points = gs.sort(points_)
means = _circle_variances(mean0, var0, sample_size, sorted_points)
return means[gs.argmin(means[:, 1]), 0]
def _circle_variances(mean, var, n_samples, points):
"""Compute the minimizer of the variance functional.
Parameters
----------
mean : float
Mean angle.
var : float
Variance of the angles.
n_samples : int
Number of samples.
points : array-like, shape=[n,]
Data set of ordered angles.
References
----------
..[HH15] Hotz, T. and S. F. Huckemann (2015), "Intrinsic means on the circle:
Uniqueness, locus and asymptotics", Annals of the Institute of
Statistical Mathematics 67 (1), 177–193.
https://arxiv.org/abs/1108.2141
"""
means = (mean + gs.linspace(0.0, 2 * gs.pi, n_samples + 1)[:-1]) % (2 * gs.pi)
means = gs.where(means >= gs.pi, means - 2 * gs.pi, means)
parts = gs.array([sum(points) / n_samples if means[0] < 0 else 0])
m_plus = means >= 0
left_sums = gs.cumsum(points)
right_sums = left_sums[-1] - left_sums
i = gs.arange(n_samples, dtype=right_sums.dtype)
j = i[1:]
parts2 = right_sums[:-1] / (n_samples - j)
first_term = parts2[:1]
parts2 = gs.where(m_plus[1:], left_sums[:-1] / j, parts2)
parts = gs.concatenate([parts, first_term, parts2[1:]])
# Formula (6) from [HH15]_
plus_vec = (4 * gs.pi * i / n_samples) * (gs.pi + parts - mean) - (
2 * gs.pi * i / n_samples
) ** 2
minus_vec = (4 * gs.pi * (n_samples - i) / n_samples) * (gs.pi - parts + mean) - (
2 * gs.pi * (n_samples - i) / n_samples
) ** 2
minus_vec = gs.where(m_plus, plus_vec, minus_vec)
means = gs.transpose(gs.vstack([means, var + minus_vec]))
return means
[docs]class FrechetMean(BaseEstimator):
r"""Empirical Frechet mean.
Parameters
----------
metric : RiemannianMetric
Riemannian metric.
max_iter : int
Maximum number of iterations for gradient descent.
Optional, default: 32.
epsilon : float
Tolerance for stopping the gradient descent.
Optional, default : 1e-4
point_type : str, {\'vector\', \'matrix\'}
Point type.
Optional, default: None.
method : str, {\'default\', \'adaptive\', \'batch\'}
Gradient descent method.
The `adaptive` method uses a Levenberg-Marquardt style adaptation of
the learning rate. The `batch` method is similar to the default
method but for batches of equal length of samples. In this case,
samples must be of shape [n_samples, n_batch, {dim, [n,n]}].
Optional, default: \'default\'.
init_point : array-like, shape=[{dim, [n, n]}]
Initial point.
Optional, default : None. In this case the first sample of the input data is
used.
init_step_size : float
Initial step size or learning rate.
verbose : bool
Verbose option.
Optional, default: False.
"""
def __init__(
self,
metric,
max_iter=32,
epsilon=EPSILON,
point_type=None,
method="default",
init_point=None,
init_step_size=1.0,
verbose=False,
):
self.metric = metric
self.max_iter = max_iter
self.epsilon = epsilon
self.point_type = point_type
self.method = method
self.init_step_size = init_step_size
self.verbose = verbose
self.init_point = init_point
self.estimate_ = None
if point_type is None:
self.point_type = metric.default_point_type
error.check_parameter_accepted_values(
self.point_type, "point_type", ["vector", "matrix"]
)
[docs] def fit(self, X, y=None, weights=None):
"""Compute the empirical Frechet mean.
Parameters
----------
X : {array-like, sparse matrix}, shape=[..., {dim, [n, n]}]
Training input samples.
y : array-like, shape=[...,] or [..., n_outputs]
Target values (class labels in classification, real numbers in
regression).
Ignored.
weights : array-like, shape=[...,]
Weights associated to the points.
Optional, default: None.
Returns
-------
self : object
Returns self.
"""
metric_str = self.metric.__str__()
is_linear_metric = (
"EuclideanMetric" in metric_str
or "MatricesMetric" in metric_str
or "MinkowskiMetric" in metric_str
)
if "HypersphereMetric" in metric_str and self.metric.dim == 1:
mean = Hypersphere.angle_to_extrinsic(_circle_mean(X))
error.check_parameter_accepted_values(
self.method, "method", ["default", "adaptive", "batch"]
)
if is_linear_metric:
mean = linear_mean(points=X, weights=weights, point_type=self.point_type)
elif self.method == "default":
mean = _default_gradient_descent(
points=X,
weights=weights,
metric=self.metric,
max_iter=self.max_iter,
init_step_size=self.init_step_size,
point_type=self.point_type,
epsilon=self.epsilon,
verbose=self.verbose,
init_point=self.init_point,
)
elif self.method == "adaptive":
mean = _adaptive_gradient_descent(
points=X,
metric=self.metric,
weights=weights,
max_iter=self.max_iter,
epsilon=self.epsilon,
init_step_size=self.init_step_size,
init_point=self.init_point,
point_type=self.point_type,
verbose=self.verbose,
)
elif self.method == "batch":
mean = _batch_gradient_descent(
points=X,
metric=self.metric,
weights=weights,
max_iter=self.max_iter,
init_step_size=self.init_step_size,
epsilon=self.epsilon,
point_type=self.point_type,
verbose=self.verbose,
init_point=self.init_point,
)
self.estimate_ = mean
return self