# Source code for geomstats.learning.frechet_mean

```"""Frechet mean.

Lead authors: Nicolas Guigui and Nina Miolane.
"""

import abc
import logging
import math

from sklearn.base import BaseEstimator

import geomstats.backend as gs
import geomstats.errors as error
from geomstats.geometry.discrete_curves import ElasticMetric, SRVMetric
from geomstats.geometry.euclidean import EuclideanMetric
from geomstats.geometry.hypersphere import HypersphereMetric

ELASTIC_METRICS = [SRVMetric, ElasticMetric]

def _is_linear_metric(metric):
return isinstance(metric, EuclideanMetric)

def _is_elastic_metric(metric):
return isinstance(metric, tuple(ELASTIC_METRICS))

def _scalarmul(scalar, array):
return gs.einsum("n,n...->n...", scalar, array)

def _scalarmulsum(scalar, array):
return gs.einsum("n,n...->...", scalar, array)

def _batchscalarmulsum(array_1, array_2):
return gs.einsum("ni,ni...->i...", array_1, array_2)

[docs]
def variance(space, points, base_point, weights=None):
"""Variance of (weighted) points wrt a base point.

Parameters
----------
space : Manifold
Equipped manifold.
points : array-like, shape=[n_samples, dim]
Points.
weights : array-like, shape=[n_samples,]
Weights associated to the points.
Optional, default: None.

Returns
-------
var : float
Weighted variance of the points.
"""
if weights is None:
n_points = gs.shape(points)[0]
weights = gs.ones((n_points,))

sum_weights = gs.sum(weights)
sq_dists = space.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):
"""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=[n_samples, dim]
Points to be averaged.
weights : array-like, shape=[n_samples,]
Weights associated to the points.
Optional, default: None.

Returns
-------
mean : array-like, shape=[dim,]
Weighted linear mean of the points.
"""
if weights is None:
n_points = gs.shape(points)[0]
weights = gs.ones(n_points)
sum_weights = gs.sum(weights)

weighted_points = _scalarmul(weights, points)

mean = gs.sum(weighted_points, axis=0) / sum_weights
return mean

[docs]

Parameters
----------
max_iter : int, optional
Maximum number of iterations for the gradient descent.
epsilon : float, optional
Tolerance for stopping the gradient descent.
init_point : array-like, shape=[*metric.shape]
Initial point.
Optional, default : None. In this case the first sample of the input
data is used.
init_step_size : float
Learning rate in the gradient descent.
Optional, default: 1.
verbose : bool
Level of verbosity to inform about convergence.
Optional, default: False.
"""

def __init__(
self,
max_iter=32,
epsilon=1e-4,
init_point=None,
init_step_size=1.0,
verbose=False,
):
self.max_iter = max_iter
self.epsilon = epsilon
self.init_step_size = init_step_size
self.verbose = verbose
self.init_point = init_point

[docs]
@abc.abstractmethod
def minimize(self, space, points, weights=None):
pass

[docs]

[docs]
def minimize(self, space, points, weights=None):
n_points = gs.shape(points)[0]
if weights is None:
weights = gs.ones((n_points,))

mean = points[0] if self.init_point is None else self.init_point

if n_points == 1:
return mean

sum_weights = gs.sum(weights)
iteration = 0
sq_dist = 0.0
var = 0.0

norm_old = gs.linalg.norm(points)
step_size = self.init_step_size

while iteration < self.max_iter:
logs = space.metric.log(point=points, base_point=mean)

var = gs.sum(space.metric.squared_norm(logs, mean) * weights) / sum_weights

tangent_mean = _scalarmulsum(weights, logs)
tangent_mean /= sum_weights
norm = gs.linalg.norm(tangent_mean)

sq_dist = space.metric.squared_norm(tangent_mean, mean)

var_is_0 = gs.isclose(var, 0.0)

sq_dist_is_small = gs.less_equal(sq_dist, self.epsilon * space.dim)

condition = ~gs.logical_or(var_is_0, sq_dist_is_small)
if not (condition or iteration == 0):
break

estimate_next = space.metric.exp(step_size * tangent_mean, mean)
mean = estimate_next
iteration += 1

if norm < norm_old:
norm_old = norm
elif norm > norm_old:
step_size = step_size / 2.0

if iteration == self.max_iter:
logging.warning(
"Maximum number of iterations %d reached. The mean may be inaccurate",
self.max_iter,
)

if self.verbose:
logging.info(
"n_iter: {}, final variance: {}, final dist: {}".format(
iteration, var, sq_dist
)
)

return mean

[docs]

[docs]
def minimize(self, space, points, weights=None):
shape = points.shape
n_points, n_batch = shape[:2]
point_shape = shape[2:]

if n_points == 1:
return points[0]

if weights is None:
weights = gs.ones((n_points, n_batch))

flat_shape = (n_batch * n_points,) + point_shape
estimates = points[0] if self.init_point is None else self.init_point
points_flattened = gs.reshape(points, (n_points * n_batch,) + point_shape)
convergence = math.inf
iteration = 0
convergence_old = convergence

step_size = self.init_step_size

while convergence > self.epsilon and self.max_iter > iteration:
iteration += 1

tangent_mean = _batchscalarmulsum(weights, tangent_grad) / n_points

next_estimates = space.metric.exp(step_size * tangent_mean, estimates)
convergence = gs.sum(space.metric.squared_norm(tangent_mean, estimates))
estimates = next_estimates

if convergence < convergence_old:
convergence_old = convergence
elif convergence > convergence_old:
step_size = step_size / 2.0

if iteration == self.max_iter:
logging.warning(
"Maximum number of iterations %d reached. The mean may be inaccurate",
self.max_iter,
)

if self.verbose:
logging.info(
"n_iter: %d, final dist: %e, final step size: %e",
iteration,
convergence,
step_size,
)

return estimates

[docs]

[docs]
def minimize(self, space, points, weights=None):

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=[n_samples, *metric.shape]
Points to be averaged.
weights : array-like, shape=[n_samples,], optional
Weights associated to the points.

Returns
-------
current_mean: array-like, shape=[*metric.shape]
Weighted Frechet mean of the points.
"""
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 self.init_point is None else self.init_point

if weights is None:
weights = gs.ones((n_points,))
sum_weights = gs.sum(weights)

tau = self.init_step_size
iteration = 0

logs = space.metric.log(point=points, base_point=current_mean)
var = (
gs.sum(space.metric.squared_norm(logs, current_mean) * weights)
/ sum_weights
)

current_tangent_mean = _scalarmulsum(weights, logs)
current_tangent_mean /= sum_weights
sq_norm_current_tangent_mean = space.metric.squared_norm(
current_tangent_mean, base_point=current_mean
)

while (
sq_norm_current_tangent_mean > self.epsilon**2 and iteration < self.max_iter
):
iteration += 1

shooting_vector = tau * current_tangent_mean
next_mean = space.metric.exp(
tangent_vec=shooting_vector, base_point=current_mean
)

logs = space.metric.log(point=points, base_point=next_mean)
var = (
gs.sum(space.metric.squared_norm(logs, current_mean) * weights)
/ sum_weights
)

next_tangent_mean = _scalarmulsum(weights, logs)
next_tangent_mean /= sum_weights
sq_norm_next_tangent_mean = space.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 == self.max_iter:
logging.warning(
"Maximum number of iterations %d reached. The mean may be inaccurate",
self.max_iter,
)

if self.verbose:
logging.info(
"n_iter: %d, final variance: %e, final dist: %e, final_step_size: %e",
iteration,
var,
sq_norm_current_tangent_mean,
tau,
)

return current_mean

[docs]
class LinearMean(BaseEstimator):
"""Linear mean.

Parameters
----------
space : Manifold
Equipped manifold.

Attributes
----------
estimate_ : array-like, shape=[*space.shape]
If fit, Frechet mean.
"""

def __init__(self, space):
self.space = space
self.estimate_ = None

[docs]
def fit(self, X, y=None, weights=None):
"""Compute the Euclidean mean.

Parameters
----------
X : array-like, shape=[n_samples, *metric.shape]
Training input samples.
y : None
Target values. Ignored.
weights : array-like, shape=[n_samples,]
Weights associated to the samples.
Optional, default: None, in which case it is equally weighted.

Returns
-------
self : object
Returns self.
"""
self.estimate_ = linear_mean(points=X, weights=weights)
return self

[docs]
class ElasticMean(BaseEstimator):
"""Elastic mean.

Parameters
----------
space : Manifold
Equipped manifold.

Attributes
----------
estimate_ : array-like, shape=[*space.shape]
If fit, Frechet mean.
"""

def __init__(self, space):
self.space = space
self.estimate_ = None

def _elastic_mean(self, points, weights=None):
"""Compute the weighted mean of elastic curves.

SRV: Square Root Velocity.

SRV curves are a special case of Elastic curves.

The computation of the mean goes as follows:

- Transform the curves into their SRVs/F-transform representations,
- Compute the linear mean of the SRVs/F-transform representations,
- Inverse-transform the mean in curve space.

Parameters
----------
points : array-like, shape=[n_samples, k_sampling_points, dim]
Points on the manifold of curves (i.e. curves) to be averaged.
weights : array-like, shape=[n_samples,]
Weights associated to the points (i.e. curves).
Optional, default: None.

Returns
-------
mean : array-like, shape=[k_sampling_points, dim]
Weighted linear mean of the points (i.e. of the curves).
"""
diffeo = self.space.metric.diffeo
transformed = diffeo(points)
transformed_linear_mean = linear_mean(transformed, weights=weights)

return diffeo.inverse(transformed_linear_mean)

[docs]
def fit(self, X, y=None, weights=None):
"""Compute the elastic mean.

Parameters
----------
X : array-like, shape=[n_samples, *metric.shape]
Training input samples.
y : None
Target values. Ignored.
weights : array-like, shape=[n_samples,]
Weights associated to the samples.
Optional, default: None, in which case it is equally weighted.

Returns
-------
self : object
Returns self.
"""
self.estimate_ = self._elastic_mean(X, weights=weights)
return self

[docs]
class CircleMean(BaseEstimator):
"""Circle mean.

Parameters
----------
space : Manifold
Equipped manifold.

Attributes
----------
estimate_ : array-like, shape=[2,]
If fit, Frechet mean.
"""

def __init__(self, space):
self.space = space
self.estimate_ = None

def _circle_mean(self, 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.

Parameters
----------
points : array-like, shape=[n_samples, 1]
Data set of angles (intrinsic coordinates).

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
"""
sample_size = points.shape[0]
mean0 = gs.mean(points)
var0 = gs.sum((points - mean0) ** 2)
sorted_points = gs.sort(points, axis=0)
means = self._circle_variances(mean0, var0, sample_size, sorted_points)
return means[gs.argmin(means[:, 1]), 0]

@staticmethod
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_samples,]
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([gs.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]
def fit(self, X, y=None):
"""Compute the circle mean.

Parameters
----------
X : array-like, shape=[n_samples, 2]
Training input samples.
y : None
Target values. Ignored.
weights : array-like, shape=[n_samples,]
Weights associated to the samples.
Optional, default: None, in which case it is equally weighted.

Returns
-------
self : object
Returns self.
"""
self.estimate_ = self.space.angle_to_extrinsic(
self._circle_mean(self.space.extrinsic_to_angle(X))
)
return self

[docs]
class FrechetMean(BaseEstimator):
r"""Empirical Frechet mean.

Parameters
----------
space : Manifold
Equipped manifold.
method : str, {\'default\', \'adaptive\', \'batch\'}
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, *space.shape].
Optional, default: \'default\'.

Attributes
----------
estimate_ : array-like, shape=[*space.shape]
If fit, Frechet mean.

Notes
-----
* Required metric methods for general case:
* `log`, `exp`, `squared_norm` (for convergence criteria)
"""

def __new__(cls, space, **kwargs):
"""Interface for instantiating proper algorithm."""
if isinstance(space.metric, HypersphereMetric) and space.dim == 1:
return CircleMean(space, **kwargs)

elif _is_linear_metric(space.metric):
return LinearMean(space, **kwargs)

elif _is_elastic_metric(space.metric):
return ElasticMean(space, **kwargs)

return super().__new__(cls)

def __init__(self, space, method="default"):
self.space = space

self._method = None
self.method = method

self.estimate_ = None

[docs]
def set(self, **kwargs):
"""Set optimizer parameters.

Especially useful for one line instantiations.
"""
for param_name, value in kwargs.items():
if not hasattr(self.optimizer, param_name):
raise ValueError(f"Unknown parameter {param_name}.")

setattr(self.optimizer, param_name, value)
return self

@property
def method(self):
return self._method

@method.setter
def method(self, value):
error.check_parameter_accepted_values(
)
if value == self._method:
return

self._method = value
MAP_OPTIMIZER = {
}
self.optimizer = MAP_OPTIMIZER[value]()

[docs]
def fit(self, X, y=None, weights=None):
"""Compute the empirical weighted Frechet mean.

Parameters
----------
X : array-like, shape=[n_samples, *metric.shape]
Training input samples.
y : None
Target values. Ignored.
weights : array-like, shape=[n_samples,]
Weights associated to the samples.
Optional, default: None, in which case it is equally weighted.

Returns
-------
self : object
Returns self.
"""
self.estimate_ = self.optimizer.minimize(
space=self.space,
points=X,
weights=weights,
)
return self

```