def _iterate_over_manifolds(self, func, args, intrinsic=False):
cum_index = (
gs.cumsum(self.dims)[:-1]
if intrinsic
else gs.cumsum([k + 1 for k in self.dims])
)
arguments = {}
float_args = {}
for key, value in args.items():
if not isinstance(value, float):
arguments[key] = gs.split(value, cum_index, axis=-1)
else:
float_args[key] = value
args_list = [
{key: arguments[key][j] for key in arguments}
for j in range(len(self.manifolds))
]
pool = joblib.Parallel(n_jobs=self.n_jobs)
out = pool(
joblib.delayed(self._get_method)(
self.manifolds[i], func, {**args_list[i], **float_args}
)
for i in range(len(self.manifolds))
)
return out
def test_cumsum(self):
result = gs.cumsum(gs.arange(10))
expected = gs.array(([0, 1, 3, 6, 10, 15, 21, 28, 36, 45]))
self.assertAllClose(result, expected)
result = gs.cumsum(gs.arange(10).reshape(2, 5), axis=1)
expected = gs.array(([[0, 1, 3, 6, 10], [5, 11, 18, 26, 35]]))
self.assertAllClose(result, expected)
def _iterate_over_metrics(
self, func, args, intrinsic=False):
cum_index = gs.cumsum(self.dims, axis=0)[:-1] if intrinsic else \
gs.cumsum(gs.array([k + 1 for k in self.dims]), axis=0)
arguments = {
key: gs.split(args[key], cum_index, axis=1) for key in args.keys()}
args_list = [{key: arguments[key][j] for key in args.keys()} for j in
range(self.n_metrics)]
pool = joblib.Parallel(n_jobs=self.n_jobs, prefer='threads')
out = pool(
joblib.delayed(self._get_method)(
self.metrics[i], func, args_list[i]) for i in range(
self.n_metrics))
return out
def _iterate_over_manifolds(
self, func, args, intrinsic=False):
cum_index = gs.cumsum(self.dims)[:-1] if intrinsic else \
gs.cumsum([k + 1 for k in self.dims])
arguments = {key: gs.split(
args[key], cum_index, axis=1) for key in args.keys()}
args_list = [{key: arguments[key][j] for key in args.keys()} for j in
range(len(self.manifolds))]
pool = joblib.Parallel(n_jobs=self.n_jobs)
out = pool(
joblib.delayed(self._get_method)(
self.manifolds[i], func, args_list[i]) for i in range(
len(self.manifolds)))
return out
def square_root_velocity_inverse(self, srv, starting_point):
"""Retrieve a curve from sqrt velocity rep and starting point.
Parameters
----------
srv :
starting_point :
Returns
-------
curve :
"""
if not isinstance(self.ambient_metric, EuclideanMetric):
raise AssertionError('The square root velocity inverse is only '
'implemented for dicretized curves embedded '
'in a Euclidean space.')
if gs.ndim(srv) != gs.ndim(starting_point):
starting_point = gs.transpose(gs.tile(starting_point, (1, 1, 1)),
axes=(1, 0, 2))
srv_shape = srv.shape
srv = gs.to_ndarray(srv, to_ndim=3)
n_curves, n_sampling_points_minus_one, n_coords = srv.shape
srv = gs.reshape(srv,
(n_curves * n_sampling_points_minus_one, n_coords))
srv_norm = self.ambient_metric.norm(srv)
delta_points = 1 / n_sampling_points_minus_one * srv_norm * srv
delta_points = gs.reshape(delta_points, srv_shape)
curve = gs.concatenate((starting_point, delta_points), -2)
curve = gs.cumsum(curve, -2)
return curve
def test_fit_to_target_explained_variance(self):
X = self.spd.random_point(n_samples=5)
target = 0.90
tpca = TangentPCA(self.spd_metric, n_components=target)
tpca.fit(X)
result = gs.cumsum(tpca.explained_variance_ratio_)[-1] > target
expected = True
self.assertAllClose(result, expected)
def log(self, point, base_point):
"""Compute Riemannian logarithm of a curve wrt a base curve.
Parameters
----------
point : array-like, shape=[..., n_sampling_points, ambient_dim]
Discrete curve.
base_point : array-like, shape=[..., n_sampling_points, ambient_dim]
Discrete curve to use as base point.
Returns
-------
log : array-like, shape=[..., n_sampling_points, ambient_dim]
Tangent vector to a discrete curve.
"""
if not isinstance(self.ambient_metric, EuclideanMetric):
raise AssertionError('The logarithm map is only implemented '
'for discrete curves embedded in a '
'Euclidean space.')
point = gs.to_ndarray(point, to_ndim=3)
base_point = gs.to_ndarray(base_point, to_ndim=3)
n_curves, n_sampling_points, n_coords = point.shape
curve_srv = self.square_root_velocity(point)
base_curve_srv = self.square_root_velocity(base_point)
base_curve_velocity = (n_sampling_points - 1) * (base_point[:, 1:, :] -
base_point[:, :-1, :])
base_curve_velocity_norm = self.pointwise_norm(base_curve_velocity,
base_point[:, :-1, :])
inner_prod = self.pointwise_inner_product(curve_srv - base_curve_srv,
base_curve_velocity,
base_point[:, :-1, :])
coef_1 = gs.sqrt(base_curve_velocity_norm)
coef_2 = 1 / base_curve_velocity_norm**(3 / 2) * inner_prod
term_1 = gs.einsum('ij,ijk->ijk', coef_1, curve_srv - base_curve_srv)
term_2 = gs.einsum('ij,ijk->ijk', coef_2, base_curve_velocity)
log_derivative = term_1 + term_2
log_starting_points = self.ambient_metric.log(
point=point[:, 0, :], base_point=base_point[:, 0, :])
log_starting_points = gs.to_ndarray(
log_starting_points, to_ndim=3, axis=1)
log_cumsum = gs.hstack(
[gs.zeros((n_curves, 1, n_coords)),
gs.cumsum(log_derivative, -2)])
log = log_starting_points + 1 / (n_sampling_points - 1) * log_cumsum
return log
def log(self, curve, base_curve):
"""Compute Riemannian logarithm of a curve wrt a base curve.
Parameters
----------
curve :
base_curve :
Returns
-------
log :
"""
if not isinstance(self.ambient_metric, EuclideanMetric):
raise AssertionError('The logarithm map is only implemented '
'for dicretized curves embedded in a '
'Euclidean space.')
curve = gs.to_ndarray(curve, to_ndim=3)
base_curve = gs.to_ndarray(base_curve, to_ndim=3)
n_curves, n_sampling_points, n_coords = curve.shape
curve_srv = self.square_root_velocity(curve)
base_curve_srv = self.square_root_velocity(base_curve)
base_curve_velocity = (n_sampling_points - 1) * (base_curve[:, 1:, :] -
base_curve[:, :-1, :])
base_curve_velocity_norm = self.pointwise_norm(base_curve_velocity,
base_curve[:, :-1, :])
inner_prod = self.pointwise_inner_product(curve_srv - base_curve_srv,
base_curve_velocity,
base_curve[:, :-1, :])
coef_1 = gs.sqrt(base_curve_velocity_norm)
coef_2 = 1 / base_curve_velocity_norm**(3 / 2) * inner_prod
term_1 = gs.einsum('ij,ijk->ijk', coef_1, curve_srv - base_curve_srv)
term_2 = gs.einsum('ij,ijk->ijk', coef_2, base_curve_velocity)
log_derivative = term_1 + term_2
log_starting_points = self.ambient_metric.log(
point=curve[:, 0, :], base_point=base_curve[:, 0, :])
log_starting_points = gs.transpose(
gs.tile(log_starting_points, (1, 1, 1)), (1, 0, 2))
log_cumsum = gs.hstack(
[gs.zeros((n_curves, 1, n_coords)),
gs.cumsum(log_derivative, -2)])
log = log_starting_points + 1 / (n_sampling_points - 1) * log_cumsum
return log
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
def square_root_velocity_inverse(self, srv, starting_point):
"""Retrieve a curve from sqrt velocity rep and starting point.
Parameters
----------
srv : array-like, shape=[..., n_sampling_points - 1, ambient_dim]
Square-root velocity representation of a discrete curve.
starting_point : array-like, shape=[..., ambient_dim]
Point of the ambient manifold to use as start of the retrieved
curve.
Returns
-------
curve : array-like, shape=[..., n_sampling_points, ambient_dim]
Curve retrieved from its square-root velocity.
"""
if not isinstance(self.ambient_metric, EuclideanMetric):
raise AssertionError('The square root velocity inverse is only '
'implemented for discrete curves embedded '
'in a Euclidean space.')
if gs.ndim(srv) != gs.ndim(starting_point):
starting_point = gs.to_ndarray(
starting_point, to_ndim=srv.ndim, axis=1)
srv_shape = srv.shape
srv = gs.to_ndarray(srv, to_ndim=3)
n_curves, n_sampling_points_minus_one, n_coords = srv.shape
srv = gs.reshape(srv,
(n_curves * n_sampling_points_minus_one, n_coords))
srv_norm = self.ambient_metric.norm(srv)
delta_points = gs.einsum(
'...,...i->...i', 1 / n_sampling_points_minus_one * srv_norm, srv)
delta_points = gs.reshape(delta_points, srv_shape)
curve = gs.concatenate((starting_point, delta_points), -2)
curve = gs.cumsum(curve, -2)
return curve
