def test_estimate_hyperbolic(self):
point = gs.array([2., 1., 1., 1.])
points = gs.array([point, point])
mean = FrechetMean(metric=self.hyperbolic.metric)
mean.fit(X=points)
expected = point
result = mean.estimate_
self.assertAllClose(result, expected)
def test_estimate_and_belongs_adaptive_gradient_descent_so_matrix(self):
point = self.so_matrix.random_uniform(10)
mean = FrechetMean(metric=self.so_matrix.bi_invariant_metric,
method='adaptive',
verbose=True,
lr=.5)
mean.fit(point)
result = self.so_matrix.belongs(mean.estimate_)
self.assertTrue(result)
def test_stiefel_n_samples(self):
space = Stiefel(3, 2)
metric = space.metric
point = space.random_point(2)
mean = FrechetMean(metric,
method="default",
init_step_size=0.5,
verbose=True)
mean.fit(point)
result = space.belongs(mean.estimate_)
self.assertTrue(result)
def test_estimate_shape_adaptive_gradient_descent_sphere(self):
dim = 5
point_a = gs.array([1., 0., 0., 0., 0.])
point_b = gs.array([0., 1., 0., 0., 0.])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method='adaptive')
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim, ))
def test_estimate_adaptive_gradient_descent_sphere(self):
point = gs.array([0., 0., 0., 0., 1.])
points = gs.array([point, point])
mean = FrechetMean(metric=self.sphere.metric, method='adaptive')
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_estimate_and_belongs_default_gradient_descent_sphere(self):
point_a = gs.array([1., 0., 0., 0., 0.])
point_b = gs.array([0., 1., 0., 0., 0.])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method='default')
mean.fit(points)
result = self.sphere.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_estimate_curves_2d(self):
point = self.curves_2d.random_point(n_samples=1)
points = gs.array([point, point])
mean = FrechetMean(metric=self.curves_2d.srv_metric)
mean.fit(X=points)
result = mean.estimate_
expected = point
self.assertAllClose(expected, result)
def test_weighted_frechet_mean(self):
"""Test for weighted mean."""
data = gs.array([[0.1, 0.2], [0.25, 0.35]])
weights = gs.array([3., 1.])
mean_o = FrechetMean(metric=self.metric, point_type='vector', lr=1.)
mean_o.fit(data, weights=weights)
result = mean_o.estimate_
expected = self.metric.exp(
weights[1] / gs.sum(weights) * self.metric.log(data[1], data[0]),
data[0])
self.assertAllClose(result, expected)
def main():
"""Perform tangent PCA at the mean."""
fig = plt.figure(figsize=(15, 5))
hyperbolic_plane = Hyperbolic(dimension=2)
data = hyperbolic_plane.random_uniform(n_samples=140)
mean = FrechetMean(metric=hyperbolic_plane.metric)
mean.fit(data)
mean_estimate = mean.estimate_
tpca = TangentPCA(metric=hyperbolic_plane.metric, n_components=2)
tpca = tpca.fit(data, base_point=mean_estimate)
tangent_projected_data = tpca.transform(data)
geodesic_0 = hyperbolic_plane.metric.geodesic(
initial_point=mean_estimate,
initial_tangent_vec=tpca.components_[0])
geodesic_1 = hyperbolic_plane.metric.geodesic(
initial_point=mean_estimate,
initial_tangent_vec=tpca.components_[1])
n_steps = 100
t = np.linspace(-1, 1, n_steps)
geodesic_points_0 = geodesic_0(t)
geodesic_points_1 = geodesic_1(t)
logging.info(
'Coordinates of the Log of the first 5 data points at the mean, '
'projected on the principal components:')
logging.info('\n{}'.format(tangent_projected_data[:5]))
ax_var = fig.add_subplot(121)
xticks = np.arange(1, 2 + 1, 1)
ax_var.xaxis.set_ticks(xticks)
ax_var.set_title('Explained variance')
ax_var.set_xlabel('Number of Principal Components')
ax_var.set_ylim((0, 1))
ax_var.plot(xticks, tpca.explained_variance_ratio_)
ax = fig.add_subplot(122)
visualization.plot(
mean_estimate, ax, space='H2_poincare_disk', color='darkgreen', s=10)
visualization.plot(
geodesic_points_0, ax, space='H2_poincare_disk', linewidth=2)
visualization.plot(
geodesic_points_1, ax, space='H2_poincare_disk', linewidth=2)
visualization.plot(
data, ax, space='H2_poincare_disk', color='black', alpha=0.7)
plt.show()
def test_estimate_shape_default_gradient_descent_sphere(self):
dim = 5
point_a = gs.array([1.0, 0.0, 0.0, 0.0, 0.0])
point_b = gs.array([0.0, 1.0, 0.0, 0.0, 0.0])
points = gs.array([point_a, point_b])
mean = FrechetMean(metric=self.sphere.metric, method="default", verbose=True)
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim,))
def test_estimate_default_gradient_descent_so3(self):
points = self.so3.random_uniform(2)
mean_vec = FrechetMean(metric=self.so3.bi_invariant_metric,
method='default')
mean_vec.fit(points)
logs = self.so3.bi_invariant_metric.log(points, mean_vec.estimate_)
result = gs.sum(logs, axis=0)
expected = gs.zeros_like(points[0])
self.assertAllClose(result, expected)
def test_mean_euclidean_shape(self):
dim = 2
point = gs.array([1., 4.])
mean = FrechetMean(metric=self.euclidean.metric)
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (dim, ))
def test_coincides_with_frechet_so(self):
gs.random.seed(0)
point = self.so.random_uniform(self.n_samples)
estimator = ExponentialBarycenter(self.so, max_iter=40, epsilon=1e-10)
estimator.fit(point)
result = estimator.estimate_
frechet_estimator = FrechetMean(self.so.bi_invariant_metric,
max_iter=40,
epsilon=1e-10)
frechet_estimator.fit(point)
expected = frechet_estimator.estimate_
self.assertAllClose(result, expected)
def test_mean_matrices_shape(self):
m, n = (2, 2)
point = gs.array([[1., 4.], [2., 3.]])
metric = MatricesMetric(m, n)
mean = FrechetMean(metric=metric, point_type='matrix')
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
self.assertAllClose(gs.shape(result), (m, n))
def test_estimate_and_belongs_hyperbolic(self):
point_a = self.hyperbolic.random_point()
point_b = self.hyperbolic.random_point()
point_c = self.hyperbolic.random_point()
points = gs.stack([point_a, point_b, point_c], axis=0)
mean = FrechetMean(metric=self.hyperbolic.metric)
mean.fit(X=points)
result = self.hyperbolic.belongs(mean.estimate_)
expected = True
self.assertAllClose(result, expected)
def test_estimate_sphere(self):
point = gs.array([0., 0., 0., 0., 1.])
points = gs.zeros((2, point.shape[0]))
points[0, :] = point
points[1, :] = point
mean = FrechetMean(metric=self.sphere.metric)
mean.fit(X=points)
result = mean.estimate_
expected = helper.to_vector(point)
self.assertAllClose(expected, result)
def test_estimate_and_belongs_sphere(self):
point_a = gs.array([1., 0., 0., 0., 0.])
point_b = gs.array([0., 1., 0., 0., 0.])
points = gs.zeros((2, point_a.shape[0]))
points[0, :] = point_a
points[1, :] = point_b
mean = FrechetMean(metric=self.sphere.metric)
mean.fit(points)
result = self.sphere.belongs(mean.estimate_)
expected = gs.array([[True]])
self.assertAllClose(result, expected)
def test_estimate_and_belongs_hyperbolic(self):
point_a = self.hyperbolic.random_uniform()
point_b = self.hyperbolic.random_uniform()
point_c = self.hyperbolic.random_uniform()
points = gs.concatenate([point_a, point_b, point_c], axis=0)
mean = FrechetMean(metric=self.hyperbolic.metric)
mean.fit(X=points)
result = self.hyperbolic.belongs(mean.estimate_)
expected = gs.array([[True]])
self.assertAllClose(result, expected)
def test_mean_matrices(self):
m, n = (2, 2)
point = gs.array([[1.0, 4.0], [2.0, 3.0]])
metric = MatricesMetric(m, n)
mean = FrechetMean(metric=metric, point_type="matrix")
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
def test_fit(self):
X = self.data
clustering = OnlineKMeans(
metric=self.metric, n_clusters=1, n_repetitions=10)
clustering.fit(X)
center = clustering.cluster_centers_
mean = FrechetMean(metric=self.metric, lr=1.)
mean.fit(X)
result = self.metric.dist(center, mean.estimate_)
expected = 0.
self.assertAllClose(expected, result, atol=TOLERANCE)
def __init__(self,
manifold,
metric,
bandwidth,
tol=1e-2,
**FrechetMean_kwargs):
self.manifold = manifold
self.metric = metric
self.bandwidth = bandwidth
self.tol = tol
self.mean = FrechetMean(self.metric, **FrechetMean_kwargs)
self.centers = None
def main():
"""Perform tangent PCA at the mean on the sphere."""
fig = plt.figure(figsize=(15, 5))
sphere = Hypersphere(dim=2)
data = sphere.random_von_mises_fisher(kappa=15, n_samples=140)
mean = FrechetMean(metric=sphere.metric)
mean.fit(data)
mean_estimate = mean.estimate_
tpca = TangentPCA(metric=sphere.metric, n_components=2)
tpca = tpca.fit(data, base_point=mean_estimate)
tangent_projected_data = tpca.transform(data)
geodesic_0 = sphere.metric.geodesic(
initial_point=mean_estimate, initial_tangent_vec=tpca.components_[0]
)
geodesic_1 = sphere.metric.geodesic(
initial_point=mean_estimate, initial_tangent_vec=tpca.components_[1]
)
n_steps = 100
t = np.linspace(-1, 1, n_steps)
geodesic_points_0 = geodesic_0(t)
geodesic_points_1 = geodesic_1(t)
logging.info(
"Coordinates of the Log of the first 5 data points at the mean, "
"projected on the principal components:"
)
logging.info("\n{}".format(tangent_projected_data[:5]))
ax_var = fig.add_subplot(121)
xticks = np.arange(1, 2 + 1, 1)
ax_var.xaxis.set_ticks(xticks)
ax_var.set_title("Explained variance")
ax_var.set_xlabel("Number of Principal Components")
ax_var.set_ylim((0, 1))
ax_var.plot(xticks, tpca.explained_variance_ratio_)
ax = fig.add_subplot(122, projection="3d")
visualization.plot(mean_estimate, ax, space="S2", color="darkgreen", s=10)
visualization.plot(geodesic_points_0, ax, space="S2", linewidth=2)
visualization.plot(geodesic_points_1, ax, space="S2", linewidth=2)
visualization.plot(data, ax, space="S2", color="black", alpha=0.7)
plt.show()
def test_estimate_default_gradient_descent_so_matrix(self):
points = self.so_matrix.random_uniform(2)
mean_vec = FrechetMean(
metric=self.so_matrix.bi_invariant_metric,
method="default",
init_step_size=1.0,
)
mean_vec.fit(points)
logs = self.so_matrix.bi_invariant_metric.log(points,
mean_vec.estimate_)
result = gs.sum(logs, axis=0)
expected = gs.zeros_like(points[0])
self.assertAllClose(result, expected, atol=1e-5)
def test_mean_minkowski(self):
point = gs.array([2., -math.sqrt(3)])
points = [point, point, point]
mean = FrechetMean(metric=self.minkowski.metric)
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
points = gs.array([
[1., 0.],
[2., math.sqrt(3)],
[3., math.sqrt(8)],
[4., math.sqrt(24)]])
weights = gs.array([1., 2., 1., 2.])
mean = FrechetMean(metric=self.minkowski.metric)
mean.fit(points, weights=weights)
result = mean.estimate_
result = self.minkowski.belongs(result)
expected = gs.array(True)
self.assertAllClose(result, expected)
def test_mean_euclidean(self):
point = gs.array([1., 4.])
mean = FrechetMean(metric=self.euclidean.metric)
points = [point, point, point]
mean.fit(points)
result = mean.estimate_
expected = point
self.assertAllClose(result, expected)
points = gs.array([
[1., 2.],
[2., 3.],
[3., 4.],
[4., 5.]])
weights = [1., 2., 1., 2.]
mean = FrechetMean(metric=self.euclidean.metric)
mean.fit(points, weights=weights)
result = mean.estimate_
expected = gs.array([16. / 6., 22. / 6.])
self.assertAllClose(result, expected)
def test_batched(self):
space = SPDMatrices(3)
metric = SPDMetricAffine(3)
point = space.random_point(4)
mean_batch = FrechetMean(metric, method="batch", verbose=True)
data = gs.stack([point[:2], point[2:]], axis=1)
mean_batch.fit(data)
result = mean_batch.estimate_
mean = FrechetMean(metric)
mean.fit(data[:, 0])
expected_1 = mean.estimate_
mean.fit(data[:, 1])
expected_2 = mean.estimate_
expected = gs.stack([expected_1, expected_2])
self.assertAllClose(expected, result)
def test_spd_kmeans_fit(self):
gs.random.seed(0)
dim = 3
n_points = 2
space = spd_matrices.SPDMatrices(dim)
data = space.random_point(n_samples=n_points)
metric = spd_matrices.SPDMetricAffine(dim)
kmeans = RiemannianKMeans(metric, n_clusters=1, lr=1.0)
kmeans.fit(data)
result = kmeans.centroids
mean = FrechetMean(metric=metric, point_type="matrix", max_iter=100)
mean.fit(data)
expected = mean.estimate_
self.assertAllClose(result, expected)
def score(self, X, y, weights=None):
"""Compute training score.
Compute the training score defined as R^2.
Parameters
----------
X : {array-like, sparse matrix}, shape=[...,}]
Training input samples.
y : array-like, shape=[..., {dim, [n,n]}]
Training target values.
weights : array-like, shape=[...,]
Weights associated to the points.
Optional, default: None.
Returns
-------
_ : float
Training score.
"""
y_pred = self.predict(X)
if weights is None:
weights = 1.0
mean = FrechetMean(self.metric, verbose=self.verbose).fit(y).estimate_
numerator = gs.sum(weights * self.metric.squared_dist(y, y_pred))
denominator = gs.sum(weights * self.metric.squared_dist(y, mean))
return 1 - numerator / denominator if denominator != 0 else 0.0
def update_means(self, data, posterior_probabilities):
"""Update means."""
n_gaussians = posterior_probabilities.shape[-1]
mean = FrechetMean(metric=self.metric,
method=self.mean_method,
lr=self.lr_mean,
epsilon=self.tol_mean,
max_iter=self.max_iter_mean,
point_type=self.point_type)
data_expand = gs.expand_dims(data, 1)
data_expand = gs.repeat(data_expand, n_gaussians, axis=1)
mean.fit(data_expand, weights=posterior_probabilities)
self.means = gs.squeeze(mean.estimate_)
def test_logs_at_mean_adaptive_gradient_descent_sphere(self):
n_tests = 100
estimator = FrechetMean(metric=self.sphere.metric, method='adaptive')
result = 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)
estimator.fit(points)
mean = estimator.estimate_
logs = self.sphere.metric.log(point=points, base_point=mean)
result[i] = gs.linalg.norm(logs[1, :] + logs[0, :])
expected = gs.zeros(n_tests)
self.assertAllClose(expected, result, rtol=1e-10, atol=1e-10)
