def test_christoffels(self):
"""Test Christoffel symbols in dimension 2.
Check the Christoffel symbols in dimension 2.
"""
gs.random.seed(123)
dirichlet2 = DirichletDistributions(2)
points = dirichlet2.random_point(self.n_points)
result = dirichlet2.metric.christoffels(points)
def coefficients(param_a, param_b):
"""Christoffel coefficients for the beta distributions."""
poly1a = gs.polygamma(1, param_a)
poly2a = gs.polygamma(2, param_a)
poly1b = gs.polygamma(1, param_b)
poly2b = gs.polygamma(2, param_b)
poly1ab = gs.polygamma(1, param_a + param_b)
poly2ab = gs.polygamma(2, param_a + param_b)
metric_det = 2 * (poly1a * poly1b - poly1ab * (poly1a + poly1b))
c1 = (poly2a * (poly1b - poly1ab) - poly1b * poly2ab) / metric_det
c2 = -poly1b * poly2ab / metric_det
c3 = (poly2b * poly1ab - poly1b * poly2ab) / metric_det
return c1, c2, c3
param_a, param_b = points[:, 0], points[:, 1]
c1, c2, c3 = coefficients(param_a, param_b)
c4, c5, c6 = coefficients(param_b, param_a)
vector_0 = gs.stack([c1, c2, c3], axis=-1)
vector_1 = gs.stack([c6, c5, c4], axis=-1)
gamma_0 = SymmetricMatrices.from_vector(vector_0)
gamma_1 = SymmetricMatrices.from_vector(vector_1)
expected = gs.stack([gamma_0, gamma_1], axis=-3)
self.assertAllClose(result, expected)
def setUp(self):
warnings.simplefilter('ignore', category=UserWarning)
self.dim = 3
self.dirichlet = DirichletDistributions(self.dim)
self.metric = DirichletMetric(self.dim)
self.n_points = 10
self.n_samples = 20
def setUp(self):
"""Define the parameters of the tests."""
warnings.simplefilter('ignore', category=UserWarning)
self.dim = 3
self.dirichlet = DirichletDistributions(self.dim)
self.metric = DirichletMetric(self.dim)
self.n_points = 10
self.n_samples = 20
def test_metric_matrix_dim2(self):
"""Test metric matrix in dimension 2.
Check the metric matrix in dimension 2.
"""
dirichlet2 = DirichletDistributions(2)
points = dirichlet2.random_point(self.n_points)
result = dirichlet2.metric.metric_matrix(points)
param_a = points[:, 0]
param_b = points[:, 1]
polygamma_ab = gs.polygamma(1, param_a + param_b)
polygamma_a = gs.polygamma(1, param_a)
polygamma_b = gs.polygamma(1, param_b)
vector = gs.stack([
polygamma_a - polygamma_ab, -polygamma_ab,
polygamma_b - polygamma_ab
],
axis=-1)
expected = SymmetricMatrices.from_vector(vector)
self.assertAllClose(result, expected)
class TestDirichletDistributions(geomstats.tests.TestCase):
"""Class defining the Dirichlet distributions tests."""
def setUp(self):
"""Define the parameters of the tests."""
warnings.simplefilter('ignore', category=UserWarning)
self.dim = 3
self.dirichlet = DirichletDistributions(self.dim)
self.metric = DirichletMetric(self.dim)
self.n_points = 10
self.n_samples = 20
def test_random_uniform_and_belongs(self):
"""Test random_uniform and belongs.
Test that the random uniform method samples
on the Dirichlet distribution space.
"""
point = self.dirichlet.random_point()
result = self.dirichlet.belongs(point)
expected = True
self.assertAllClose(expected, result)
def test_random_uniform_and_belongs_vectorization(self):
"""Test random_uniform and belongs.
Test that the random uniform method samples
on the Dirichlet distribution space.
"""
points = self.dirichlet.random_point(self.n_points)
result = self.dirichlet.belongs(points)
expected = gs.array([True] * self.n_points)
self.assertAllClose(expected, result)
def test_random_uniform(self):
"""Test random_uniform.
Test that the random uniform method samples points of the right shape.
"""
points = self.dirichlet.random_point(self.n_points)
self.assertAllClose(gs.shape(points), (self.n_points, self.dim))
@geomstats.tests.np_only
def test_sample(self):
"""Test sample.
Check that the samples have the right shape.
"""
point = self.dirichlet.random_point()
samples = self.dirichlet.sample(point, self.n_samples)
result = samples.shape
expected = (self.n_samples, self.dim)
self.assertAllClose(expected, result)
points = self.dirichlet.random_point(self.n_points)
samples = self.dirichlet.sample(points, self.n_samples)
result = samples.shape
expected = (self.n_points, self.n_samples, self.dim)
self.assertAllClose(expected, result)
@geomstats.tests.np_only
def test_sample_belong(self):
"""Test that sample samples in the simplex.
Check that samples belong to the simplex,
i.e. that their components sum up to one.
"""
points = self.dirichlet.random_point(self.n_points)
samples = self.dirichlet.sample(points, self.n_samples)
result = gs.sum(samples, -1)
expected = gs.ones((self.n_points, self.n_samples))
self.assertAllClose(expected, result)
@geomstats.tests.np_only
def test_point_to_pdf(self):
"""Test point_to_pdf.
Check vectorization of the computation of the pdf.
"""
n_points = 2
points = self.dirichlet.random_point(n_points)
pdfs = self.dirichlet.point_to_pdf(points)
alpha = gs.ones(self.dim)
samples = self.dirichlet.sample(alpha, self.n_samples)
result = pdfs(samples)
pdf1 = [dirichlet.pdf(x, points[0, :]) for x in samples]
pdf2 = [dirichlet.pdf(x, points[1, :]) for x in samples]
expected = gs.stack([gs.array(pdf1), gs.array(pdf2)], axis=0)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_metric_matrix_vectorization(self):
"""Test metric matrix vectorization..
Check vectorization of the metric matrix.
"""
points = self.dirichlet.random_point(self.n_points)
mat = self.dirichlet.metric.metric_matrix(points)
result = mat.shape
expected = (self.n_points, self.dim, self.dim)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_pytorch_only
def test_metric_matrix_dim2(self):
"""Test metric matrix in dimension 2.
Check the metric matrix in dimension 2.
"""
dirichlet2 = DirichletDistributions(2)
points = dirichlet2.random_point(self.n_points)
result = dirichlet2.metric.metric_matrix(points)
param_a = points[:, 0]
param_b = points[:, 1]
polygamma_ab = gs.polygamma(1, param_a + param_b)
polygamma_a = gs.polygamma(1, param_a)
polygamma_b = gs.polygamma(1, param_b)
vector = gs.stack([
polygamma_a - polygamma_ab, -polygamma_ab,
polygamma_b - polygamma_ab
],
axis=-1)
expected = SymmetricMatrices.from_vector(vector)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_christoffels(self):
"""Test Christoffel symbols in dimension 2.
Check the Christoffel symbols in dimension 2.
"""
gs.random.seed(123)
dirichlet2 = DirichletDistributions(2)
points = dirichlet2.random_point(self.n_points)
result = dirichlet2.metric.christoffels(points)
def coefficients(param_a, param_b):
"""Christoffel coefficients for the beta distributions."""
poly1a = gs.polygamma(1, param_a)
poly2a = gs.polygamma(2, param_a)
poly1b = gs.polygamma(1, param_b)
poly2b = gs.polygamma(2, param_b)
poly1ab = gs.polygamma(1, param_a + param_b)
poly2ab = gs.polygamma(2, param_a + param_b)
metric_det = 2 * (poly1a * poly1b - poly1ab * (poly1a + poly1b))
c1 = (poly2a * (poly1b - poly1ab) - poly1b * poly2ab) / metric_det
c2 = -poly1b * poly2ab / metric_det
c3 = (poly2b * poly1ab - poly1b * poly2ab) / metric_det
return c1, c2, c3
param_a, param_b = points[:, 0], points[:, 1]
c1, c2, c3 = coefficients(param_a, param_b)
c4, c5, c6 = coefficients(param_b, param_a)
vector_0 = gs.stack([c1, c2, c3], axis=-1)
vector_1 = gs.stack([c6, c5, c4], axis=-1)
gamma_0 = SymmetricMatrices.from_vector(vector_0)
gamma_1 = SymmetricMatrices.from_vector(vector_1)
expected = gs.stack([gamma_0, gamma_1], axis=-3)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_christoffels_vectorization(self):
"""Test Christoffel synbols.
Check vectorization of Christoffel symbols.
"""
n_points = 2
points = self.dirichlet.random_point(n_points)
christoffel_1 = self.metric.christoffels(points[0, :])
christoffel_2 = self.metric.christoffels(points[1, :])
christoffels = self.metric.christoffels(points)
result = christoffels.shape
expected = gs.array([n_points, self.dim, self.dim, self.dim])
self.assertAllClose(result, expected)
expected = gs.stack((christoffel_1, christoffel_2), axis=0)
self.assertAllClose(christoffels, expected)
@geomstats.tests.np_only
def test_exp(self):
"""Test Exp.
Test that the Riemannian exponential at points on the planes
xk = xj in the direction of that plane stays in the plane.
"""
n_points = 2
gs.random.seed(123)
points = self.dirichlet.random_point(n_points)
vectors = self.dirichlet.random_point(n_points)
initial_vectors = gs.array([[vec_x, vec_x, vec_x]
for vec_x in vectors[:, 0]])
base_points = gs.array([[param_x, param_x, param_x]
for param_x in points[:, 0]])
result = self.metric.exp(initial_vectors, base_points)
expected = gs.transpose(gs.tile(result[:, 0], (self.dim, 1)))
self.assertAllClose(expected, result)
initial_vectors[:, 2] = gs.random.rand(n_points)
base_points[:, 2] = gs.random.rand(n_points)
result_points = self.metric.exp(initial_vectors, base_points)
result = gs.isclose(result_points[:, 0], result_points[:, 1]).all()
expected = gs.array([True] * n_points)
self.assertAllClose(expected, result)
@geomstats.tests.np_only
def test_log_and_exp(self):
"""Test Log and Exp.
Test that the Riemannian exponential
and the Riemannian logarithm are inverse.
Expect their composition to give the identity function.
"""
gs.random.seed(123)
base_points = self.dirichlet.random_point(self.n_points)
points = self.dirichlet.random_point(self.n_points)
log = self.metric.log(points, base_points, n_steps=500)
expected = points
result = self.metric.exp(tangent_vec=log, base_point=base_points)
# print(gs.abs(result - expected) / expected)
self.assertAllClose(result, expected, rtol=1e-2)
@geomstats.tests.np_only
def test_exp_vectorization(self):
"""Test vectorization of Exp.
Test the case with one initial point and several tangent vectors.
"""
point = self.dirichlet.random_point()
tangent_vec = gs.array([1., 0.5, 2.])
n_tangent_vecs = 10
t = gs.linspace(0., 1., n_tangent_vecs)
tangent_vecs = gs.einsum('i,...k->...ik', t, tangent_vec)
end_points = self.metric.exp(tangent_vec=tangent_vecs,
base_point=point)
result = end_points.shape
expected = (n_tangent_vecs, self.dim)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_log_vectorization(self):
"""Test vectorization of Log.
Test the case with several base points and one end point.
"""
base_points = self.dirichlet.random_point(self.n_points)
point = self.dirichlet.random_point()
tangent_vecs = self.metric.log(base_point=base_points, point=point)
result = tangent_vecs.shape
expected = (self.n_points, self.dim)
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def tests_geodesic_ivp_and_bvp(self):
"""Test geodesic intial and boundary value problems.
Check the shape of the geodesic.
"""
n_steps = 50
t = gs.linspace(0., 1., n_steps)
initial_points = self.dirichlet.random_point(self.n_points)
initial_tangent_vecs = self.dirichlet.random_point(self.n_points)
geodesic = self.metric._geodesic_ivp(initial_points,
initial_tangent_vecs)
geodesic_at_t = geodesic(t)
result = geodesic_at_t.shape
expected = (self.n_points, n_steps, self.dim)
self.assertAllClose(result, expected)
end_points = self.dirichlet.random_point(self.n_points)
geodesic = self.metric._geodesic_bvp(initial_points, end_points)
geodesic_at_t = geodesic(t)
result = geodesic_at_t.shape
self.assertAllClose(result, expected)
@geomstats.tests.np_only
def test_geodesic(self):
"""Test geodesic.
Check that the norm of the velocity is constant.
"""
initial_point = self.dirichlet.random_point()
end_point = self.dirichlet.random_point()
n_steps = 10000
geod = self.metric.geodesic(initial_point=initial_point,
end_point=end_point)
t = gs.linspace(0., 1., n_steps)
geod_at_t = geod(t)
velocity = n_steps * (geod_at_t[1:, :] - geod_at_t[:-1, :])
velocity_norm = self.metric.norm(velocity, geod_at_t[:-1, :])
result = 1 / velocity_norm.min() * (velocity_norm.max() -
velocity_norm.min())
expected = 0.
self.assertAllClose(expected, result, atol=1e-4, rtol=1.)
@geomstats.tests.np_only
def test_geodesic_vectorization(self):
"""Check vectorization of geodesic.
Check the shape of geodesic at time t for
different scenarios.
"""
initial_point = self.dirichlet.random_point()
initial_tangent_vec = self.dirichlet.random_point()
geod = self.metric.geodesic(initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
time = 0.5
result = geod(time).shape
expected = (self.dim, )
self.assertAllClose(expected, result)
n_vecs = 5
n_times = 10
initial_tangent_vecs = self.dirichlet.random_point(n_vecs)
geod = self.metric.geodesic(initial_point=initial_point,
initial_tangent_vec=initial_tangent_vecs)
times = gs.linspace(0., 1., n_times)
result = geod(times).shape
expected = (n_vecs, n_times, self.dim)
self.assertAllClose(result, expected)
end_points = self.dirichlet.random_point(self.n_points)
geod = self.metric.geodesic(initial_point=initial_point,
end_point=end_points)
time = 0.5
result = geod(time).shape
expected = (self.n_points, self.dim)
self.assertAllClose(expected, result)