def test_parallel_transport(self, n):
space = SPDMatrices(*n)
metric = self.metric(*n)
shape = (2, *n, *n)
point = space.random_point(2)
end_point = space.random_point(2)
tan_b = gs.random.rand(*shape)
tan_b = space.to_tangent(tan_b, point)
# use a vector orthonormal to tan_b
tan_a = gs.random.rand(*shape)
tan_a = space.to_tangent(tan_a, point)
# orthonormalize and move to base_point
tan_a -= gs.einsum(
"...,...ij->...ij",
metric.inner_product(tan_a, tan_b, point) /
metric.squared_norm(tan_b, point),
tan_b,
)
tan_b = gs.einsum("...ij,...->...ij", tan_b,
1.0 / metric.norm(tan_b, point))
tan_a = gs.einsum("...ij,...->...ij", tan_a,
1.0 / metric.norm(tan_a, point))
transported = metric.parallel_transport(tan_a,
point,
end_point=end_point,
n_steps=15,
step="rk4")
result = metric.norm(transported, end_point)
expected = metric.norm(tan_a, point)
self.assertAllClose(result, expected)
is_tangent = space.is_tangent(transported, end_point)
self.assertTrue(gs.all(is_tangent))
transported = metric.parallel_transport(tan_a,
point,
tan_b,
n_steps=15,
step="rk4")
end_point = metric.exp(tan_b, point)
result = metric.norm(transported, end_point)
expected = metric.norm(tan_a, point)
self.assertAllClose(result, expected)
is_tangent = space.is_tangent(transported, end_point)
self.assertTrue(gs.all(is_tangent))
class TestSPDMatrices(geomstats.tests.TestCase):
"""Test of SPDMatrices methods."""
def setUp(self):
"""Set up the test."""
warnings.simplefilter('ignore', category=ImportWarning)
gs.random.seed(1234)
self.n = 3
self.space = SPDMatrices(n=self.n)
self.metric_affine = SPDMetricAffine(n=self.n)
self.metric_bureswasserstein = SPDMetricBuresWasserstein(n=self.n)
self.metric_euclidean = SPDMetricEuclidean(n=self.n)
self.metric_logeuclidean = SPDMetricLogEuclidean(n=self.n)
self.n_samples = 4
def test_belongs(self):
"""Test of belongs method."""
mats = gs.array([[3., -1.], [-1., 3.]])
result = SPDMatrices(2).belongs(mats)
expected = True
self.assertAllClose(result, expected)
mats = gs.array([[-1., -1.], [-1., 3.]])
result = SPDMatrices(2).belongs(mats)
expected = False
self.assertAllClose(result, expected)
mats = gs.eye(3)
result = SPDMatrices(2).belongs(mats)
expected = False
self.assertAllClose(result, expected)
def test_belongs_vectorization(self):
"""Test of belongs method."""
mats = gs.array([[[1., 0], [0, 1.]], [[1., 2.], [2., 1.]],
[[1., 0.], [1., 1.]]])
result = SPDMatrices(2).belongs(mats)
expected = gs.array([True, False, False])
self.assertAllClose(result, expected)
def test_random_point_and_belongs(self):
"""Test of random_point and belongs methods."""
point = self.space.random_point()
result = self.space.belongs(point)
expected = True
self.assertAllClose(result, expected)
def test_random_point_and_belongs_vectorization(self):
"""Test of random_point and belongs methods."""
points = self.space.random_point(4)
result = self.space.belongs(points)
expected = gs.array([True] * 4)
self.assertAllClose(result, expected)
def test_vector_from_symmetric_matrix_and_symmetric_matrix_from_vector(
self):
"""Test for matrix to vector and vector to matrix conversions."""
sym_mat_1 = gs.array([[1., 0.6, -3.], [0.6, 7., 0.], [-3., 0., 8.]])
vector_1 = self.space.to_vector(sym_mat_1)
result_1 = self.space.from_vector(vector_1)
expected_1 = sym_mat_1
self.assertTrue(gs.allclose(result_1, expected_1))
vector_2 = gs.array([1., 2., 3., 4., 5., 6.])
sym_mat_2 = self.space.from_vector(vector_2)
result_2 = self.space.to_vector(sym_mat_2)
expected_2 = vector_2
self.assertTrue(gs.allclose(result_2, expected_2))
def test_vector_and_symmetric_matrix_vectorization(self):
"""Test of vectorization."""
n_samples = self.n_samples
vector = gs.random.rand(n_samples, 6)
sym_mat = self.space.from_vector(vector)
result = self.space.to_vector(sym_mat)
expected = vector
self.assertTrue(gs.allclose(result, expected))
sym_mat = self.space.random_point(n_samples)
vector = self.space.to_vector(sym_mat)
result = self.space.from_vector(vector)
expected = sym_mat
self.assertTrue(gs.allclose(result, expected))
def test_logm(self):
"""Test of logm method."""
expected = gs.array([[[0., 1., 0.], [1., 0., 0.], [0., 0., 1.]]])
c = math.cosh(1)
s = math.sinh(1)
e = math.exp(1)
v = gs.array([[[c, s, 0.], [s, c, 0.], [0., 0., e]]])
result = self.space.logm(v)
self.assertAllClose(result, expected)
def test_differential_power(self):
"""Test of differential_power method."""
base_point = gs.array([[1., 0., 0.], [0., 2.5, 1.5], [0., 1.5, 2.5]])
tangent_vec = gs.array([[2., 1., 1.], [1., .5, .5], [1., .5, .5]])
power = .5
result = self.space.differential_power(power=power,
tangent_vec=tangent_vec,
base_point=base_point)
expected = gs.array([[1., 1 / 3, 1 / 3], [1 / 3, .125, .125],
[1 / 3, .125, .125]])
self.assertAllClose(result, expected)
def test_inverse_differential_power(self):
"""Test of inverse_differential_power method."""
base_point = gs.array([[1., 0., 0.], [0., 2.5, 1.5], [0., 1.5, 2.5]])
tangent_vec = gs.array([[1., 1 / 3, 1 / 3], [1 / 3, .125, .125],
[1 / 3, .125, .125]])
power = .5
result = self.space.inverse_differential_power(power=power,
tangent_vec=tangent_vec,
base_point=base_point)
expected = gs.array([[2., 1., 1.], [1., .5, .5], [1., .5, .5]])
self.assertAllClose(result, expected)
def test_differential_log(self):
"""Test of differential_log method."""
base_point = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., 4.]])
tangent_vec = gs.array([[1., 1., 3.], [1., 1., 3.], [3., 3., 4.]])
result = self.space.differential_log(tangent_vec, base_point)
x = 2 * gs.log(2.)
expected = gs.array([[1., 1., x], [1., 1., x], [x, x, 1]])
self.assertAllClose(result, expected)
def test_inverse_differential_log(self):
"""Test of inverse_differential_log method."""
base_point = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., 4.]])
x = 2 * gs.log(2.)
tangent_vec = gs.array([[1., 1., x], [1., 1., x], [x, x, 1]])
result = self.space.inverse_differential_log(tangent_vec, base_point)
expected = gs.array([[1., 1., 3.], [1., 1., 3.], [3., 3., 4.]])
self.assertAllClose(result, expected)
def test_differential_exp(self):
"""Test of differential_exp method."""
base_point = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., -1.]])
tangent_vec = gs.array([[1., 1., 1.], [1., 1., 1.], [1., 1., 1.]])
result = self.space.differential_exp(tangent_vec, base_point)
x = gs.exp(1.)
y = gs.sinh(1.)
expected = gs.array([[x, x, y], [x, x, y], [y, y, 1 / x]])
self.assertAllClose(result, expected)
def test_inverse_differential_exp(self):
"""Test of inverse_differential_exp method."""
base_point = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., -1.]])
x = gs.exp(1.)
y = gs.sinh(1.)
tangent_vec = gs.array([[x, x, y], [x, x, y], [y, y, 1. / x]])
result = self.space.inverse_differential_exp(tangent_vec, base_point)
expected = gs.array([[1., 1., 1.], [1., 1., 1.], [1., 1., 1.]])
self.assertAllClose(result, expected)
def test_bureswasserstein_inner_product(self):
"""Test of SPDMetricBuresWasserstein.inner_product method."""
base_point = gs.array([[1., 0., 0.], [0., 1.5, .5], [0., .5, 1.5]])
tangent_vec_a = gs.array([[2., 1., 1.], [1., .5, .5], [1., .5, .5]])
tangent_vec_b = gs.array([[1., 2., 4.], [2., 3., 8.], [4., 8., 5.]])
metric = SPDMetricBuresWasserstein(3)
result = metric.inner_product(tangent_vec_a, tangent_vec_b, base_point)
expected = gs.array(4.)
self.assertAllClose(result, expected)
def test_power_affine_inner_product(self):
"""Test of SPDMetricAffine.inner_product method."""
base_point = gs.array([[1., 0., 0.], [0., 2.5, 1.5], [0., 1.5, 2.5]])
tangent_vec = gs.array([[2., 1., 1.], [1., .5, .5], [1., .5, .5]])
metric = SPDMetricAffine(3, power_affine=.5)
result = metric.inner_product(tangent_vec, tangent_vec, base_point)
expected = 713 / 144
self.assertAllClose(result, expected)
def test_power_euclidean_inner_product(self):
"""Test of SPDMetricEuclidean.inner_product method."""
base_point = gs.array([[1., 0., 0.], [0., 2.5, 1.5], [0., 1.5, 2.5]])
tangent_vec = gs.array([[2., 1., 1.], [1., .5, .5], [1., .5, .5]])
metric = SPDMetricEuclidean(3, power_euclidean=.5)
result = metric.inner_product(tangent_vec, tangent_vec, base_point)
expected = 3472 / 576
self.assertAllClose(result, expected)
result = self.metric_euclidean.inner_product(tangent_vec, tangent_vec,
base_point)
expected = MatricesMetric(3, 3).inner_product(tangent_vec, tangent_vec)
self.assertAllClose(result, expected)
@geomstats.tests.np_and_tf_only
def test_euclidean_exp_domain(self):
"""Test of SPDMetricEuclidean.exp_domain method."""
base_point = gs.array([[1., 0., 0.], [0., 2., 0.], [0., 0., 3.]])
tangent_vec = gs.array([[-1., 0., 0.], [0., -.5, 0.], [0., 0., 1.]])
metric = self.metric_euclidean
result = metric.exp_domain(tangent_vec, base_point)
expected = gs.array([-3, 1])
self.assertAllClose(result, expected)
def test_log_euclidean_inner_product(self):
"""Test of SPDMetricLogEuclidean.inner_product method."""
base_point = gs.array([[1., 0., 0.], [0., 1., 0.], [0., 0., 4.]])
tangent_vec = gs.array([[1., 1., 3.], [1., 1., 3.], [3., 3., 4.]])
metric = self.metric_logeuclidean
result = metric.inner_product(tangent_vec, tangent_vec, base_point)
x = 2 * gs.log(2.)
expected = 5. + 4. * x**2
self.assertAllClose(result, expected)
def test_log_and_exp_affine_invariant(self):
"""Test of SPDMetricAffine.log and exp methods with power=1."""
base_point = gs.array([[5., 0., 0.], [0., 7., 2.], [0., 2., 8.]])
point = gs.array([[9., 0., 0.], [0., 5., 0.], [0., 0., 1.]])
metric = self.metric_affine
log = metric.log(point=point, base_point=base_point)
result = metric.exp(tangent_vec=log, base_point=base_point)
expected = point
self.assertAllClose(result, expected)
def test_log_and_exp_power_affine(self):
"""Test of SPDMetricAffine.log and exp methods with power!=1."""
base_point = gs.array([[5., 0., 0.], [0., 7., 2.], [0., 2., 8.]])
point = gs.array([[9., 0., 0.], [0., 5., 0.], [0., 0., 1.]])
metric = SPDMetricAffine(3, power_affine=.5)
log = metric.log(point, base_point)
result = metric.exp(log, base_point)
expected = point
self.assertAllClose(result, expected)
def test_log_and_exp_bureswasserstein(self):
"""Test of SPDMetricBuresWasserstein.log and exp methods."""
base_point = gs.array([[5., 0., 0.], [0., 7., 2.], [0., 2., 8.]])
point = gs.array([[9., 0., 0.], [0., 5., 0.], [0., 0., 1.]])
metric = self.metric_bureswasserstein
log = metric.log(point=point, base_point=base_point)
result = metric.exp(tangent_vec=log, base_point=base_point)
expected = point
self.assertAllClose(result, expected)
def test_log_and_exp_logeuclidean(self):
"""Test of SPDMetricLogEuclidean.log and exp methods."""
base_point = gs.array([[5., 0., 0.], [0., 7., 2.], [0., 2., 8.]])
point = gs.array([[9., 0., 0.], [0., 5., 0.], [0., 0., 1.]])
metric = self.metric_logeuclidean
log = metric.log(point=point, base_point=base_point)
result = metric.exp(tangent_vec=log, base_point=base_point)
expected = point
self.assertAllClose(result, expected)
def test_exp_and_belongs(self):
"""Test of SPDMetricAffine.exp with power=1 and belongs methods."""
n_samples = self.n_samples
base_point = self.space.random_point(n_samples=1)
tangent_vec = self.space.random_tangent_vec(n_samples=n_samples,
base_point=base_point)
metric = self.metric_affine
exps = metric.exp(tangent_vec, base_point)
result = self.space.belongs(exps)
expected = gs.array([True] * n_samples)
self.assertAllClose(result, expected)
def test_exp_vectorization(self):
"""Test of SPDMetricAffine.exp with power=1 and vectorization."""
n_samples = self.n_samples
one_base_point = self.space.random_point(n_samples=1)
n_base_point = self.space.random_point(n_samples=n_samples)
n_tangent_vec_same_base = self.space.random_tangent_vec(
n_samples=n_samples, base_point=one_base_point)
n_tangent_vec = self.space.random_tangent_vec(n_samples=n_samples,
base_point=n_base_point)
metric = self.metric_affine
# Test with the 1 base_point, and several different tangent_vecs
result = metric.exp(n_tangent_vec_same_base, one_base_point)
self.assertAllClose(gs.shape(result),
(n_samples, self.space.n, self.space.n))
# Test with the same number of base_points and tangent_vecs
result = metric.exp(n_tangent_vec, n_base_point)
self.assertAllClose(gs.shape(result),
(n_samples, self.space.n, self.space.n))
def test_log_vectorization(self):
"""Test of SPDMetricAffine.log with power 1 and vectorization."""
n_samples = self.n_samples
one_base_point = self.space.random_point(n_samples=1)
n_base_point = self.space.random_point(n_samples=n_samples)
one_point = self.space.random_point(n_samples=1)
n_point = self.space.random_point(n_samples=n_samples)
metric = self.metric_affine
# Test with different points, one base point
result = metric.log(n_point, one_base_point)
self.assertAllClose(gs.shape(result),
(n_samples, self.space.n, self.space.n))
# Test with the same number of points and base points
result = metric.log(n_point, n_base_point)
self.assertAllClose(gs.shape(result),
(n_samples, self.space.n, self.space.n))
# Test with the one point and n base points
result = metric.log(one_point, n_base_point)
self.assertAllClose(gs.shape(result),
(n_samples, self.space.n, self.space.n))
def test_geodesic_and_belongs(self):
"""Test of SPDMetricAffine.geodesic with power 1 and belongs."""
initial_point = self.space.random_point()
initial_tangent_vec = self.space.random_tangent_vec(
n_samples=1, base_point=initial_point)
metric = self.metric_affine
geodesic = metric.geodesic(initial_point=initial_point,
initial_tangent_vec=initial_tangent_vec)
n_points = 10
t = gs.linspace(start=0., stop=1., num=n_points)
points = geodesic(t)
result = self.space.belongs(points)
self.assertTrue(gs.all(result))
def test_squared_dist_is_symmetric(self):
"""Test of SPDMetricAffine.squared_dist (power=1) and is_symmetric."""
n_samples = self.n_samples
point_1 = self.space.random_point(n_samples=1)
point_2 = self.space.random_point(n_samples=1)
point_1 = gs.cast(point_1, gs.float64)
point_2 = gs.cast(point_2, gs.float64)
metric = self.metric_affine
sq_dist_1_2 = metric.squared_dist(point_1, point_2)
sq_dist_2_1 = metric.squared_dist(point_2, point_1)
self.assertAllClose(sq_dist_1_2, sq_dist_2_1)
point_2 = self.space.random_point(n_samples=n_samples)
point_2 = gs.cast(point_2, gs.float64)
sq_dist_1_2 = metric.squared_dist(point_1, point_2)
sq_dist_2_1 = metric.squared_dist(point_2, point_1)
self.assertAllClose(sq_dist_1_2, sq_dist_2_1)
point_1 = self.space.random_point(n_samples=n_samples)
point_2 = self.space.random_point(n_samples=1)
point_1 = gs.cast(point_1, gs.float64)
point_2 = gs.cast(point_2, gs.float64)
sq_dist_1_2 = metric.squared_dist(point_1, point_2)
sq_dist_2_1 = metric.squared_dist(point_2, point_1)
self.assertAllClose(sq_dist_1_2, sq_dist_2_1)
sq_dist_1_2 = metric.squared_dist(point_1, point_2)
sq_dist_2_1 = metric.squared_dist(point_2, point_1)
self.assertAllClose(sq_dist_1_2, sq_dist_2_1)
def test_squared_dist_vectorization(self):
"""Test of SPDMetricAffine.squared_dist (power=1) and vectorization."""
n_samples = self.n_samples
point_1 = self.space.random_point(n_samples=n_samples)
point_2 = self.space.random_point(n_samples=n_samples)
metric = self.metric_affine
result = metric.squared_dist(point_1, point_2)
self.assertAllClose(gs.shape(result), (n_samples, ))
point_1 = self.space.random_point(n_samples=1)
point_2 = self.space.random_point(n_samples=n_samples)
result = metric.squared_dist(point_1, point_2)
self.assertAllClose(gs.shape(result), (n_samples, ))
point_1 = self.space.random_point(n_samples=n_samples)
point_2 = self.space.random_point(n_samples=1)
result = metric.squared_dist(point_1, point_2)
self.assertAllClose(gs.shape(result), (n_samples, ))
point_1 = self.space.random_point(n_samples=1)
point_2 = self.space.random_point(n_samples=1)
result = metric.squared_dist(point_1, point_2)
self.assertAllClose(gs.shape(result), ())
def test_parallel_transport_affine_invariant(self):
"""Test of SPDMetricAffine.parallel_transport method with power=1."""
n_samples = self.n_samples
gs.random.seed(1)
point = self.space.random_point(n_samples)
tan_a = self.space.random_tangent_vec(n_samples, point)
tan_b = self.space.random_tangent_vec(n_samples, point)
point = gs.cast(point, gs.float64)
tan_a = gs.cast(tan_a, gs.float64)
tan_b = gs.cast(tan_b, gs.float64)
metric = self.metric_affine
expected = metric.norm(tan_a, point)
end_point = metric.exp(tan_b, point)
transported = metric.parallel_transport(tan_a, tan_b, point)
result = metric.norm(transported, end_point)
self.assertAllClose(expected, result)
def test_squared_dist_bureswasserstein(self):
"""Test of SPDMetricBuresWasserstein.squared_dist method."""
point_a = gs.array([[5., 0., 0.], [0., 7., 2.], [0., 2., 8.]])
point_b = gs.array([[9., 0., 0.], [0., 5., 0.], [0., 0., 1.]])
metric = self.metric_bureswasserstein
result = metric.squared_dist(point_a, point_b)
log = metric.log(point=point_b, base_point=point_a)
expected = metric.squared_norm(vector=log, base_point=point_a)
self.assertAllClose(result, expected)
def test_squared_dist_bureswasserstein_vectorization(self):
"""Test of SPDMetricBuresWasserstein.squared_dist method."""
point_a = self.space.random_point(2)
point_b = gs.array([[9., 0., 0.], [0., 5., 0.], [0., 0., 1.]])
point_a = gs.cast(point_a, gs.float64)
point_b = gs.cast(point_b, gs.float64)
metric = self.metric_bureswasserstein
result = metric.squared_dist(point_a, point_b)
log = metric.log(point=point_b, base_point=point_a)
expected = metric.squared_norm(vector=log, base_point=point_a)
self.assertAllClose(result, expected)
def test_to_tangent_and_is_tangent(self):
mat = gs.random.rand(3, 3)
projection = self.space.to_tangent(mat)
result = self.space.is_tangent(projection)
self.assertTrue(result)