def parallel_transport_test_data(self):
k_landmarks = 4
m_ambient = 3
n_samples = 10
space = PreShapeSpace(4, 3)
base_point = space.projection(gs.eye(4)[:, :3])
vec_a = gs.random.rand(n_samples, k_landmarks, m_ambient)
tangent_vec_a = space.to_tangent(space.center(vec_a), base_point)
vec_b = gs.random.rand(n_samples, k_landmarks, m_ambient)
tangent_vec_b = space.to_tangent(space.center(vec_b), base_point)
smoke_data = [
dict(
k_landmarks=k_landmarks,
m_ambient=m_ambient,
tangent_vec_a=tangent_vec_a,
tangent_vec_b=tangent_vec_b,
base_point=base_point,
)
]
return self.generate_tests(smoke_data)
class TestPreShapeSpace(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.k_landmarks = 4
self.m_ambient = 3
self.space = PreShapeSpace(self.k_landmarks, self.m_ambient)
self.matrices = self.space.embedding_manifold
self.n_samples = 10
self.shape_metric = KendallShapeMetric(self.k_landmarks,
self.m_ambient)
def test_belongs(self):
point = gs.random.rand(self.m_ambient - 1, self.k_landmarks)
result = self.space.belongs(point)
self.assertFalse(result)
point = gs.random.rand(self.n_samples, self.m_ambient - 1,
self.k_landmarks)
result = self.space.belongs(point)
self.assertFalse(gs.all(result))
def test_random_point_and_belongs(self):
"""Test random uniform and belongs.
Test that the random uniform method samples
on the pre-shape space.
"""
n_samples = self.n_samples
point = self.space.random_point(n_samples)
result = self.space.belongs(point)
expected = gs.array([True] * n_samples)
self.assertAllClose(expected, result)
def test_random_point_shape(self):
point = self.space.random_point()
result = gs.shape(point)
expected = (
self.k_landmarks,
self.m_ambient,
)
self.assertAllClose(result, expected)
point = self.space.random_point(self.n_samples)
result = gs.shape(point)
expected = (
self.n_samples,
self.k_landmarks,
self.m_ambient,
)
self.assertAllClose(result, expected)
def test_projection_and_belongs(self):
point = Matrices.transpose(
gs.array([[1., 0., 0., 1.], [0., 1., 0., 1.], [0., 0., 1., 1.]]))
proj = self.space.projection(point)
result = self.space.belongs(proj)
expected = True
self.assertAllClose(expected, result)
def test_is_centered(self):
point = gs.ones((self.k_landmarks, self.m_ambient))
result = self.space.is_centered(point)
self.assertFalse(result)
point = gs.zeros((self.k_landmarks, self.m_ambient))
result = self.space.is_centered(point)
self.assertTrue(result)
def test_to_center_is_center(self):
point = gs.ones((self.k_landmarks, self.m_ambient))
point = self.space.center(point)
result = self.space.is_centered(point)
self.assertTrue(result)
def test_to_center_is_centered_vectorization(self):
point = gs.ones((self.n_samples, self.k_landmarks, self.m_ambient))
point = self.space.center(point)
result = gs.all(self.space.is_centered(point))
self.assertTrue(result)
def test_is_tangent_to_tangent(self):
point, vector = self.matrices.random_point(2)
point = self.space.projection(point)
result = self.space.is_tangent(vector, point)
self.assertFalse(result)
tangent_vec = self.space.to_tangent(vector, point)
result = self.space.is_tangent(tangent_vec, point)
self.assertTrue(result)
vec = gs.array([tangent_vec, vector])
result = self.space.is_tangent(vec, point)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
def test_vertical_projection(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
vertical = self.space.vertical_projection(tan, point)
transposed_point = Matrices.transpose(point)
tmp_expected = gs.matmul(transposed_point, tan)
expected = Matrices.transpose(tmp_expected) - tmp_expected
tmp_result = gs.matmul(transposed_point, vertical)
result = Matrices.transpose(tmp_result) - tmp_result
self.assertAllClose(result, expected)
def test_vertical_projection_vectorization(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
vertical = self.space.vertical_projection(tan, point)
transposed_point = Matrices.transpose(point)
tmp_expected = gs.matmul(transposed_point, tan)
expected = Matrices.transpose(tmp_expected) - tmp_expected
tmp_result = gs.matmul(transposed_point, vertical)
result = Matrices.transpose(tmp_result) - tmp_result
self.assertAllClose(result, expected)
def test_horizontal_projection(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
transposed_point = Matrices.transpose(point)
result = gs.matmul(transposed_point, horizontal)
expected = Matrices.transpose(result)
self.assertAllClose(result, expected)
def test_horizontal_projection_vectorized(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
transposed_point = Matrices.transpose(point)
result = gs.matmul(transposed_point, horizontal)
expected = Matrices.transpose(result)
self.assertAllClose(result, expected)
def test_horizontal_and_is_tangent(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
horizontal = gs.stack([horizontal, vector])
result = self.space.is_tangent(horizontal, point)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
def test_align(self):
point, base_point = self.space.random_point(2)
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(result)
def test_align_vectorization(self):
base_point = self.space.random_point()
point = self.space.random_point(2)
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(gs.all(result))
base_point = self.space.random_point(2)
point = self.space.random_point()
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(gs.all(result))
def test_inner_product_shape(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
inner = self.space.ambient_metric.inner_product(tan, tan, point)
self.assertAllClose(inner.shape, (self.n_samples, ))
def test_exp_and_belongs(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(result)
exp = self.space.ambient_metric.exp(gs.zeros_like(point), point)
result = gs.isclose(point, exp)
self.assertTrue(gs.all(result))
def test_exp_and_belongs_vectorization(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(gs.all(result))
point = point[0]
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(gs.all(result))
def test_log_and_exp(self):
point, base_point = self.space.random_point(2)
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
exp = self.space.ambient_metric.exp(log, base_point)
self.assertAllClose(exp, point)
def test_exp_and_log(self):
base_point = self.space.random_point()
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
tangent_vec = self.space.to_tangent(vector, base_point)
point = self.space.ambient_metric.exp(tangent_vec, base_point)
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
self.assertAllClose(tangent_vec, log)
def test_log_vectorization(self):
point = self.space.random_point(self.n_samples)
base_point = self.space.random_point()
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(gs.all(result))
exp = self.space.ambient_metric.exp(log, base_point)
self.assertAllClose(exp, point)
log = self.space.ambient_metric.log(base_point, point)
result = self.space.is_tangent(log, point)
self.assertTrue(gs.all(result))
exp = self.space.ambient_metric.exp(log, point)
expected = gs.stack([base_point] * self.n_samples)
self.assertAllClose(exp, expected)
def test_kendall_inner_product_shape(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
inner = self.shape_metric.inner_product(tan, tan, point)
self.assertAllClose(inner.shape, (self.n_samples, ))
def test_kendall_log_and_exp(self):
point, base_point = self.space.random_point(2)
expected = self.space.align(point, base_point)
log = self.shape_metric.log(expected, base_point)
result = self.space.is_horizontal(log, base_point)
self.assertTrue(result)
exp = self.shape_metric.exp(log, base_point)
self.assertAllClose(exp, expected)
def test_kendall_exp_and_log(self):
base_point = self.space.random_point()
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
tangent_vec = self.space.to_tangent(vector, base_point)
point = self.shape_metric.exp(tangent_vec, base_point)
log = self.shape_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
expected = self.space.horizontal_projection(tangent_vec, base_point)
self.assertAllClose(expected, log)
def test_dist_extreme_case(self):
point = self.space.projection(gs.eye(self.k_landmarks, self.m_ambient))
result = self.shape_metric.dist(point, point)
expected = 0.
self.assertAllClose(result, expected)
def test_dist(self):
point, base_point = self.space.random_point(2)
result = self.shape_metric.dist(point, base_point)
log = self.shape_metric.log(point, base_point)
expected = self.shape_metric.norm(log, base_point)
self.assertAllClose(result, expected)
def test_dist_vectorization(self):
point = self.space.random_point(self.n_samples)
base_point = self.space.random_point(self.n_samples)
aligned = self.space.align(point, base_point)
result = self.shape_metric.dist(aligned, base_point)
log = self.shape_metric.log(aligned, base_point)
expected = self.shape_metric.norm(log, base_point)
self.assertAllClose(result, expected)
def test_curvature_is_skew_operator(self):
space = self.space
base_point = space.random_point(2)
vector = gs.random.rand(4, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[:2], base_point)
tangent_vec_b = space.to_tangent(vector[2:], base_point)
result = self.shape_metric.curvature(tangent_vec_a, tangent_vec_a,
tangent_vec_b, base_point)
expected = gs.zeros_like(result)
self.assertAllClose(result, expected)
def test_curvature_bianchi_identity(self):
space = self.space
base_point = space.random_point()
vector = gs.random.rand(3, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[0], base_point)
tangent_vec_b = space.to_tangent(vector[1], base_point)
tangent_vec_c = space.to_tangent(vector[2], base_point)
curvature_1 = self.shape_metric.curvature(tangent_vec_a, tangent_vec_b,
tangent_vec_c, base_point)
curvature_2 = self.shape_metric.curvature(tangent_vec_b, tangent_vec_c,
tangent_vec_a, base_point)
curvature_3 = self.shape_metric.curvature(tangent_vec_c, tangent_vec_a,
tangent_vec_b, base_point)
result = curvature_1 + curvature_2 + curvature_3
expected = gs.zeros_like(result)
self.assertAllClose(result, expected)
def test_integrability_tensor(self):
space = self.space
base_point = space.random_point()
vector = gs.random.rand(2, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[0], base_point)
tangent_vec_b = space.to_tangent(vector[1], base_point)
result_ab = space.integrability_tensor(tangent_vec_a, tangent_vec_b,
base_point)
result = space.ambient_metric.inner_product(tangent_vec_b, result_ab,
base_point)
expected = 0.
self.assertAllClose(result, expected)
horizontal_b = space.horizontal_projection(tangent_vec_b, base_point)
horizontal_a = space.horizontal_projection(tangent_vec_a, base_point)
result = space.integrability_tensor(horizontal_a, horizontal_b,
base_point)
expected = -space.integrability_tensor(horizontal_b, horizontal_a,
base_point)
self.assertAllClose(result, expected)
is_vertical = space.is_vertical(result, base_point)
self.assertTrue(is_vertical)
vertical_b = tangent_vec_b - horizontal_b
result = space.integrability_tensor(horizontal_a, vertical_b,
base_point)
is_horizontal = space.is_horizontal(result, base_point)
self.assertTrue(is_horizontal)
def test_kendall_directional_curvature(self):
space = self.space
kendall = self.shape_metric
n_samples = 4 * self.k_landmarks * self.m_ambient
base_point = self.space.random_point(1)
vec_a = gs.random.rand(n_samples, self.k_landmarks, self.m_ambient)
tg_vec_a = space.to_tangent(space.center(vec_a), base_point)
hor_a = space.horizontal_projection(tg_vec_a, base_point)
vec_b = gs.random.rand(n_samples, self.k_landmarks, self.m_ambient)
tg_vec_b = space.to_tangent(space.center(vec_b), base_point)
hor_b = space.horizontal_projection(tg_vec_b, base_point)
tidal = kendall.directional_curvature(hor_a, hor_b, base_point)
numerator = kendall.inner_product(tidal, hor_b, base_point)
denominator = \
kendall.inner_product(hor_a, hor_a, base_point) * \
kendall.inner_product(hor_b, hor_b, base_point) - \
kendall.inner_product(hor_a, hor_b, base_point) ** 2
condition = ~gs.isclose(denominator, 0.)
kappa = numerator[condition] / denominator[condition]
kappa_direct = \
kendall.sectional_curvature(hor_a, hor_b, base_point)[condition]
self.assertAllClose(kappa, kappa_direct)
result = (kappa > 1.0 - 1e-12)
self.assertTrue(gs.all(result))
def test_parallel_transport(self):
space = self.space
metric = self.shape_metric
shape = (self.n_samples, self.k_landmarks, self.m_ambient)
point = space.projection(gs.eye(4)[:, :3])
tan_b = gs.random.rand(*shape)
tan_b = space.to_tangent(tan_b, point)
tan_b = space.horizontal_projection(tan_b, point)
# use a vector orthonormal to tan_b
tan_a = gs.random.rand(*shape)
tan_a = space.to_tangent(tan_a, point)
tan_a = space.horizontal_projection(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. / metric.norm(tan_b, point))
tan_a = gs.einsum('...ij,...->...ij', tan_a,
1. / metric.norm(tan_a, point))
transported = metric.parallel_transport(tan_a,
tan_b,
point,
n_steps=100,
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)
is_horizontal = space.is_horizontal(transported, end_point)
self.assertTrue(gs.all(is_tangent))
self.assertTrue(gs.all(is_horizontal))
class TestPreShapeSpace(geomstats.tests.TestCase):
def setUp(self):
gs.random.seed(1234)
self.k_landmarks = 4
self.m_ambient = 3
self.space = PreShapeSpace(self.k_landmarks, self.m_ambient)
self.matrices = self.space.embedding_manifold
self.n_samples = 10
self.shape_metric = KendallShapeMetric(self.k_landmarks,
self.m_ambient)
def test_belongs(self):
point = gs.random.rand(self.m_ambient - 1, self.k_landmarks)
result = self.space.belongs(point)
self.assertFalse(result)
point = gs.random.rand(self.n_samples, self.m_ambient - 1,
self.k_landmarks)
result = self.space.belongs(point)
self.assertFalse(gs.all(result))
def test_random_point_and_belongs(self):
"""Test random uniform and belongs.
Test that the random uniform method samples
on the pre-shape space.
"""
n_samples = self.n_samples
point = self.space.random_point(n_samples)
result = self.space.belongs(point)
expected = gs.array([True] * n_samples)
self.assertAllClose(expected, result)
def test_random_point_shape(self):
point = self.space.random_point()
result = gs.shape(point)
expected = (
self.k_landmarks,
self.m_ambient,
)
self.assertAllClose(result, expected)
point = self.space.random_point(self.n_samples)
result = gs.shape(point)
expected = (
self.n_samples,
self.k_landmarks,
self.m_ambient,
)
self.assertAllClose(result, expected)
def test_projection_and_belongs(self):
point = Matrices.transpose(
gs.array([[1., 0., 0., 1.], [0., 1., 0., 1.], [0., 0., 1., 1.]]))
proj = self.space.projection(point)
result = self.space.belongs(proj)
expected = True
self.assertAllClose(expected, result)
def test_is_centered(self):
point = gs.ones((self.k_landmarks, self.m_ambient))
result = self.space.is_centered(point)
self.assertFalse(result)
point = gs.zeros((self.k_landmarks, self.m_ambient))
result = self.space.is_centered(point)
self.assertTrue(result)
def test_to_center_is_center(self):
point = gs.ones((self.k_landmarks, self.m_ambient))
point = self.space.center(point)
result = self.space.is_centered(point)
self.assertTrue(result)
def test_to_center_is_centered_vectorization(self):
point = gs.ones((self.n_samples, self.k_landmarks, self.m_ambient))
point = self.space.center(point)
result = gs.all(self.space.is_centered(point))
self.assertTrue(result)
def test_is_tangent_to_tangent(self):
point, vector = self.matrices.random_point(2)
point = self.space.projection(point)
result = self.space.is_tangent(vector, point)
self.assertFalse(result)
tangent_vec = self.space.to_tangent(vector, point)
result = self.space.is_tangent(tangent_vec, point)
self.assertTrue(result)
vec = gs.array([tangent_vec, vector])
result = self.space.is_tangent(vec, point)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
def test_vertical_projection(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
vertical = self.space.vertical_projection(tan, point)
transposed_point = Matrices.transpose(point)
tmp_expected = gs.matmul(transposed_point, tan)
expected = Matrices.transpose(tmp_expected) - tmp_expected
tmp_result = gs.matmul(transposed_point, vertical)
result = Matrices.transpose(tmp_result) - tmp_result
self.assertAllClose(result, expected)
def test_vertical_projection_vectorization(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
vertical = self.space.vertical_projection(tan, point)
transposed_point = Matrices.transpose(point)
tmp_expected = gs.matmul(transposed_point, tan)
expected = Matrices.transpose(tmp_expected) - tmp_expected
tmp_result = gs.matmul(transposed_point, vertical)
result = Matrices.transpose(tmp_result) - tmp_result
self.assertAllClose(result, expected)
def test_horizontal_projection(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
transposed_point = Matrices.transpose(point)
result = gs.matmul(transposed_point, horizontal)
expected = Matrices.transpose(result)
self.assertAllClose(result, expected)
def test_horizontal_projection_vectorized(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
transposed_point = Matrices.transpose(point)
result = gs.matmul(transposed_point, horizontal)
expected = Matrices.transpose(result)
self.assertAllClose(result, expected)
def test_horizontal_and_is_tangent(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
horizontal = gs.stack([horizontal, vector])
result = self.space.is_tangent(horizontal, point)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
def test_align(self):
point, base_point = self.space.random_point(2)
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(result)
def test_align_vectorization(self):
base_point = self.space.random_point()
point = self.space.random_point(2)
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(gs.all(result))
base_point = self.space.random_point(2)
point = self.space.random_point()
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(gs.all(result))
def test_inner_product_shape(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
inner = self.space.ambient_metric.inner_product(tan, tan, point)
self.assertAllClose(inner.shape, (self.n_samples, ))
def test_exp_and_belongs(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(result)
exp = self.space.ambient_metric.exp(gs.zeros_like(point), point)
result = gs.isclose(point, exp)
self.assertTrue(gs.all(result))
def test_exp_and_belongs_vectorization(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(gs.all(result))
point = point[0]
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(gs.all(result))
def test_log_and_exp(self):
point, base_point = self.space.random_point(2)
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
exp = self.space.ambient_metric.exp(log, base_point)
self.assertAllClose(exp, point)
def test_exp_and_log(self):
base_point = self.space.random_point()
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
tangent_vec = self.space.to_tangent(vector, base_point)
point = self.space.ambient_metric.exp(tangent_vec, base_point)
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
self.assertAllClose(tangent_vec, log)
def test_log_vectorization(self):
point = self.space.random_point(self.n_samples)
base_point = self.space.random_point()
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(gs.all(result))
exp = self.space.ambient_metric.exp(log, base_point)
self.assertAllClose(exp, point)
log = self.space.ambient_metric.log(base_point, point)
result = self.space.is_tangent(log, point)
self.assertTrue(gs.all(result))
exp = self.space.ambient_metric.exp(log, point)
expected = gs.stack([base_point] * self.n_samples)
self.assertAllClose(exp, expected)
def test_kendall_inner_product_shape(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
inner = self.shape_metric.inner_product(tan, tan, point)
self.assertAllClose(inner.shape, (self.n_samples, ))
def test_kendall_log_and_exp(self):
point, base_point = self.space.random_point(2)
expected = self.space.align(point, base_point)
log = self.shape_metric.log(expected, base_point)
result = self.space.is_horizontal(log, base_point)
self.assertTrue(result)
exp = self.shape_metric.exp(log, base_point)
self.assertAllClose(exp, expected)
def test_kendall_exp_and_log(self):
base_point = self.space.random_point()
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
tangent_vec = self.space.to_tangent(vector, base_point)
point = self.shape_metric.exp(tangent_vec, base_point)
log = self.shape_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
expected = self.space.horizontal_projection(tangent_vec, base_point)
self.assertAllClose(expected, log)
def test_dist_extreme_case(self):
point = self.space.projection(gs.eye(self.k_landmarks, self.m_ambient))
result = self.shape_metric.dist(point, point)
expected = 0.
self.assertAllClose(result, expected)
def test_dist(self):
point, base_point = self.space.random_point(2)
result = self.shape_metric.dist(point, base_point)
log = self.shape_metric.log(point, base_point)
expected = self.shape_metric.norm(log, base_point)
self.assertAllClose(result, expected)
def test_dist_vectorization(self):
point = self.space.random_point(self.n_samples)
base_point = self.space.random_point(self.n_samples)
aligned = self.space.align(point, base_point)
result = self.shape_metric.dist(aligned, base_point)
log = self.shape_metric.log(aligned, base_point)
expected = self.shape_metric.norm(log, base_point)
self.assertAllClose(result, expected)
def test_curvature_is_skew_operator(self):
space = self.space
base_point = space.random_point(2)
vector = gs.random.rand(4, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[:2], base_point)
tangent_vec_b = space.to_tangent(vector[2:], base_point)
result = self.shape_metric.curvature(tangent_vec_a, tangent_vec_a,
tangent_vec_b, base_point)
expected = gs.zeros_like(result)
self.assertAllClose(result, expected)
def test_curvature_bianchi_identity(self):
space = self.space
base_point = space.random_point()
vector = gs.random.rand(3, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[0], base_point)
tangent_vec_b = space.to_tangent(vector[1], base_point)
tangent_vec_c = space.to_tangent(vector[2], base_point)
curvature_1 = self.shape_metric.curvature(tangent_vec_a, tangent_vec_b,
tangent_vec_c, base_point)
curvature_2 = self.shape_metric.curvature(tangent_vec_b, tangent_vec_c,
tangent_vec_a, base_point)
curvature_3 = self.shape_metric.curvature(tangent_vec_c, tangent_vec_a,
tangent_vec_b, base_point)
result = curvature_1 + curvature_2 + curvature_3
expected = gs.zeros_like(result)
self.assertAllClose(result, expected)
def test_integrability_tensor(self):
space = self.space
base_point = space.random_point()
vector = gs.random.rand(2, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[0], base_point)
tangent_vec_b = space.to_tangent(vector[1], base_point)
result_ab = space.integrability_tensor(tangent_vec_a, tangent_vec_b,
base_point)
result = space.ambient_metric.inner_product(tangent_vec_b, result_ab,
base_point)
expected = 0.
self.assertAllClose(result, expected)
horizontal_b = space.horizontal_projection(tangent_vec_b, base_point)
horizontal_a = space.horizontal_projection(tangent_vec_a, base_point)
result = space.integrability_tensor(horizontal_a, horizontal_b,
base_point)
expected = -space.integrability_tensor(horizontal_b, horizontal_a,
base_point)
self.assertAllClose(result, expected)
is_vertical = space.is_vertical(result, base_point)
self.assertTrue(is_vertical)
vertical_b = tangent_vec_b - horizontal_b
result = space.integrability_tensor(horizontal_a, vertical_b,
base_point)
is_horizontal = space.is_horizontal(result, base_point)
self.assertTrue(is_horizontal)
class TestPreShapeSpace(geomstats.tests.TestCase):
def setup_method(self):
gs.random.seed(1234)
self.k_landmarks = 4
self.m_ambient = 3
self.space = PreShapeSpace(self.k_landmarks, self.m_ambient)
self.matrices = self.space.embedding_space
self.n_samples = 10
self.shape_metric = KendallShapeMetric(self.k_landmarks,
self.m_ambient)
self.base_point = self.space.random_point()
vector = gs.random.rand(11, self.k_landmarks, self.m_ambient)
tg_vec_0 = self.space.to_tangent(vector[0], self.base_point)
self.hor_x = self.space.horizontal_projection(tg_vec_0,
self.base_point)
tg_vec_1 = self.space.to_tangent(vector[1], self.base_point)
self.hor_y = self.space.horizontal_projection(tg_vec_1,
self.base_point)
tg_vec_2 = self.space.to_tangent(vector[2], self.base_point)
self.hor_z = self.space.horizontal_projection(tg_vec_2,
self.base_point)
tg_vec_3 = self.space.to_tangent(vector[3], self.base_point)
self.hor_h = self.space.horizontal_projection(tg_vec_3,
self.base_point)
tg_vec_4 = self.space.to_tangent(vector[4], self.base_point)
self.ver_v = self.space.vertical_projection(tg_vec_4, self.base_point)
tg_vec_5 = self.space.to_tangent(vector[5], self.base_point)
self.ver_w = self.space.vertical_projection(tg_vec_5, self.base_point)
tg_vec_6 = self.space.to_tangent(vector[6], self.base_point)
hor_dy = self.space.horizontal_projection(tg_vec_6, self.base_point)
tg_vec_7 = self.space.to_tangent(vector[7], self.base_point)
hor_dz = self.space.horizontal_projection(tg_vec_7, self.base_point)
tg_vec_8 = self.space.to_tangent(vector[8], self.base_point)
ver_dv = self.space.vertical_projection(tg_vec_8, self.base_point)
tg_vec_9 = self.space.to_tangent(vector[9], self.base_point)
ver_dw = self.space.vertical_projection(tg_vec_9, self.base_point)
tg_vec_10 = self.space.to_tangent(vector[10], self.base_point)
hor_dh = self.space.horizontal_projection(tg_vec_10, self.base_point)
# generate valid derivatives of horizontal / vertical vector fields.
a_x_y = self.space.integrability_tensor(self.hor_x, self.hor_y,
self.base_point)
self.nabla_x_y = hor_dy + a_x_y
a_x_z = self.space.integrability_tensor(self.hor_x, self.hor_z,
self.base_point)
self.nabla_x_z = hor_dz + a_x_z
a_x_v = self.space.integrability_tensor(self.hor_x, self.ver_v,
self.base_point)
self.nabla_x_v = ver_dv + a_x_v
a_x_w = self.space.integrability_tensor(self.hor_x, self.ver_w,
self.base_point)
self.nabla_x_w = ver_dw + a_x_w
a_x_h = self.space.integrability_tensor(self.hor_x, self.hor_h,
self.base_point)
self.nabla_x_h = hor_dh + a_x_h
def test_belongs(self):
point = gs.random.rand(self.m_ambient - 1, self.k_landmarks)
result = self.space.belongs(point)
self.assertFalse(result)
point = gs.random.rand(self.n_samples, self.m_ambient - 1,
self.k_landmarks)
result = self.space.belongs(point)
self.assertFalse(gs.all(result))
def test_random_point_and_belongs(self):
"""Test random uniform and belongs.
Test that the random uniform method samples
on the pre-shape space.
"""
n_samples = self.n_samples
point = self.space.random_point(n_samples)
result = self.space.belongs(point)
expected = gs.array([True] * n_samples)
self.assertAllClose(expected, result)
def test_random_point_shape(self):
point = self.space.random_point()
result = gs.shape(point)
expected = (
self.k_landmarks,
self.m_ambient,
)
self.assertAllClose(result, expected)
point = self.space.random_point(self.n_samples)
result = gs.shape(point)
expected = (
self.n_samples,
self.k_landmarks,
self.m_ambient,
)
self.assertAllClose(result, expected)
def test_projection_and_belongs(self):
point = Matrices.transpose(
gs.array([
[1.0, 0.0, 0.0, 1.0],
[0.0, 1.0, 0.0, 1.0],
[0.0, 0.0, 1.0, 1.0],
]))
proj = self.space.projection(point)
result = self.space.belongs(proj)
expected = True
self.assertAllClose(expected, result)
def test_is_centered(self):
point = gs.ones((self.k_landmarks, self.m_ambient))
result = self.space.is_centered(point)
self.assertFalse(result)
point = gs.zeros((self.k_landmarks, self.m_ambient))
result = self.space.is_centered(point)
self.assertTrue(result)
def test_to_center_is_center(self):
point = gs.ones((self.k_landmarks, self.m_ambient))
point = self.space.center(point)
result = self.space.is_centered(point)
self.assertTrue(result)
def test_to_center_is_centered_vectorization(self):
point = gs.ones((self.n_samples, self.k_landmarks, self.m_ambient))
point = self.space.center(point)
result = gs.all(self.space.is_centered(point))
self.assertTrue(result)
def test_is_tangent_to_tangent(self):
point, vector = self.matrices.random_point(2)
point = self.space.projection(point)
result = self.space.is_tangent(vector, point)
self.assertFalse(result)
tangent_vec = self.space.to_tangent(vector, point)
result = self.space.is_tangent(tangent_vec, point)
self.assertTrue(result)
vec = gs.array([tangent_vec, vector])
result = self.space.is_tangent(vec, point)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
def test_vertical_projection(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
vertical = self.space.vertical_projection(tan, point)
transposed_point = Matrices.transpose(point)
tmp_expected = gs.matmul(transposed_point, tan)
expected = Matrices.transpose(tmp_expected) - tmp_expected
tmp_result = gs.matmul(transposed_point, vertical)
result = Matrices.transpose(tmp_result) - tmp_result
self.assertAllClose(result, expected)
def test_vertical_projection_vectorization(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
vertical = self.space.vertical_projection(tan, point)
transposed_point = Matrices.transpose(point)
tmp_expected = gs.matmul(transposed_point, tan)
expected = Matrices.transpose(tmp_expected) - tmp_expected
tmp_result = gs.matmul(transposed_point, vertical)
result = Matrices.transpose(tmp_result) - tmp_result
self.assertAllClose(result, expected)
def test_horizontal_projection(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
transposed_point = Matrices.transpose(point)
result = gs.matmul(transposed_point, horizontal)
expected = Matrices.transpose(result)
self.assertAllClose(result, expected)
def test_horizontal_projection_vectorized(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
transposed_point = Matrices.transpose(point)
result = gs.matmul(transposed_point, horizontal)
expected = Matrices.transpose(result)
self.assertAllClose(result, expected)
def test_horizontal_and_is_tangent(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
horizontal = self.space.horizontal_projection(tan, point)
horizontal = gs.stack([horizontal, vector])
result = self.space.is_tangent(horizontal, point)
expected = gs.array([True, False])
self.assertAllClose(result, expected)
def test_align(self):
point, base_point = self.space.random_point(2)
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(result)
def test_align_vectorization(self):
base_point = self.space.random_point()
point = self.space.random_point(2)
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(gs.all(result))
base_point = self.space.random_point(2)
point = self.space.random_point()
aligned = self.space.align(point, base_point)
alignment = gs.matmul(Matrices.transpose(aligned), base_point)
result = Matrices.is_symmetric(alignment)
self.assertTrue(gs.all(result))
def test_inner_product_shape(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
inner = self.space.ambient_metric.inner_product(tan, tan, point)
self.assertAllClose(inner.shape, (self.n_samples, ))
def test_exp_and_belongs(self):
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(result)
exp = self.space.ambient_metric.exp(gs.zeros_like(point), point)
result = gs.isclose(point, exp)
self.assertTrue(gs.all(result))
def test_exp_and_belongs_vectorization(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point(self.n_samples)
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(gs.all(result))
point = point[0]
tan = self.space.to_tangent(vector, point)
exp = self.space.ambient_metric.exp(tan, point)
result = self.space.belongs(exp)
self.assertTrue(gs.all(result))
def test_log_and_exp(self):
point, base_point = self.space.random_point(2)
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
exp = self.space.ambient_metric.exp(log, base_point)
self.assertAllClose(exp, point)
def test_exp_and_log(self):
base_point = self.space.random_point()
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
tangent_vec = self.space.to_tangent(vector, base_point)
point = self.space.ambient_metric.exp(tangent_vec, base_point)
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
self.assertAllClose(tangent_vec, log)
def test_log_vectorization(self):
point = self.space.random_point(self.n_samples)
base_point = self.space.random_point()
log = self.space.ambient_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(gs.all(result))
exp = self.space.ambient_metric.exp(log, base_point)
self.assertAllClose(exp, point)
log = self.space.ambient_metric.log(base_point, point)
result = self.space.is_tangent(log, point)
self.assertTrue(gs.all(result))
exp = self.space.ambient_metric.exp(log, point)
expected = gs.stack([base_point] * self.n_samples)
self.assertAllClose(exp, expected)
def test_kendall_inner_product_shape(self):
vector = gs.random.rand(self.n_samples, self.k_landmarks,
self.m_ambient)
point = self.space.random_point()
tan = self.space.to_tangent(vector, point)
inner = self.shape_metric.inner_product(tan, tan, point)
self.assertAllClose(inner.shape, (self.n_samples, ))
def test_kendall_log_and_exp(self):
point, base_point = self.space.random_point(2)
expected = self.space.align(point, base_point)
log = self.shape_metric.log(expected, base_point)
result = self.space.is_horizontal(log, base_point)
self.assertTrue(result)
exp = self.shape_metric.exp(log, base_point)
self.assertAllClose(exp, expected)
def test_kendall_exp_and_log(self):
base_point = self.space.random_point()
vector = gs.random.rand(self.k_landmarks, self.m_ambient)
tangent_vec = self.space.to_tangent(vector, base_point)
point = self.shape_metric.exp(tangent_vec, base_point)
log = self.shape_metric.log(point, base_point)
result = self.space.is_tangent(log, base_point)
self.assertTrue(result)
expected = self.space.horizontal_projection(tangent_vec, base_point)
self.assertAllClose(expected, log, rtol=1e-3)
def test_dist_extreme_case(self):
point = self.space.projection(gs.eye(self.k_landmarks, self.m_ambient))
result = self.shape_metric.dist(point, point)
expected = 0.0
self.assertAllClose(result, expected)
def test_dist(self):
point, base_point = self.space.random_point(2)
result = self.shape_metric.dist(point, base_point)
log = self.shape_metric.log(point, base_point)
expected = self.shape_metric.norm(log, base_point)
self.assertAllClose(result, expected)
def test_dist_vectorization(self):
point = self.space.random_point(self.n_samples)
base_point = self.space.random_point(self.n_samples)
aligned = self.space.align(point, base_point)
result = self.shape_metric.dist(aligned, base_point)
log = self.shape_metric.log(aligned, base_point)
expected = self.shape_metric.norm(log, base_point)
self.assertAllClose(result, expected)
def test_curvature_is_skew_operator(self):
"""Pre-shape space curvature tensor is skew in the first two arguments.
:math:`R(X,X)Y = 0`.
"""
space = self.space
base_point = space.random_point(2)
vector = gs.random.rand(4, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[:2], base_point)
tangent_vec_b = space.to_tangent(vector[2:], base_point)
result = self.shape_metric.curvature(tangent_vec_a, tangent_vec_a,
tangent_vec_b, base_point)
expected = gs.zeros_like(result)
self.assertAllClose(result, expected)
def test_curvature_bianchi_identity(self):
"""First Bianchi identity on curvature in pre-shape space.
:math:`R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0`.
"""
space = self.space
base_point = space.random_point()
vector = gs.random.rand(3, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[0], base_point)
tangent_vec_b = space.to_tangent(vector[1], base_point)
tangent_vec_c = space.to_tangent(vector[2], base_point)
curvature_1 = self.shape_metric.curvature(tangent_vec_a, tangent_vec_b,
tangent_vec_c, base_point)
curvature_2 = self.shape_metric.curvature(tangent_vec_b, tangent_vec_c,
tangent_vec_a, base_point)
curvature_3 = self.shape_metric.curvature(tangent_vec_c, tangent_vec_a,
tangent_vec_b, base_point)
result = curvature_1 + curvature_2 + curvature_3
expected = gs.zeros_like(result)
self.assertAllClose(result, expected)
def test_integrability_tensor(self):
"""Identities of integrability tensor in kendall pre-shape space.
The integrability tensor A_X E is skew-symmetric with respect to the
pre-shape metric, :math:`< A_X E, F> + <E, A_X F> = 0`. By
polarization, this is equivalent to :math:`< A_X E, E> = 0`.
The integrability tensor is also alternating (:math:`A_X Y =
- A_Y X`) for horizontal vector fields :math:'X,Y', and it is
exchanging horizontal and vertical vector spaces.
"""
space = self.space
base_point = space.random_point()
vector = gs.random.rand(2, self.k_landmarks, self.m_ambient)
tangent_vec_a = space.to_tangent(vector[0], base_point)
tangent_vec_b = space.to_tangent(vector[1], base_point)
result_ab = space.integrability_tensor(tangent_vec_a, tangent_vec_b,
base_point)
result = space.ambient_metric.inner_product(tangent_vec_b, result_ab,
base_point)
expected = 0.0
self.assertAllClose(result, expected)
horizontal_b = space.horizontal_projection(tangent_vec_b, base_point)
horizontal_a = space.horizontal_projection(tangent_vec_a, base_point)
result = space.integrability_tensor(horizontal_a, horizontal_b,
base_point)
expected = -space.integrability_tensor(horizontal_b, horizontal_a,
base_point)
self.assertAllClose(result, expected)
is_vertical = space.is_vertical(result, base_point)
self.assertTrue(is_vertical)
vertical_b = tangent_vec_b - horizontal_b
result = space.integrability_tensor(horizontal_a, vertical_b,
base_point)
is_horizontal = space.is_horizontal(result, base_point)
self.assertTrue(is_horizontal)
def test_integrability_tensor_old(self):
"""Test if old and new implementation give the same result."""
space = self.space
base_point = space.random_point()
vector = gs.random.rand(2, self.k_landmarks, self.m_ambient)
tangent_vec_x = space.to_tangent(vector[0], base_point)
tangent_vec_e = space.to_tangent(vector[1], base_point)
result = space.integrability_tensor_old(tangent_vec_x, tangent_vec_e,
base_point)
expected = space.integrability_tensor(tangent_vec_x, tangent_vec_e,
base_point)
self.assertAllClose(result, expected)
def test_kendall_sectional_curvature(self):
"""Sectional curvature of Kendall shape space is larger than 1.
The sectional curvature always increase by taking the quotient in a
Riemannian submersion. Thus, it should larger in kendall shape space
thane the sectional curvature of the pre-shape space which is 1 as it
a hypersphere.
The sectional curvature is computed here with the generic
directional_curvature and sectional curvature methods.
"""
space = self.space
metric = self.shape_metric
n_samples = 4 * self.k_landmarks * self.m_ambient
base_point = self.space.random_point(1)
vec_a = gs.random.rand(n_samples, self.k_landmarks, self.m_ambient)
tg_vec_a = space.to_tangent(space.center(vec_a), base_point)
hor_a = space.horizontal_projection(tg_vec_a, base_point)
vec_b = gs.random.rand(n_samples, self.k_landmarks, self.m_ambient)
tg_vec_b = space.to_tangent(space.center(vec_b), base_point)
hor_b = space.horizontal_projection(tg_vec_b, base_point)
tidal_force = metric.directional_curvature(hor_a, hor_b, base_point)
numerator = metric.inner_product(tidal_force, hor_a, base_point)
denominator = (metric.inner_product(hor_a, hor_a, base_point) *
metric.inner_product(hor_b, hor_b, base_point) -
metric.inner_product(hor_a, hor_b, base_point)**2)
condition = ~gs.isclose(denominator, 0.0)
kappa = numerator[condition] / denominator[condition]
kappa_direct = metric.sectional_curvature(hor_a, hor_b,
base_point)[condition]
self.assertAllClose(kappa, kappa_direct)
result = kappa > 1.0 - 1e-12
self.assertTrue(gs.all(result))
def test_integrability_tensor_derivative_is_alternate(self):
r"""Integrability tensor derivatives is alternate in pre-shape.
For two horizontal vector fields :math:`X,Y` the integrability
tensor (hence its derivatives) is alternate:
:math:`\nabla_X ( A_Y Z + A_Z Y ) = 0`.
"""
nabla_x_a_y_z, a_y_z = self.space.integrability_tensor_derivative(
self.hor_x,
self.hor_y,
self.nabla_x_y,
self.hor_z,
self.nabla_x_z,
self.base_point,
)
nabla_x_a_z_y, a_z_y = self.space.integrability_tensor_derivative(
self.hor_x,
self.hor_z,
self.nabla_x_z,
self.hor_y,
self.nabla_x_y,
self.base_point,
)
result = nabla_x_a_y_z + nabla_x_a_z_y
self.assertAllClose(a_y_z + a_z_y, gs.zeros_like(result))
self.assertAllClose(result, gs.zeros_like(result))
def test_integrability_tensor_derivative_is_skew_symmetric(self):
r"""Integrability tensor derivatives is skew-symmetric in pre-shape.
For :math:`X,Y` horizontal and :math:`V,W` vertical:
:math:`\nabla_X (< A_Y Z , V > + < A_Y V , Z >) = 0`.
"""
scal = self.space.ambient_metric.inner_product
nabla_x_a_y_z, a_y_z = self.space.integrability_tensor_derivative(
self.hor_x,
self.hor_y,
self.nabla_x_y,
self.hor_z,
self.nabla_x_z,
self.base_point,
)
nabla_x_a_y_v, a_y_v = self.space.integrability_tensor_derivative(
self.hor_x,
self.hor_y,
self.nabla_x_y,
self.ver_v,
self.nabla_x_v,
self.base_point,
)
result = (scal(nabla_x_a_y_z, self.ver_v) +
scal(a_y_z, self.nabla_x_v) +
scal(nabla_x_a_y_v, self.hor_z) +
scal(a_y_v, self.nabla_x_z))
self.assertAllClose(result, gs.zeros_like(result))
def test_integrability_tensor_derivative_reverses_hor_ver(self):
r"""Integrability tensor derivatives exchanges hor & ver in pre-shape.
For :math:`X,Y,Z` horizontal and :math:`V,W` vertical, the
integrability tensor (and thus its derivative) reverses horizontal
and vertical subspaces: :math:`\nabla_X < A_Y Z, H > = 0` and
:math:`nabla_X < A_Y V, W > = 0`.
"""
scal = self.space.ambient_metric.inner_product
nabla_x_a_y_z, a_y_z = self.space.integrability_tensor_derivative(
self.hor_x,
self.hor_y,
self.nabla_x_y,
self.hor_z,
self.nabla_x_z,
self.base_point,
)
result = scal(nabla_x_a_y_z, self.hor_h) + scal(a_y_z, self.nabla_x_h)
self.assertAllClose(result, gs.zeros_like(result))
nabla_x_a_y_v, a_y_v = self.space.integrability_tensor_derivative(
self.hor_x,
self.hor_y,
self.nabla_x_y,
self.ver_v,
self.nabla_x_v,
self.base_point,
)
result = scal(nabla_x_a_y_v, self.ver_w) + scal(a_y_v, self.nabla_x_w)
self.assertAllClose(result, gs.zeros_like(result))
def test_integrability_tensor_derivative_parallel(self):
"""Test optimized integrability tensor derivatives in pre-shape space.
Optimized version for quotient-parallel vector fields should equal
the general implementation.
"""
(
nabla_x_a_y_z_qp,
a_y_z_qp,
) = self.space.integrability_tensor_derivative_parallel(
self.hor_x, self.hor_y, self.hor_z, self.base_point)
a_x_y = self.space.integrability_tensor(self.hor_x, self.hor_y,
self.base_point)
a_x_z = self.space.integrability_tensor(self.hor_x, self.hor_z,
self.base_point)
nabla_x_a_y_z, a_y_z = self.space.integrability_tensor_derivative(
self.hor_x, self.hor_y, a_x_y, self.hor_z, a_x_z, self.base_point)
self.assertAllClose(a_y_z, a_y_z_qp)
self.assertAllClose(nabla_x_a_y_z, nabla_x_a_y_z_qp)
def test_iterated_integrability_tensor_derivative_parallel(self):
"""Test optimized iterated integrability tensor derivatives.
The optimized version of the iterated integrability tensor
:math:`A_X A_Y A_X Y`, computed with the horizontal lift of
quotient-parallel vector fields extending the tangent vectors
:math:`X,Y` of Kendall shape spaces (identified to horizontal vectors
of the pre-shape space), is the recursive application of two general
integrability tensor derivatives with proper derivatives.
Intermediate computations returned are also verified.
"""
a_x_y = self.space.integrability_tensor(self.hor_x, self.hor_y,
self.base_point)
nabla_x_v, a_x_y = self.space.integrability_tensor_derivative(
self.hor_x,
self.hor_x,
gs.zeros_like(self.hor_x),
self.hor_y,
a_x_y,
self.base_point,
)
(
nabla_x_a_y_a_x_y,
a_y_a_x_y,
) = self.space.integrability_tensor_derivative(self.hor_x, self.hor_y,
a_x_y, a_x_y, nabla_x_v,
self.base_point)
a_x_a_y_a_x_y = self.space.integrability_tensor(
self.hor_x, a_y_a_x_y, self.base_point)
(
nabla_x_a_y_a_x_y_qp,
a_x_a_y_a_x_y_qp,
nabla_x_v_qp,
a_y_a_x_y_qp,
ver_v_qp,
) = self.space.iterated_integrability_tensor_derivative_parallel(
self.hor_x, self.hor_y, self.base_point)
self.assertAllClose(a_x_y, ver_v_qp)
self.assertAllClose(a_y_a_x_y, a_y_a_x_y_qp)
self.assertAllClose(nabla_x_v, nabla_x_v_qp)
self.assertAllClose(a_x_a_y_a_x_y, a_x_a_y_a_x_y_qp)
self.assertAllClose(nabla_x_a_y_a_x_y, nabla_x_a_y_a_x_y_qp)
def test_kendall_curvature_derivative_bianchi_identity(self):
r"""2nd Bianchi identity on curvature derivative in kendall space.
For any 3 tangent vectors horizontally lifted from kendall shape
space to Kendall pre-shape space, :math:`(\nabla_X R)(Y, Z)
+ (\nabla_Y R)(Z,X) + (\nabla_Z R)(X, Y) = 0`.
"""
term_x = self.shape_metric.curvature_derivative(
self.hor_x, self.hor_y, self.hor_z, self.hor_h, self.base_point)
term_y = self.shape_metric.curvature_derivative(
self.hor_y, self.hor_z, self.hor_x, self.hor_h, self.base_point)
term_z = self.shape_metric.curvature_derivative(
self.hor_z, self.hor_x, self.hor_y, self.hor_h, self.base_point)
result = term_x + term_y + term_z
self.assertAllClose(result, gs.zeros_like(result))
def test_curvature_derivative_is_skew_operator(self):
r"""Derivative of a skew operator is skew.
For any 3 tangent vectors horizontally lifted from kendall shape space
to Kendall pre-shape space, :math:`(\nabla_X R)(Y,Y)Z = 0`.
"""
result = self.shape_metric.curvature_derivative(
self.hor_x, self.hor_y, self.hor_y, self.hor_z, self.base_point)
self.assertAllClose(result, gs.zeros_like(result))
def test_directional_curvature_derivative(self):
"""Test equality of directional curvature derivative implementations.
General formula based on curvature derivative, optimized method of
KendallShapeMetric class, method from the QuotientMetric class and
method from the Connection class have to give identical results.
"""
metric = self.shape_metric
# General formula based on curvature derivative
expected = metric.curvature_derivative(self.hor_x, self.hor_y,
self.hor_x, self.hor_y,
self.base_point)
# Optimized method of KendallShapeMetric class
result_kendall_shape_metric = metric.directional_curvature_derivative(
self.hor_x, self.hor_y, self.base_point)
self.assertAllClose(result_kendall_shape_metric, expected)
# Method from the QuotientMetric class
result_quotient_metric = super(
KendallShapeMetric,
metric).directional_curvature_derivative(self.hor_x, self.hor_y,
self.base_point)
self.assertAllClose(result_quotient_metric, expected)
# Method from the Connection class
from geomstats.geometry.quotient_metric import QuotientMetric
result_connection = super(QuotientMetric,
metric).directional_curvature_derivative(
self.hor_x, self.hor_y, self.base_point)
self.assertAllClose(result_connection, expected)
def test_directional_curvature_derivative_is_quadratic(self):
"""Directional curvature derivative is quadratic in both variables."""
coef_x = -2.5
coef_y = 1.5
result = self.shape_metric.directional_curvature_derivative(
coef_x * self.hor_x, coef_y * self.hor_y, self.base_point)
expected = (coef_x**2 * coef_y**2 *
self.shape_metric.directional_curvature_derivative(
self.hor_x, self.hor_y, self.base_point))
self.assertAllClose(result, expected)
@geomstats.tests.np_autograd_and_torch_only
def test_parallel_transport(self):
space = self.space
metric = self.shape_metric
shape = (self.n_samples, self.k_landmarks, self.m_ambient)
point = space.projection(gs.eye(4)[:, :3])
tan_b = gs.random.rand(*shape)
tan_b = space.to_tangent(tan_b, point)
tan_b = space.horizontal_projection(tan_b, point)
# use a vector orthonormal to tan_b
tan_a = gs.random.rand(*shape)
tan_a = space.to_tangent(tan_a, point)
tan_a = space.horizontal_projection(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,
tan_b,
n_steps=150,
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)
is_horizontal = space.is_horizontal(transported, end_point)
self.assertTrue(gs.all(is_tangent))
self.assertTrue(gs.all(is_horizontal))
transported = metric.parallel_transport(tan_a[0],
point,
end_point=end_point[0])
result = metric.norm(transported, end_point[0])
self.assertAllClose(result, expected[0])