def test_srv_inner_product_elastic(self, dim, n_sampling_points, curve_a):
"""Test inner product of SRVMetric.
Check that the pullback metric gives an elastic metric
with parameters a=1, b=1/2.
"""
tangent_vec_a = gs.random.rand(n_sampling_points, dim)
tangent_vec_b = gs.random.rand(n_sampling_points, dim)
r3 = Euclidean(dim)
srv_metric_r3 = SRVMetric(r3)
result = srv_metric_r3.inner_product(tangent_vec_a, tangent_vec_b, curve_a)
d_vec_a = (n_sampling_points - 1) * (
tangent_vec_a[1:, :] - tangent_vec_a[:-1, :]
)
d_vec_b = (n_sampling_points - 1) * (
tangent_vec_b[1:, :] - tangent_vec_b[:-1, :]
)
velocity_vec = (n_sampling_points - 1) * (curve_a[1:, :] - curve_a[:-1, :])
velocity_norm = r3.metric.norm(velocity_vec)
unit_velocity_vec = gs.einsum("ij,i->ij", velocity_vec, 1 / velocity_norm)
a_param = 1
b_param = 1 / 2
integrand = (
a_param**2 * gs.sum(d_vec_a * d_vec_b, axis=1)
- (a_param**2 - b_param**2)
* gs.sum(d_vec_a * unit_velocity_vec, axis=1)
* gs.sum(d_vec_b * unit_velocity_vec, axis=1)
) / velocity_norm
expected = gs.sum(integrand) / n_sampling_points
self.assertAllClose(result, expected)
def test_split_horizontal_vertical(
self, times, n_discretized_curves, curve_a, curve_b
):
"""Test split horizontal vertical.
Check that horizontal and vertical parts of any tangent
vector are othogonal with respect to the SRVMetric inner
product, and check vectorization.
"""
srv_metric_r3 = SRVMetric(r3)
quotient_srv_metric_r3 = DiscreteCurves(ambient_manifold=r3).quotient_srv_metric
geod = srv_metric_r3.geodesic(initial_curve=curve_a, end_curve=curve_b)
geod = geod(times)
tangent_vec = n_discretized_curves * (geod[1, :, :] - geod[0, :, :])
(
tangent_vec_hor,
tangent_vec_ver,
_,
) = quotient_srv_metric_r3.split_horizontal_vertical(tangent_vec, curve_a)
result = srv_metric_r3.inner_product(tangent_vec_hor, tangent_vec_ver, curve_a)
expected = 0.0
self.assertAllClose(result, expected, atol=1e-4)
tangent_vecs = n_discretized_curves * (geod[1:] - geod[:-1])
_, _, result = quotient_srv_metric_r3.split_horizontal_vertical(
tangent_vecs, geod[:-1]
)
expected = []
for i in range(n_discretized_curves - 1):
_, _, res = quotient_srv_metric_r3.split_horizontal_vertical(
tangent_vecs[i], geod[i]
)
expected.append(res)
expected = gs.stack(expected)
self.assertAllClose(result, expected)
def test_srv_inner_product_vectorization(self, dim, n_sampling_points,
curve_a, curve_b):
"""Test inner product of SRVMetric.
Check vectorization.
"""
curves = gs.stack((curve_a, curve_b))
tangent_vecs_1 = gs.random.rand(2, n_sampling_points, dim)
tangent_vecs_2 = gs.random.rand(2, n_sampling_points, dim)
srv_metric_r3 = SRVMetric(r3)
result = srv_metric_r3.inner_product(tangent_vecs_1, tangent_vecs_2,
curves)
res_a = srv_metric_r3.inner_product(tangent_vecs_1[0],
tangent_vecs_2[0], curve_a)
res_b = srv_metric_r3.inner_product(tangent_vecs_1[1],
tangent_vecs_2[1], curve_b)
expected = gs.stack((res_a, res_b))
self.assertAllClose(result, expected)
