def test_integrated_exp_at_id(self):
group = self.matrix_so3
metric = InvariantMetric(group=group)
basis = metric.normal_basis(group.lie_algebra.basis)
vector = gs.random.rand(2, len(basis))
tangent_vec = gs.einsum("...j,jkl->...kl", vector, basis)
identity = self.matrix_so3.identity
result = metric.exp(tangent_vec, identity, n_steps=100, step="rk4")
expected = group.exp(tangent_vec, identity)
self.assertAllClose(expected, result, atol=1e-5)
result = metric.exp(tangent_vec, identity, n_steps=100, step="rk2")
self.assertAllClose(expected, result, atol=1e-5)
def test_integrated_se3_exp_at_id(self, group):
lie_algebra = group.lie_algebra
metric = InvariantMetric(group=group)
canonical_metric = group.left_canonical_metric
basis = metric.normal_basis(lie_algebra.basis)
vector = gs.random.rand(len(basis))
tangent_vec = gs.einsum("...j,jkl->...kl", vector, basis)
identity = group.identity
result = metric.exp(tangent_vec, identity, n_steps=100, step="rk4")
expected = canonical_metric.exp(tangent_vec, identity)
self.assertAllClose(expected, result, atol=1e-4)
result = metric.exp(tangent_vec, identity, n_steps=100, step="rk2")
self.assertAllClose(expected, result, atol=1e-4)
def test_integrated_parallel_transport(self, group, n, n_samples):
metric = InvariantMetric(group=group)
point = group.identity
tan_b = Matrices(n + 1, n + 1).random_point(n_samples)
tan_b = group.to_tangent(tan_b)
# use a vector orthonormal to tan_b
tan_a = Matrices(n + 1, n + 1).random_point(n_samples)
tan_a = group.to_tangent(tan_a)
coef = metric.inner_product(tan_a, tan_b) / metric.squared_norm(tan_b)
tan_a -= gs.einsum("...,...ij->...ij", coef, tan_b)
tan_b = gs.einsum("...ij,...->...ij", tan_b,
1.0 / metric.norm(tan_b, base_point=point))
tan_a = gs.einsum("...ij,...->...ij", tan_a,
1.0 / metric.norm(tan_a, base_point=point))
expected = group.left_canonical_metric.parallel_transport(
tan_a, point, tan_b)
result, end_point_result = metric.parallel_transport(
tan_a, point, tan_b, n_steps=20, step="rk4", return_endpoint=True)
expected_end_point = metric.exp(tan_b, point, n_steps=20)
self.assertAllClose(end_point_result,
expected_end_point,
atol=gs.atol * 1000)
self.assertAllClose(expected, result, atol=gs.atol * 1000)
def test_integrated_se3_exp_at_id(self):
group = self.matrix_se3
lie_algebra = group.lie_algebra
metric = InvariantMetric(group=group)
canonical_metric = group.left_canonical_metric
basis = metric.orthonormal_basis(lie_algebra.basis)
vector = gs.random.rand(len(basis))
tangent_vec = gs.einsum('...j,jkl->...kl', vector, basis)
identity = self.matrix_se3.identity
result = metric.exp(tangent_vec, identity, n_steps=100, step='rk4')
expected = canonical_metric.exp(tangent_vec, identity)
self.assertAllClose(expected, result, atol=1e-5)
result = metric.exp(tangent_vec, identity, n_steps=100, step='rk2')
self.assertAllClose(expected, result, atol=1e-5)
def test_integrated_se3_exp(self):
group = self.matrix_se3
lie_algebra = group.lie_algebra
metric = InvariantMetric(group=group)
canonical_metric = group.left_canonical_metric
basis = metric.normal_basis(lie_algebra.basis)
point = group.random_point()
vector = gs.random.rand(len(basis))
tangent_vec = gs.einsum("...j,jkl->...kl", vector, basis)
tangent_vec = group.tangent_translation_map(point)(tangent_vec)
result = metric.exp(tangent_vec, point, n_steps=100, step="rk4")
expected = canonical_metric.exp(tangent_vec, point)
self.assertAllClose(expected, result)
result = metric.exp(tangent_vec, point, n_steps=100, step="rk2")
self.assertAllClose(expected, result, atol=4e-5)
def test_integrated_exp_and_log_at_id(self):
group = self.matrix_so3
metric = InvariantMetric(group=group)
basis = group.lie_algebra.basis
vector = gs.random.rand(2, len(basis))
tangent_vec = gs.einsum("...j,jkl->...kl", vector, basis)
identity = self.matrix_so3.identity
exp = metric.exp(tangent_vec, identity, n_steps=100, step="rk4")
result = metric.log(exp, identity, n_steps=15, step="rk4", verbose=False)
self.assertAllClose(tangent_vec, result, atol=1e-5)