def test_horizontal_lift_and_tangent_submersion(self):
mat = self.bundle.total_space.random_uniform()
tangent_vec = GeneralLinear.to_symmetric(
self.bundle.total_space.random_uniform())
horizontal = self.bundle.horizontal_lift(tangent_vec, mat)
result = self.bundle.tangent_submersion(horizontal, mat)
self.assertAllClose(result, tangent_vec)
def test_is_horizontal(self):
mat = self.bundle.total_space.random_uniform()
tangent_vec = GeneralLinear.to_symmetric(
self.bundle.total_space.random_uniform())
horizontal = self.bundle.horizontal_lift(tangent_vec, mat)
result = self.bundle.is_horizontal(horizontal, mat)
self.assertTrue(result)
def test_exp(self):
mat = self.bundle.total_space.random_uniform()
point = self.bundle.submersion(mat)
tangent_vec = GeneralLinear.to_symmetric(
self.bundle.total_space.random_uniform()) / 5
result = self.quotient_metric.exp(tangent_vec, point)
expected = self.base_metric.exp(tangent_vec, point)
self.assertAllClose(result, expected)
def test_integrability_tensor(self):
mat = self.bundle.total_space.random_point()
point = self.bundle.submersion(mat)
tangent_vec = GeneralLinear.to_symmetric(
self.bundle.total_space.random_point()) / 5
self.assertRaises(
NotImplementedError, lambda: self.bundle.integrability_tensor(
tangent_vec, tangent_vec, point))
def test_inner_product(self):
mat = self.bundle.total_space.random_point()
point = self.bundle.submersion(mat)
tangent_vecs = GeneralLinear.to_symmetric(
self.bundle.total_space.random_point(2)) / 10
result = self.quotient_metric.inner_product(
tangent_vecs[0], tangent_vecs[1], point=mat)
expected = self.base_metric.inner_product(
tangent_vecs[0], tangent_vecs[1], point)
self.assertAllClose(result, expected)
def _aux_log(point, sqrt_base_point, inv_sqrt_base_point):
"""Compute the log (auxiliary function).
Parameters
----------
point : array-like, shape=[..., n, n]
sqrt_base_point : array-like, shape=[..., n, n]
inv_sqrt_base_point : array-like, shape=[.., n, n]
Returns
-------
log : array-like, shape=[..., n, n]
"""
point_near_id = Matrices.mul(
inv_sqrt_base_point, point, inv_sqrt_base_point)
point_near_id = GeneralLinear.to_symmetric(point_near_id)
log_at_id = SPDMatrices.logm(point_near_id)
log = Matrices.mul(sqrt_base_point, log_at_id, sqrt_base_point)
return log
def _aux_exp(tangent_vec, sqrt_base_point, inv_sqrt_base_point):
"""Compute the exponential map (auxiliary function).
Parameters
----------
tangent_vec : array-like, shape=[..., n, n]
sqrt_base_point : array-like, shape=[..., n, n]
inv_sqrt_base_point : array-like, shape=[..., n, n]
Returns
-------
exp : array-like, shape=[..., n, n]
"""
tangent_vec_at_id = Matrices.mul(inv_sqrt_base_point, tangent_vec,
inv_sqrt_base_point)
tangent_vec_at_id = GeneralLinear.to_symmetric(tangent_vec_at_id)
exp_from_id = SymmetricMatrices.expm(tangent_vec_at_id)
exp = Matrices.mul(sqrt_base_point, exp_from_id, sqrt_base_point)
return exp
def _aux_log(point, sqrt_base_point, inv_sqrt_base_point):
"""Compute the log (auxiliary function).
Parameters
----------
point
sqrt_base_point
inv_sqrt_base_point
Returns
-------
log
"""
point_near_id = gs.einsum('...ij,...jk->...ik', inv_sqrt_base_point,
point)
point_near_id = gs.einsum('...ij,...jk->...ik', point_near_id,
inv_sqrt_base_point)
point_near_id = GeneralLinear.to_symmetric(point_near_id)
log_at_id = SPDMatrices.logm(point_near_id)
log = gs.einsum('...ij,...jk->...ik', sqrt_base_point, log_at_id)
log = gs.einsum('...ij,...jk->...ik', log, sqrt_base_point)
return log
def _aux_exp(tangent_vec, sqrt_base_point, inv_sqrt_base_point):
"""Compute the exponential map (auxiliary function).
Parameters
----------
tangent_vec : array-like, shape=[..., n, n]
sqrt_base_point
inv_sqrt_base_point
Returns
-------
exp
"""
tangent_vec_at_id = gs.einsum('...ij,...jk->...ik',
inv_sqrt_base_point, tangent_vec)
tangent_vec_at_id = gs.einsum('...ij,...jk->...ik', tangent_vec_at_id,
inv_sqrt_base_point)
tangent_vec_at_id = GeneralLinear.to_symmetric(tangent_vec_at_id)
exp_from_id = SymmetricMatrices.expm(tangent_vec_at_id)
exp = gs.einsum('...ij,...jk->...ik', exp_from_id, sqrt_base_point)
exp = gs.einsum('...ij,...jk->...ik', sqrt_base_point, exp)
return exp
def tangent_submersion(tangent_vec, base_point):
product = GeneralLinear.mul(base_point,
GeneralLinear.transpose(tangent_vec))
return 2 * GeneralLinear.to_symmetric(product)