def exp(self, tangent_vec, base_point, **kwargs):
"""Compute the Log-Euclidean exponential map.
Compute the Riemannian exponential at point base_point
of tangent vector tangent_vec wrt the Log-Euclidean metric.
This gives a symmetric positive definite matrix.
Parameters
----------
tangent_vec : array-like, shape=[..., n, n]
Tangent vector at base point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
exp : array-like, shape=[..., n, n]
Riemannian exponential.
"""
log_base_point = SPDMatrices.logm(base_point)
dlog_tangent_vec = SPDMatrices.differential_log(
tangent_vec, base_point)
exp = SymmetricMatrices.expm(log_base_point + dlog_tangent_vec)
return exp
def test_expm(self):
"""Test of expm method."""
sym_n = SymmetricMatrices(self.n)
v = gs.array([[0.0, 1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 0.0, 1.0]])
result = sym_n.expm(v)
c = math.cosh(1)
s = math.sinh(1)
e = math.exp(1)
expected = gs.array([[c, s, 0.0], [s, c, 0.0], [0.0, 0.0, e]])
four_dim_v = gs.broadcast_to(v, (2, 2) + v.shape)
four_dim_expected = gs.broadcast_to(expected, (2, 2) + expected.shape)
four_dim_result = sym_n.expm(four_dim_v)
self.assertAllClose(result, expected)
self.assertAllClose(four_dim_result, four_dim_expected)
def test_expm(self):
"""Test of expm method."""
sym_n = SymmetricMatrices(self.n)
v = gs.array([[0., 1., 0.], [1., 0., 0.], [0., 0., 1.]])
result = sym_n.expm(v)
c = math.cosh(1)
s = math.sinh(1)
e = math.exp(1)
expected = gs.array([[c, s, 0.], [s, c, 0.], [0., 0., e]])
self.assertAllClose(result, expected)
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 = Matrices.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 exp(self, tangent_vec, base_point):
"""Compute the Log-Euclidean exponential map.
Compute the Riemannian exponential at point base_point
of tangent vector tangent_vec wrt the Log-Euclidean metric.
This gives a symmetric positive definite matrix.
Parameters
----------
tangent_vec : array-like, shape=[n_samples, n, n]
base_point : array-like, shape=[n_samples, n, n]
Returns
-------
exp : array-like, shape=[n_samples, n, n]
"""
log_base_point = self.space.logm(base_point)
dlog_tangent_vec = self.space.differential_log(tangent_vec, base_point)
exp = SymmetricMatrices.expm(log_base_point + dlog_tangent_vec)
return exp
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 = 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 test_expm(self, mat, expected):
result = SymmetricMatrices.expm(gs.array(mat))
self.assertAllClose(result, gs.array(expected))
