def inner_product(self, tangent_vec_a, tangent_vec_b, base_point):
"""Compute the Log-Euclidean inner-product.
Compute the inner-product of tangent_vec_a and tangent_vec_b
at point base_point using the log-Euclidean metric.
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
Tangent vector at base point.
tangent_vec_b : array-like, shape=[..., n, n]
Tangent vector at base point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
inner_product : array-like, shape=[...,]
Inner-product.
"""
spd_space = self.space
modified_tangent_vec_a = spd_space.differential_log(
tangent_vec_a, base_point)
modified_tangent_vec_b = spd_space.differential_log(
tangent_vec_b, base_point)
product = Matrices.trace_product(
modified_tangent_vec_a, modified_tangent_vec_b)
return product
def inner_product(self, tangent_vec_a, tangent_vec_b, base_point):
r"""Compute the inner-product of two tangent vectors at a base point.
Canonical inner-product on the tangent space at `base_point`,
which is different from the inner-product induced by the embedding
(see [RLSMRZ2017]_).
.. math::
\langle\Delta, \tilde{\Delta}\rangle_{U}=\operatorname{tr}
\left(\Delta^{T}\left(I-\frac{1}{2} U U^{T}\right)
\tilde{\Delta}\right)
References
----------
.. [RLSMRZ2017] R Zimmermann. A matrix-algebraic algorithm for the
Riemannian logarithm on the Stiefel manifold under the canonical
metric. SIAM Journal on Matrix Analysis and Applications 38 (2),
322-342, 2017. https://epubs.siam.org/doi/pdf/10.1137/16M1074485
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, p]
First tangent vector at base point.
tangent_vec_b : array-like, shape=[..., n, p]
Second tangent vector at base point.
base_point : array-like, shape=[..., n, p]
Point in the Stiefel manifold.
Returns
-------
inner_prod : array-like, shape=[..., 1]
Inner-product of the two tangent vectors.
"""
base_point_transpose = Matrices.transpose(base_point)
aux = gs.matmul(
Matrices.transpose(tangent_vec_a),
gs.eye(self.n) - 0.5 * gs.matmul(base_point, base_point_transpose),
)
inner_prod = Matrices.trace_product(aux, tangent_vec_b)
return inner_prod
def _aux_inner_product(tangent_vec_a, tangent_vec_b, inv_base_point):
"""Compute the inner-product (auxiliary).
Parameters
----------
tangent_vec_a : array-like, shape=[..., n, n]
tangent_vec_b : array-like, shape=[..., n, n]
inv_base_point : array-like, shape=[..., n, n]
Returns
-------
inner_product : array-like, shape=[...]
"""
aux_a = Matrices.mul(inv_base_point, tangent_vec_a)
aux_b = Matrices.mul(inv_base_point, tangent_vec_b)
# Use product instead of matrix product and trace to save time
inner_product = Matrices.trace_product(aux_a, aux_b)
return inner_product
def test_trace_product(self, mat_a, mat_b, expected):
self.assertAllClose(
Matrices.trace_product(gs.array(mat_a), gs.array(mat_b)),
gs.array(expected))
