def log(self, point, base_point, **kwargs):
"""Compute the Cholesky logarithm map.
Compute the Riemannian logarithm at point base_point,
of point wrt the Cholesky metric.
This gives a tangent vector at point base_point.
Parameters
----------
point : array-like, shape=[..., n, n]
Point.
base_point : array-like, shape=[..., n, n]
Base point.
Returns
-------
log : array-like, shape=[..., n, n]
Riemannian logarithm.
"""
sl_base_point = Matrices.to_strictly_lower_triangular(base_point)
sl_point = Matrices.to_strictly_lower_triangular(point)
diag_base_point = Matrices.diagonal(base_point)
diag_point = Matrices.diagonal(point)
diag_product_logm = gs.log(gs.divide(diag_point, diag_base_point))
sl_log = sl_point - sl_base_point
diag_log = gs.vec_to_diag(diag_base_point * diag_product_logm)
log = sl_log + diag_log
return log
def squared_dist(self, point_a, point_b, **kwargs):
"""Compute the Cholesky Metric squared distance.
Compute the Riemannian squared distance between point_a and point_b.
Parameters
----------
point_a : array-like, shape=[..., n, n]
Point.
point_b : array-like, shape=[..., n, n]
Point.
Returns
-------
_ : array-like, shape=[...]
Riemannian squared distance.
"""
log_diag_a = gs.log(Matrices.diagonal(point_a))
log_diag_b = gs.log(Matrices.diagonal(point_b))
diag_diff = log_diag_a - log_diag_b
squared_dist_diag = gs.sum((diag_diff) ** 2, axis=-1)
sl_a = Matrices.to_strictly_lower_triangular(point_a)
sl_b = Matrices.to_strictly_lower_triangular(point_b)
sl_diff = sl_a - sl_b
squared_dist_sl = Matrices.frobenius_product(sl_diff, sl_diff)
return squared_dist_sl + squared_dist_diag
def exp(self, tangent_vec, base_point, **kwargs):
"""Compute the Cholesky exponential map.
Compute the Riemannian exponential at point base_point
of tangent vector tangent_vec wrt the Cholesky metric.
This gives a lower triangular matrix with positive elements.
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.
"""
sl_base_point = Matrices.to_strictly_lower_triangular(base_point)
sl_tangent_vec = Matrices.to_strictly_lower_triangular(tangent_vec)
diag_base_point = Matrices.diagonal(base_point)
diag_tangent_vec = Matrices.diagonal(tangent_vec)
diag_product_expm = gs.exp(gs.divide(diag_tangent_vec, diag_base_point))
sl_exp = sl_base_point + sl_tangent_vec
diag_exp = gs.vec_to_diag(diag_base_point * diag_product_expm)
exp = sl_exp + diag_exp
return exp
