def summand(k):
exp_arg = -((dim - 1 - 2 * k)**2) / 2 / variances
erf_arg_2 = (gs.pi * variances -
(dim - 1 - 2 * k) * 1j) / gs.sqrt(2 * variances)
sign = (-1.0)**k
comb_2 = gs.comb(k, dim - 1)
return sign * comb_2 * gs.exp(exp_arg) * gs.real(gs.erf(erf_arg_2))
def group_log_from_identity(self, point, point_type=None):
"""
Group logarithm of the Lie group of
all invertible matrices at the identity.
"""
point = gs.to_ndarray(point, to_ndim=3)
group_log = gs.linalg.logm(point)
return gs.real(group_log)
def group_exp_from_identity(self, tangent_vec, point_type=None):
"""
Group exponential of the Lie group of
all invertible matrices at the identity.
"""
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=3)
group_exp = gs.linalg.expm(tangent_vec)
return gs.real(group_exp)
def group_exp_from_identity(self, tangent_vec, point_type=None):
"""Compute group exponential at the identity.
Group exponential of the Lie group of
all invertible matrices at the identity.
Parameters
----------
tangent_vec
point_type
Returns
-------
group_exp
"""
tangent_vec = gs.to_ndarray(tangent_vec, to_ndim=3)
group_exp = gs.linalg.expm(tangent_vec)
return gs.real(group_exp)
def group_log_from_identity(self, point, point_type=None):
"""Compute group logarithm at the identity.
Group logarithm of the Lie group of
all invertible matrices at the identity.
Parameters
----------
point
point_type
Returns
-------
group_log
"""
point = gs.to_ndarray(point, to_ndim=3)
group_log = gs.linalg.logm(point)
return gs.real(group_log)