def rhess(cost, proj):
"""
Generates the Riemannian hessian of the cost. Specifically, rhess(cost,
proj)(x, u) is the directional derivatative of cost at point X on the
manifold, in direction u.
cost and proj must be defined using autograd.numpy.
See http://sites.uclouvain.be/absil/2013-01/Weingarten_07PA_techrep.pdf
for some discussion.
proj and cost must be defined using autograd.
Currently this is correct but not efficient, because of the jacobian-
vector product. Hopefully this can be fixed in future.
"""
return lambda x, u: proj(x, np.tensordot(jacobian(rgrad(cost, proj))(x), u,
axes=u.ndim))
def rhess(cost, proj):
"""
Generates the Riemannian hessian of the cost. Specifically, rhess(cost,
proj)(x, u) is the directional derivatative of cost at point X on the
manifold, in direction u.
cost and proj must be defined using autograd.numpy.
See http://sites.uclouvain.be/absil/2013-01/Weingarten_07PA_techrep.pdf
for some discussion.
proj and cost must be defined using autograd.
Currently this is correct but not efficient, because of the jacobian-
vector product. Hopefully this can be fixed in future.
"""
return lambda x, u: proj(
x, np.tensordot(jacobian(rgrad(cost, proj))(x), u, axes=u.ndim))
def ehess2rhess(proj):
"""
Generates an ehess2rhess function for a manifold which is a sub-manifold
of Euclidean space.
ehess2rhess(proj)(x, egrad, ehess, u) converts the Euclidean hessian ehess
at the point x to a Riemannian hessian. That is the directional
derivatative of the gradient in the direction u.
proj must be defined using autograd.numpy.
This will not be an efficient implementation because of missing support
for efficient jacobian-vector products in autograd.
"""
# Differentiate proj w.r.t. the first argument
d_proj = jacobian(proj)
return lambda x, egrad, ehess, u: proj(
x, ehess + np.tensordot(d_proj(x, egrad), u, axes=u.ndim))
def ehess2rhess(proj):
"""
Generates an ehess2rhess function for a manifold which is a sub-manifold
of Euclidean space.
ehess2rhess(proj)(x, egrad, ehess, u) converts the Euclidean hessian ehess
at the point x to a Riemannian hessian. That is the directional
derivatative of the gradient in the direction u.
proj must be defined using autograd.numpy.
This will not be an efficient implementation because of missing support
for efficient jacobian-vector products in autograd.
"""
# Differentiate proj w.r.t. the first argument
d_proj = jacobian(proj)
return lambda x, egrad, ehess, u: proj(x, ehess +
np.tensordot(d_proj(x, egrad), u,
axes=u.ndim))
