def calc_stiefel():
# Y is a matrix point
eta = matrices('eta')
Y = stiefels('Y')
a = sm.sym_symb('a')
al0, al1 = scalars('al0 al1')
# scalars are symmetric
sm.g_symms.update((al0, al1))
def J(Y, eta):
return mat_spfy(t(Y) * eta + t(eta) * Y).doit()
def J_adj(Y, a):
dY = symbols('dY', commutative=False)
return xtrace(trace(mat_spfy(J(Y, dY) * a)), dY)
def g(Y, eta):
return al0*eta+(al1-al0)*Y*t(Y)*eta
def g_inv(Y, eta):
return mat_spfy(1/al0*eta + (1/al1-1/al0)*Y*t(Y)*eta)
J_giv_J_adj = J(Y, g_inv(Y, J_adj(Y, a)))
print(J_giv_J_adj)
def proj(Y, omg):
jo = mat_spfy(J(Y, omg))
ifactor = al1/Integer(4)
return omg - mat_spfy(
g_inv(Y, mat_spfy(J_adj(Y, ifactor*jo))))
def r_gradient(Y, omg):
return mat_spfy(
proj(Y, mat_spfy(g_inv(Y, omg))))
print(r_gradient(Y, eta))
xi, phi = matrices('xi phi')
trilinear = mat_spfy(trace(DDR(g(Y, eta), Y, phi) * t(xi)))
xcross = xtrace(trilinear, phi)
K = (Integer(1)/Integer(2))*(DDR(g(Y, eta), Y, xi) +
DDR(g(Y, xi), Y, eta) - xcross)
def d_proj(Y, xi, omg):
e = matrices('e')
r = mat_spfy(proj(Y, e))
expr = DDR(r, Y, xi)
return expr.xreplace({e: omg})
dp_xi_eta = d_proj(Y, xi, eta)
prK = simplify_stiefel_tangent(proj(Y, g_inv(Y, K)), Y, (xi, eta))
Gamma = mat_spfy(
simplify_stiefel_tangent(prK - dp_xi_eta, Y, (xi, eta)))
print("This is the Christoffel function:")
pprint(Gamma)
fY, fYY = matrices('fY fYY')
rhess02 = trace(mat_spfy(t(eta)*fYY*xi-Gamma * t(fY)))
rhess11_bf_gr = xtrace(rhess02, eta)
print("This is the Riemannian Hessian Vector Product:")
pprint(rhess11_bf_gr)
def EN_inner(Y, P, Ba, Da, Bb, Db):
return trace(mat_spfy(Da * Db) + mat_spfy(Ba * t(Bb)))
def ambient_inner(Y, P, omg_Y, omg_P, xi_Y, xi_P):
return mat_spfy(
trace(
mat_spfy((al0 * omg_Y +
(al1 - al0) * Y * t(Y) * omg_Y) * t(xi_Y))) +
trace(mat_spfy(bt * inv(P) * omg_P * inv(P) * t(xi_P))))
def base_ambient_inner(omg_Y, omg_P, xi_Y, xi_P):
return mat_spfy(
trace(mat_spfy(omg_Y * t(xi_Y))) +
trace(mat_spfy(omg_P * t(xi_P))))
def EJ_inner(Y, P, a_P, a_YP, b_P, b_YP):
return trace(mat_spfy(-a_P * b_P) + mat_spfy(a_YP * t(b_YP)))
def J_adj(Y, a):
dY = symbols('dY', commutative=False)
return xtrace(trace(mat_spfy(J(Y, dY) * a)), dY)
def calc_pd():
""" For positive definite matrices
Y is a matrix point, a positive definite matrix
eta is an ambient point, same size with Y not necessarily
symmetric or invertible
b is a point in E_J. b is antisymmetric
"""
# eta is an ambient
Y = sm.sym_symb('Y')
eta = matrices('eta')
b = sm.asym_symb('b')
def J(Y, eta):
return eta - t(eta)
def J_adj(Y, a):
dY = symbols('dY', commutative=False)
return xtrace(trace(mat_spfy(J(Y, dY) * a)), dY)
def g(Y, eta):
return inv(Y)*eta*inv(Y)
def g_inv(Y, eta):
return Y*eta*Y
J_g_inv_J_adj = J(Y, g_inv(Y, J_adj(Y, b)))
pprint(J_g_inv_J_adj)
def solve_JginvJadj(Y, a):
return Integer(-1)/Integer(4)*inv(Y)*a*inv(Y)
def proj(Y, omg):
jo = mat_spfy(J(Y, omg))
cJinvjo = solve_JginvJadj(Y, jo)
return mat_spfy(omg - mat_spfy(
g_inv(Y, mat_spfy(J_adj(Y, cJinvjo)))))
def r_gradient(Y, omg):
return mat_spfy(
proj(Y, mat_spfy(g_inv(Y, omg))))
print(proj(Y, eta))
print(r_gradient(Y, eta))
xi, phi = matrices('xi phi')
xcross = xtrace(mat_spfy(trace(DDR(g(Y, eta), Y, phi) * t(xi))), phi)
K = (Integer(1)/Integer(2))*(
DDR(g(Y, eta), Y, xi) + DDR(g(Y, xi), Y, eta) - xcross)
def d_proj(Y, xi, omg):
e = matrices('e')
r = mat_spfy(proj(Y, e))
expr = DDR(r, Y, xi)
return expr.xreplace({e: omg})
dp_xi_eta = d_proj(Y, xi, eta)
prK = simplify_pd_tangent(proj(Y, mat_spfy(g_inv(Y, K))), Y, (xi, eta))
Gamma = mat_spfy(
simplify_pd_tangent(-dp_xi_eta+prK, Y, (xi, eta)))
print("This is the Christoffel function:")
pprint(Gamma)
fY, fYY = matrices('fY fYY')
rhess02 = trace(mat_spfy(t(eta)*fYY*xi-Gamma * t(fY)))
rhess11_bf_gr = xtrace(rhess02, eta)
print("This is the Riemannian Hessian Vector Product:")
pprint(r_gradient(Y, rhess11_bf_gr))