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 NT(Y, P, omg_Y, omg_P):
nB, nD = matrices('nB nD')
ipt = mat_spfy(base_ambient_inner(*N(Y, P, nB, nD), omg_Y, omg_P))
ntB = mat_spfy(xtrace(ipt, nB))
ntD1 = mat_spfy(xtrace(ipt, nD))
ntD = mat_spfy(Integer(1) / Integer(2) * (ntD1 + t(ntD1)))
return ntB, ntD
def calc_psd_range():
# in this method, the ambient is still
# R(n times p) + Symmetric(p)
# Horizontal is still
# al1*t(Y)*omg_Y + bt*omg_P*inv(R*R) - bt*inv(R*R)*omg_P)
# manifold is on pair (Y, P)
# Use pair B, D. B size n*(n-p), D size p(p+1)/2
# Use the embedding
# omg_Y = bt(Y*(inv(P)*D - D*inv(P))
# omg_P = al1*D
# so need to set up a new variable Y_0 and relation YY_0=0
Y, Y0 = stiefels('Y Y0')
sm.g_cstiefels[Y] = Y0
sm.g_cstiefels[Y0] = Y
B, eta_Y, eta_P = matrices('B eta_Y eta_P')
P, D = sym_symb('P D')
al0, al1, bt = scalars('al0 al1 bt')
def g(Y, P, omg_Y, omg_P):
return al0 * omg_Y + (al1 - al0) * Y * t(Y) * omg_Y, bt * inv(
P) * omg_P * inv(P)
def ginv(Y, P, omg_Y, omg_P):
return 1 / al0 * omg_Y + (
1 / al1 - 1 / al0) * Y * t(Y) * omg_Y, 1 / bt * P * omg_P * P
# check that ginv \circ g is id
e1, e2 = ginv(Y, P, *(g(Y, P, eta_Y, eta_P)))
e1 = mat_spfy(e1)
e2 = mat_spfy(e2)
print(e1, e2)
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 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 EN_inner(Y, P, Ba, Da, Bb, Db):
return trace(mat_spfy(Da * Db) + mat_spfy(Ba * t(Bb)))
qat = asym_symb('qat')
ipr = ambient_inner(Y, P, eta_Y, eta_P, Y * qat, P * qat - qat * P)
dqat = mat_spfy(xtrace(ipr, qat))
print(dqat)
def N(Y, P, B, D):
N_Y = mat_spfy(bt * Y * (inv(P) * D - D * inv(P))) + Y0 * B
N_P = mat_spfy(al1 * D)
return N_Y, N_P
def NT(Y, P, omg_Y, omg_P):
nB, nD = matrices('nB nD')
ipt = mat_spfy(base_ambient_inner(*N(Y, P, nB, nD), omg_Y, omg_P))
ntB = mat_spfy(xtrace(ipt, nB))
ntD1 = mat_spfy(xtrace(ipt, nD))
ntD = mat_spfy(Integer(1) / Integer(2) * (ntD1 + t(ntD1)))
return ntB, ntD
# check that image of N is horizontal:
print(
mat_spfy(
xtrace(
mat_spfy(
ambient_inner(Y, P, *N(Y, P, B, D), Y * qat,
P * qat - qat * P)), qat)))
NTe_B, NTe_D = NT(Y, P, eta_Y, eta_P)
mdict = {'bt': r'\beta', 'al': r'\alpha'}
print(latex_map(mat_latex(NTe_B), mdict))
print(latex_map(mat_latex(NTe_D), mdict))
gN = g(Y, P, *N(Y, P, B, D))
gN_B = mat_spfy(gN[0])
gN_D = mat_spfy(gN[1])
print(latex_map(mat_latex(gN_B), mdict))
print(latex_map(mat_latex(gN_D), mdict))
NTgN_B, NTgN_D = NT(Y, P, *gN)
print(latex_map(mat_latex(NTgN_B), mdict))
print(latex_map(mat_latex(NTgN_D), mdict))
NTg_B, NTg_D = NT(Y, P, *g(Y, P, eta_Y, eta_P))
print(latex_map(mat_latex(NTg_B), mdict))
# print(latex_map(sp.latex(NTgN_P), mdict))
print(latex_map(mat_latex(NTg_D), mdict))
def sym(x):
return mat_spfy(Integer(1) / Integer(2) * (x + t(x)))
xi_Y, xi_P, phi_Y, phi_P = matrices('xi_Y xi_P phi_Y phi_P')
gYPeta = g(Y, P, eta_Y, eta_P)
Dgxieta_Y = DDR(gYPeta[0], Y, xi_Y)
Dgxieta_P = DDR(gYPeta[1], P, xi_P)
gYPxi = g(Y, P, xi_Y, xi_P)
Dgetaxi_Y = DDR(gYPxi[0], Y, eta_Y)
Dgetaxi_P = DDR(gYPxi[1], P, eta_P)
Dgxiphi_Y = DDR(gYPxi[0], Y, phi_Y)
Dgxiphi_P = DDR(gYPxi[1], P, phi_P)
tr3 = mat_spfy(base_ambient_inner(Y, P, Dgxiphi_Y, Dgxiphi_P, eta_Y,
eta_P))
xcross_Y = xtrace(tr3, phi_Y)
xcross_P = xtrace(tr3, phi_P)
K_Y = (Integer(1) / Integer(2)) * (Dgxieta_Y + Dgetaxi_Y - xcross_Y)
K_P = (Integer(1) / Integer(2)) * (Dgxieta_P + Dgetaxi_P - xcross_P)
pprint(K_Y)
pprint(K_P)
def JT(Y, P, a_P, a_YP):
dY, dP = matrices('dY dP')
ipt = mat_spfy(EJ_inner(Y, P, *J(Y, P, dY, dP), a_P, a_YP))
jty = mat_spfy(xtrace(ipt, dY))
jtp = mat_spfy(xtrace(ipt, dP))
return jty, jtp
def calc_psd():
# manifold is on pair (Y, P)
Y = stiefels('Y')
P = sym_symb('P')
a_P = asym_symb('a_P')
a_YP, eta_Y, eta_P = matrices('a_YP eta_Y eta_P')
al0, al1, bt = scalars('al0 al1 bt')
def g(Y, P, omg_Y, omg_P):
return al0 * omg_Y + (al1 - al0) * Y * t(Y) * omg_Y, bt * inv(
P) * omg_P * inv(P)
def ginv(Y, P, omg_Y, omg_P):
return 1 / al0 * omg_Y + (
1 / al1 - 1 / al0) * Y * t(Y) * omg_Y, 1 / bt * P * omg_P * P
# check that ginv \circ g is id
e1, e2 = ginv(Y, P, *(g(Y, P, eta_Y, eta_P)))
e1 = mat_spfy(e1)
e2 = mat_spfy(e2)
print(e1, e2)
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(Y, P, 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)))
qat = matrices('qat')
ipr = ambient_inner(Y, P, eta_Y, eta_P, Y * qat, P * qat - qat * P)
dqat = mat_spfy(xtrace(ipr, qat))
print(dqat)
def J(Y, P, omg_Y, omg_P):
J_P = mat_spfy(omg_P - t(omg_P))
J_YP = mat_spfy(al1 * t(Y) * omg_Y + bt * omg_P * inv(P) -
bt * inv(P) * omg_P)
return J_P, J_YP
def JT(Y, P, a_P, a_YP):
dY, dP = matrices('dY dP')
ipt = mat_spfy(EJ_inner(Y, P, *J(Y, P, dY, dP), a_P, a_YP))
jty = mat_spfy(xtrace(ipt, dY))
jtp = mat_spfy(xtrace(ipt, dP))
return jty, jtp
JTa_Y, JTa_P = JT(Y, P, a_P, a_YP)
ginvJT = ginv(Y, P, JTa_Y, JTa_P)
ginvJT_Y = mat_spfy(ginvJT[0])
ginvJT_P = mat_spfy(ginvJT[1])
pprint(ginvJT_Y)
pprint(ginvJT_P)
Jginv_P, Jginv_YP = J(Y, P, *ginv(Y, P, eta_Y, eta_P))
pprint(Jginv_P)
pprint(Jginv_YP)
b_P, b_YP = J(Y, P, *ginvJT)
pprint(b_P)
pprint(b_YP)
# even part of a_YP
aYPev = sym(a_YP)
exp1 = bt * (Integer(1) / Integer(2) * b_P - P * aYPev + aYPev * P)
exp1 = mat_spfy(exp1)
pprint(exp1)
print('check the formula for even part of a_YP recover b_YP')
exp1 = al1 * aYPev + bt / 2 * (b_P * inv(P) - inv(P) * b_P) - sym(b_YP)
pprint(mat_spfy(exp1))
aYPodd = mat_spfy(a_YP - aYPev)
exp2 = (al1-2*bt)*aYPodd + bt*P*aYPodd*inv(P) + bt*inv(P)*aYPodd*P -\
Integer(1)/Integer(2)*(b_YP-t(b_YP))
# this is the equation to solve a_YPodd
print('this is the equation to solve a_YPodd')
pprint(exp2)
def even_solve(Y, P, b_P, b_YP):
return Integer(1)/al1*sym(b_YP) +\
bt/(Integer(2)*al1)*(inv(P)*b_P - b_P*inv(P))
print(mat_spfy(even_solve(Y, P, b_P, b_YP)))
xi_Y, xi_P, phi_Y, phi_P = matrices('xi_Y xi_P phi_Y phi_P')
gYPeta = g(Y, P, eta_Y, eta_P)
Dgxieta_Y = DDR(gYPeta[0], Y, xi_Y)
Dgxieta_P = DDR(gYPeta[1], P, xi_P)
gYPxi = g(Y, P, xi_Y, xi_P)
Dgetaxi_Y = DDR(gYPxi[0], Y, eta_Y)
Dgetaxi_P = DDR(gYPxi[1], P, eta_P)
Dgxiphi_Y = DDR(gYPeta[0], Y, phi_Y)
Dgxiphi_P = DDR(gYPeta[1], P, phi_P)
tr3 = mat_spfy(base_ambient_inner(Y, P, Dgxiphi_Y, Dgxiphi_P, xi_Y, xi_P))
xcross_Y = xtrace(tr3, phi_Y)
xcross_P = xtrace(tr3, phi_P)
K_Y = (Integer(1) / Integer(2)) * (Dgxieta_Y + Dgetaxi_Y - xcross_Y)
K_P = (Integer(1) / Integer(2)) * (Dgxieta_P + Dgetaxi_P - xcross_P)
K_Y = simplify_stiefel_tangent(K_Y, Y, (eta_Y, xi_Y))
K_P = simplify_pd_tangent(K_P, P, (eta_P, xi_P))
pprint(K_Y)
pprint(K_P)
giKY, giKP = ginv(Y, P, K_Y, K_P)
giKY1 = simplify_stiefel_tangent(giKY, Y, (eta_Y, xi_Y))
giKP1 = simplify_pd_tangent(giKP, P, (eta_P, xi_P))
jK1 = J(Y, P, giKY1, giKP1)
jKP = simplify_pd_tangent(jK1[0], P, (eta_P, xi_P))
jKY = simplify_stiefel_tangent(jK1[1], Y, (eta_Y, xi_Y))
jKY1 = simplify_pd_tangent(jKY, P, (eta_P, xi_P))
pprint(jKP)
pprint(jKY1)
def DJ(Y, P, xi_Y, xi_P, eta_Y, eta_P):
expr_P, expr_YP = J(Y, P, eta_Y, eta_P)
der_P = DDR(expr_P, Y, xi_Y) + DDR(expr_P, P, xi_P)
der_YP = DDR(expr_YP, Y, xi_Y) + DDR(expr_YP, P, xi_P)
return mat_spfy(der_P), mat_spfy(der_YP)
pprint(DJ(Y, P, xi_Y, xi_P, eta_Y, eta_P))
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))