def optim_test():
n, d = (1000, 50)
# n, d = (10, 3)
# simple function. Distance to a given matrix
# || S - A||_F^2
Y0, _ = np.linalg.qr(crandn(n, d))
P0 = np.diag(randint(1, 1000, d) * .001)
A00 = Y0 @ P0 @ Y0.T.conjugate()
A0 = hsym(A00)
A = (hsym(crandn(n, n)) * 1e-2 + A0)
alpha = np.array([1, 1])
print("alpha %s" % str(alpha))
beta = alpha[1] * .1
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
XInit = man.rand()
opt_pre = solve_dist_with_man(man, A, X0=XInit, maxiter=20)
beta = alpha[1] * 1
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
opt_mid = solve_dist_with_man(man, A, X0=opt_pre, maxiter=20)
# opt_mid = opt_pre
beta = alpha[1] * 30
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
opt = solve_dist_with_man(man, A, X0=opt_mid, maxiter=500)
opt_mat = opt.Y @ opt.P @ opt.Y.T.conjugate()
if False:
print(A0)
print(opt_mat)
print(np.max(np.abs(A0 - opt_mat)))
def test_christ_flat():
"""now test that christofel preserve metrics:
on the flat space
d_xi <v M v> = 2 <v M nabla_xi v>
v = proj(W) @ (aa W + b)
"""
alpha = randint(1, 10, 2) * .1
gamma = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .1
m = 4
n = 5
p = 3
man = ComplexFixedRank(m, n, p, alpha=alpha, beta=beta, gamma=gamma)
S = man.rand()
xi = man.randvec(S)
aaU = crandn(m*p, m*p)
bbU = crandn(m*p)
aaV = crandn(n*p, n*p)
bbV = crandn(n*p)
cc = crandn(p*p, p*p)
dd = hsym(crandn(p, p))
def v_func_flat(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ S.P.reshape(-1)).reshape(p, p))
return fr_ambient(
(aaU @ S.U.reshape(-1) + bbU).reshape(m, p),
(aaV @ S.V.reshape(-1) + bbV).reshape(n, p),
csp + dd)
vv = v_func_flat(S) # vv is not horizontal
dlt = 1e-7
Snew = fr_point(
S.U + dlt * xi.tU,
S.V + dlt * xi.tV,
S.P + dlt * xi.tP)
vnew = v_func_flat(Snew)
val = man.inner(S, vv)
valnew = man.inner(Snew, vnew)
d1 = (valnew - val)/dlt
dv = (vnew - vv).scalar_mul(1/dlt)
nabla_xi_v = dv + man.g_inv(
S, man.christoffel_form(S, xi, vv))
# not equal bc vv is not horizontal:
nabla_xi_va = dv + man.g_inv(
S, super(ComplexFixedRank, man).christoffel_form(S, xi, vv))
print(check_zero(man._vec(nabla_xi_v) - man._vec(nabla_xi_va)))
d2 = man.inner(S, vv, nabla_xi_v)
print(d1)
print(2*d2)
def get_D2(omg):
UTomg = X.U.T.conj()@omg.tU
VTomg = X.V.T.conj()@omg.tV
Piv = X.Pinv
D2 = hsym([email protected]@X.Pinv + 1/(al[1]+gm[1])*(
al[1]*(Piv@UTomg - UTomg@Piv) +
gm[1]*(Piv@VTomg - VTomg@Piv)))
return D2
def NTgN_opt(X, B, C, D):
Piv = X.Pinv
Dp, Dm = hsym(D), ahsym(D)
Dp_ = (1/bt-2/(al[1]+gm[1]))*Piv@Dp@Piv + 1/(al[1]+gm[1])*(
[email protected]@[email protected]@X.Pinv)
Dm_ = al[1]*gm[1]*(al[1]+gm[1])*Dm
return al[0]*B, gm[0]*C, Dp_ + Dm_
def v_func_flat(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ S.P.reshape(-1)).reshape(p, p))
return fr_ambient(
(aaU @ S.U.reshape(-1) + bbU).reshape(m, p),
(aaV @ S.V.reshape(-1) + bbV).reshape(n, p),
csp + dd)
def xi_func(S):
csp_xi = hsym((cc_xi @ (S.P-S0.P).reshape(-1)).reshape(p, p))
xi_amb = fr_ambient(
(aa_xiU @ (S.U-S0.U).reshape(-1) +
inct_xi.tU.reshape(-1)).reshape(m, p),
(aa_xiV @ (S.V-S0.V).reshape(-1) +
inct_xi.tV.reshape(-1)).reshape(n, p),
csp_xi + inct_xi.tP)
return man.proj(S, xi_amb)
def v_func(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ (S.P-S0.P).reshape(-1)).reshape(p, p))
return man.proj(S, fr_ambient(
(aaU @ (S.U-S0.U).reshape(-1) + intcU).reshape(m, p),
(aaV @ (S.V-S0.V).reshape(-1) + intcV).reshape(n, p),
csp + p_intc))
def v_func(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ (S.P - S0.P).reshape(-1)).reshape(d, d))
return man.proj(
S,
psd_ambient((aa @ (S.Y - S0.Y).reshape(-1) + intc).reshape(n, d),
csp + p_intc))
def test_chris_vectorfields():
# now test that it works on embedded metrics
# we test that D_xi (eta g eta) = 2(eta g nabla_xi eta)
n, d = (20, 3)
alpha = randint(1, 10, 2) * .1
beta = randint(1, 10, 1)[0] * .1
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
S0 = man.rand()
aa = crandn(n * d, n * d)
intc = crandn(n * d)
cc = crandn(d * d, d * d)
p_intc = hsym(crandn(d, d))
inct_xi = man._rand_ambient()
aa_xi = crandn(n * d, n * d)
cc_xi = crandn(d * d, d * d)
def v_func(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ (S.P - S0.P).reshape(-1)).reshape(d, d))
return man.proj(
S,
psd_ambient((aa @ (S.Y - S0.Y).reshape(-1) + intc).reshape(n, d),
csp + p_intc))
SS = psd_point(S0.Y, S0.P)
xi = man.proj(SS, inct_xi)
nabla_xi_v, dv, cxv = calc_covar_numeric(man, SS, xi, v_func)
def xi_func(S):
csp_xi = hsym((cc_xi @ (S.P - S0.P).reshape(-1)).reshape(d, d))
xi_amb = psd_ambient((aa_xi @ (S.Y - S0.Y).reshape(-1) +
inct_xi.tY.reshape(-1)).reshape(n, d),
csp_xi + inct_xi.tP)
return man.proj(S, xi_amb)
vv = v_func(SS)
nabla_v_xi, dxi, cxxi = calc_covar_numeric(man, SS, vv, xi_func)
diff = nabla_xi_v - nabla_v_xi
print(diff.tY, diff.tP)
# now do Lie bracket:
dlt = 1e-7
SnewXi = psd_point(SS.Y + dlt * xi.tY, SS.P + dlt * xi.tP)
Snewvv = psd_point(SS.Y + dlt * vv.tY, SS.P + dlt * vv.tP)
vnewxi = v_func(SnewXi)
xnewv = xi_func(Snewvv)
dxiv = (vnewxi - vv).scalar_mul(1 / dlt)
dvxi = (xnewv - xi).scalar_mul(1 / dlt)
diff2 = man.proj(SS, dxiv - dvxi)
print(check_zero(man._vec(diff) - man._vec(diff2)))
def N(man, X, B, C, D):
al, bt, gm = (man.alpha, man.beta, man.gamma)
U, V, Piv = (X.U, X.V, X.Pinv)
Dm = ahsym(D)
Dp = hsym(D)
bkPivDp = Piv @ Dp - Dp @ Piv
U0 = null_space(X._U.T.conj())
V0 = null_space(X._V.T.conj())
U0B = U0 @ B
V0C = V0 @ C
return fr_ambient(
U @ (-gm[1]*Dm + 1/(al[1]+gm[1])*bkPivDp)+U0B,
V @(al[1]*Dm + 1/(al[1]+gm[1])*bkPivDp)+V0C,
1/bt*Dp)
def test_christ_flat():
"""now test that christofel preserve metrics:
on the flat space
d_xi <v M v> = 2 <v M nabla_xi v>
v = proj(W) @ (aa W + b)
"""
alpha = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .1
n = 20
d = 3
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
S = man.rand()
xi = man.randvec(S)
xi = man.randvec(S)
aa = crandn(n * d, n * d)
bb = crandn(n * d)
cc = crandn(d * d, d * d)
dd = hsym(crandn(d, d))
def v_func_flat(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ S.P.reshape(-1)).reshape(d, d))
return psd_ambient((aa @ S.Y.reshape(-1) + bb).reshape(n, d), csp + dd)
vv = v_func_flat(S)
dlt = 1e-7
Snew = psd_point(S.Y + dlt * xi.tY, S.P + dlt * xi.tP)
vnew = v_func_flat(Snew)
val = man.inner_product_amb(S, vv)
valnew = man.inner_product_amb(Snew, vnew)
d1 = (valnew - val) / dlt
dv = (vnew - vv).scalar_mul(1 / dlt)
nabla_xi_v = dv + man.g_inv(S, man.christoffel_form(S, xi, vv))
nabla_xi_va = dv + man.g_inv(
S,
super(ComplexPositiveSemidefinite, man).christoffel_form(S, xi, vv))
print(check_zero(man._vec(nabla_xi_v) - man._vec(nabla_xi_va)))
d2 = man.inner(S, vv, nabla_xi_v)
print(d1)
print(2 * d2)
def test_covariance_deriv():
# now test full:
# do covariant derivatives
# check that it works, preseving everything
n, d = (5, 3)
alpha = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .01
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
S = man.rand()
aa = crandn(n * d, n * d)
cc = crandn(d * d, d * d)
icpt = man._rand_ambient()
def omg_func(S):
csp = hsym((cc @ S.P.reshape(-1)).reshape(d, d))
return psd_ambient(
(aa @ S.Y.reshape(-1) + icpt.tY.reshape(-1)).reshape(n, d),
csp + icpt.tP)
xi = man.randvec(S)
egrad = omg_func(S)
ecsp = hsym((cc @ xi.tP.reshape(-1)).reshape(d, d))
ehess = psd_ambient((aa @ xi.tY.reshape(-1)).reshape(n, d), ecsp)
val1 = man.ehess2rhess(S, egrad, ehess, xi)
def rgrad_func(W):
return man.proj_g_inv(W, omg_func(W))
if False:
first = ehess
a = man.J_g_inv(S, egrad)
rgrad = man.proj_g_inv(S, egrad)
second = man.D_g(S, xi, man.g_inv(S, egrad)).scalar_mul(-1)
aout = man.solve_J_g_inv_Jst(S, a)
third = man.proj(S, man.D_g_inv_Jst(S, xi, aout)).scalar_mul(-1)
fourth = man.christoffel_form(S, xi, rgrad)
val1a1 = man.proj_g_inv(S, first + second + fourth) + third
print(check_zero(man._vec(val1 - val1a1)))
elif True:
d_xi_rgrad = num_deriv_amb(man, S, xi, rgrad_func)
rgrad = man.proj_g_inv(S, egrad)
fourth = man.christoffel_form(S, xi, rgrad)
val1a = man.proj(S, d_xi_rgrad) + man.proj_g_inv(S, fourth)
print(check_zero(man._vec(val1 - val1a)))
# nabla_v_xi, dxi, cxxi
val2a, _, _ = calc_covar_numeric(man, S, xi, omg_func)
val2, _, _ = calc_covar_numeric(man, S, xi, rgrad_func)
# val2_p = project(prj, val2)
val2_p = man.proj(S, val2)
# print(val1)
# print(val2_p)
print(check_zero(man._vec(val1) - man._vec(val2_p)))
if True:
H = xi
valrangeA_ = ehess + man.g(S, man.D_proj(
S, H, man.g_inv(S, egrad))) - man.D_g(
S, H, man.g_inv(S, egrad)) +\
man.christoffel_form(S, H, man.proj_g_inv(S, egrad))
valrangeB = man.proj_g_inv(S, valrangeA_)
valrange = man.ehess2rhess_alt(S, egrad, ehess, xi)
print(check_zero(man._vec(valrange) - man._vec(val2_p)))
print(check_zero(man._vec(valrange) - man._vec(val1)))
print(check_zero(man._vec(valrange) - man._vec(valrangeB)))
def xi_func(S):
csp_xi = hsym((cc_xi @ (S.P - S0.P).reshape(-1)).reshape(d, d))
xi_amb = psd_ambient((aa_xi @ (S.Y - S0.Y).reshape(-1) +
inct_xi.tY.reshape(-1)).reshape(n, d),
csp_xi + inct_xi.tP)
return man.proj(S, xi_amb)
def v_func_flat(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ S.P.reshape(-1)).reshape(d, d))
return psd_ambient((aa @ S.Y.reshape(-1) + bb).reshape(n, d), csp + dd)
def eta_field(Sin):
return man.proj(S, fr_ambient(
mU1 @ (Sin.U - S.U) @ m2,
mV1 @ (Sin.V - S.V) @ m2,
hsym((m_p @ (Sin.P - S.P).reshape(-1)).reshape(p, p)))) + eeta
def solveNTgN(X, Bo, Co, Do):
Dp, Dm = hsym(Do), ahsym(Do)
Dm_ = 1/(al[1]*gm[1]*(al[1]+gm[1]))*Dm
Dp_ = complex_extended_lyapunov(1/bt, 1/(al[1]+gm[1]), X.P, X.P@[email protected])
return 1/al[0]*Bo, 1/gm[0]*Co, Dp_ + Dm_
def eta_field(Sin):
return man.proj(
S,
psd_ambient(m1 @ (Sin.Y - S.Y) @ m2,
hsym((m_p @ (Sin.P - S.P).reshape(-1)).reshape(
d, d)))) + eeta
def omg_func(S):
csp = hsym((cc @ S.P.reshape(-1)).reshape(d, d))
return psd_ambient(
(aa @ S.Y.reshape(-1) + icpt.tY.reshape(-1)).reshape(n, d),
csp + icpt.tP)
def test_chris_vectorfields():
# now test that it works on embedded metrics
# we test that D_xi (eta g eta) = 2(eta g nabla_xi eta)
alpha = randint(1, 10, 2) * .1
gamma = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .1
m = 4
n = 5
p = 3
man = ComplexFixedRank(m, n, p, alpha=alpha, beta=beta, gamma=gamma)
S0 = man.rand()
aaU = crandn(m*p, m*p)
intcU = crandn(m*p)
aaV = crandn(n*p, n*p)
intcV = crandn(n*p)
cc = crandn(p*p, p*p)
p_intc = hsym(crandn(p, p))
inct_xi = man._rand_ambient()
aa_xiU = crandn(m*p, m*p)
aa_xiV = crandn(n*p, n*p)
cc_xi = crandn(p*p, p*p)
def v_func(S):
# a function from the manifold
# to ambient
csp = hsym((cc @ (S.P-S0.P).reshape(-1)).reshape(p, p))
return man.proj(S, fr_ambient(
(aaU @ (S.U-S0.U).reshape(-1) + intcU).reshape(m, p),
(aaV @ (S.V-S0.V).reshape(-1) + intcV).reshape(n, p),
csp + p_intc))
SS = fr_point(S0.U, S0.V, S0.P)
xi = man.proj(SS, inct_xi)
nabla_xi_v, dv, cxv = calc_covar_numeric(
man, SS, xi, v_func)
def xi_func(S):
csp_xi = hsym((cc_xi @ (S.P-S0.P).reshape(-1)).reshape(p, p))
xi_amb = fr_ambient(
(aa_xiU @ (S.U-S0.U).reshape(-1) +
inct_xi.tU.reshape(-1)).reshape(m, p),
(aa_xiV @ (S.V-S0.V).reshape(-1) +
inct_xi.tV.reshape(-1)).reshape(n, p),
csp_xi + inct_xi.tP)
return man.proj(S, xi_amb)
vv = v_func(SS)
nabla_v_xi, dxi, cxxi = calc_covar_numeric(
man, SS, vv, xi_func)
diff = nabla_xi_v - nabla_v_xi
print(diff.tU, diff.tV, diff.tP)
# now do Lie bracket:
dlt = 1e-7
SnewXi = fr_point(SS.U+dlt*xi.tU,
SS.V+dlt*xi.tV,
SS.P+dlt*xi.tP)
Snewvv = fr_point(SS.U+dlt*vv.tU,
SS.V+dlt*vv.tV,
SS.P+dlt*vv.tP)
vnewxi = v_func(SnewXi)
xnewv = xi_func(Snewvv)
dxiv = (vnewxi - vv).scalar_mul(1/dlt)
dvxi = (xnewv - xi).scalar_mul(1/dlt)
diff2 = man.proj(SS, dxiv-dvxi)
print(check_zero(man._vec(diff) - man._vec(diff2)))
def test_rhess_02():
n, d = (50, 3)
alpha = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .1
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
S = man.rand()
# simple function. Distance to a given matrix
# || S - A||_F^2
A = hsym(crandn(n, n))
def f(S):
diff = (A - S.Y @ S.P @ S.Y.T.conjugate())
return trace(diff @ diff.T.conjugate())
def df(S):
return psd_ambient(-4 * A @ S.Y @ S.P,
2 * (S.P - S.Y.T.conjugate() @ A @ S.Y))
def ehess_form(S, xi, eta):
return (
trace(-4 * A @ (xi.tY @ S.P + S.Y @ xi.tP) @ eta.tY.T.conjugate())
+ 2 * trace(
(xi.tP - xi.tY.T.conjugate() @ A @ S.Y -
S.Y.T.conjugate() @ A @ xi.tY) @ eta.tP.T.conjugate())).real
def ehess_vec(S, xi):
return psd_ambient(
-4 * A @ (xi.tY @ S.P + S.Y @ xi.tP),
2 * (xi.tP - xi.tY.T.conjugate() @ A @ S.Y -
S.Y.T.conjugate() @ A @ xi.tY))
xxi = man.randvec(S)
dlt = 1e-8
Snew = psd_point(S.Y + dlt * xxi.tY, S.P + dlt * xxi.tP)
d1 = (f(Snew) - f(S)) / dlt
d2 = df(S)
print(d1 - man.base_inner_ambient(d2, xxi))
eeta = man.randvec(S)
d1 = man.base_inner_ambient((df(Snew) - df(S)), eeta) / dlt
ehess_val = ehess_form(S, xxi, eeta)
dv2 = ehess_vec(S, xxi)
print(man.base_inner_ambient(dv2, eeta))
print(d1, ehess_val, d1 - ehess_val)
# now check the formula: ehess = xi (eta_func(f)) - <D_xi eta, df(Y)>
# promote eta to a vector field.
m1 = crandn(n, n)
m2 = crandn(d, d)
m_p = crandn(d * d, d * d)
def eta_field(Sin):
return man.proj(
S,
psd_ambient(m1 @ (Sin.Y - S.Y) @ m2,
hsym((m_p @ (Sin.P - S.P).reshape(-1)).reshape(
d, d)))) + eeta
# xietaf: should go to ehess(xi, eta) + df(Y) @ etafield)
xietaf = (man.base_inner_ambient(df(Snew), eta_field(Snew)) -
man.base_inner_ambient(df(S), eta_field(S))) / dlt
# appy eta_func to f: should go to tr(m1 @ xxi @ m2 @ df(Y).T)
Dxietaf = man.base_inner_ambient(
(eta_field(Snew) - eta_field(S)), df(S)) / dlt
# this is ehess. should be same as d1 or ehess_val
print(xietaf - Dxietaf)
print(xietaf - Dxietaf - ehess_val)
# now check: rhess. Need to make sure xi, eta in the tangent space.
# first compare this with numerical differentiation
xi1 = man.proj(S, xxi)
eta1 = man.proj(S, eeta)
egvec = df(S)
ehvec = ehess_vec(S, xi1)
rhessvec = man.ehess2rhess(S, egvec, ehvec, xi1)
# check it numerically:
def rgrad_func(Y):
return man.proj_g_inv(Y, df(Y))
# val2a, _, _ = calc_covar_numeric_raw(man, W, xi1, df)
val2, _, _ = calc_covar_numeric(man, S, xi1, rgrad_func)
val2_p = man.proj(S, val2)
# print(rhessvec)
# print(val2_p)
print(man._vec(rhessvec - val2_p))
rhessval = man.inner_product_amb(S, rhessvec, eta1)
print(man.inner_product_amb(S, val2, eta1))
print(rhessval)
# check symmetric:
ehvec_e = ehess_vec(S, eta1)
rhessvec_e = man.ehess2rhess(S, egvec, ehvec_e, eta1)
rhessval_e = man.inner_product_amb(S, rhessvec_e, xi1)
print(rhessval_e)
# the above computed inner_prod(Nabla_xi Pi * df, eta)
# in the following check. Extend eta1 to eta_proj
# (Pi Nabla_hat Pi g_inv df, g eta)
# = D_xi (Pi g_inv df, g eta) - (Pi g_inv df g Pi Nabla_hat eta)
def eta_proj(S):
return man.proj(S, eta_field(S))
print(check_zero(man._vec(eta1 - eta_proj(S))))
e1 = man.inner(S, man.proj_g_inv(S, df(S)), eta_proj(S))
e1a = man.base_inner_ambient(df(S), eta_proj(S))
print(e1, e1a, e1 - e1a)
Snew = psd_point(S.Y + dlt * xi1.tY, S.P + dlt * xi1.tP)
e2 = man.inner(Snew, man.proj_g_inv(Snew, df(Snew)), eta_proj(Snew))
e2a = man.base_inner_ambient(df(Snew), eta_proj(Snew))
print(e2, e2a, e2 - e2a)
first = (e2 - e1) / dlt
first1 = (man.base_inner_ambient(df(Snew), eta_proj(Snew)) -
man.base_inner_ambient(df(S), eta_proj(S))) / dlt
print(first - first1)
val3, _, _ = calc_covar_numeric(man, S, xi1, eta_proj)
second = man.inner(S, man.proj_g_inv(S, df(S)), man.proj(S, val3))
second2 = man.inner(S, man.proj_g_inv(S, df(S)), val3)
print(second, second2, second - second2)
print('same as rhess_val %f' % (first - second))
def test_covariance_deriv():
# now test full:
# do covariant derivatives
alpha = randint(1, 10, 2) * .1
gamma = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .1
m = 4
n = 5
p = 3
man = ComplexFixedRank(m, n, p, alpha=alpha, beta=beta, gamma=gamma)
S = man.rand()
aaU = crandn(m*p, m*p)
aaV = crandn(n*p, n*p)
cc = crandn(p*p, p*p)
icpt = man._rand_ambient()
def omg_func(S):
csp = hsym((cc @ S.P.reshape(-1)).reshape(p, p))
return fr_ambient(
(aaU @ S.U.reshape(-1) + icpt.tU.reshape(-1)).reshape(m, p),
(aaV @ S.V.reshape(-1) + icpt.tV.reshape(-1)).reshape(n, p),
csp + icpt.tP)
xi = man.randvec(S)
egrad = omg_func(S)
ecsp = hsym((cc @ xi.tP.reshape(-1)).reshape(p, p))
ehess = fr_ambient(
(aaU @ xi.tU.reshape(-1)).reshape(m, p),
(aaV @ xi.tV.reshape(-1)).reshape(n, p),
ecsp)
val1 = man.ehess2rhess(S, egrad, ehess, xi)
def rgrad_func(W):
return man.proj_g_inv(W, omg_func(W))
if False:
first = ehess
a = man.J_g_inv(S, egrad)
rgrad = man.proj_g_inv(S, egrad)
second = man.D_g(
S, xi, man.g_inv(S, egrad)).scalar_mul(-1)
aout = man.solve_J_g_inv_Jst(S, a)
third = man.proj(S, man.D_g_inv_Jst(S, xi, aout)).scalar_mul(-1)
fourth = man.christoffel_form(S, xi, rgrad)
val1a1 = man.proj_g_inv(S, first + second + fourth) + third
print(check_zero(man._vec(val1-val1a1)))
elif True:
d_xi_rgrad = num_deriv_amb(man, S, xi, rgrad_func)
rgrad = man.proj_g_inv(S, egrad)
fourth = man.christoffel_form(S, xi, rgrad)
val1a = man.proj(S, d_xi_rgrad) + man.proj_g_inv(S, fourth)
print(check_zero(man._vec(val1-val1a)))
# nabla_v_xi, dxi, cxxi
val2a, _, _ = calc_covar_numeric(man, S, xi, omg_func)
val2, _, _ = calc_covar_numeric(man, S, xi, rgrad_func)
# val2_p = project(prj, val2)
val2_p = man.proj(S, val2)
# print(val1)
# print(val2_p)
print(check_zero(man._vec(val1)-man._vec(val2_p)))
if True:
H = xi
valrangeA_ = ehess + man.g(S, man.D_proj(
S, H, man.g_inv(S, egrad))) - man.D_g(
S, H, man.g_inv(S, egrad)) +\
man.christoffel_form(S, H, man.proj_g_inv(S, egrad))
valrangeB = man.proj_g_inv(S, valrangeA_)
valrange = man.ehess2rhess(S, egrad, ehess, xi)
print(check_zero(man._vec(valrange)-man._vec(val2_p)))
print(check_zero(man._vec(valrange)-man._vec(val1)))
print(check_zero(man._vec(valrange)-man._vec(valrangeB)))
def omg_func(S):
csp = hsym((cc @ S.P.reshape(-1)).reshape(p, p))
return fr_ambient(
(aaU @ S.U.reshape(-1) + icpt.tU.reshape(-1)).reshape(m, p),
(aaV @ S.V.reshape(-1) + icpt.tV.reshape(-1)).reshape(n, p),
csp + icpt.tP)
def test_J(man, X):
from scipy.linalg import null_space
al = man.alpha
bt = man.beta
gm = man.gamma
p = man.p
# U, V, P, Piv = (X.U, X.V, X.P, X.Pinv)
jjmat = np.zeros((8*p*p, man.tdim))
# this map is not full onto - but just check tangent maps to zero
for i in range(man.tdim):
Ux = zeros(man.tdim)
Ux[i] = 1
omg = man._unvec(Ux)
jjmat[:, i] = np.concatenate(
[cvec(stU(X, omg)),
cvec(stV(X, omg)),
cvec(symP(X, omg)),
cvec(Hz(man, X, omg))])
# nsp = null_space(jjmat)
# prj = nsp @ la.solve(nsp.T.conj() @ nsp, nsp.T.conj())
omg = man._rand_ambient()
eta = man.proj(X, omg)
def rand_vertical():
oo = crandn(man.p, man.p)
oo = oo - oo.T.conj()
return fr_ambient(
X.U @ oo,
X.V @ oo,
-oo @ X.P + X.P @oo)
vv = rand_vertical()
print(man.inner(X, vv, eta))
nmat = make_N_mat(man, X)
gmat = make_g_mat(man, X)
NTgN = nmat.T @ gmat @ nmat
bcd = nmat.T @ gmat @ man._vec(omg)
m, n, p = (man.m, man.n, man.p)
B = cunvec(bcd[:2*(m-p)*p], (m-p, p))
C = cunvec(bcd[2*(m-p)*p:2*(m+n-2*p)*p], (n-p, p))
D = cunvec(bcd[2*(m+n-2*p)*p:], (p, p))
Dp = hsym(D)
Dm = ahsym(D)
Dm2 = al[1]*gm[1]*ahsym(X.V.T.conj()@omg.tV - X.U.T.conj()@omg.tU)
pprint(Dm - Dm2)
U0 = null_space(X._U.T.conj())
V0 = null_space(X._V.T.conj())
B2 = al[0]*U0.T.conj()@omg.tU
pprint(B-B2)
C2 = gm[0]*V0.T.conj()@omg.tV
pprint(C-C2)
Dmsolve = Dm2/(al[1]+gm[1])/al[1]/gm[1]
pprint(Dmsolve - (1/(al[1]+gm[1]))*(
ahsym(X.V.T.conj()@omg.tV - X.U.T.conj()@omg.tU)))
Dpsolve = complex_extended_lyapunov(1/bt, 1/(al[1]+gm[1]), X.P, X.P@[email protected])
Drecv = (1/bt-2/(al[1]+gm[1]))*X.Pinv@[email protected] + 1/(al[1]+gm[1])*(
[email protected]@[email protected]@X.Pinv)
pprint(Drecv - Dp)
def get_D2(omg):
UTomg = X.U.T.conj()@omg.tU
VTomg = X.V.T.conj()@omg.tV
Piv = X.Pinv
D2 = hsym([email protected]@X.Pinv + 1/(al[1]+gm[1])*(
al[1]*(Piv@UTomg - UTomg@Piv) +
gm[1]*(Piv@VTomg - VTomg@Piv)))
return D2
Dp2 = get_D2(omg)
print(check_zero(Dp - Dp2))
def NTgN_opt(X, B, C, D):
Piv = X.Pinv
Dp, Dm = hsym(D), ahsym(D)
Dp_ = (1/bt-2/(al[1]+gm[1]))*Piv@Dp@Piv + 1/(al[1]+gm[1])*(
[email protected]@[email protected]@X.Pinv)
Dm_ = al[1]*gm[1]*(al[1]+gm[1])*Dm
return al[0]*B, gm[0]*C, Dp_ + Dm_
def solveNTgN(X, Bo, Co, Do):
Dp, Dm = hsym(Do), ahsym(Do)
Dm_ = 1/(al[1]*gm[1]*(al[1]+gm[1]))*Dm
Dp_ = complex_extended_lyapunov(1/bt, 1/(al[1]+gm[1]), X.P, X.P@[email protected])
return 1/al[0]*Bo, 1/gm[0]*Co, Dp_ + Dm_
def testNTgN(man, X):
m, n, p = (man.m, man.n, man.p)
B0 = crandn(m-p, p)
C0 = crandn(n-p, p)
D0 = crandn(p, p)
out1 = NTgN @ np.concatenate(
[cvec(B0), cvec(C0), cvec(D0)])
out2a = NTgN_opt(X, B0, C0, D0)
out2 = np.concatenate(
[cvec(out2a[0]), cvec(out2a[1]), cvec(out2a[2])])
print(check_zero(out1-out2))
out2b = solveNTgN(X, *out2a)
print(check_zero(out2b[2]-D0))
print(check_zero(out2b[1]-C0))
print(check_zero(out2b[0]-B0))
Bs, Cs, Ds = solveNTgN(X, B, C, D)
# the formulas Using the orthogonal complements of U and V
# Dsm = ahsym(Ds)
# Dsp = hsym(Ds)
Bf = U0.T.conj() @ omg.tU
Cf = V0.T.conj() @ omg.tV
print(check_zero(Bf-Bs))
print(check_zero(Cf-Cs))
Dfm = 1/(al[1]+gm[1])*ahsym(X.V.T.conj()@omg.tV - X.U.T.conj()@omg.tU)
UTomg = X.U.T.conj()@omg.tU
VTomg = X.V.T.conj()@omg.tV
Dfp = complex_extended_lyapunov(
1/bt, 1/(al[1]+gm[1]), X.P,
hsym(omg.tP + 1/(al[1]+gm[1])*(
al[1]*([email protected] - X.P@UTomg) +
gm[1]*([email protected] - X.P@VTomg))), X.evl, X.evec)
ee1 = N(man, X, Bf, Cf, Dfp+Dfm)
print(check_zero(Ds - Dfm - Dfp))
print(check_zero(man._vec(ee1-eta)))
