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)
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_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 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_geodesics():
from scipy.linalg import expm
alpha = randint(1, 10, 2) * .1
gamma = randint(1, 10, 2) * .1
beta = randint(1, 10, 2)[0] * .1
m = 5
n = 6
p = 3
man = ComplexFixedRank(m, n, p, alpha=alpha, beta=beta, gamma=gamma)
X = man.rand()
alf = alpha[1]/alpha[0]
gmm = gamma[1]/gamma[0]
def calc_Christoffel_Gamma(man, X, xi, eta):
dprj = man.D_proj(X, xi, eta)
proj_christoffel = man.proj_g_inv(
X, man.christoffel_form(X, xi, eta))
return proj_christoffel - dprj
eta = man.randvec(X)
g1 = calc_Christoffel_Gamma(man, X, eta, eta)
g2 = man.christoffel_gamma(X, eta, eta)
print(man._vec(g1-g2))
egrad = man._rand_ambient()
print(man.base_inner_ambient(g1, egrad))
print(man.rhess02_alt(X, eta, eta, egrad, 0))
print(man.rhess02(X, eta, eta, egrad, man.zerovec(X)))
t = 2
AU = X.U.T.conj() @ eta.tU
KU = eta.tU - X.U @ (X.U.T.conj() @ eta.tU)
Up, RU = np.linalg.qr(KU)
xU_mat = np.bmat([[2*alf*AU, -RU.T.conj()], [RU, zeros((p, p))]])
Ut = np.bmat([X.U, Up]) @ expm(t*xU_mat)[:, :p] @ \
expm(t*(1-2*alf)*AU)
xU_d_mat = xU_mat[:, :p].copy()
xU_d_mat[:p, :] += (1-2*alf) * AU
Udt = np.bmat([X.U, Up]) @ expm(t*xU_mat) @ xU_d_mat @\
expm(t*(1-2*alf)*AU)
xU_dd_mat = xU_mat @ xU_d_mat + xU_d_mat @ ((1-2*alf)*AU)
Uddt = np.bmat([X.U, Up]) @ expm(t*xU_mat) @ xU_dd_mat @\
expm(t*(1-2*alf)*AU)
AV = X.V.T.conj() @ eta.tV
KV = eta.tV - X.V @ (X.V.T.conj() @ eta.tV)
Vp, RV = np.linalg.qr(KV)
xV_mat = np.bmat([[2*gmm*AV, -RV.T.conj()], [RV, zeros((p, p))]])
Vt = np.bmat([X.V, Vp]) @ expm(t*xV_mat)[:, :p] @ \
expm(t*(1-2*gmm)*AV)
xV_d_mat = xV_mat[:, :p].copy()
xV_d_mat[:p, :] += (1-2*gmm) * AV
Vdt = np.bmat([X.V, Vp]) @ expm(t*xV_mat) @ xV_d_mat @\
expm(t*(1-2*gmm)*AV)
xV_dd_mat = xV_mat @ xV_d_mat + xV_d_mat @ ((1-2*gmm)*AV)
Vddt = np.bmat([X.V, Vp]) @ expm(t*xV_mat) @ xV_dd_mat @\
expm(t*(1-2*gmm)*AV)
sqrtP = X.evec @ np.diag(np.sqrt(X.evl)) @ X.evec.T.conj()
isqrtP = X.evec @ np.diag(1/np.sqrt(X.evl)) @ X.evec.T.conj()
Pinn = t*[email protected]@isqrtP
ePinn = expm(Pinn)
Pt = sqrtP@ePinn@sqrtP
Pdt = eta.tP@isqrtP@ePinn@sqrtP
Pddt = eta.tP@isqrtP@ ePinn@[email protected]
Xt = fr_point(np.array(Ut),
np.array(Vt),
np.array(Pt))
Xdt = fr_ambient(np.array(Udt),
np.array(Vdt),
np.array(Pdt))
Xddt = fr_ambient(np.array(Uddt),
np.array(Vddt),
np.array(Pddt))
gcheck = Xddt + calc_Christoffel_Gamma(man, Xt, Xdt, Xdt)
print(man._vec(gcheck))
Xt1 = man.exp(X, t*eta)
print((Xt1.U - Xt.U))
print((Xt1.V - Xt.V))
print((Xt1.P - Xt.P))
def ehess(S, xi):
return fr_ambient(-2*A @ (xi.tV @ S.P + S.V @ xi.tP),
-2*A.T.conj() @ (xi.tU @S.P + [email protected]),
2*(xi.tP - xi.tU.T.conj()@[email protected] -
S.U.T.conj()@[email protected]))
def egrad(S):
return fr_ambient(-2*A @ S.V @ S.P,
-2*A.T.conj() @ S.U @S.P,
2*(S.P-S.U.T.conj() @ A @ S.V))
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 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_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)))