def test_vectorize():
from ManNullRange.manifolds.tools import cvech, cunvech, cvecah, cunvecah
p = 10
mat = crandn(p, p)
mat += mat.T.conjugate()
v = cvech(mat)
mat2 = cunvech(v)
print(check_zero(mat - mat2))
# check inner product:
for i in range(p * p):
v = zeros(p * p)
v[i] = 1
h = cunvech(v)
assert (np.abs(trace(h @ h.T.conjugate()).real - 1) < 1e-10)
# now do skew
mat = crandn(p, p)
mat -= mat.T.conjugate()
v = cvecah(mat)
mat2 = cunvecah(v)
print(check_zero(mat - mat2))
for i in range(p * p):
v = zeros(p * p)
v[i] = 1
h = cunvecah(v)
assert (np.abs(trace(h @ h.T.conjugate()).real - 1) < 1e-10)
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
n = 5
d = 3
man = ComplexStiefel(n, d, alpha=alpha)
Y = man.rand()
xi = man.randvec(Y)
aa = crandn(n * d, n * d)
bb = crandn(n * d)
def v_func_flat(Y):
return (aa @ Y.reshape(-1) + bb).reshape(n, d)
vv = v_func_flat(Y)
dlt = 1e-7
Ynew = Y + dlt * xi
vnew = v_func_flat(Ynew)
val = man.inner(Y, vv, vv)
valnew = man.inner(Ynew, vnew, vnew)
d1 = (valnew - val) / dlt
dv = (vnew - vv) / dlt
nabla_xi_v = dv + man.g_inv(Y, man.christoffel_form(Y, xi, vv))
d2 = man.inner(Y, vv, nabla_xi_v)
print(d1)
print(2 * d2)
def optim_test():
m, n, p = (1000, 500, 50)
# m, n, p = (10, 3, 2)
# simple function. Distance to a given matrix
# || S - A||_F^2
U0, _ = la.qr(crandn(m, p))
V0, _ = la.qr(crandn(n, p))
P0 = np.diag(randint(1, 1000, p)*.001)
A0 = U0 @ P0 @ V0.T.conj()
A = crandn(m, n)*1e-2 + A0
alpha = np.array([1, 1])
gamma = np.array([1, 1])
print("alpha %s" % str(alpha))
beta = alpha[1] * .1
man = ComplexFixedRank(m, n, p, alpha=alpha, beta=beta, gamma=gamma)
XInit = man.rand()
opt_pre = solve_dist_with_man(man, A, X0=XInit, maxiter=20)
beta = alpha[1] * 1
man = ComplexFixedRank(m, n, p, alpha=alpha, beta=beta, gamma=gamma)
opt_mid = solve_dist_with_man(man, A, X0=opt_pre, maxiter=20)
# opt_mid = opt_pre
beta = alpha[1] * 30
man = ComplexFixedRank(m, n, p, alpha=alpha, beta=beta, gamma=gamma)
opt = solve_dist_with_man(man, A, X0=opt_mid, maxiter=50)
opt_mat = opt.U @ opt.P @ opt.V.T.conj()
if False:
print(A0)
print(opt_mat)
print(np.max(np.abs(A0-opt_mat)))
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)
"""
dvec = np.array([10, 3, 2, 3])
p = dvec.shape[0] - 1
alpha = randint(1, 10, (p, p + 1)) * .1
man = ComplexFlag(dvec, alpha=alpha)
Y = man.rand()
n = man.n
d = man.d
xi = man.randvec(Y)
aa = crandn(n * d, n * d)
bb = crandn(n * d)
def v_func_flat(Y):
return (aa @ Y.reshape(-1) + bb).reshape(n, d)
vv = v_func_flat(Y)
dlt = 1e-7
Ynew = Y + dlt * xi
vnew = v_func_flat(Ynew)
val = man.inner(Y, vv, vv)
valnew = man.inner(Ynew, vnew, vnew)
d1 = (valnew - val) / dlt
dv = (vnew - vv) / dlt
nabla_xi_v = dv + man.g_inv(Y, man.christoffel_form(Y, xi, vv))
d2 = man.inner(Y, 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
man = ComplexStiefel(n, d, alpha=alpha)
Y = man.rand()
slp = crandn(n * d)
aa = crandn(n * d, n * d)
def omg_func(Y):
return (aa @ Y.reshape(-1) + slp).reshape(n, d)
xi = man.randvec(Y)
egrad = omg_func(Y)
ehess = (aa @ xi.reshape(-1)).reshape(n, d)
val1 = man.ehess2rhess(Y, egrad, ehess, xi)
if False:
val1a = man.ehess2rhess_alt(Y, egrad, ehess, xi)
print(check_zero(val1 - val1a))
def rgrad_func(W):
return man.proj_g_inv(W, omg_func(W))
if False:
first = ehess
a = man.J(Y, man.g_inv(Y, egrad))
rgrad = man.proj_g_inv(Y, egrad)
second = -man.D_g(Y, xi, man.g_inv(Y, egrad))
aout = man.solve_J_g_inv_Jst(Y, a)
third = -man.proj(Y, man.D_g_inv_Jst(Y, xi, aout))
fourth = man.christoffel_form(Y, xi, rgrad)
val1a = man.proj_g_inv(Y, first + second + fourth) + third
d_xi_rgrad = num_deriv(man, Y, xi, rgrad_func)
rgrad = man.proj_g_inv(Y, egrad)
fourth = man.christoffel_form(Y, xi, rgrad)
val1b = man.proj(Y, d_xi_rgrad) + man.proj_g_inv(Y, fourth)
print(check_zero(val1 - val1b))
# nabla_v_xi, dxi, cxxi
val2a, _, _ = calc_covar_numeric(man, Y, xi, omg_func)
val2, _, _ = calc_covar_numeric(man, Y, xi, rgrad_func)
# val2_p = project(prj, val2)
val2_p = man.proj(Y, val2)
# print(val1)
# print(val2_p)
print(val1 - val2_p)
def testN(man, X):
m, n, p = (man.m, man.n, man.p)
B = crandn(m-p, p)
C = crandn(n-p, p)
D = crandn(p, p)
ee = N(man, X, B, C, D)
print(check_zero(stU(X, ee)))
print(check_zero(stV(X, ee)))
print(check_zero(symP(X, ee)))
print(check_zero(Hz(man, X, ee)))
nmat = make_N_mat(man, X)
ee2 = man._unvec(nmat @ np.concatenate(
[cvec(B), cvec(C), cvec(D)]))
print(check_zero(man._vec(ee - ee2)))
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))
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 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_chris_vectorfields():
# now test that it works on embedded metrics
# we test that D_xi (eta g eta) = 2(eta g nabla_xi eta)
dvec = np.array([10, 3, 2, 3])
p = dvec.shape[0] - 1
alpha = randint(1, 10, (p, p + 1)) * .1
man = ComplexFlag(dvec, alpha=alpha)
n = man.n
d = man.d
slp = crandn(n * d)
Y0 = man.rand()
slpxi = crandn(n * d)
aa = crandn(n * d, n * d)
aaxi = crandn(n * d, n * d)
def v_func(Y):
return man.proj(Y, (aa @ (Y - Y0).reshape(-1) + slp).reshape(n, d))
YY = Y0.copy()
xi = man.proj(YY, slpxi.reshape(n, d))
nabla_xi_v, dv, cxv = calc_covar_numeric(man, YY, xi, v_func)
def xi_func(Y):
return man.proj(Y, (aaxi @ (Y - Y0).reshape(-1) + slpxi).reshape(n, d))
vv = v_func(YY)
nabla_v_xi, dxi, cxxi = calc_covar_numeric(man, YY, vv, xi_func)
diff = nabla_xi_v - nabla_v_xi
# print(diff)
# now do Lie bracket:
dlt = 1e-7
YnewXi = YY + dlt * xi
Ynewvv = YY + dlt * vv
vnewxi = v_func(YnewXi)
xnewv = xi_func(Ynewvv)
dxiv = (vnewxi - vv) / dlt
dvxi = (xnewv - xi) / dlt
diff2 = man.proj(YY, dxiv - dvxi)
print(check_zero(diff - diff2))
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 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 test_lyapunov():
alpha = randint(1, 10, 2) * .1
beta = randint(1, 10, 1)[0] * .02
n = 5
d = 3
man = ComplexPositiveSemidefinite(n, d, alpha=alpha, beta=beta)
S = man.rand()
P = S.P
B = crandn(d, d)
alpha1 = alpha[1]
def L(X, P):
Piv = la.inv(P)
return (alpha1 - 2 * beta) * X + beta * (P @ X @ Piv + Piv @ X @ P)
X = extended_lyapunov(alpha1, beta, P, B)
# L(X, P)
print(check_zero(B - L(X, P)))
def test_rhess_02():
n, d = (5, 3)
alpha = randint(1, 10, 2) * .1
man = ComplexStiefel(n, d, alpha=alpha)
Y = man.rand()
UU = make_sym_pos(n)
def f(Y):
return rtrace(UU @ Y @ Y.T.conj())
def df(Y):
return 2 * UU @ Y
def ehess_form(Y, xi, eta):
return 2 * rtrace(UU @ xi @ eta.T.conj())
def ehess_vec(Y, xi):
return 2 * UU @ xi
xxi = crandn(n, d)
dlt = 1e-8
Ynew = Y + dlt * xxi
d1 = (f(Ynew) - f(Y)) / dlt
d2 = df(Y)
print(d1 - rtrace(d2 @ xxi.T.conj()))
eeta = crandn(n, d)
d1 = rtrace((df(Ynew) - df(Y)) @ eeta.T.conj()) / dlt
ehess_val = ehess_form(Y, xxi, eeta)
dv2 = ehess_vec(Y, xxi)
print(rtrace(dv2 @ eeta.T.conj()))
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)
def eta_field(Yin):
return m1 @ (Yin - Y) @ m2 + eeta
# xietaf: should go to ehess(xi, eta) + df(Y) @ etafield)
xietaf = rtrace(
df(Ynew) @ eta_field(Ynew).T.conj() -
df(Y) @ eta_field(Y).T.conj()) / dlt
# appy eta_func to f: should go to tr(m1 @ xxi @ m2 @ df(Y).T.conj())
Dxietaf = rtrace((eta_field(Ynew) - eta_field(Y)) @ df(Y).T.conj()) / 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(Y, xxi)
eta1 = man.proj(Y, eeta)
egvec = df(Y)
ehvec = ehess_vec(Y, xi1)
rhessvec = man.ehess2rhess(Y, egvec, ehvec, xi1)
# check it numerically:
def rgrad_func(Y):
return man.proj_g_inv(Y, df(Y))
val2, _, _ = calc_covar_numeric(man, Y, xi1, rgrad_func)
val2_p = man.proj(Y, val2)
# print(rhessvec)
# print(val2_p)
print(check_zero(rhessvec - val2_p))
rhessval = man.inner(Y, rhessvec, eta1)
print(man.inner(Y, val2, eta1))
print(rhessval)
# check symmetric:
ehvec_e = ehess_vec(Y, eta1)
rhessvec_e = man.ehess2rhess(Y, egvec, ehvec_e, eta1)
rhessval_e = man.inner(Y, rhessvec_e, xi1)
rhessval_e1 = man.rhess02(Y, xi1, eta1, egvec, ehvec)
rhessval_e2 = man.rhess02_alt(Y, xi1, eta1, egvec,
rtrace(ehvec @ eta1.T.conj()))
print(rhessval_e, rhessval_e1, rhessval_e2)
print('rhessval_e %f ' % 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(Y):
return man.proj(Y, eta_field(Y))
print(check_zero(eta1 - eta_proj(Y)))
e1 = man.inner(Y, man.proj_g_inv(Y, df(Y)), eta_proj(Y))
e1a = rtrace(df(Y) @ eta_proj(Y).T.conj())
print(e1, e1a, e1 - e1a)
Ynew = Y + xi1 * dlt
e2 = man.inner(Ynew, man.proj_g_inv(Ynew, df(Ynew)), eta_proj(Ynew))
e2a = rtrace(df(Ynew) @ eta_proj(Ynew).T.conj())
print(e2, e2a, e2 - e2a)
first = (e2 - e1) / dlt
first1 = rtrace(
df(Ynew) @ eta_proj(Ynew).T.conj() -
df(Y) @ eta_proj(Y).T.conj()) / dlt
print(first - first1)
val3, _, _ = calc_covar_numeric(man, Y, xi1, eta_proj)
second = man.inner(Y, man.proj_g_inv(Y, df(Y)), man.proj(Y, val3))
second2 = man.inner(Y, man.proj_g_inv(Y, df(Y)), val3)
print(second, second2, second - second2)
print('same as rhess_val %f' % (first - second))
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
# 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 test_rhess_02():
np.random.seed(0)
dvec = np.array([10, 3, 2, 3])
p = dvec.shape[0] - 1
alpha = randint(1, 10, (p, p + 1)) * .1
man = ComplexFlag(dvec, alpha=alpha)
n = man.n
d = man.d
Y = man.rand()
UU = {}
p = alpha.shape[0]
VV = {}
gidx = man._g_idx
for rr in range(p):
UU[rr] = make_sym_pos(n)
VV[rr] = crandn(n, dvec[rr + 1])
def f(Y):
ss = 0
for rr in range(p):
br, er = gidx[rr + 1]
wr = Y[:, br:er]
ss += trace(UU[rr] @ wr @ wr.T.conjugate()).real
return ss
def df(W):
ret = np.zeros_like(W)
for rr in range(p):
br, er = gidx[rr + 1]
wr = W[:, br:er]
ret[:, br:er] += 2 * UU[rr] @ wr
return ret
def ehess_form(W, xi, eta):
ss = 0
for rr in range(p):
br, er = gidx[rr + 1]
ss += 2 * trace(
UU[rr] @ xi[:, br:er] @ eta[:, br:er].T.conjugate()).real
return ss
def ehess_vec(W, xi):
ret = np.zeros_like(W)
for rr in range(p):
br, er = gidx[rr + 1]
ret[:, br:er] += 2 * UU[rr] @ xi[:, br:er]
return ret
xxi = crandn(n, d)
dlt = 1e-8
Ynew = Y + dlt * xxi
d1 = (f(Ynew) - f(Y)) / dlt
d2 = df(Y)
print(d1 - trace(d2 @ xxi.T.conjugate()).real)
eeta = crandn(n, d)
d1 = trace((df(Ynew) - df(Y)) @ eeta.T.conjugate()).real / dlt
ehess_val = ehess_form(Y, xxi, eeta)
# ehess_val2 = ehess_form(Y, eeta, xxi)
dv2 = ehess_vec(Y, xxi)
print(trace(dv2 @ eeta.T.conjugate()).real)
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)
def eta_field(Yin):
return m1 @ (Yin - Y) @ m2 + eeta
# xietaf: should go to ehess(xi, eta) + df(Y) @ etafield)
xietaf = trace(
df(Ynew) @ eta_field(Ynew).T.conjugate() -
df(Y) @ eta_field(Y).T.conjugate()).real / dlt
# appy eta_func to f: should go to tr(m1 @ xxi @ m2 @ df(Y).T.conjugate())
Dxietaf = trace(
(eta_field(Ynew) - eta_field(Y)) @ df(Y).T.conjugate()).real / 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(Y, xxi)
eta1 = man.proj(Y, eeta)
egvec = df(Y)
ehvec = ehess_vec(Y, xi1)
rhessvec = man.ehess2rhess(Y, egvec, ehvec, xi1)
# check it numerically:
def rgrad_func(Y):
return man.proj_g_inv(Y, df(Y))
val2, _, _ = calc_covar_numeric(man, Y, xi1, rgrad_func)
val2_p = man.proj(Y, val2)
# print(rhessvec)
# print(val2_p)
print(check_zero(rhessvec - val2_p))
rhessval = man.inner(Y, rhessvec, eta1)
print(man.inner(Y, val2, eta1))
print(rhessval)
# check symmetric:
ehvec_e = ehess_vec(Y, eta1)
ehess_valp = ehess_form(Y, xi1, eta1)
rhessvec_e = man.ehess2rhess(Y, egvec, ehvec_e, eta1)
rhessval_e = man.inner(Y, rhessvec_e, xi1)
rhessval_e1 = man.rhess02(Y, xi1, eta1, egvec, ehess_valp)
# rhessval_e2 = man.rhess02_alt(Y, xi1, eta1, egvec,
# trace([email protected]()).real)
# print(rhessval_e, rhessval_e1, rhessval_e2)
print(rhessval_e, rhessval_e1, rhessval_e - rhessval_e1)
print('rhessval_e %f ' % 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(Y):
return man.proj(Y, eta_field(Y))
print(check_zero(eta1 - eta_proj(Y)))
e1 = man.inner(Y, man.proj_g_inv(Y, df(Y)), eta_proj(Y))
e1a = trace(df(Y) @ eta_proj(Y).T.conjugate()).real
print(e1, e1a, e1 - e1a)
Ynew = Y + xi1 * dlt
e2 = man.inner(Ynew, man.proj_g_inv(Ynew, df(Ynew)), eta_proj(Ynew))
e2a = trace(df(Ynew) @ eta_proj(Ynew).T.conjugate()).real
print(e2, e2a, e2 - e2a)
first = (e2 - e1) / dlt
first1 = trace(
df(Ynew) @ eta_proj(Ynew).T.conjugate() -
df(Y) @ eta_proj(Y).T.conjugate()).real / dlt
print(first - first1)
val3, _, _ = calc_covar_numeric(man, Y, xi1, eta_proj)
second = man.inner(Y, man.proj_g_inv(Y, df(Y)), man.proj(Y, val3))
second2 = man.inner(Y, man.proj_g_inv(Y, df(Y)), val3)
print(second, second2, second - second2)
print('same as rhess_val %f' % (first - second))
def test_geodesics():
from scipy.linalg import expm
alpha = np.random.randint(1, 10, (2)) * .1
# alpha = np.array([1, .5])
m, d = (5, 3)
man = ComplexStiefel(m, d, alpha=alpha)
Y = man.rand()
alf = alpha[1] / alpha[0]
def calc_gamma(man, Y, eta):
etaxiy = 2 * eta @ (eta.T.conj() @ Y)
egcoef = Y @ (eta.T.conj() @ eta)
ft = 1 - alf
egcoef += ft * (etaxiy - Y @ (Y.T.conj() @ etaxiy))
return egcoef
eta = man.randvec(Y)
g1 = calc_gamma(man, Y, eta)
g2 = man.christoffel_gamma(Y, eta, eta)
print(g1 - g2)
egrad = crandn(m, d)
print(rtrace(g1 @ egrad.T.conj()))
print(man.rhess02_alt(Y, eta, eta, egrad, 0))
# try to see if the solution is good:
A = Y.T.conj() @ eta
S0 = eta.T.conj() @ eta
e_mat = np.bmat([[(2 * alf - 1) * A, -S0 - 2 * (1 - alf) * A @ A],
[np.eye(d), A]])
init_c = np.bmat([Y, eta])
def ff(t):
v1 = init_c @ expm(t * e_mat)
v2 = expm(t * (1 - 2 * alf) * A)
return v1[:, :d] @ v2, v1[:, d:] @ v2, None
t = 100
dlt = 1e-8
fval, fd, fdd = ff(t)
fval1, fd1, fdd1 = ff(t + dlt)
fval2, fd2, fdd2 = ff(t - dlt)
ffdot = (fval1 - fval) / dlt
print(check_zero(ffdot - fd))
ffddot_0 = (fval1 - 2 * fval + fval2) / (dlt * dlt)
ffddot = (fd1 - fd) / dlt
print(ffddot_0)
print(ffddot)
print(ffddot + calc_gamma(man, fval, ffdot))
# second solution:
K = eta - Y @ (Y.T.conj() @ eta)
Yp, R = np.linalg.qr(K)
x_mat = np.bmat([[2 * alf * A, -R.T.conj()], [R, zeros((d, d))]])
Yt = np.bmat([Y, Yp]) @ expm(t*x_mat)[:, :d] @ \
expm(t*(1-2*alf)*A)
x_d_mat = x_mat[:, :d].copy()
x_d_mat[:d, :] += (1 - 2 * alf) * A
Ydt = np.bmat([Y, Yp]) @ expm(t*x_mat) @ x_d_mat @\
expm(t*(1-2*alf)*A)
x_dd_mat = x_mat @ x_d_mat + x_d_mat @ ((1 - 2 * alf) * A)
Yddt = np.bmat([Y, Yp]) @ expm(t*x_mat) @ x_dd_mat @\
expm(t*(1-2*alf)*A)
print(Yddt + calc_gamma(man, Yt, Ydt))
print(man.exp(Y, t * eta) - Yt)
def test_rhess_02():
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()
# simple function. Distance to a given matrix
# || S - A||_F^2 Basically SVD
A = crandn(m, n)
def f(S):
diff = (A - S.U @ S.P @ S.V.T.conj())
return rtrace(diff @ diff.T.conj())
def df(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 ehess_vec(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 ehess_form(S, xi, eta):
ev = ehess_vec(S, xi)
return rtrace(ev.tU.T.conj() @ eta.tU) +\
rtrace(ev.tV.T.conj() @ eta.tV) +\
rtrace(ev.tP.T.conj() @ eta.tP)
xxi = man.randvec(S)
dlt = 1e-8
Snew = fr_point(
S.U+dlt*xxi.tU,
S.V+dlt*xxi.tV,
S.P + dlt*xxi.tP)
d1 = (f(Snew) - f(S))/dlt
d2 = df(S)
print(d1 - man.base_inner_ambient(d2, xxi))
dv1 = (df(Snew) - df(S)).scalar_mul(1/dlt)
dv2 = ehess_vec(S, xxi)
print(man._vec(dv1-dv2))
eeta = man.randvec(S)
d1 = man.base_inner_ambient((df(Snew) - df(S)), eeta) / dlt
ehess_val = ehess_form(S, xxi, eeta)
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.
mU1 = crandn(m, m)
mV1 = crandn(n, n)
m2 = crandn(p, p)
m_p = crandn(p*p, p*p)
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
# 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.conj())
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(S, rhessvec, eta1)
print(man.inner(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(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 = fr_point(
S.U + dlt*xi1.tU,
S.V + dlt*xi1.tV,
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 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)))
