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_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_gamma(): dvec = np.array([0, 30, 2, 1]) p = dvec.shape[0] - 1 alpha = randint(1, 10, (p, p + 1)) * .1 dvec[0] = 1000 - dvec[1:].sum() man = ComplexFlag(dvec, alpha=alpha) X = man.rand() eta = man.randvec(X) g2 = man.christoffel_gamma(X, eta, eta) egrad = man._rand_ambient() ehess = man._rand_ambient() ehess_val = man.base_inner_ambient(ehess, eta) print(man.rhess02_alt(X, eta, eta, egrad, ehess_val)) print(man.rhess02(X, eta, eta, egrad, ehess)) print(ehess_val - man.base_inner_ambient(g2, egrad)) print(man.exp(X, eta))
def optim_test3(): from pymanopt import Problem from pymanopt.solvers import TrustRegions from pymanopt.function import Callable n = 1000 # problem Tr(AXBX^T) for i in range(1): # Stiefel manifold dvec = np.array([0, 50]) dvec[0] = n - dvec[1:].sum() d = dvec[1:].sum() p = dvec.shape[0] - 1 alpha = randint(1, 10, (p, p + 1)) * .1 alpha0 = randint(1, 10, (p)) # alpha0 = randint(1, 2, (p)) alpha = make_simple_alpha(alpha0, 0) B = np.diag(np.concatenate([randint(1, 10, d), np.zeros(n - d)])) D = randint(1, 10, n) * 0.02 + 1 OO = random_orthogonal(n) A = OO @ np.diag(D) @ OO.T man = ComplexFlag(dvec, alpha) cf = 10 B2 = B @ B @Callable def cost(X): return cf * trace( B @ X @ st(X) @ B2 @ X @ st(X) @ B) +\ trace(st(X) @ A @ X) @Callable def egrad(X): R = cf * 4 * B2 @ X @ st(X) @ B2 @ X + 2 * A @ X return R @Callable def ehess(X, H): return 4*cf*B2 @ H @ st(X) @ B2 @ X +\ 4*cf*B2 @ X @ st(H) @ B2 @ X +\ 4*cf*B2 @ X @ st(X) @ B2 @ H + 2*A @ H if False: X = man.rand() xi = man.randvec(X) d1 = num_deriv(man, X, xi, cost) d2 = trace(egrad(X) @ st(xi)).real print(check_zero(d1 - d2)) d3 = num_deriv(man, X, xi, egrad) d4 = ehess(X, xi) print(check_zero(d3 - d4)) XInit = man.rand() prob = Problem(man, cost, egrad=egrad, ehess=ehess) solver = TrustRegions(maxtime=100000, maxiter=100) opt = solver.solve(prob, x=XInit, Delta_bar=2500) print(cost(opt)) if False: # print(opt) # double check: # print(cost(opt)) min_val = 1e190 # min_X = None for i in range(100): Xi = man.rand() c = cost(Xi) if c < min_val: # min_X = Xi min_val = c if i % 1000 == 0: print('i=%d min=%f' % (i, min_val)) print(min_val) alpha0 = randint(1, 2, (p)) alpha1 = make_simple_alpha(alpha0, 0) man1 = ComplexFlag(dvec, alpha=alpha1) prob = Problem(man1, cost, egrad=egrad, ehess=ehess) solver = TrustRegions(maxtime=100000, maxiter=100) opt = solver.solve(prob, x=XInit, Delta_bar=250) alpha0 = randint(1, 2, (p)) alpha2 = make_simple_alpha(alpha0, -1) man1 = ComplexFlag(dvec, alpha=alpha2) # man1 = RealStiefel(n, d, alpha=np.array([1, 1])) prob = Problem(man1, cost, egrad=egrad, ehess=ehess) solver = TrustRegions(maxtime=100000, maxiter=100) opt = solver.solve(prob, x=XInit, Delta_bar=250)
def optim_test(): from pymanopt import Problem from pymanopt.solvers import TrustRegions from pymanopt.function import Callable n = 300 # problem Tr(AXBX^T) for i in range(1): dvec = np.array([0, 5, 4, 10]) dvec[0] = n - dvec[1:].sum() d = dvec[1:].sum() alpha = randint(1, 10, 2) * .1 D = randint(1, 10, n) * 0.02 + 1 OO = random_orthogonal(n) A = OO @ np.diag(D) @ OO.T.conjugate() B = make_sym_pos(d) p = dvec.shape[0] - 1 alpha = randint(1, 10, (p, p + 1)) * .1 alpha0 = randint(1, 10, (p)) # alpha0 = randint(1, 2, (p)) alpha = make_simple_alpha(alpha0, 0) man = ComplexFlag(dvec, alpha) @Callable def cost(X): return trace(A @ X @ B @ X.T.conjugate()).real @Callable def egrad(X): return 2 * A @ X @ B @Callable def ehess(X, H): return 2 * A @ H @ B X = man.rand() if False: xi = man.randvec(X) d1 = num_deriv(man, X, xi, cost) d2 = trace(egrad(X) @ xi.T.conjugate()).real print(check_zero(d1 - d2)) prob = Problem(man, cost, egrad=egrad) XInit = man.rand() prob = Problem(man, cost, egrad=egrad, ehess=ehess) solver = TrustRegions(maxtime=100000, maxiter=100) opt = solver.solve(prob, x=XInit, Delta_bar=250) print(cost(opt)) alpha0 = randint(1, 2, (p)) alpha1 = make_simple_alpha(alpha0, 0) man1 = ComplexFlag(dvec, alpha=alpha1) prob = Problem(man1, cost, egrad=egrad, ehess=ehess) solver = TrustRegions(maxtime=100000, maxiter=100) opt = solver.solve(prob, x=XInit, Delta_bar=250) alpha0 = randint(1, 2, (p)) alpha2 = make_simple_alpha(alpha0, -1) man1 = ComplexFlag(dvec, alpha=alpha2) prob = Problem(man1, cost, egrad=egrad, ehess=ehess) solver = TrustRegions(maxtime=100000, maxiter=100) opt = solver.solve(prob, x=XInit, Delta_bar=250)
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_covariance_deriv(): # now test full: # do covariant derivatives # check that it works, preseving everything 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() slp = np.random.randn(n * d) aa = np.random.randn(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) def rgrad_func(W): return man.proj_g_inv(W, omg_func(W)) if False: d_xi_rgrad = num_deriv(man, Y, xi, rgrad_func) rgrad = man.proj_g_inv(Y, egrad) fourth = man.christoffel_form(Y, xi, rgrad) val1c = man.proj(Y, d_xi_rgrad) + man.proj_g_inv(Y, fourth) 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(check_zero(val1 - val2_p))
def test_all_projections(): 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() U = man._rand_ambient() Upr = man.proj(Y, U) test_inner(man, Y) test_J(man, Y) # now check metric, Jst etc # check Jst: vectorize the operator J then compare Jst with jmat.T.conjugate() jmat = make_j_mat(man, Y) test_Jst(man, Y, jmat) ginv_mat = make_g_inv_mat(man, Y) # test g_inv_Jst for ii in range(10): a = man._rand_range_J() avec = man._vec_range_J(a) jtout = man._unvec(ginv_mat @ jmat.T @ avec) jtout2 = man.g_inv_Jst(Y, a) diff = check_zero(jtout - jtout2) print(diff) # test projection test_projection(man, Y) for i in range(20): Uran = man._rand_ambient() Upr = man.proj(Y, man.g_inv(Y, Uran)) Upr2 = man.proj_g_inv(Y, Uran) print(check_zero(Upr - Upr2)) for ii in range(10): a = man._rand_range_J() xi = man._rand_ambient() jtout2 = man.Jst(Y, a) dlt = 1e-7 Ynew = Y + dlt * xi jtout2a = man.Jst(Ynew, a) d1 = (jtout2a - jtout2) / dlt d2 = man.D_Jst(Y, xi, a) print(check_zero(d2 - d1)) for ii in range(10): Y = man.rand() eta = man._rand_ambient() xi = man.randvec(Y) a1 = man.J(Y, eta) dlt = 1e-7 Ynew = Y + dlt * xi a2 = man.J(Ynew, eta) d1 = (man._vec_range_J(a2) - man._vec_range_J(a1)) / dlt d2 = man._vec_range_J(man.D_J(Y, xi, eta)) print(check_zero(d2 - d1)) for ii in range(10): a = man._rand_range_J() xi = man._rand_ambient() jtout2 = man.g_inv_Jst(Y, a) dlt = 1e-7 Ynew = Y + dlt * xi jtout2a = man.g_inv_Jst(Ynew, a) d1 = (jtout2a - jtout2) / dlt d2 = man.D_g_inv_Jst(Y, xi, a) print(check_zero(d2 - d1)) for ii in range(10): arand = man._rand_range_J() a2 = man.solve_J_g_inv_Jst(Y, arand) a1 = man.J(Y, man.g_inv_Jst(Y, a2)) print(check_zero(man._vec_range_J(a1) - man._vec_range_J(arand))) # derives for ii in range(10): Y1 = man.rand() xi = man.randvec(Y1) omg1 = man._rand_ambient() omg2 = man._rand_ambient() dlt = 1e-7 Y2 = Y1 + dlt * xi p1 = man.inner(Y1, omg1, omg2) p2 = man.inner(Y2, omg1, omg2) der1 = (p2 - p1) / dlt der2 = man.base_inner_ambient(man.D_g(Y1, xi, omg2), omg1) print(check_zero(der1 - der2)) # cross term for christofel for i in range(10): Y1 = man.rand() xi = man.randvec(Y1) omg1 = man._rand_ambient() omg2 = man._rand_ambient() dr1 = man.D_g(Y1, xi, omg1) x12 = man.contract_D_g(Y1, omg1, omg2) p1 = trace(dr1 @ omg2.T.conjugate()).real p2 = trace(x12 @ xi.T.conjugate()).real print(p1, p2, p1 - p2) # now test christofel: # two things: symmetric on vector fields # and christofel relation # in the case metric for i in range(10): Y1 = man.rand() xi = man.randvec(Y1) eta1 = man.randvec(Y1) eta2 = man.randvec(Y1) p1 = man.proj_g_inv(Y1, man.christoffel_form(Y1, xi, eta1)) p2 = man.proj_g_inv(Y1, man.christoffel_form(Y1, eta1, xi)) print(check_zero(p1 - p2)) v1 = man.base_inner_ambient(man.christoffel_form(Y1, eta1, eta2), xi) v2 = man.base_inner_ambient(man.D_g(Y1, eta1, eta2), xi) v3 = man.base_inner_ambient(man.D_g(Y1, eta2, eta1), xi) v4 = man.base_inner_ambient(man.D_g(Y1, xi, eta1), eta2) print(v1, 0.5 * (v2 + v3 - v4), v1 - 0.5 * (v2 + v3 - v4)) """