Example #1
0
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)
Example #2
0
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))
Example #3
0
def test_all_projections():
    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)
    print(man)
    S = man.rand()

    test_inner(man, S)
    test_J(man, S)

    # now check metric, Jst etc
    # check Jst: vectorize the operator J then compare Jst with jmat.T
    jmat = make_j_mat(man, S)
    test_Jst(man, S, jmat)
    ginv_mat = make_g_inv_mat(man, S)
    ee = man._rand_ambient()
    g1 = man._unvec(ginv_mat @ man._vec(ee))
    print(check_zero(man.g_inv(S, ee).tP - g1.tP))
    print(check_zero(man.g_inv(S, ee).tY - g1.tY))
    # test g_inv_Jst
    for ii in range(10):
        a = man._rand_range_J()
        avec = man._vec_range_J(a)
        jtout = ginv_mat @ jmat.T @ avec
        jtout2 = man._vec(man.g_inv_Jst(S, a))
        diff = check_zero(jtout - jtout2)
        print(diff)
    # test projection
    test_projection(man, S)
    # now diff projection

    for i in range(1):
        e = man._rand_ambient()
        S1 = man.rand()
        xi = man.randvec(S1)
        dlt = 1e-7
        S2 = psd_point(S1.Y + dlt * xi.tY, S1.P + dlt * xi.tP)

        # S = psd_point(S1.Y, S1.P)
        d1P = (man.proj(S2, e).tP - man.proj(S1, e).tP) / dlt
        d1Y = (man.proj(S2, e).tY - man.proj(S1, e).tY) / dlt
        d2 = man.D_proj(S1, xi, e)
        print(check_zero(d1P - d2.tP) + check_zero(d1Y - d2.tY))

    for i in range(10):
        a = man._rand_range_J()
        eta = man._rand_ambient()
        print(man.base_inner_ambient(eta, man.Jst(S, a)))
        print((trace((eta.tP.T.conjugate() - eta.tP) @ a['P'] +
                     (man.alpha[1] * eta.tY.T.conjugate() @ S.Y + man.beta *
                      (S.Pinv @ eta.tP.T.conjugate() -
                       eta.tP.T.conjugate() @ S.Pinv)) @ a['YP'])).real)

        print((trace(2 * eta.tP.T.conjugate() @ a['P']) + trace(
            (man.alpha[1] * eta.tY.T.conjugate() @ S.Y + man.beta *
             (S.Pinv @ eta.tP.T.conjugate() - eta.tP.T.conjugate() @ S.Pinv))
            @ a['YP'])).real)

        print((trace(
            eta.tP.T.conjugate() @ (2 * a['P'] + man.beta *
                                    (a['YP'] @ S.Pinv - S.Pinv @ a['YP']))) +
               trace(eta.tY.T.conjugate() @ (man.alpha[1] * S.Y @ a['YP']))
               ).real)

        print(man.base_inner_E_J(man.J(S, eta), a))
        print((
            trace((eta.tP - eta.tP.T.conjugate()).T.conjugate() @ a['P'] +
                  (man.alpha[1] * S.Y.T.conjugate() @ eta.tY + man.beta *
                   (eta.tP @ S.Pinv - S.Pinv @ eta.tP)).T.conjugate() @ a['YP']
                  )).real)

    for i in range(10):
        a = man._rand_range_J()
        beta = man.beta
        alf = man.alpha
        anew1 = man.J(S, man.g_inv_Jst(S, a))

        anew = {}
        saYP = a['YP'] + a['YP'].T.conjugate()
        anew['P'] = 4 / beta * S.P @ a['P'] @ S.P + S.P @ saYP - saYP @ S.P
        anew['YP'] = alf[1] * a['YP'] + beta * (
            ((2 / man.beta) * S.P @ a['P'] @ S.P + S.P @ a['YP'] -
             a['YP'] @ S.P) @ S.Pinv - S.Pinv @ (
                 (2 / man.beta) * S.P @ a['P'] @ S.P + S.P @ a['YP'] -
                 a['YP'] @ S.P))

        anew['YP'] = alf[1] * a['YP'] + (
            (2 * S.P @ a['P'] + beta * S.P @ a['YP'] @ S.Pinv - beta * a['YP'])
            - (2 * a['P'] @ S.P + beta * a['YP'] -
               beta * S.Pinv @ a['YP'] @ S.P))

        anew['YP'] = (alf[1] - 2 * beta) * a['YP'] + (
            (2 * S.P @ a['P'] + beta * S.P @ a['YP'] @ S.Pinv) -
            (2 * a['P'] @ S.P - beta * S.Pinv @ a['YP'] @ S.P))

        anew['YP'] = (alf[1] - 2 * beta) * a['YP'] + (
            (2 * S.P @ a['P'] - 2 * a['P'] @ S.P +
             beta * S.P @ a['YP'] @ S.Pinv + beta * S.Pinv @ a['YP'] @ S.P))
        print(check_zero(man._vec_range_J(anew1) - man._vec_range_J(anew)))

    for i in range(10):
        a = man._rand_range_J()
        b1 = man.J(S, man.g_inv_Jst(S, a))
        b2 = man.J_g_inv_Jst(S, a)
        print(check_zero(man._vec_range_J(b1) - man._vec_range_J(b2)))
        a1 = man.solve_J_g_inv_Jst(S, b1)
        print(check_zero(man._vec_range_J(a) - man._vec_range_J(a1)))

    for ii in range(10):
        E = man._rand_ambient()
        a2 = man.J_g_inv(S, E)
        a1 = man.J(S, man.g_inv(S, E))
        print(check_zero(man._vec_range_J(a1) - man._vec_range_J(a2)))

    for i in range(20):
        Uran = man._rand_ambient()
        Upr = man.proj(S, man.g_inv(S, Uran))
        Upr2 = man.proj_g_inv(S, Uran)
        print(check_zero(man._vec(Upr) - man._vec(Upr2)))

    for ii in range(10):
        a = man._rand_range_J()
        xi = man.randvec(S)
        jtout2 = man.Jst(S, a)
        dlt = 1e-7
        Snew = psd_point(S.Y + dlt * xi.tY, S.P + dlt * xi.tP)
        jtout2a = man.Jst(Snew, a)
        d1 = (jtout2a - jtout2).scalar_mul(1 / dlt)
        d2 = man.D_Jst(S, xi, a)
        print(check_zero(man._vec(d2) - man._vec(d1)))

    for ii in range(10):
        S1 = man.rand()
        eta = man._rand_ambient()
        xi = man.randvec(S1)
        a1 = man.J(S1, eta)
        dlt = 1e-8
        Snew = psd_point(S1.Y + dlt * xi.tY, S1.P + dlt * xi.tP)
        a2 = man.J(Snew, eta)
        d1 = {
            'P': (a2['P'] - a1['P']) / dlt,
            'YP': (a2['YP'] - a1['YP']) / dlt
        }
        d2 = man.D_J(S1, xi, eta)
        print(check_zero(man._vec_range_J(d2) - man._vec_range_J(d1)))

    # derives metrics
    for ii in range(10):
        S1 = man.rand()
        xi = man.randvec(S1)
        omg1 = man._rand_ambient()
        omg2 = man._rand_ambient()
        dlt = 1e-7
        S2 = psd_point(S1.Y + dlt * xi.tY, S1.P + dlt * xi.tP)
        p1 = man.inner(S1, omg1, omg2)
        p2 = man.inner(S2, omg1, omg2)
        der1 = (p2 - p1) / dlt
        der2 = man.base_inner_ambient(man.D_g(S1, xi, omg2), omg1)
        print(der1 - der2)
        if np.abs(der1 - der2) > 1e-4:
            print(der1, der2)
            break

    # cross term for christofel
    for i in range(10):
        S1 = man.rand()
        xi = man.randvec(S1)
        eta1 = man.randvec(S1)
        eta2 = man.randvec(S1)
        dr1 = man.D_g(S1, xi, eta1)
        x12 = man.contract_D_g(S1, eta1, eta2)

        p1 = man.base_inner_ambient(dr1, eta2)
        p2 = man.base_inner_ambient(x12, xi)
        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):
        S1 = man.rand()
        xi = man.randvec(S1)
        eta1 = man.randvec(S1)
        eta2 = man.randvec(S1)
        p1 = man.proj_g_inv(S1, man.christoffel_form(S1, xi, eta1))
        p2 = man.proj_g_inv(S1, man.christoffel_form(S1, eta1, xi))
        print(check_zero(man._vec(p1) - man._vec(p2)))
        v1 = man.base_inner_ambient(man.christoffel_form(S1, eta1, eta2), xi)
        v2 = man.base_inner_ambient(man.D_g(S1, eta1, eta2), xi)
        v3 = man.base_inner_ambient(man.D_g(S1, eta2, eta1), xi)
        v4 = man.base_inner_ambient(man.D_g(S1, xi, eta1), eta2)
        print(v1, 0.5 * (v2 + v3 - v4), v1 - 0.5 * (v2 + v3 - v4))