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#include <eigen3/Eigen/Dense>
#include <eigen3/unsupported/Eigen/CXX11/Tensor>
#include <eigen3/unsupported/Eigen/CXX11/TensorSymmetry>
#include <getopt.h>
#include <iomanip>
#include "pcg-cpp/include/pcg_random.hpp"
#include "randutils/randutils.hpp"
using Rng = randutils::random_generator<pcg32>;
using Real = double;
using Vector = Eigen::Matrix<Real, Eigen::Dynamic, 1>;
using Matrix = Eigen::Matrix<Real, Eigen::Dynamic, Eigen::Dynamic>;
class Tensor : public Eigen::Tensor<Real, 3> {
using Eigen::Tensor<Real, 3>::Tensor;
public:
Matrix operator*(const Vector& x) const {
std::array<Eigen::IndexPair<int>, 1> ip20 = {Eigen::IndexPair<int>(2, 0)};
const Eigen::Tensor<Real, 1> xT = Eigen::TensorMap<const Eigen::Tensor<Real, 1>>(x.data(), x.size());
const Eigen::Tensor<Real, 2> JxT = contract(xT, ip20);
return Eigen::Map<const Matrix>(JxT.data(), dimension(0), dimension(1));
}
};
Matrix operator*(const Eigen::Matrix<Real, 1, Eigen::Dynamic>& x, const Tensor& J) {
std::array<Eigen::IndexPair<int>, 1> ip00 = {Eigen::IndexPair<int>(0, 0)};
const Eigen::Tensor<Real, 1> xT = Eigen::TensorMap<const Eigen::Tensor<Real, 1>>(x.data(), x.size());
const Eigen::Tensor<Real, 2> JxT = J.contract(xT, ip00);
return Eigen::Map<const Matrix>(JxT.data(), J.dimension(1), J.dimension(2));
}
Vector normalize(const Vector& x) {
return x * sqrt(x.size() / x.squaredNorm());
}
Vector ∂HFromV∂V(const Vector& V, const Matrix& ∂V) {
return V.transpose() * ∂V;
}
Vector VFromABJx(const Vector& b, const Matrix& A, const Matrix& Jx, const Vector& x) {
return b + (A + 0.5 * Jx) * x;
}
class QuadraticModel {
private:
Tensor J;
Matrix A;
Vector b;
std::tuple<Vector, Matrix, const Tensor&> V_∂V_∂∂V(const Vector& x) const {
Matrix Jx = J * x;
Vector V = VFromABJx(b, A, Jx, x);
Matrix ∂V = A + Jx;
return {V, ∂V, J};
}
std::tuple<Vector, Matrix> ∂H_∂∂H(const Vector& x) const {
auto [V, ∂V, ∂∂V] = V_∂V_∂∂V(x);
Vector ∂H = ∂HFromV∂V(V, ∂V);
Matrix ∂∂H = V.transpose() * ∂∂V + ∂V.transpose() * ∂V;
return {∂H, ∂∂H};
}
public:
unsigned N;
unsigned M;
QuadraticModel(unsigned N, unsigned M, Real σ², Real μA, Real μJ, Rng& r) : J(M, N, N), A(M, N), b(M), N(N), M(M) {
Eigen::StaticSGroup<Eigen::Symmetry<1,2>> sym23;
for (unsigned k = 0; k < N; k++) {
for (unsigned j = k; j < N; j++) {
for (unsigned i = 0; i < M; i++) {
sym23(J, i, j, k) = r.variate<Real, std::normal_distribution>(0, sqrt(2 * μJ) / N);
}
}
}
for (Real& Aij : A.reshaped()) {
Aij = r.variate<Real, std::normal_distribution>(0, sqrt(μA / N));
}
for (Real& bi : b) {
bi = r.variate<Real, std::normal_distribution>(0, sqrt(σ²));
}
}
Real getHamiltonian(const Vector& x) const {
Vector V = VFromABJx(b, A, J * x, x);
return 0.5 * V.squaredNorm();
}
Vector getGradient(const Vector& x) const {
auto [V, ∂V, ∂∂V] = V_∂V_∂∂V(x);
Vector ∂H = ∂HFromV∂V(V, ∂V);
return ∂H - (∂H.dot(x) / x.squaredNorm()) * x;
}
Matrix getHessian(const Vector& x) const {
auto [∂H, ∂∂H] = ∂H_∂∂H(x);
Matrix P = Matrix::Identity(N, N) - x * x.transpose() / x.squaredNorm();
Matrix HessH = P * ∂∂H * P.transpose() - (x.dot(∂H) / N) * Matrix::Identity(N, N);
return HessH;
}
Vector spectrum(const Vector& x) const {
Matrix HessH = getHessian(x);
Eigen::EigenSolver<Matrix> eigenS(HessH);
return eigenS.eigenvalues().real();
}
Real maxEigenvalue(const Vector& x) const {
return spectrum(x).maxCoeff();
}
};
Vector gradientAscent(const QuadraticModel& M, const Vector& x0, Real ε = 1e-13) {
Vector x = x0;
Real α = 1;
Real H = M.getHamiltonian(x0);
Real m;
Vector ∇H;
while (
∇H = M.getGradient(x),
m = ∇H.squaredNorm(),
m / M.N > ε
) {
Real HNew;
Vector xNew;
while(
xNew = normalize(x + α * ∇H),
HNew = M.getHamiltonian(xNew),
HNew < H + 0.5 * α * m
) {
α /= 2;
}
x = xNew;
H = HNew;
α *= 1.25;
}
return x;
}
Vector subagAlgorithm(const QuadraticModel& M, const Vector& x0) {
Vector σ = x0 / x0.norm();
for (unsigned i = 0; i < M.N; i++) {
Vector ∇H = M.getGradient(σ);
Vector v = ∇H / ∇H.norm();
σ += v;
}
return normalize(σ);
}
int main(int argc, char* argv[]) {
unsigned N = 10;
Real α = 1;
Real σ² = 1;
Real μA = 1;
Real μJ = 1;
unsigned samples = 10;
int opt;
while ((opt = getopt(argc, argv, "N:a:s:A:J:n:")) != -1) {
switch (opt) {
case 'N':
N = (unsigned)atof(optarg);
break;
case 'a':
α = atof(optarg);
break;
case 's':
σ² = atof(optarg);
break;
case 'A':
μA = atof(optarg);
break;
case 'J':
μJ = atof(optarg);
break;
case 'n':
samples = atoi(optarg);
break;
default:
exit(1);
}
}
unsigned M = std::round(α * N);
Rng r;
Vector x = Vector::Zero(N);
x(0) = sqrt(N);
std::cout << std::setprecision(15);
for (unsigned sample = 0; sample < samples; sample++) {
QuadraticModel* ls = new QuadraticModel(N, M, σ², μA, μJ, r);
Vector xGD = gradientAscent(*ls, x);
std::cout << ls->getHamiltonian(xGD) / N << " " << ls->maxEigenvalue(xGD) << " ";
delete ls;
ls = new QuadraticModel(N, M, σ², μA, μJ, r);
Vector xMP = subagAlgorithm(*ls, x);
xMP = gradientAscent(*ls, xMP);
std::cout << ls->getHamiltonian(xMP) / N << " " << ls->maxEigenvalue(xMP) << std::endl;
delete ls;
}
return 0;
}
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