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#include <eigen3/Eigen/Dense>
#include <getopt.h>
#include <eigen3/unsupported/Eigen/CXX11/Tensor>
#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>;
/* Eigen tensor manipulations are quite annoying, especially with the need to convert other types
* into tensors beforehand. Here I overload multiplication operators to allow contraction between
* vectors and the first or last index of a tensor.
*/
class Tensor : public Eigen::Tensor<Real, 3> {
using Eigen::Tensor<Real, 3>::Tensor;
public:
Matrix operator*(const Vector& x) const {
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) {
const 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((Real)x.size() / x.squaredNorm());
}
class QuadraticModel {
private:
Tensor J;
Matrix A;
Vector b;
public:
unsigned N;
unsigned M;
template <class Generator>
QuadraticModel(unsigned N, unsigned M, Generator& r, double μ1, double μ2, double μ3) : N(N), M(M), J(M, N, N), A(M, N), b(M) {
std::normal_distribution<Real> distribution(0, 1);
for (unsigned i = 0; i < M; i++) {
for (unsigned j = 0; j < N; j++) {
for (unsigned k = 0; k < N; k++) {
J(i, j, k) = (2 * μ3 / N) * distribution(r);
}
}
}
for (unsigned i = 0; i < M; i++) {
for (unsigned j = 0; j < N; j++) {
A(i, j) = (μ2 / sqrt(N)) * distribution(r);
}
}
for (unsigned i = 0; i < M; i++) {
b(i) = μ1 * distribution(r);
}
}
std::tuple<Vector, Matrix, const Tensor&> VdVddV(const Vector& x) const {
Matrix Jx = J * x;
Vector V1 = (A + 0.5 * Jx) * x;
Matrix dV = A + Jx;
return {b + V1, dV, J};
}
std::tuple<Real, Vector, Matrix> HdHddH(const Vector& x) const {
auto [V, dV, ddV] = VdVddV(x);
Real H = 0.5 * V.squaredNorm();
Vector dH = V.transpose() * dV;
Matrix ddH = V.transpose() * ddV + dV.transpose() * dV;
return {H, dH, ddH};
}
std::tuple<Real, Vector, Matrix> hamGradHess(const Vector& x) const {
auto [H, dH, ddH] = HdHddH(x);
Vector gradH = dH - dH.dot(x) * x / (Real)N;
Matrix hessH = ddH - (dH * x.transpose() + x.dot(dH) * Matrix::Identity(N, N) + (ddH * x) * x.transpose()) / (Real)N + 2.0 * x * x.transpose();
return {H, gradH, hessH};
}
Vector spectrum(const Vector& x) const {
Matrix hessH;
std::tie(std::ignore, std::ignore, hessH) = hamGradHess(x);
Eigen::EigenSolver<Matrix> eigenS(hessH);
return eigenS.eigenvalues().real();
}
};
Vector findMinimum(const QuadraticModel& M, const Vector& x0, Real ε) {
Vector x = x0;
Real λ = 100;
auto [H, g, m] = M.hamGradHess(x0);
while (g.norm() / x.size() > ε && λ < 1e8) {
Vector dx = (m + λ * (Matrix)abs(m.diagonal().array()).matrix().asDiagonal()).partialPivLu().solve(g);
dx -= x.dot(dx) * x / M.N;
Vector xNew = normalize(x - dx);
auto [HNew, gNew, mNew] = M.hamGradHess(xNew);
if (HNew * 1.0001 <= H) {
x = xNew;
H = HNew;
g = gNew;
m = mNew;
λ /= 2;
} else {
λ *= 1.5;
}
}
return x;
}
int main(int argc, char* argv[]) {
unsigned N = 10;
Real α = 1;
Real σ = 1;
Real A = 1;
Real J = 1;
int opt;
while ((opt = getopt(argc, argv, "N:a:s:A:J:")) != -1) {
switch (opt) {
case 'N':
N = (unsigned)atof(optarg);
break;
case 'a':
α = atof(optarg);
break;
case 's':
σ = atof(optarg);
break;
case 'A':
A = atof(optarg);
break;
case 'J':
J = atof(optarg);
break;
default:
exit(1);
}
}
unsigned M = (unsigned)(α * N);
Rng r;
QuadraticModel leastSquares(N, M, r.engine(), σ, A, J);
Vector x = Vector::Zero(N);
x(0) = sqrt(N);
double energy;
std::tie(energy, std::ignore, std::ignore) = leastSquares.hamGradHess(x);
std::cout << energy / N << std::endl;
Vector xMin = findMinimum(leastSquares, x, 1e-12);
std::tie(energy, std::ignore, std::ignore) = leastSquares.hamGradHess(xMin);
std::cout << energy / N << std::endl;
std::cout << leastSquares.spectrum(xMin)(1) / N << std::endl;
return 0;
}
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