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#include <getopt.h>
#include <iomanip>
#include <random>
#include "pcg-cpp/include/pcg_random.hpp"
#include "randutils/randutils.hpp"
#include "eigen/Eigen/Dense"
using Rng = randutils::random_generator<pcg32>;
using Real = double;
using Vector = Eigen::Matrix<Real, Eigen::Dynamic, 1>;
Vector normalizeVector(const Vector& x) {
return x * sqrt(x.size() / x.squaredNorm());
}
Vector randomVector(unsigned N, Rng& r, Real σ = 1) {
Vector v(N);
for (Real& vᵢ : v) {
vᵢ = r.variate<Real, std::normal_distribution>(0, σ);
}
return v;
}
Vector projectionOfOn(const Vector& v, const Vector& u) {
return (v.dot(u) / u.squaredNorm()) * u;
}
Real wignerCDF(Real λ) {
return 0.5 + (λ * sqrt(4 - pow(λ, 2)) / 4 + atan(λ / sqrt(4 - pow(λ, 2)))) / M_PI;
}
Real wignerInverse(Real p, Real ε = 1e-14) {
Real a = -2;
Real b = 2;
while (b - a > ε) {
Real c = (a + b) / 2;
if ((wignerCDF(a) - p) * (wignerCDF(c) - p) > 0) {
a = c;
} else {
b = c;
}
}
return (a + b) / 2;
}
class QuadraticModel {
private:
Vector J;
public:
unsigned N;
QuadraticModel(unsigned N, Rng& r) : J(N), N(N) {
for (Real& Jᵢ : J) {
Jᵢ = wignerInverse(r.uniform(0.0, 1.0));
}
}
Real H(const Vector& x) const {
return 0.5 * (J.cwiseProduct(x)).dot(x);
}
Vector ∇H(const Vector& x) const {
Vector ∂H = J.cwiseProduct(x);
return ∂H - projectionOfOn(∂H, x);
}
};
Vector gradientDescent(const QuadraticModel& M, const Vector& x₀, Real E, Real ε = 1e-14) {
Vector xₜ = x₀;
Real Hₜ = M.H(x₀);
Real α = 1.0 / M.N;
Real m;
Vector ∇H;
while (
∇H = (Hₜ / M.N - E) * M.∇H(xₜ) / M.N, m = ∇H.squaredNorm(),
m > ε
) {
Vector xₜ₊₁;
Real Hₜ₊₁;
while (
xₜ₊₁ = normalizeVector(xₜ - α * ∇H), Hₜ₊₁ = M.H(xₜ₊₁),
pow(Hₜ₊₁ / M.N - E, 2) > pow(Hₜ / M.N - E, 2) - α * m
) {
α /= 2;
}
xₜ = xₜ₊₁;
Hₜ = Hₜ₊₁;
α *= 1.25;
}
return xₜ;
}
Vector randomStep(const QuadraticModel& M, const Vector& x₀, Real E, Rng& r, Real Δt = 1e-4) {
Vector η = randomVector(M.N, r, sqrt(2 * Δt));
η -= projectionOfOn(η, x₀);
η -= projectionOfOn(η, M.∇H(x₀));
return gradientDescent(M, normalizeVector(x₀ + η), E);
}
int main(int argc, char* argv[]) {
unsigned N = 10;
Real T = 10;
Real T₀ = 2;
Real E = 0;
Real Δt = 1e-4;
int opt;
while ((opt = getopt(argc, argv, "N:E:T:t:I:")) != -1) {
switch (opt) {
case 'N':
N = (unsigned)atof(optarg);
break;
case 'E':
E = atof(optarg);
break;
case 'T':
T = atof(optarg);
break;
case 'I':
T₀ = atof(optarg);
break;
case 't':
Δt = atof(optarg);
break;
default:
exit(1);
}
}
Rng r;
QuadraticModel model(N, r);
Vector x₀ = normalizeVector(randomVector(N, r));
x₀ = gradientDescent(model, x₀, E);
std::cout << std::setprecision(15);
for (Real t = 0; t < T₀; t += Δt) {
x₀ = randomStep(model, x₀, E, r, Δt);
}
Vector x = x₀;
for (Real t = 0; t < T; t += Δt) {
std::cout << t << " " << x.dot(x₀) / N << std::endl;
x = randomStep(model, x, E, r, Δt);
}
return 0;
}
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