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#include <eigen3/Eigen/Dense>
#include <random>
#include <getopt.h>
#include "pcg-cpp/include/pcg_random.hpp"
#include "randutils/randutils.hpp"
template <class Real>
using Vector = Eigen::Matrix<Real, Eigen::Dynamic, 1>;
template <class Real>
using Matrix = Eigen::Matrix<Real, Eigen::Dynamic, Eigen::Dynamic>;
template <class Real>
class Model {
private:
Matrix<Real> A;
Vector<Real> b;
public:
template <class Generator>
Model(Real σ, unsigned N, unsigned M, Generator& r) : A(M, N), b(M) {
std::normal_distribution<Real> aDistribution(0, 1 / sqrt(N));
for (unsigned i = 0; i < M; i++) {
for (unsigned j =0; j < N; j++) {
A(i, j) = aDistribution(r);
}
}
std::normal_distribution<Real> bDistribution(0, σ);
for (unsigned i = 0; i < M; i++) {
b(i) = bDistribution(r);
}
}
unsigned N() const {
return A.cols();
}
unsigned M() const {
return A.rows();
}
Vector<Real> V(const Vector<Real>& x) const {
return A * x + b;
}
Matrix<Real> dV(const Vector<Real>& x) const {
return A;
}
// const Real ddV(const Vector<Real>& x) {
// return Matrix::Zero(;
// }
Real H(const Vector<Real>& x) const {
return V(x).squaredNorm();
}
Vector<Real> dH(const Vector<Real>& x) const {
return dV(x).transpose() * V(x);
}
Matrix<Real> ddH(const Vector<Real>& x) const {
return dV(x).transpose() * dV(x);
}
Vector<Real> ∇H(const Vector<Real>& x) const {
return dH(x) - dH(x).dot(x) * x / x.squaredNorm();
}
Matrix<Real> HessH(const Vector<Real>& x) const {
Matrix<Real> hess = ddH(x) - x.dot(dH(x)) * Matrix<Real>::Identity(N(), N());
return hess - (hess * x) * x.transpose() / x.squaredNorm();
}
Vector<Real> HessSpectrum(const Vector<Real>& x) const {
Eigen::EigenSolver<Matrix<Real>> eigenS(HessH(x));
return eigenS.eigenvalues().real();
}
};
template <typename Derived>
Vector<typename Derived::Scalar> normalize(const Eigen::MatrixBase<Derived>& z) {
return z * sqrt((double)z.size() / (typename Derived::Scalar)(z.transpose() * z));
}
template <class Real>
Vector<Real> findMinimum(const Model<Real>& M, const Vector<Real>& x0, Real ε) {
Vector<Real> x = x0;
Real λ = 100;
Real H = M.H(x);
Vector<Real> dH = M.dH(x);
Matrix<Real> ddH = M.ddH(x);
Vector<Real> g = dH - x.dot(dH) * x / x.squaredNorm();
Matrix<Real> m = ddH - (dH * x.transpose() + x.dot(dH) * Matrix<Real>::Identity(M.N(), M.N()) + (ddH * x) * x.transpose()) / x.squaredNorm() + 2.0 * x * x.transpose();
while (g.norm() / x.size() > ε && λ < 1e8) {
Vector<Real> dz = (m + λ * (Matrix<Real>)abs(m.diagonal().array()).matrix().asDiagonal()).partialPivLu().solve(g);
dz -= x.dot(dz) * x / x.squaredNorm();
Vector<Real> zNew = normalize(x - dz);
Real HNew = M.H(zNew);
Vector<Real> dHNew = M.dH(zNew);
Matrix<Real> ddHNew = M.ddH(zNew);
if (HNew * 1.0001 <= H) {
x = zNew;
H = HNew;
dH = dHNew;
ddH = ddHNew;
g = dH - x.dot(dH) * x / (Real)x.size();
m = ddH - (dH * x.transpose() + x.dot(dH) * Matrix<Real>::Identity(x.size(), x.size()) + (ddH * x) * x.transpose()) / (Real)x.size() + 2.0 * x * x.transpose();
λ /= 2;
} else {
λ *= 1.5;
}
}
return x;
}
using Rng = randutils::random_generator<pcg32>;
using Real = double;
int main(int argc, char* argv[]) {
unsigned N = 10;
Real α = 1;
Real σ = 1;
int opt;
while ((opt = getopt(argc, argv, "N:a:s:")) != -1) {
switch (opt) {
case 'N':
N = (unsigned)atof(optarg);
break;
case 'a':
α = atof(optarg);
break;
case 's':
σ = atof(optarg);
break;
default:
exit(1);
}
}
unsigned M = (unsigned)(α * N);
Rng r;
Model<Real> leastSquares(σ, N, M, r.engine());
Vector<Real> x = Vector<Real>::Zero(N);
x(0) = sqrt(N);
std::cout << leastSquares.H(x) / N << std::endl;
Vector<Real> xMin = findMinimum(leastSquares, x, 1e-12);
std::cout << leastSquares.H(xMin) / N << std::endl;
std::cout << leastSquares.HessSpectrum(xMin)(1) / N << std::endl;
return 0;
}
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