GMMExpectationMaximization::Real GMMExpectationMaximization::gauss(const VectorX & mean,
const MatrixX & cov,const VectorX & pt) const
{
Real det = cov.determinant();
uint dim = mean.size();
// check that the covariance matrix is invertible
if (std::abs(det) < std::pow(m_epsilon,dim) * 0.1)
return 0.0; // the gaussian has approximately zero width: the probability of any point falling into it is approximately 0.
// else, compute pdf
MatrixX inverse_cov = cov.inverse();
VectorX dist = pt - mean;
Real exp = - (dist.dot(inverse_cov * dist)) / 2.0;
Real den = std::sqrt(std::pow(2.0 * M_PI,dim) * std::abs(det));
return std::exp(exp) / den;
}
bool cpu_linesearch(VectorX &xk, VectorX &z,
cpu_cost_function *cpucf, floatdouble &fk, VectorX &gk,
size_t &evals, const VectorX &gkm1, const floatdouble &fkm1,
lbfgs::status &stat, floatdouble &step, size_t maxEvals)
{
const floatdouble c1 = 1e-4f;
const floatdouble c2 = 0.9f;
const floatdouble alpha_0 = 0.0;
const floatdouble phi_0 = fk;
const floatdouble phi_prime_0 = z.dot(gk);
if (phi_prime_0 >= 0.0)
{
stat = lbfgs::LBFGS_LINE_SEARCH_FAILED;
return false;
}
const floatdouble alpha_max = 1e8;
floatdouble alpha = 1.0;
floatdouble alpha_old = 0.0;
bool second_iter = false;
floatdouble alpha_low, alpha_high;
floatdouble phi_low, phi_high;
floatdouble phi_prime_low, phi_prime_high;
for (;;)
{
xk += (alpha - alpha_old) * z;
cpucf->cpu_f_gradf(xk.data(), &fk, gk.data());
++evals;
const floatdouble phi_alpha = fk;
const floatdouble phi_prime_alpha = z.dot(gk);
const bool armijo_violated = (phi_alpha > phi_0 + c1 * alpha * phi_prime_0 || (second_iter && phi_alpha >= phi_0));
const bool strong_wolfe = (std::abs(phi_prime_alpha) <= -c2 * phi_prime_0);
// If both Armijo and Strong Wolfe hold, we're done
if (!armijo_violated && strong_wolfe)
{
step = alpha;
return true;
}
if (evals >= maxEvals)
{
stat = lbfgs::LBFGS_REACHED_MAX_EVALS;
return false;
}
// If Armijio condition is violated, we've bracketed a viable minimum
// Interval is [alpha_0, alpha]
if (armijo_violated)
{
alpha_low = alpha_0;
alpha_high = alpha;
phi_low = phi_0;
phi_high = phi_alpha;
phi_prime_low = phi_prime_0;
phi_prime_high = phi_prime_alpha;
break;
}
// If the directional derivative at alpha is positive, we've bracketed a viable minimum
// Interval is [alpha, alpha_0]
if (phi_prime_alpha >= 0)
{
alpha_low = alpha;
alpha_high = alpha_0;
phi_low = phi_alpha;
phi_high = phi_0;
phi_prime_low = phi_prime_alpha;
phi_prime_high = phi_prime_0;
break;
}
// Else look to the "right" of alpha for a viable minimum
floatdouble alpha_new = alpha + 4 * (alpha - alpha_old);
alpha_old = alpha;
alpha = alpha_new;
if (alpha > alpha_max)
{
stat = lbfgs::LBFGS_LINE_SEARCH_FAILED;
return false;
}
second_iter = true;
}
// The minimum is now bracketed in [alpha_low, alpha_high]
// Find it...
size_t tries = 0;
const size_t minTries = 10;
while (true)
{
tries++;
alpha_old = alpha;
// Quadratic interpolation:
// Least-squares fit a parabola to (alpha_low, phi_low),
// (alpha_high, phi_high) with gradients phi_prime_low and
// phi_prime_high and select the minimum of that parabola as
// the new alpha
alpha = 0.5f * (alpha_low + alpha_high);
alpha += (phi_high - phi_low) / (phi_prime_low - phi_prime_high);
if (alpha < alpha_low || alpha > alpha_high)
alpha = 0.5f * (alpha_low + alpha_high);
xk += (alpha - alpha_old) * z;
cpucf->cpu_f_gradf(xk.data(), &fk, gk.data());
++evals;
const floatdouble phi_j = fk;
const floatdouble phi_prime_j = z.dot(gk);
const bool armijo_violated = (phi_j > phi_0 + c1 * alpha * phi_prime_0 || phi_j >= phi_low);
const bool strong_wolfe = (std::abs(phi_prime_j) <= -c2 * phi_prime_0);
if (!armijo_violated && strong_wolfe)
{
// The Armijo and Strong Wolfe conditions hold
step = alpha;
break;
}
else if (std::abs(alpha_high - alpha_low) < 1e-5 && tries > minTries)
{
// The search interval has become too small
stat = lbfgs::LBFGS_LINE_SEARCH_FAILED;
return false;
}
else if (armijo_violated)
{
alpha_high = alpha;
phi_high = phi_j;
phi_prime_high = phi_prime_j;
}
else
{
if (phi_prime_j * (alpha_high - alpha_low) >= 0)
{
alpha_high = alpha_low;
phi_high = phi_low;
phi_prime_high = phi_prime_low;
}
alpha_low = alpha;
phi_low = phi_j;
phi_prime_low = phi_prime_j;
}
if (evals >= maxEvals)
{
stat = lbfgs::LBFGS_REACHED_MAX_EVALS;
return false;
}
}
return true;
}