double SoftmaxErrorFunction<MetricType>::Evaluate(const arma::mat& coordinates)
{
// Calculate the denominators and numerators, if necessary.
Precalculate(coordinates);
return -accu(p); // Sum of p_i for all i. We negate because our solver
// minimizes, not maximizes.
};
size_t MaxVarianceNewCluster::EmptyCluster(const MatType& data,
const size_t emptyCluster,
const arma::mat& oldCentroids,
arma::mat& newCentroids,
arma::Col<size_t>& clusterCounts,
MetricType& metric,
const size_t iteration)
{
// If necessary, calculate the variances and assignments.
if (iteration != this->iteration || assignments.n_elem != data.n_cols)
Precalculate(data, oldCentroids, clusterCounts, metric);
this->iteration = iteration;
// Now find the cluster with maximum variance.
arma::uword maxVarCluster;
variances.max(maxVarCluster);
// Now, inside this cluster, find the point which is furthest away.
size_t furthestPoint = data.n_cols;
double maxDistance = -DBL_MAX;
for (size_t i = 0; i < data.n_cols; ++i)
{
if (assignments[i] == maxVarCluster)
{
const double distance = std::pow(metric.Evaluate(data.col(i),
newCentroids.col(maxVarCluster)), 2.0);
if (distance > maxDistance)
{
maxDistance = distance;
furthestPoint = i;
}
}
}
// Take that point and add it to the empty cluster.
newCentroids.col(maxVarCluster) *= (double(clusterCounts[maxVarCluster]) /
double(clusterCounts[maxVarCluster] - 1));
newCentroids.col(maxVarCluster) -= (1.0 / (clusterCounts[maxVarCluster] - 1.0)) *
arma::vec(data.col(furthestPoint));
clusterCounts[maxVarCluster]--;
clusterCounts[emptyCluster]++;
newCentroids.col(emptyCluster) = arma::vec(data.col(furthestPoint));
assignments[furthestPoint] = emptyCluster;
// Modify the variances, as necessary.
variances[emptyCluster] = 0;
// One has already been subtracted from clusterCounts[maxVarCluster].
variances[maxVarCluster] = (1.0 / (clusterCounts[maxVarCluster])) *
((clusterCounts[maxVarCluster] + 1) * variances[maxVarCluster] - maxDistance);
// Output some debugging information.
Log::Debug << "Point " << furthestPoint << " assigned to empty cluster " <<
emptyCluster << ".\n";
return 1; // We only changed one point.
}
void SoftmaxErrorFunction<MetricType>::Gradient(const arma::mat& coordinates,
arma::mat& gradient)
{
// Calculate the denominators and numerators, if necessary.
Precalculate(coordinates);
// Now, we handle the summation over i:
// sum_i (p_i sum_k (p_ik x_ik x_ik^T) -
// sum_{j in class of i} (p_ij x_ij x_ij^T)
// We can algebraically manipulate the whole thing to produce a more
// memory-friendly way to calculate this. Looping over each i and k (again
// O((n * (n + 1)) / 2) as with the last step, we can add the following to the
// sum:
//
// if class of i is the same as the class of k, add
// (((p_i - (1 / p_i)) p_ik) + ((p_k - (1 / p_k)) p_ki)) x_ik x_ik^T
// otherwise, add
// (p_i p_ik + p_k p_ki) x_ik x_ik^T
arma::mat sum;
sum.zeros(stretchedDataset.n_rows, stretchedDataset.n_rows);
for (size_t i = 0; i < stretchedDataset.n_cols; i++)
{
for (size_t k = (i + 1); k < stretchedDataset.n_cols; k++)
{
// Calculate p_ik and p_ki first.
double eval = exp(-metric.Evaluate(stretchedDataset.unsafe_col(i),
stretchedDataset.unsafe_col(k)));
double p_ik = 0, p_ki = 0;
p_ik = eval / denominators(i);
p_ki = eval / denominators(k);
// Subtract x_i from x_k. We are not using stretched points here.
arma::vec x_ik = dataset.col(i) - dataset.col(k);
arma::mat secondTerm = (x_ik * trans(x_ik));
if (labels[i] == labels[k])
sum += ((p[i] - 1) * p_ik + (p[k] - 1) * p_ki) * secondTerm;
else
sum += (p[i] * p_ik + p[k] * p_ki) * secondTerm;
}
}
// Assemble the final gradient.
gradient = -2 * coordinates * sum;
}