SparseWeightMatrix klle_weight_matrix(RandomAccessIterator begin, RandomAccessIterator end,
const Neighbors& neighbors, PairwiseCallback callback, DefaultScalarType shift)
{
timed_context context("KLLE weight computation");
const unsigned int k = neighbors[0].size();
SparseTriplets sparse_triplets;
sparse_triplets.reserve((k*k+2*k+1)*(end-begin));
RandomAccessIterator iter;
RandomAccessIterator iter_begin = begin, iter_end = end;
DenseMatrix gram_matrix = DenseMatrix::Zero(k,k);
DenseVector dots(k);
DenseVector rhs = DenseVector::Ones(k);
DenseVector weights;
for (RandomAccessIterator iter=iter_begin; iter!=iter_end; ++iter)
{
DefaultScalarType kernel_value = callback(*iter,*iter);
const LocalNeighbors& current_neighbors = neighbors[iter-begin];
for (unsigned int i=0; i<k; ++i)
dots[i] = callback(*iter, begin[current_neighbors[i]]);
for (unsigned int i=0; i<k; ++i)
{
for (unsigned int j=i; j<k; ++j)
gram_matrix(i,j) = kernel_value - dots(i) - dots(j) + callback(begin[current_neighbors[i]],begin[current_neighbors[j]]);
}
DefaultScalarType trace = gram_matrix.trace();
gram_matrix.diagonal().array() += 1e-3*trace;
weights = gram_matrix.selfadjointView<Eigen::Upper>().ldlt().solve(rhs);
weights /= weights.sum();
sparse_triplets.push_back(SparseTriplet(iter-begin,iter-begin,1.0+shift));
for (unsigned int i=0; i<k; ++i)
{
sparse_triplets.push_back(SparseTriplet(current_neighbors[i],iter-begin,
-weights[i]));
sparse_triplets.push_back(SparseTriplet(iter-begin,current_neighbors[i],
-weights[i]));
for (unsigned int j=0; j<k; ++j)
sparse_triplets.push_back(SparseTriplet(current_neighbors[i],current_neighbors[j],
+weights(i)*weights(j)));
}
}
SparseWeightMatrix weight_matrix(end-begin,end-begin);
weight_matrix.setFromTriplets(sparse_triplets.begin(),sparse_triplets.end());
return weight_matrix;
}
Laplacian compute_laplacian(RandomAccessIterator begin,
RandomAccessIterator end,const Neighbors& neighbors,
DistanceCallback callback, ScalarType width)
{
SparseTriplets sparse_triplets;
timed_context context("Laplacian computation");
const IndexType k = neighbors[0].size();
sparse_triplets.reserve((k+1)*(end-begin));
DenseVector D = DenseVector::Zero(end-begin);
for (RandomAccessIterator iter=begin; iter!=end; ++iter)
{
const LocalNeighbors& current_neighbors = neighbors[iter-begin];
for (IndexType i=0; i<k; ++i)
{
ScalarType distance = callback(*iter,begin[current_neighbors[i]]);
ScalarType heat = exp(-distance*distance/width);
D(iter-begin) += heat;
D(current_neighbors[i]) += heat;
sparse_triplets.push_back(SparseTriplet(current_neighbors[i],(iter-begin),-heat));
sparse_triplets.push_back(SparseTriplet((iter-begin),current_neighbors[i],-heat));
}
}
for (IndexType i=0; i<(end-begin); ++i)
sparse_triplets.push_back(SparseTriplet(i,i,D(i)));
#ifdef EIGEN_YES_I_KNOW_SPARSE_MODULE_IS_NOT_STABLE_YET
Eigen::DynamicSparseMatrix<ScalarType> dynamic_weight_matrix(end-begin,end-begin);
dynamic_weight_matrix.reserve(sparse_triplets.size());
for (SparseTriplets::const_iterator it=sparse_triplets.begin(); it!=sparse_triplets.end(); ++it)
dynamic_weight_matrix.coeffRef(it->col(),it->row()) += it->value();
SparseWeightMatrix weight_matrix(dynamic_weight_matrix);
#else
SparseWeightMatrix weight_matrix(end-begin,end-begin);
weight_matrix.setFromTriplets(sparse_triplets.begin(),sparse_triplets.end());
#endif
return Laplacian(weight_matrix,DenseDiagonalMatrix(D));
}
SparseWeightMatrix kltsa_weight_matrix(RandomAccessIterator begin, RandomAccessIterator end,
const Neighbors& neighbors, PairwiseCallback callback,
unsigned int target_dimension, DefaultScalarType shift,
bool partial_eigendecomposer=false)
{
timed_context context("KLTSA weight matrix computation");
const unsigned int k = neighbors[0].size();
SparseTriplets sparse_triplets;
sparse_triplets.reserve((k*k+k+1)*(end-begin));
RandomAccessIterator iter;
RandomAccessIterator iter_begin = begin, iter_end = end;
DenseMatrix gram_matrix = DenseMatrix::Zero(k,k);
DenseVector col_means(k), row_means(k);
DenseVector rhs = DenseVector::Ones(k);
DenseMatrix G = DenseMatrix::Zero(k,target_dimension+1);
G.col(0).setConstant(1/sqrt(DefaultScalarType(k)));
DefaultDenseSelfAdjointEigenSolver solver;
RESTRICT_ALLOC;
//#pragma omp parallel for private(iter,gram_matrix,G)
for (iter=iter_begin; iter<iter_end; ++iter)
{
const LocalNeighbors& current_neighbors = neighbors[iter-begin];
for (unsigned int i=0; i<k; ++i)
{
for (unsigned int j=i; j<k; ++j)
{
DefaultScalarType kij = callback(begin[current_neighbors[i]],begin[current_neighbors[j]]);
gram_matrix(i,j) = kij;
gram_matrix(j,i) = kij;
}
}
col_means = gram_matrix.colwise().mean();
DefaultScalarType grand_mean = gram_matrix.mean();
gram_matrix.array() += grand_mean;
gram_matrix.rowwise() -= col_means.transpose();
gram_matrix.colwise() -= col_means;
UNRESTRICT_ALLOC;
if (partial_eigendecomposer)
{
G.rightCols(target_dimension).noalias() = eigen_embedding<DenseMatrix,DenseMatrixOperation>(ARPACK,gram_matrix,target_dimension,0).first;
}
else
{
solver.compute(gram_matrix);
G.rightCols(target_dimension).noalias() = solver.eigenvectors().rightCols(target_dimension);
}
RESTRICT_ALLOC;
gram_matrix.noalias() = G * G.transpose();
sparse_triplets.push_back(SparseTriplet(iter-begin,iter-begin,shift));
for (unsigned int i=0; i<k; ++i)
{
sparse_triplets.push_back(SparseTriplet(current_neighbors[i],current_neighbors[i],1.0));
for (unsigned int j=0; j<k; ++j)
sparse_triplets.push_back(SparseTriplet(current_neighbors[i],current_neighbors[j],
-gram_matrix(i,j)));
}
}
UNRESTRICT_ALLOC;
SparseWeightMatrix weight_matrix(end-begin,end-begin);
weight_matrix.setFromTriplets(sparse_triplets.begin(),sparse_triplets.end());
return weight_matrix;
}
SparseWeightMatrix hlle_weight_matrix(RandomAccessIterator begin, RandomAccessIterator end,
const Neighbors& neighbors, PairwiseCallback callback, unsigned int target_dimension)
{
timed_context context("KLTSA weight matrix computation");
const unsigned int k = neighbors[0].size();
SparseTriplets sparse_triplets;
sparse_triplets.reserve(k*k*(end-begin));
RandomAccessIterator iter_begin = begin, iter_end = end;
DenseMatrix gram_matrix = DenseMatrix::Zero(k,k);
DenseVector col_means(k), row_means(k);
DenseVector rhs = DenseVector::Ones(k);
DenseMatrix G = DenseMatrix::Zero(k,target_dimension+1);
RandomAccessIterator iter;
for (iter=iter_begin; iter!=iter_end; ++iter)
{
const LocalNeighbors& current_neighbors = neighbors[iter-begin];
for (unsigned int i=0; i<k; ++i)
{
for (unsigned int j=i; j<k; ++j)
{
DefaultScalarType kij = callback(begin[current_neighbors[i]],begin[current_neighbors[j]]);
gram_matrix(i,j) = kij;
gram_matrix(j,i) = kij;
}
}
for (unsigned int i=0; i<k; ++i)
{
col_means[i] = gram_matrix.col(i).mean();
row_means[i] = gram_matrix.row(i).mean();
}
DefaultScalarType grand_mean = gram_matrix.mean();
gram_matrix.array() += grand_mean;
gram_matrix.rowwise() -= col_means.transpose();
gram_matrix.colwise() -= row_means;
DefaultDenseSelfAdjointEigenSolver sae_solver;
sae_solver.compute(gram_matrix);
G.col(0).setConstant(1/sqrt(DefaultScalarType(k)));
G.rightCols(target_dimension).noalias() = sae_solver.eigenvectors().rightCols(target_dimension);
gram_matrix = G * G.transpose();
sparse_triplets.push_back(SparseTriplet(iter-begin,iter-begin,1e-8));
for (unsigned int i=0; i<k; ++i)
{
sparse_triplets.push_back(SparseTriplet(current_neighbors[i],current_neighbors[i],1.0));
for (unsigned int j=0; j<k; ++j)
sparse_triplets.push_back(SparseTriplet(current_neighbors[i],current_neighbors[j],
-gram_matrix(i,j)));
}
}
SparseWeightMatrix weight_matrix(end-begin,end-begin);
weight_matrix.setFromTriplets(sparse_triplets.begin(),sparse_triplets.end());
return weight_matrix;
};