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;
}
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;
};
SparseWeightMatrix tangent_weight_matrix(RandomAccessIterator begin, RandomAccessIterator end,
const Neighbors& neighbors, PairwiseCallback callback,
const IndexType target_dimension, const ScalarType shift,
const bool partial_eigendecomposer=false)
{
timed_context context("KLTSA weight matrix computation");
const IndexType k = neighbors[0].size();
SparseTriplets sparse_triplets;
sparse_triplets.reserve((k*k+2*k+1)*(end-begin));
#pragma omp parallel shared(begin,end,neighbors,callback,sparse_triplets) default(none)
{
IndexType index_iter;
DenseMatrix gram_matrix = DenseMatrix::Zero(k,k);
DenseVector rhs = DenseVector::Ones(k);
DenseMatrix G = DenseMatrix::Zero(k,target_dimension+1);
G.col(0).setConstant(1/sqrt(static_cast<ScalarType>(k)));
DenseSelfAdjointEigenSolver solver;
SparseTriplets local_triplets;
local_triplets.reserve(k*k+2*k+1);
#pragma omp for nowait
for (index_iter=0; index_iter<static_cast<IndexType>(end-begin); index_iter++)
{
const LocalNeighbors& current_neighbors = neighbors[index_iter];
for (IndexType i=0; i<k; ++i)
{
for (IndexType j=i; j<k; ++j)
{
ScalarType kij = callback.kernel(begin[current_neighbors[i]],begin[current_neighbors[j]]);
gram_matrix(i,j) = kij;
gram_matrix(j,i) = kij;
}
}
centerMatrix(gram_matrix);
//UNRESTRICT_ALLOC;
#ifdef TAPKEE_WITH_ARPACK
if (partial_eigendecomposer)
{
G.rightCols(target_dimension).noalias() =
eigendecomposition<DenseMatrix,DenseMatrixOperation>(Arpack,gram_matrix,target_dimension,0).first;
}
else
#endif
{
solver.compute(gram_matrix);
G.rightCols(target_dimension).noalias() = solver.eigenvectors().rightCols(target_dimension);
}
//RESTRICT_ALLOC;
gram_matrix.noalias() = G * G.transpose();
SparseTriplet diagonal_triplet(index_iter,index_iter,shift);
local_triplets.push_back(diagonal_triplet);
for (IndexType i=0; i<k; ++i)
{
SparseTriplet neighborhood_diagonal_triplet(current_neighbors[i],current_neighbors[i],1.0);
local_triplets.push_back(neighborhood_diagonal_triplet);
for (IndexType j=0; j<k; ++j)
{
SparseTriplet tangent_triplet(current_neighbors[i],current_neighbors[j],-gram_matrix(i,j));
local_triplets.push_back(tangent_triplet);
}
}
#pragma omp critical
{
copy(local_triplets.begin(),local_triplets.end(),back_inserter(sparse_triplets));
}
local_triplets.clear();
}
}
return sparse_matrix_from_triplets(sparse_triplets, end-begin, end-begin);
}
SparseWeightMatrix hessian_weight_matrix(RandomAccessIterator begin, RandomAccessIterator end,
const Neighbors& neighbors, PairwiseCallback callback,
const IndexType target_dimension)
{
timed_context context("Hessian weight matrix computation");
const IndexType k = neighbors[0].size();
SparseTriplets sparse_triplets;
sparse_triplets.reserve(k*k*(end-begin));
const IndexType dp = target_dimension*(target_dimension+1)/2;
#pragma omp parallel shared(begin,end,neighbors,callback,sparse_triplets) default(none)
{
IndexType index_iter;
DenseMatrix gram_matrix = DenseMatrix::Zero(k,k);
DenseMatrix Yi(k,1+target_dimension+dp);
SparseTriplets local_triplets;
local_triplets.reserve(k*k+2*k+1);
#pragma omp for nowait
for (index_iter=0; index_iter<static_cast<IndexType>(end-begin); index_iter++)
{
const LocalNeighbors& current_neighbors = neighbors[index_iter];
for (IndexType i=0; i<k; ++i)
{
for (IndexType j=i; j<k; ++j)
{
ScalarType kij = callback.kernel(begin[current_neighbors[i]],begin[current_neighbors[j]]);
gram_matrix(i,j) = kij;
gram_matrix(j,i) = kij;
}
}
centerMatrix(gram_matrix);
DenseSelfAdjointEigenSolver sae_solver;
sae_solver.compute(gram_matrix);
Yi.col(0).setConstant(1.0);
Yi.block(0,1,k,target_dimension).noalias() = sae_solver.eigenvectors().rightCols(target_dimension);
IndexType ct = 0;
for (IndexType j=0; j<target_dimension; ++j)
{
for (IndexType p=0; p<target_dimension-j; ++p)
{
Yi.col(ct+p+1+target_dimension).noalias() = Yi.col(j+1).cwiseProduct(Yi.col(j+p+1));
}
ct += ct + target_dimension - j;
}
for (IndexType i=0; i<static_cast<IndexType>(Yi.cols()); i++)
{
for (IndexType j=0; j<i; j++)
{
ScalarType r = Yi.col(i).dot(Yi.col(j));
Yi.col(i) -= r*Yi.col(j);
}
ScalarType norm = Yi.col(i).norm();
Yi.col(i) *= (1.f / norm);
}
for (IndexType i=0; i<dp; i++)
{
ScalarType colsum = Yi.col(1+target_dimension+i).sum();
if (colsum > 1e-4)
Yi.col(1+target_dimension+i).array() /= colsum;
}
// reuse gram matrix storage m'kay?
gram_matrix.noalias() = Yi.rightCols(dp)*(Yi.rightCols(dp).transpose());
for (IndexType i=0; i<k; ++i)
{
for (IndexType j=0; j<k; ++j)
{
SparseTriplet hessian_triplet(current_neighbors[i],current_neighbors[j],gram_matrix(i,j));
local_triplets.push_back(hessian_triplet);
}
}
#pragma omp critical
{
copy(local_triplets.begin(),local_triplets.end(),back_inserter(sparse_triplets));
}
local_triplets.clear();
}
}
return sparse_matrix_from_triplets(sparse_triplets, end-begin, end-begin);
}
SparseWeightMatrix linear_weight_matrix(const RandomAccessIterator& begin, const RandomAccessIterator& end,
const Neighbors& neighbors, PairwiseCallback callback,
const ScalarType shift, const ScalarType trace_shift)
{
timed_context context("KLLE weight computation");
const IndexType k = neighbors[0].size();
SparseTriplets sparse_triplets;
sparse_triplets.reserve((k*k+2*k+1)*(end-begin));
#pragma omp parallel shared(begin,end,neighbors,callback,sparse_triplets) default(none)
{
IndexType index_iter;
DenseMatrix gram_matrix = DenseMatrix::Zero(k,k);
DenseVector dots(k);
DenseVector rhs = DenseVector::Ones(k);
DenseVector weights;
SparseTriplets local_triplets;
local_triplets.reserve(k*k+2*k+1);
//RESTRICT_ALLOC;
#pragma omp for nowait
for (index_iter=0; index_iter<static_cast<IndexType>(end-begin); index_iter++)
{
ScalarType kernel_value = callback.kernel(begin[index_iter],begin[index_iter]);
const LocalNeighbors& current_neighbors = neighbors[index_iter];
for (IndexType i=0; i<k; ++i)
dots[i] = callback.kernel(begin[index_iter], begin[current_neighbors[i]]);
for (IndexType i=0; i<k; ++i)
{
for (IndexType j=i; j<k; ++j)
gram_matrix(i,j) = kernel_value - dots(i) - dots(j) +
callback.kernel(begin[current_neighbors[i]],begin[current_neighbors[j]]);
}
ScalarType trace = gram_matrix.trace();
gram_matrix.diagonal().array() += trace_shift*trace;
weights = gram_matrix.selfadjointView<Eigen::Upper>().ldlt().solve(rhs);
weights /= weights.sum();
SparseTriplet diagonal_triplet(index_iter,index_iter,1.0+shift);
local_triplets.push_back(diagonal_triplet);
for (IndexType i=0; i<k; ++i)
{
SparseTriplet row_side_triplet(current_neighbors[i],index_iter,-weights[i]);
SparseTriplet col_side_triplet(index_iter,current_neighbors[i],-weights[i]);
local_triplets.push_back(row_side_triplet);
local_triplets.push_back(col_side_triplet);
for (IndexType j=0; j<k; ++j)
{
SparseTriplet cross_triplet(current_neighbors[i],current_neighbors[j],weights(i)*weights(j));
local_triplets.push_back(cross_triplet);
}
}
#pragma omp critical
{
copy(local_triplets.begin(),local_triplets.end(),back_inserter(sparse_triplets));
}
local_triplets.clear();
}
//UNRESTRICT_ALLOC;
}
return sparse_matrix_from_triplets(sparse_triplets, end-begin, end-begin);
}