SparseMatrix NeighborhoodBuilder::buildMatrix_(std::vector<T>& entries, const DenseMatrix& mat) const
{
// Make sure, that there are no duplicate entries
std::sort(entries.begin(), entries.begin() + k_ * mat.rows(), triple_less);
auto new_end = std::unique(entries.begin(), entries.begin() + k_ * mat.rows(), triple_equal);
// An iterator for inserting the transposed entries
// This will also serve as the new end pointer of the
// entries array.
auto new_it = new_end;
// Symmetrize the matrix
for(auto old_it = entries.begin(); old_it != new_end; ++old_it, ++new_it) {
*new_it = T(old_it->col(), old_it->row(), old_it->value());
}
// Build the temporary matrix for the results
SparseMatrix result(mat.rowNames(), mat.rowNames());
// Fill the matrix and convert to CCS
// new_it points to the end of valid matrix entries.
result.matrix().setFromTriplets(entries.begin(), new_it);
result.matrix().makeCompressed();
return result;
}
DenseMatrix<FloatType> DenseMatrix<FloatType>::subtract(const DenseMatrix<FloatType> &A, const DenseMatrix<FloatType> &B)
{
MLIB_ASSERT_STR(A.rows() == B.rows() && A.cols() == B.cols(), "invalid matrix dimensions");
const UINT rows = A.m_rows;
const UINT cols = A.m_cols;
DenseMatrix<FloatType> result(A.m_rows, A.m_cols);
for(UINT row = 0; row < rows; row++)
for(UINT col = 0; col < cols; col++)
result.m_dataPtr[row * cols + col] = A.m_dataPtr[row * cols + col] - B.m_dataPtr[row * cols + col];
return result;
}
GPUDenseMatrixOperation(const DenseMatrix& matrix)
{
mat = viennacl::matrix<ScalarType>(matrix.cols(),matrix.rows());
vec = viennacl::matrix<ScalarType>(matrix.cols(),1);
res = viennacl::matrix<ScalarType>(matrix.cols(),1);
viennacl::copy(matrix,mat);
}
DenseVector ConstraintBSpline::evalHessian(const DenseVector &x) const
{
DenseVector xa = adjustToDomainBounds(x);
DenseVector ddx = DenseVector::Zero(nnzHessian);
// Get x-values
DenseVector xx = xa.block(0,0,bspline.getNumVariables(),1);
// Calculate Hessian
DenseMatrix H = bspline.evalHessian(xx);
// H is symmetric so fill out lower left triangle only
int idx = 0;
for (int row = 0; row < H.rows(); row++)
{
for (int col = 0; col <= row; col++)
{
//if (H(row,col) != 0)
//{
ddx(idx++) = H(row,col);
//}
}
}
return ddx;
}
DenseVector ConstraintBSpline::evalJacobian(const DenseVector &x) const
{
DenseVector xa = adjustToDomainBounds(x);
DenseVector dx = DenseVector::Zero(nnzJacobian);
//return centralDifference(xa);
// Get x-values
DenseVector xx = xa.block(0,0,bspline.getNumVariables(),1);
// Evaluate B-spline Jacobian
DenseMatrix jac = bspline.evalJacobian(xx);
// Derivatives on inputs x
int k = 0;
for (int i = 0; i < jac.rows(); i++)
{
for (int j = 0; j < jac.cols(); j++)
{
dx(k++) = jac(i,j);
}
}
// Derivatives on outputs y
for (unsigned int i = 0; i < numConstraints; i++)
{
dx(k++) = -1;
}
return dx;
}
GPUDenseImplicitSquareMatrixOperation(const DenseMatrix& matrix)
{
timed_context c("Storing matrices");
mat = viennacl::matrix<ScalarType>(matrix.cols(),matrix.rows());
vec = viennacl::matrix<ScalarType>(matrix.cols(),1);
res = viennacl::matrix<ScalarType>(matrix.cols(),1);
viennacl::copy(matrix,mat);
}
void BSpline::setControlPoints(const DenseMatrix &controlPoints)
{
if (controlPoints.cols() != numVariables + 1)
throw Exception("BSpline::setControlPoints: Incompatible size of control point matrix.");
int nc = controlPoints.rows();
knotaverages = controlPoints.block(0, 0, nc, numVariables);
coefficients = controlPoints.block(0, numVariables, nc, 1);
checkControlPoints();
}
DenseMatrix<FloatType> DenseMatrix<FloatType>::multiply(const DenseMatrix<FloatType> &A, const DenseMatrix<FloatType> &B)
{
MLIB_ASSERT_STR(A.cols() == B.rows(), "invalid dimensions");
const UINT rows = A.rows();
const UINT cols = B.cols();
const UINT innerCount = A.cols();
DenseMatrix<FloatType> result(rows, cols);
for(UINT row = 0; row < rows; row++)
for(UINT col = 0; col < cols; col++)
{
FloatType sum = 0.0;
for(UINT inner = 0; inner < innerCount; inner++)
sum += A(row, inner) * B(inner, col);
result(row, col) = sum;
}
return result;
}
SparseMatrix NeighborhoodBuilder::build(DenseMatrix mat) const
{
/*
* This matrix contains the k_ best neighbors for every vertex.
* The candidates are sorted in ascending order.
*
* The matrix is updated for entry as correlations are computed.
* This allows us to cut the computation time in half!
*
* We allocate twice the (worst-case) storage needed, as we have
* to symmetrize the matrix afterwards. Alternatively we could
* resize later, which might lead to a full copy.
*/
std::vector<T> entries(2 * k_ * mat.rows());
std::vector<DenseMatrix::value_type> sd(mat.rows());
DenseMatrix::Vector mu = mat.matrix().rowwise().mean();
for(unsigned int i = 0; i < mat.rows(); ++i) {
mat.row(i) = mat.row(i).array() - mu[i];
sd[i] = mat.row(i).norm();
}
for(unsigned int i = 0; i < mat.rows(); ++i) {
for(unsigned int j = i + 1; j < mat.rows(); ++j) {
double cov = mat.row(i).dot(mat.row(j));
cov = fabs(cov) / (sd[i] * sd[j]);
// Insert into the entries vector
insert_(T(i, j, cov), entries.begin() + i * k_);
insert_(T(i, j, cov), entries.begin() + j * k_);
}
}
return buildMatrix_(entries, mat);
}
/*
* Calculate coefficients of B-spline representing a multivariate polynomial
*
* The polynomial f(x), with x in R^n, has m terms on the form
* f(x) = c(0)*x(0)^E(0,0)*x(1)^E(0,1)*...*x(n-1)^E(0,n-1)
* +c(1)*x(0)^E(1,0)*x(1)^E(1,1)*...*x(n-1)^E(1,n-1)
* +...
* +c(m-1)*x(0)^E(m-1,0)*x(1)^E(m-1,1)*...*x(n-1)^E(m-1,n-1)
* where c in R^m is a vector with coefficients for each of the m terms,
* and E in N^(mxn) is a matrix with the exponents of each variable in each of the m terms,
* e.g. the first row of E defines the first term with variable exponents E(0,0) to E(0,n-1).
*
* Note: E must be a matrix of nonnegative integers
*/
DenseMatrix getBSplineBasisCoefficients(DenseVector c, DenseMatrix E, std::vector<double> lb, std::vector<double> ub)
{
unsigned int dim = E.cols();
unsigned int terms = E.rows();
assert(dim >= 1); // At least one variable
assert(terms >= 1); // At least one term (assumes that c is a column vector)
assert(terms == c.rows());
assert(dim == lb.size());
assert(dim == ub.size());
// Get highest power of each variable
DenseVector powers = E.colwise().maxCoeff();
// Store in std vector
std::vector<unsigned int> powers2;
for (unsigned int i = 0; i < powers.size(); ++i)
powers2.push_back(powers(i));
// Calculate tensor product transformation matrix T
DenseMatrix T = getTransformationMatrix(powers2, lb, ub);
// Compute power basis coefficients (lambda vector)
SparseMatrix L(T.cols(),1);
L.setZero();
for (unsigned int i = 0; i < terms; i++)
{
SparseMatrix Li(1,1);
Li.insert(0,0) = 1;
for (unsigned int j = 0; j < dim; j++)
{
int e = E(i,j);
SparseVector li(powers(j)+1);
li.reserve(1);
li.insert(e) = 1;
SparseMatrix temp = Li;
Li = kroneckerProduct(temp, li);
}
L += c(i)*Li;
}
// Compute B-spline coefficients
DenseMatrix C = T*L;
return C;
}
ConstraintQuadratic::ConstraintQuadratic(std::vector<VariablePtr> variables, DenseMatrix A, DenseMatrix b, double c, double lb, double ub)
: Constraint(variables), A(A), b(b), c(c)
{
assert(A.cols() == (int)variables.size());
assert(A.rows() == b.rows());
assert(b.cols() == 1);
numConstraints = 1;
this->lb.push_back(lb);
this->ub.push_back(ub);
jacobianCalculated = true;
hessianCalculated = true;
constraintLinear = false;
constraintConvex = false;
convexRelaxationAvailable = true;
nnzJacobian = A.rows();
nnzHessian = 0;
constraintName = "Constraint Quadratic";
// // Check for parameters for NaN
// for (int i = 0; i < A.rows(); i++)
// {
// for (int j = 0; j < A.cols(); j++)
// {
// bool nanA = false;
// if (A(i,j) != A(i,j)) nanA = true;
// assert(nanA == false);
// }
// bool nanb = false;
// if (b(i) != b(i)) nanb = true;
// assert(nanb == false);
// }
// Calculate and store Hessian
H = A.transpose() + A;
// H is symmetric so fill out lower left triangle only
for (int row = 0; row < H.rows(); row++)
{
for (int col = 0; col <= row; col++)
{
if (H(row,col) != 0)
{
nnzHessian++;
}
}
}
// Check convexity using Hessian
Eigen::EigenSolver<DenseMatrix> es(H);
DenseVector eigs = es.eigenvalues().real();
double minEigVal = 0;
for (int i = 0; i < eigs.rows(); i++)
{
if (eigs(i) < minEigVal) minEigVal = eigs(i);
}
if (minEigVal >= 0 && lb <= -INF)
{
constraintConvex = true;
}
// Note could also check that max. eigen value <= 0 and ub = INF
checkConstraintSanity();
}
void manifold_sculpting_embed(RandomAccessIterator begin, RandomAccessIterator end,
DenseMatrix& data, IndexType target_dimension,
const Neighbors& neighbors, DistanceCallback callback,
IndexType max_iteration, ScalarType squishing_rate)
{
/* Step 1: Get initial distances to each neighbor and initial
* angles between the point Pi, each neighbor Nij, and the most
* collinear neighbor of Nij.
*/
ScalarType initial_average_distance;
SparseMatrix distances_to_neighbors =
neighbors_distances_matrix(begin, end, neighbors, callback, initial_average_distance);
SparseMatrixNeighborsPair angles_and_neighbors =
angles_matrix_and_neighbors(neighbors, data);
/* Step 2: Optionally preprocess the data using PCA
* (skipped for now).
*/
ScalarType no_improvement_counter = 0, normal_counter = 0;
ScalarType current_multiplier = squishing_rate;
ScalarType learning_rate = initial_average_distance;
ScalarType best_error = DBL_MAX, current_error, point_error;
/* Step 3: Do until no improvement is made for some period
* (or until max_iteration number is reached):
*/
while (((no_improvement_counter++ < max_number_of_iterations_without_improvement)
|| (current_multiplier > multiplier_treshold))
&& (normal_counter++ < max_iteration))
{
/* Step 3a: Scale the data in non-preserved dimensions
* by a factor of squishing_rate.
*/
data.bottomRows(data.rows() - target_dimension) *= squishing_rate;
while (average_neighbor_distance(data, neighbors) < initial_average_distance)
{
data.topRows(target_dimension) /= squishing_rate;
}
current_multiplier *= squishing_rate;
/* Step 3b: Restore the previously computed relationships
* (distances to neighbors and angles to ...) by adjusting
* data points in first target_dimension dimensions.
*/
/* Start adjusting from a random point */
IndexType start_point_index = std::rand() % data.cols();
std::deque<IndexType> points_to_adjust;
points_to_adjust.push_back(start_point_index);
ScalarType steps_made = 0;
current_error = 0;
std::set<IndexType> adjusted_points;
while (!points_to_adjust.empty())
{
IndexType current_point_index = points_to_adjust.front();
points_to_adjust.pop_front();
if (adjusted_points.count(current_point_index) == 0)
{
DataForErrorFunc error_func_data = {
distances_to_neighbors,
angles_and_neighbors.first,
neighbors,
angles_and_neighbors.second,
adjusted_points,
initial_average_distance
};
adjust_point_at_index(current_point_index, data, target_dimension,
learning_rate, error_func_data, point_error);
current_error += point_error;
/* Insert all neighbors into deque */
std::copy(neighbors[current_point_index].begin(),
neighbors[current_point_index].end(),
std::back_inserter(points_to_adjust));
adjusted_points.insert(current_point_index);
}
}
if (steps_made > data.cols())
learning_rate *= learning_rate_grow_factor;
else
learning_rate *= learning_rate_shrink_factor;
if (current_error < best_error)
{
best_error = current_error;
no_improvement_counter = 0;
}
}
data.conservativeResize(target_dimension, Eigen::NoChange);
data.transposeInPlace();
}
void EigenSolverVTK<FloatType>::eigenSystemInternal(const DenseMatrix<FloatType> &M, FloatType **eigenvectors, FloatType *eigenvalues) const
{
MLIB_ASSERT_STR(M.isSymmetric(), "can only handle symmetric matrices");
const unsigned int rows = M.rows();
MLIB_ASSERT_STR(M.square() && M.rows() >= 2, "invalid matrix dimensions in EigenSolverVTK<T>::eigenSystem");
int i, j, k, iq, ip, numPos, n = int(rows);
FloatType tresh, theta, tau, t, sm, s, h, g, c, tmp;
FloatType bspace[4], zspace[4];
FloatType *b = bspace;
FloatType *z = zspace;
//
// Jacobi iteration destroys the matrix so create a temporary copy
//
DenseMatrix<FloatType> a = M;
//
// only allocate memory if the matrix is large
//
if (n > 4)
{
b = new FloatType[n];
z = new FloatType[n];
}
//
// initialize
//
for (ip = 0; ip<n; ip++)
{
for (iq = 0; iq<n; iq++)
{
eigenvectors[ip][iq] = 0.0;
}
eigenvectors[ip][ip] = 1.0;
}
for (ip = 0; ip<n; ip++)
{
b[ip] = a(ip, ip);
eigenvalues[ip] = FloatType(a(ip, ip));
z[ip] = 0.0;
}
// begin rotation sequence
for (i = 0; i<VTK_MAX_ROTATIONS; i++)
{
sm = 0.0;
for (ip = 0; ip<n - 1; ip++)
{
for (iq = ip + 1; iq<n; iq++)
{
sm += fabs(a(ip, iq));
}
}
if (sm == 0.0)
{
break;
}
if (i < 3) // first 3 sweeps
{
tresh = (FloatType)0.2*sm / (n*n);
}
else
{
tresh = (FloatType)0.0;
}
for (ip = 0; ip<n - 1; ip++)
{
for (iq = ip + 1; iq<n; iq++)
{
g = FloatType(100.0*fabs(a(ip, iq)));
// after 4 sweeps
if (i > 3 && (fabs(eigenvalues[ip]) + g) == fabs(eigenvalues[ip])
&& (fabs(eigenvalues[iq]) + g) == fabs(eigenvalues[iq]))
{
a(ip, iq) = 0.0;
}
else if (fabs(a(ip, iq)) > tresh)
{
h = eigenvalues[iq] - eigenvalues[ip];
if ((fabs(h) + g) == fabs(h))
{
t = (a(ip, iq)) / h;
}
else
{
theta = (FloatType)0.5*h / (a(ip, iq));
t = (FloatType)1.0 / (fabs(theta) + sqrt((FloatType)1.0 + theta*theta));
if (theta < 0.0)
{
t = -t;
}
}
c = (FloatType)1.0 / sqrt(1 + t*t);
s = t*c;
tau = s / ((FloatType)1.0 + c);
h = t*a(ip, iq);
z[ip] -= h;
z[iq] += h;
eigenvalues[ip] -= FloatType(h);
eigenvalues[iq] += FloatType(h);
a(ip, iq) = (FloatType)0.0;
// ip already shifted left by 1 unit
for (j = 0; j <= ip - 1; j++)
{
VTK_ROTATE(a, j, ip, j, iq);
}
// ip and iq already shifted left by 1 unit
for (j = ip + 1; j <= iq - 1; j++)
{
VTK_ROTATE(a, ip, j, j, iq);
}
// iq already shifted left by 1 unit
for (j = iq + 1; j<n; j++)
{
VTK_ROTATE(a, ip, j, iq, j);
}
for (j = 0; j<n; j++)
{
#pragma warning ( disable : 4244 )
VTK_ROTATE2(eigenvectors, j, ip, j, iq);
#pragma warning ( default : 4244 )
}
}
}
}
for (ip = 0; ip<n; ip++)
{
b[ip] += z[ip];
eigenvalues[ip] = FloatType(b[ip]);
z[ip] = 0.0;
}
}
if (i >= VTK_MAX_ROTATIONS)
{
//return false;
}
// sort eigenfunctions these changes do not affect accuracy
for (j = 0; j<n - 1; j++) // boundary incorrect
{
k = j;
tmp = eigenvalues[k];
for (i = j + 1; i<n; i++) // boundary incorrect, shifted already
{
if (eigenvalues[i] >= tmp) // why exchage if same?
{
k = i;
tmp = eigenvalues[k];
}
}
if (k != j)
{
eigenvalues[k] = eigenvalues[j];
eigenvalues[j] = FloatType(tmp);
for (i = 0; i<n; i++)
{
tmp = eigenvectors[i][j];
eigenvectors[i][j] = eigenvectors[i][k];
eigenvectors[i][k] = FloatType(tmp);
}
}
}
//
// insure eigenvector consistency (i.e., Jacobi can compute vectors that
// are negative of one another (.707,.707,0) and (-.707,-.707,0). This can
// reek havoc in hyperstreamline/other stuff. We will select the most
// positive eigenvector.
//
int ceil_half_n = (n >> 1) + (n & 1);
for (j = 0; j<n; j++)
{
for (numPos = 0, i = 0; i<n; i++)
{
if (eigenvectors[i][j] >= 0.0)
{
numPos++;
}
}
// if ( numPos < ceil(double(n)/double(2.0)) )
if (numPos < ceil_half_n)
{
for (i = 0; i<n; i++)
{
eigenvectors[i][j] *= (FloatType)-1.0;
}
}
}
if (n > 4)
{
delete[] b;
delete[] z;
}
}
