double standard_deviation(const VectorXd& xx)
{
const double mean = xx.mean();
double accum = 0;
for (int kk=0, kk_max=xx.rows(); kk<kk_max; kk++)
accum += (xx(kk)-mean)*(xx(kk)-mean);
return sqrt(accum)/(xx.size()-1);
}
int main(int argc, char *argv[])
{
using namespace Eigen;
using namespace std;
MatrixXd V;
MatrixXi F;
// Load a mesh in OFF format
igl::readOFF("../shared/cheburashka.off", V, F);
// Read scalar function values from a file, U: #V by 1
VectorXd U;
igl::readDMAT("../shared/cheburashka-scalar.dmat",U);
// Compute gradient operator: #F*3 by #V
SparseMatrix<double> G;
igl::grad(V,F,G);
// Compute gradient of U
MatrixXd GU = Map<const MatrixXd>((G*U).eval().data(),F.rows(),3);
// Compute gradient magnitude
const VectorXd GU_mag = GU.rowwise().norm();
igl::viewer::Viewer viewer;
viewer.data.set_mesh(V, F);
// Compute pseudocolor for original function
MatrixXd C;
igl::jet(U,true,C);
// // Or for gradient magnitude
//igl::jet(GU_mag,true,C);
viewer.data.set_colors(C);
// Average edge length divided by average gradient (for scaling)
const double max_size = igl::avg_edge_length(V,F) / GU_mag.mean();
// Draw a black segment in direction of gradient at face barycenters
MatrixXd BC;
igl::barycenter(V,F,BC);
const RowVector3d black(0,0,0);
viewer.data.add_edges(BC,BC+max_size*GU, black);
// Hide wireframe
viewer.core.show_lines = false;
viewer.launch();
}
void linear::poisson(const VectorXd& lapl, VectorXd& f) {
if(stiffp1.size()==0) fill_stiff();
if(mass.size()==0) fill_mass();
VectorXd massl = mass * lapl ;
int N=massl.size();
VectorXd massll = massl.tail( N-1 );
VectorXd ff= solver_stiffp1.solve( massll );
f(0) = 0;
f.tail(N-1) = ff;
// zero mean
f = f.array() - f.mean();
return;
}
void UpdaterMean::costsToWeights(const VectorXd& costs, string weighting_method, double eliteness, VectorXd& weights) const
{
weights.resize(costs.size());
if (weighting_method.compare("PI-BB")==0)
{
// PI^2 style weighting: continuous, cost exponention
double h = eliteness; // In PI^2, eliteness parameter is known as "h"
double range = costs.maxCoeff()-costs.minCoeff();
if (range==0)
weights.fill(1);
else
weights = (-h*(costs.array()-costs.minCoeff())/range).exp();
}
else if (weighting_method.compare("CMA-ES")==0 || weighting_method.compare("CEM")==0 )
{
// CMA-ES and CEM are rank-based, so we must first sort the costs, and the assign a weight to
// each rank.
VectorXd costs_sorted = costs;
std::sort(costs_sorted.data(), costs_sorted.data()+costs_sorted.size());
// In Python this is more elegant because we have argsort.
// indices = np.argsort(costs)
// It is possible to do this with fancy lambda functions or std::pair in C++ too, but I don't
// mind writing two for loops instead ;-)
weights.fill(0.0);
int mu = eliteness; // In CMA-ES, eliteness parameter is known as "mu"
assert(mu<costs.size());
for (int ii=0; ii<mu; ii++)
{
double cur_cost = costs_sorted[ii];
for (int jj=0; jj<costs.size(); jj++)
{
if (costs[jj] == cur_cost)
{
if (weighting_method.compare("CEM")==0)
weights[jj] = 1.0/mu; // CEM
else
weights[jj] = log(mu+0.5) - log(ii+1); // CMA-ES
break;
}
}
}
// For debugging
//MatrixXd print_mat(3,costs.size());
//print_mat.row(0) = costs_sorted;
//print_mat.row(1) = costs;
//print_mat.row(2) = weights;
//cout << print_mat << endl;
}
else
{
cout << __FILE__ << ":" << __LINE__ << ":WARNING: Unknown weighting method '" << weighting_method << "'. Calling with PI-BB weighting." << endl;
costsToWeights(costs, "PI-BB", eliteness, weights);
return;
}
// Relative standard deviation of total costs
double mean = weights.mean();
double std = sqrt((weights.array()-mean).pow(2).mean());
double rel_std = std/mean;
if (rel_std<1e-10)
{
// Special case: all costs are the same
// Set same weights for all.
weights.fill(1);
}
// Normalize weights
weights = weights/weights.sum();
}
double DataPackage::calculateRowMean(const VectorXd &dataRow)
{
return dataRow.mean();
}
Matrix3d msac( const Eigen::Matrix2Xd& pointsFrom, const Eigen::Matrix2Xd& pointsTo,
int maxNumTrials, double confidence, double maxDistance ) {
double threshold = maxDistance;
int numPts = pointsFrom.cols();
int idxTrial = 1;
int numTrials = maxNumTrials;
double maxDis = threshold * numPts;
double bestDist = maxDis;
Matrix3d bestT;
bestT << 1, 0, 0, 0, 1, 0, 0, 0, 1;
int index1;
int index2;
// Get two random, different numbers in [0:pointsFrom.cols()-1]
std::uniform_int_distribution<int> distribution1( 0, pointsFrom.cols()-1 );
std::uniform_int_distribution<int> distribution2( 0, pointsFrom.cols()-2 );
while ( idxTrial <= numTrials ) {
// Get two random, different numbers in [0:pointsFrom.cols()-1]
index1 = distribution1( msacGenerator );
index2 = distribution2( msacGenerator );
if ( index2 >= index1 )
index2++;
Vector2d indices( index1, index2 );
/*std::cout << "indices: " << indices.transpose()
<< " pointsFrom.cols: " << pointsFrom.cols()
<< " pointsTo.cols: " << pointsTo.cols() << std::endl;*/
// Get T form Calculated from this set of points
Matrix3d T = computeTform( pointsFrom, pointsTo, indices );
VectorXd dis = evaluateTform( pointsFrom, pointsTo, T, threshold );
double accDis = dis.sum();
if ( accDis < bestDist ) {
bestDist = accDis;
bestT = T;
}
idxTrial++;
}
VectorXd dis = evaluateTform( pointsFrom, pointsTo, bestT, threshold );
threshold *= dis.mean();
int numInliers = 0;
for ( int i = 0; i < dis.rows(); i++ ){
if ( dis(i) < threshold )
numInliers++;
}
VectorXd inliers( numInliers );
int j = 0;
for ( int i = 0; i < dis.rows(); i++ ){
if ( dis(i) < threshold )
inliers(j++) = i;
}
Matrix3d T;
if ( numInliers >= 2 )
T = computeTform( pointsFrom, pointsTo, inliers );
else
T << 1, 0, 0, 0, 1, 0, 0, 0, 1;
return T;
}
// barebones version of the lasso for fixed lambda
// Eigen library is used for linear algebra
// x .............. predictor matrix
// y .............. response
// lambda ......... penalty parameter
// useSubset ...... logical indicating whether lasso should be computed on a
// subset
// subset ......... indices of subset on which lasso should be computed
// normalize ...... logical indicating whether predictors should be normalized
// useIntercept ... logical indicating whether intercept should be included
// eps ............ small numerical value (effective zero)
// useGram ........ logical indicating whether Gram matrix should be computed
// in advance
// useCrit ........ logical indicating whether to compute objective function
void fastLasso(const MatrixXd& x, const VectorXd& y, const double& lambda,
const bool& useSubset, const VectorXi& subset, const bool& normalize,
const bool& useIntercept, const double& eps, const bool& useGram,
const bool& useCrit,
// intercept, coefficients, residuals and objective function are returned
// through the following parameters
double& intercept, VectorXd& beta, VectorXd& residuals, double& crit) {
// data initializations
int n, p = x.cols();
MatrixXd xs;
VectorXd ys;
if(useSubset) {
n = subset.size();
xs.resize(n, p);
ys.resize(n);
int s;
for(int i = 0; i < n; i++) {
s = subset(i);
xs.row(i) = x.row(s);
ys(i) = y(s);
}
} else {
n = x.rows();
xs = x; // does this copy memory?
ys = y; // does this copy memory?
}
double rescaledLambda = n * lambda / 2;
// center data and store means
RowVectorXd meanX;
double meanY;
if(useIntercept) {
meanX = xs.colwise().mean(); // columnwise means of predictors
xs.rowwise() -= meanX; // sweep out columnwise means
meanY = ys.mean(); // mean of response
for(int i = 0; i < n; i++) {
ys(i) -= meanY; // sweep out mean
}
} else {
meanY = 0; // just to avoid warning, this is never used
// intercept = 0; // zero intercept
}
// some initializations
VectorXi inactive(p); // inactive predictors
int m = 0; // number of inactive predictors
VectorXi ignores; // indicates variables to be ignored
int s = 0; // number of ignored variables
// normalize predictors and store norms
RowVectorXd normX;
if(normalize) {
normX = xs.colwise().norm(); // columnwise norms
double epsNorm = eps * sqrt(n); // R package 'lars' uses n, not n-1
for(int j = 0; j < p; j++) {
if(normX(j) < epsNorm) {
// variance is too small: ignore variable
ignores.append(j, s);
s++;
// set norm to tolerance to avoid numerical problems
normX(j) = epsNorm;
} else {
inactive(m) = j; // add variable to inactive set
m++; // increase number of inactive variables
}
xs.col(j) /= normX(j); // sweep out norm
}
// resize inactive set and update number of variables if necessary
if(m < p) {
inactive.conservativeResize(m);
p = m;
}
} else {
for(int j = 0; j < p; j++) inactive(j) = j; // add variable to inactive set
m = p;
}
// compute Gram matrix if requested (saves time if number of variables is
// not too large)
MatrixXd Gram;
if(useGram) {
Gram.noalias() = xs.transpose() * xs;
}
// further initializations for iterative steps
RowVectorXd corY;
corY.noalias() = ys.transpose() * xs; // current correlations
beta.resize(p+s); // final coefficients
// compute lasso solution
if(p == 1) {
// special case of only one variable (with sufficiently large norm)
int j = inactive(0);
// set maximum step size in the direction of that variable
double maxStep = corY(j);
if(maxStep < 0) maxStep = -maxStep; // absolute value
// compute coefficients for least squares solution
VectorXd betaLS = xs.col(j).householderQr().solve(ys);
// compute lasso coefficients
beta.setZero();
if(rescaledLambda < maxStep) {
// interpolate towards least squares solution
beta(j) = (maxStep - rescaledLambda) * betaLS(0) / maxStep;
}
} else {
// further initializations for iterative steps
VectorXi active; // active predictors
int k = 0; // number of active predictors
// previous and current regression coefficients
VectorXd previousBeta = VectorXd::Zero(p+s), currentBeta = VectorXd::Zero(p+s);
// previous and current penalty parameter
double previousLambda = 0, currentLambda = 0;
// indicates variables to be dropped
VectorXi drops;
// keep track of sign of correlations for the active variables
// (double precision is necessary for solve())
VectorXd signs;
// Cholesky L of Gram matrix of active variables
MatrixXd L;
int rank = 0; // rank of Cholesky L
// maximum number of variables to be sequenced
int maxActive = findMaxActive(n, p, useIntercept);
// modified LARS algorithm for lasso solution
while(k < maxActive) {
// extract current correlations of inactive variables
VectorXd corInactiveY(m);
for(int j = 0; j < m; j++) {
corInactiveY(j) = corY(inactive(j));
}
// compute absolute values of correlations and find maximum
VectorXd absCorInactiveY = corInactiveY.cwiseAbs();
double maxCor = absCorInactiveY.maxCoeff();
// update current lambda
if(k == 0) { // no active variables
previousLambda = maxCor;
} else {
previousLambda = currentLambda;
}
currentLambda = maxCor;
if(currentLambda <= rescaledLambda) break;
if(drops.size() == 0) {
// new active variables
VectorXi newActive = findNewActive(absCorInactiveY, maxCor - eps);
// do calculations for new active variables
for(int j = 0; j < newActive.size(); j++) {
// update Cholesky L of Gram matrix of active variables
// TODO: put this into void function
int newJ = inactive(newActive(j));
VectorXd xNewJ;
double newX;
if(useGram) {
newX = Gram(newJ, newJ);
} else {
xNewJ = xs.col(newJ);
newX = xNewJ.squaredNorm();
}
double normNewX = sqrt(newX);
if(k == 0) { // no active variables, L is empty
L.resize(1,1);
L(0, 0) = normNewX;
rank = 1;
} else {
VectorXd oldX(k);
if(useGram) {
for(int j = 0; j < k; j++) {
oldX(j) = Gram(active(j), newJ);
}
} else {
for(int j = 0; j < k; j++) {
oldX(j) = xNewJ.dot(xs.col(active(j)));
}
}
VectorXd l = L.triangularView<Lower>().solve(oldX);
double lkk = newX - l.squaredNorm();
// check if L is machine singular
if(lkk > eps) {
// no singularity: update Cholesky L
lkk = sqrt(lkk);
rank++;
// add new row and column to Cholesky L
// this is a little trick: sometimes we need
// lower triangular matrix, sometimes upper
// hence we define quadratic matrix and use
// triangularView() to interpret matrix the
// correct way
L.conservativeResize(k+1, k+1);
for(int j = 0; j < k; j++) {
L(k, j) = l(j);
L(j, k) = l(j);
}
L(k,k) = lkk;
}
}
// add new variable to active set or drop it for good
// in case of singularity
if(rank == k) {
// singularity: drop variable for good
ignores.append(newJ, s);
s++; // increase number of ignored variables
p--; // decrease number of variables
if(p < maxActive) {
// adjust maximum number of active variables
maxActive = p;
}
} else {
// no singularity: add variable to active set
active.append(newJ, k);
// keep track of sign of correlation for new active variable
signs.append(sign(corY(newJ)), k);
k++; // increase number of active variables
}
}
// remove new active or ignored variables from inactive variables
// and corresponding vector of current correlations
inactive.remove(newActive);
corInactiveY.remove(newActive);
m = inactive.size(); // update number of inactive variables
}
// prepare for computation of step size
// here double precision of signs is necessary
VectorXd b = L.triangularView<Lower>().solve(signs);
VectorXd G = L.triangularView<Upper>().solve(b);
// correlations of active variables with equiangular vector
double corActiveU = 1/sqrt(G.dot(signs));
// coefficients of active variables in linear combination forming the
// equiangular vector
VectorXd w = G * corActiveU; // note that this has the right signs
// equiangular vector
VectorXd u;
if(!useGram) {
// we only need equiangular vector if we don't use the precomputed
// Gram matrix, otherwise we can compute the correlations directly
// from the Gram matrix
u = VectorXd::Zero(n);
for(int i = 0; i < n; i++) {
for(int j = 0; j < k; j++) {
u(i) += xs(i, active(j)) * w(j);
}
}
}
// compute step size in equiangular direction
double step;
if(k < maxActive) {
// correlations of inactive variables with equiangular vector
VectorXd corInactiveU(m);
if(useGram) {
for(int j = 0; j < m; j++) {
corInactiveU(j) = 0;
for(int i = 0; i < k; i++) {
corInactiveU(j) += w(i) * Gram(active(i), inactive(j));
}
}
} else {
for(int j = 0; j < m; j++) {
corInactiveU(j) = u.dot(xs.col(inactive(j)));
}
}
// compute step size in the direction of the equiangular vector
step = findStep(maxCor, corInactiveY, corActiveU, corInactiveU, eps);
} else {
// last step: take maximum possible step
step = maxCor/corActiveU;
}
// adjust step size if any sign changes and drop corresponding variables
drops = findDrops(currentBeta, active, w, eps, step);
// update current regression coefficients
previousBeta = currentBeta;
for(int j = 0; j < k; j++) {
currentBeta(active(j)) += step * w(j);
}
// update current correlations
if(useGram) {
// we also need to do this for active variables, since they may be
// dropped at a later stage
// TODO: computing a vector step * w in advance may save some computation time
for(int j = 0; j < Gram.cols(); j++) {
for(int i = 0; i < k; i++) {
corY(j) -= step * w(i) * Gram(active(i), j);
}
}
} else {
ys -= step * u; // take step in equiangular direction
corY.noalias() = ys.transpose() * xs; // update correlations
}
// drop variables if necessary
if(drops.size() > 0) {
// downdate Cholesky L
// TODO: put this into void function
for(int j = drops.size()-1; j >= 0; j--) {
// variables need to be dropped in descending order
int drop = drops(j); // index with respect to active set
// modify upper triangular part as in R package 'lars'
// simply deleting columns is not enough, other computations
// necessary but complicated due to Fortran code
L.removeCol(drop);
VectorXd z = VectorXd::Constant(k, 1, 1);
k--; // decrease number of active variables
for(int i = drop; i < k; i++) {
double a = L(i,i), b = L(i+1,i);
if(b != 0.0) {
// compute the rotation
double tau, s, c;
if(std::abs(b) > std::abs(a)) {
tau = -a/b;
s = 1.0/sqrt(1.0+tau*tau);
c = s * tau;
} else {
tau = -b/a;
c = 1.0/sqrt(1.0+tau*tau);
s = c * tau;
}
// update 'L' and 'z';
L(i,i) = c*a - s*b;
for(int j = i+1; j < k; j++) {
a = L(i,j);
b = L(i+1,j);
L(i,j) = c*a - s*b;
L(i+1,j) = s*a + c*b;
}
a = z(i);
b = z(i+1);
z(i) = c*a - s*b;
z(i+1) = s*a + c*b;
}
}
// TODO: removing all rows together may save some computation time
L.conservativeResize(k, NoChange);
rank--;
}
// mirror lower triangular part
for(int j = 0; j < k; j++) {
for(int i = j+1; i < k; i++) {
L(i,j) = L(j,i);
}
}
// add dropped variables to inactive set and make sure
// coefficients are 0
inactive.conservativeResize(m + drops.size());
for(int j = 0; j < drops.size(); j++) {
int newInactive = active(drops(j));
inactive(m + j) = newInactive;
currentBeta(newInactive) = 0; // make sure coefficient is 0
}
m = inactive.size(); // update number of inactive variables
// drop variables from active set and sign vector
// number of active variables is already updated above
active.remove(drops);
signs.remove(drops);
}
}
// interpolate coefficients for given lambda
if(k == 0) {
// lambda larger than largest lambda from steps of LARS algorithm
beta.setZero();
} else {
// penalty parameter within two steps
if(k == maxActive) {
// current coefficients are the least squares solution (in the
// high-dimensional case, as far along the solution path as possible)
// current and previous values of the penalty parameter need to be
// reset for interpolation
previousLambda = currentLambda;
currentLambda = 0;
}
// interpolate coefficients
beta = ((rescaledLambda - currentLambda) * previousBeta +
(previousLambda - rescaledLambda) * currentBeta) /
(previousLambda - currentLambda);
}
}
// transform coefficients back
VectorXd normedBeta;
if(normalize) {
if(useCrit) normedBeta = beta;
for(int j = 0; j < p; j++) beta(j) /= normX(j);
}
if(useIntercept) intercept = meanY - beta.dot(meanX);
// compute residuals for all observations
n = x.rows();
residuals = y - x * beta;
if(useIntercept) {
for(int i = 0; i < n; i++) residuals(i) -= intercept;
}
// compute value of objective function on the subset
if(useCrit && useSubset) {
if(normalize) crit = objective(normedBeta, residuals, subset, lambda);
else crit = objective(beta, residuals, subset, lambda);
}
}