/**
* The update rule for the basis matrix W.
* The function takes in all the matrices and only changes the
* value of the W matrix.
*
* @param V Input matrix to be factorized.
* @param W Basis matrix to be updated.
* @param H Encoding matrix.
*/
inline void WUpdate(const arma::sp_mat& V,
arma::mat& W,
const arma::mat& H)
{
if(!isStart) (*it)++;
else isStart = false;
if(*it == V.end())
{
delete it;
it = new arma::sp_mat::const_iterator(V.begin());
}
size_t currentUserIndex = it->col();
size_t currentItemIndex = it->row();
arma::mat deltaW(1, W.n_cols);
deltaW.zeros();
deltaW += (**it - arma::dot(W.row(currentItemIndex), H.col(currentUserIndex)))
* arma::trans(H.col(currentUserIndex));
if(kw != 0) deltaW -= kw * W.row(currentItemIndex);
W.row(currentItemIndex) += u*deltaW;
}
// [[Rcpp::export]]
arma::sp_mat sparseIterators(arma::sp_mat SM, double val) {
arma::sp_mat::iterator begin = SM.begin();
arma::sp_mat::iterator end = SM.end();
for (arma::sp_mat::iterator it = begin; it != end; ++it)
(*it) += val;
return SM;
}
/**
* Solve the following KKT system (2.10) of [AHO98]:
*
* [ 0 A^T I ] [ dsx ] = [ rd ]
* [ A 0 0 ] [ dy ] = [ rp ]
* [ E 0 F ] [ dsz ] = [ rc ]
* \---- M ----/
*
* where
*
* A = [ Asparse ]
* [ Adense ]
* dy = [ dysparse dydense ]
* E = Z sym I
* F = X sym I
*
*/
static inline void
SolveKKTSystem(const arma::sp_mat& Asparse,
const arma::mat& Adense,
const arma::mat& Z,
const arma::mat& M,
const arma::mat& F,
const arma::vec& rp,
const arma::vec& rd,
const arma::vec& rc,
arma::vec& dsx,
arma::vec& dysparse,
arma::vec& dydense,
arma::vec& dsz)
{
arma::mat Frd_rc_Mat, Einv_Frd_rc_Mat,
Einv_Frd_ATdy_rc_Mat, Frd_ATdy_rc_Mat;
arma::vec Einv_Frd_rc, Einv_Frd_ATdy_rc, dy;
// Note: Whenever a formula calls for E^(-1) v for some v, we solve Lyapunov
// equations instead of forming an explicit inverse.
// Compute the RHS of (2.12)
math::Smat(F * rd - rc, Frd_rc_Mat);
SolveLyapunov(Einv_Frd_rc_Mat, Z, 2. * Frd_rc_Mat);
math::Svec(Einv_Frd_rc_Mat, Einv_Frd_rc);
arma::vec rhs = rp;
const size_t numConstraints = Asparse.n_rows + Adense.n_rows;
if (Asparse.n_rows)
rhs(arma::span(0, Asparse.n_rows - 1)) += Asparse * Einv_Frd_rc;
if (Adense.n_rows)
rhs(arma::span(Asparse.n_rows, numConstraints - 1)) += Adense * Einv_Frd_rc;
// TODO(stephentu): use a more efficient method (e.g. LU decomposition)
if (!arma::solve(dy, M, rhs))
Log::Fatal << "PrimalDualSolver::SolveKKTSystem(): Could not solve KKT "
<< "system." << std::endl;
if (Asparse.n_rows)
dysparse = dy(arma::span(0, Asparse.n_rows - 1));
if (Adense.n_rows)
dydense = dy(arma::span(Asparse.n_rows, numConstraints - 1));
// Compute dx from (2.13)
math::Smat(F * (rd - Asparse.t() * dysparse - Adense.t() * dydense) - rc,
Frd_ATdy_rc_Mat);
SolveLyapunov(Einv_Frd_ATdy_rc_Mat, Z, 2. * Frd_ATdy_rc_Mat);
math::Svec(Einv_Frd_ATdy_rc_Mat, Einv_Frd_ATdy_rc);
dsx = -Einv_Frd_ATdy_rc;
// Compute dz from (2.14)
dsz = rd - Asparse.t() * dysparse - Adense.t() * dydense;
}
void CalculateEdgeLengths(vtkPolyData* mesh, arma::sp_mat& edge_lengths) {
vtkSmartPointer<vtkIdList> cellIdList = vtkSmartPointer<vtkIdList>::New();
vtkSmartPointer<vtkIdList> pointIdList = vtkSmartPointer<vtkIdList>::New();
const int npts = mesh->GetNumberOfPoints();
edge_lengths.set_size(npts, npts);
for (vtkIdType i = 0; i < npts; i++) {
cellIdList->Reset();
mesh->GetPointCells(i, cellIdList);
double *pt1 = mesh->GetPoint(i);
for (vtkIdType j = 0; j < cellIdList->GetNumberOfIds(); j++) {
pointIdList->Reset();
mesh->GetCellPoints(cellIdList->GetId(i), pointIdList);
for (vtkIdType k = 0; k < pointIdList->GetNumberOfIds(); k++) {
vtkIdType pt2_id = pointIdList->GetId(k);
if (pt2_id != i) { //not the same point
if (edge_lengths(i, pt2_id) != 0) { //is not calculated yet
double *pt2 = mesh->GetPoint(pt2_id);
const double dx = pt1[0] - pt2[0];
const double dy = pt1[1] - pt2[1];
const double dz = pt1[2] - pt2[2];
const double d = sqrt(dx * dx + dy * dy + dz * dz);
edge_lengths(i, pt2_id) = d;
edge_lengths(pt2_id, i) = d;
}
}
}
}
}
}
void Initialize(const arma::sp_mat& dataset, const size_t rank)
{
(void)rank;
n = dataset.n_rows;
m = dataset.n_cols;
it = new arma::sp_mat::const_iterator(dataset.begin());
isStart = true;
}
inline void SVDIncompleteIncrementalLearning::
WUpdate<arma::sp_mat>(const arma::sp_mat& V,
arma::mat& W,
const arma::mat& H)
{
arma::mat deltaW(n, W.n_cols);
deltaW.zeros();
for(arma::sp_mat::const_iterator it = V.begin_col(currentUserIndex);
it != V.end_col(currentUserIndex);it++)
{
double val = *it;
size_t i = it.row();
deltaW.row(i) += (val - arma::dot(W.row(i), H.col(currentUserIndex))) *
arma::trans(H.col(currentUserIndex));
if(kw != 0) deltaW.row(i) -= kw * W.row(i);
}
W += u*deltaW;
}
/**
* Create an unweighted graph laplacian from the edges.
*/
void CreateSparseGraphLaplacian(const arma::mat& edges,
arma::sp_mat& laplacian)
{
// Get the number of vertices in the problem.
const size_t vertices = max(max(edges)) + 1;
laplacian.zeros(vertices, vertices);
for (size_t i = 0; i < edges.n_cols; ++i)
{
laplacian(edges(0, i), edges(1, i)) = -1.0;
laplacian(edges(1, i), edges(0, i)) = -1.0;
}
for (size_t i = 0; i < vertices; ++i)
{
laplacian(i, i) = -arma::accu(laplacian.row(i));
}
}
void GeneralizedRosenbrockFunction::Gradient(const arma::mat& coordinates,
const size_t i,
arma::sp_mat& gradient) const
{
gradient.set_size(n);
const size_t p = visitationOrder[i];
gradient[p] = 400 * (std::pow(coordinates[p], 3) - coordinates[p] *
coordinates[p + 1]) + 2 * (coordinates[p] - 1);
gradient[p + 1] = 200 * (coordinates[p + 1] - std::pow(coordinates[p], 2));
}
inline void SVDIncompleteIncrementalLearning::
HUpdate<arma::sp_mat>(const arma::sp_mat& V,
const arma::mat& W,
arma::mat& H)
{
arma::mat deltaH(H.n_rows, 1);
deltaH.zeros();
for(arma::sp_mat::const_iterator it = V.begin_col(currentUserIndex);
it != V.end_col(currentUserIndex);it++)
{
double val = *it;
size_t i = it.row();
if((val = V(i, currentUserIndex)) != 0)
deltaH += (val - arma::dot(W.row(i), H.col(currentUserIndex))) *
arma::trans(W.row(i));
}
if(kh != 0) deltaH -= kh * H.col(currentUserIndex);
H.col(currentUserIndex++) += u * deltaH;
currentUserIndex = currentUserIndex % m;
}
void CalculateInversiveDistanceE2(const arma::sp_mat& l, const arma::vec& gamma, arma::sp_mat& I) {
const int npts = gamma.n_elem;
I.set_size(npts, npts);
//the measure is symmetric, process only upper triangle
for (int i = 0; i < npts; i++) {
for (int j = i + 1; j < npts; j++) {
if (l(i, j) > 0) { //only for existing edges
const double I_ij = (l(i, j) * l(i, j) - gamma(i) * gamma(i) - gamma(j) * gamma(j)) / (2 * gamma(i) * gamma(j));
I(i, j) = I_ij;
I(j, i) = I_ij;
}
}
}
}
void CalculateGeneralizedEdgeLengthsE2(const arma::vec& gamma, const arma::sp_mat& I, arma::sp_mat& gen_edge_lengths) {
const int npts = gamma.n_elem;
gen_edge_lengths.set_size(npts, npts);
//the measure is symmetric, process only upper triangle
for (int i = 0; i < npts; i++) {
for (int j = i + 1; j < npts; j++) {
if (I(i, j) > 0) { //only for existing edges
const double l_ij = sqrt(gamma(i) * gamma(i) + gamma(j) * gamma(j) + 2 * gamma(i) * gamma(j) * I(i, j));
gen_edge_lengths(i, j) = l_ij;
gen_edge_lengths(j, i) = l_ij;
}
}
}
}
void LogisticRegressionFunction<MatType>::PartialGradient(
const arma::mat& parameters,
const size_t j,
arma::sp_mat& gradient) const
{
const arma::rowvec diffs = responses - (1 / (1 + arma::exp(-parameters(0, 0)
- parameters.tail_cols(parameters.n_elem - 1) * predictors)));
gradient.set_size(arma::size(parameters));
if (j == 0)
{
gradient[j] = -arma::accu(diffs);
}
else
{
gradient[j] = arma::dot(-predictors.row(j - 1), diffs) + lambda *
parameters(0, j);
}
}
//' Compute ego/alter edge coordinates considering alter's size and aspect ratio
//'
//' Given a graph, vertices' positions and sizes, calculates the absolute positions
//' of the endpoints of the edges considering the plot's aspect ratio.
//'
//' @param graph A square matrix of size \eqn{n}. Adjacency matrix.
//' @param toa Integer vector of size \eqn{n}. Times of adoption.
//' @param x Numeric vector of size \eqn{n}. x-coordinta of vertices.
//' @param y Numeric vector of size \eqn{n}. y-coordinta of vertices.
//' @param vertex_cex Numeric vector of size \eqn{n}. Vertices' sizes in terms
//' of the x-axis (see \code{\link{symbols}}).
//' @param undirected Logical scalar. Whether the graph is undirected or not.
//' @param no_contemporary Logical scalar. Whether to return (compute) edges'
//' coordiantes for vertices with the same time of adoption (see details).
//' @param dev Numeric vector of size 2. Height and width of the device (see details).
//' @param ran Numeric vector of size 2. Range of the x and y axis (see details).
//' @param curved Logical vector.
//' @return A numeric matrix of size \eqn{m\times 5}{m * 5} with the following
//' columns:
//' \item{x0, y0}{Edge origin}
//' \item{x1, y1}{Edge target}
//' \item{alpha}{Relative angle between \code{(x0,y0)} and \code{(x1,y1)} in terms
//' of radians}
//' With \eqn{m} as the number of resulting edges.
//' @details
//'
//' In order to make the plot's visualization more appealing, this function provides
//' a straight forward way of computing the tips of the edges considering the
//' aspect ratio of the axes range. In particular, the following corrections are
//' made at the moment of calculating the egdes coords:
//'
//' \itemize{
//' \item{Instead of using the actual distance between ego and alter, a relative
//' one is calculated as follows
//' \deqn{d'=\left[(x_0-x_1)^2 + (y_0' - y_1')^2\right]^\frac{1}{2}}{d'=sqrt[(x0-x1)^2 + (y0'-y1')^2]}
//' where \eqn{%
//' y_i'=y_i\times\frac{\max x - \min x}{\max y - \min y} }{%
//' yi' = yi * [max(x) - min(x)]/[max(y) - min(y)]}
//' }
//' \item{Then, for the relative elevation angle, \code{alpha}, the relative distance \eqn{d'}
//' is used, \eqn{\alpha'=\arccos\left( (x_0 - x_1)/d' \right)}{\alpha' = acos[ (x0 - x1)/d' ]}}
//' \item{Finally, the edge's endpoint's (alter) coordinates are computed as follows: %
//' \deqn{%
//' x_1' = x_1 + \cos(\alpha')\times v_1}{%
//' x1' = x1 + cos(\alpha') * v1
//' }
//' \deqn{%
//' y_1' = y_1 -+ \sin(\alpha')\times v_1 \times\frac{\max y - \min y}{\max x - \min x} }{%
//' y1' = y1 -+ sin(\alpha')*[max(y) - min(y)]/[max(x) - min(x)]
//' }
//' Where \eqn{v_1}{v1} is alter's size in terms of the x-axis, and the sign of
//' the second term in \eqn{y_1'}{y1'} is negative iff \eqn{y_0 < y_1}{y0<y1}.
//' }
//' }
//'
//' The same process (with sign inverted) is applied to the edge starting piont.
//' The resulting values, \eqn{x_1',y_1'}{x1',y1'} can be used with the function
//' \code{\link{arrows}}. This is the workhorse function used in \code{\link{plot_threshold}}.
//'
//' The \code{dev} argument provides a reference to rescale the plot accordingly
//' to the device, and former, considering the size of the margins as well (this
//' can be easily fetched via \code{par("pin")}, plot area in inches).
//'
//' On the other hand, \code{ran} provides a reference for the adjustment
//' according to the range of the data, this is \code{range(x)[2] - range(x)[1]}
//' and \code{range(y)[2] - range(y)[1]} respectively.
//'
//' @keywords misc dplot
//' @examples
//' # --------------------------------------------------------------------------
//' data(medInnovationsDiffNet)
//' library(sna)
//'
//' # Computing coordinates
//' set.seed(79)
//' coords <- sna::gplot(as.matrix(medInnovationsDiffNet$graph[[1]]))
//'
//' # Getting edge coordinates
//' vcex <- rep(1.5, nnodes(medInnovationsDiffNet))
//' ecoords <- edges_coords(
//' medInnovationsDiffNet$graph[[1]],
//' diffnet.toa(medInnovationsDiffNet),
//' x = coords[,1], y = coords[,2],
//' vertex_cex = vcex,
//' dev = par("pin")
//' )
//'
//' ecoords <- as.data.frame(ecoords)
//'
//' # Plotting
//' symbols(coords[,1], coords[,2], circles=vcex,
//' inches=FALSE, xaxs="i", yaxs="i")
//'
//' with(ecoords, arrows(x0,y0,x1,y1, length=.1))
//' @export
// [[Rcpp::export]]
NumericMatrix edges_coords(
const arma::sp_mat & graph,
const arma::colvec & toa,
const arma::colvec & x,
const arma::colvec & y,
const arma::colvec & vertex_cex,
bool undirected=true,
bool no_contemporary=true,
NumericVector dev = NumericVector::create(),
NumericVector ran = NumericVector::create(),
LogicalVector curved = LogicalVector::create()
) {
// The output matrix has the following
// - x0 and y0
// - x1 and y1
// - alpha
std::vector< double > x0;
std::vector< double > y0;
std::vector< double > x1;
std::vector< double > y1;
std::vector< double > alpha;
// Rescaling the vertex sizes
arma::colvec vertex_size(vertex_cex);
// If yexpand is too small, just throw an error
if (ran.length() == 0) {
ran = NumericVector::create(2);
ran[0] = x.max() - x.min();
ran[1] = y.max() - y.min();
}
// Expansion factor for y
double yexpand = 1.0;
if ( ran[1] > 1e-5 ) yexpand = ran[1]/ran[0];
// Adjusting for device size
if (dev.length() == 0)
dev = NumericVector::create(2,1.0);
// Curved?
if (curved.length() == 0)
curved = LogicalVector::create(graph.n_nonzero, true);
yexpand = yexpand * (dev[0]/dev[1]);
for(arma::sp_mat::const_iterator it = graph.begin(); it != graph.end(); ++it) {
int i = it.row();
int j = it.col();
// Checking conditions
if (undirected && (i < j)) continue;
if (no_contemporary && (toa(i)==toa(j)) ) continue;
// Computing angle
double a = atan2((y(j) - y(i))/yexpand, x(j) - x(i));
alpha.push_back(a);
// Adding the xs and the ys
x0.push_back(x.at(i) + cos(a)*vertex_size.at(i));
x1.push_back(x.at(j) - cos(a)*vertex_size.at(j));
// The formula needs an extra help to figure out the ys
y0.push_back(y.at(i) + sin(a)*vertex_size.at(i)*yexpand);
y1.push_back(y.at(j) - sin(a)*vertex_size.at(j)*yexpand);
}
// Building up the output
int e = x0.size();
NumericMatrix out(e,5);
for(int i=0; i<e; ++i) {
out(i,0) = x0[i];
out(i,1) = y0[i];
out(i,2) = x1[i];
out(i,3) = y1[i];
out(i,4) = alpha[i];
}
colnames(out) = CharacterVector::create("x0", "y0", "x1", "y1", "alpha");
return out;
}
// [[Rcpp::export]]
arma::sp_mat sparseTranspose(arma::sp_mat SM) {
return SM.t();
}
// compute the log likelihood and its gradient w.r.t. theta
int dtq::compGrad(void)
{
// remember, everything here is for equispaced data
// we'll save the non-equispaced case for our scala + spark code :)
if ((! haveData) || (! haveMyh)) return 1;
if (spi<1) return 1;
loglikmat = arma::zeros(ltvec-1,numts);
if (spi==1) // special case
{
}
else
{
// strategy: precompute and store common elements in Mats and Cubs
// compute gradf and gradg at all spatial grid points
arma::mat gradfy = arma::zeros(ylen,curtheta.n_elem);
arma::mat gradgy = arma::zeros(ylen,curtheta.n_elem);
this->gradFGyvec(gradfy, gradgy);
// ompute gradf and gradg at all the data points
arma::cube gradfdata = arma::zeros(curtheta.n_elem, (ltvec-1), numts);
arma::cube gradgdata = arma::zeros(curtheta.n_elem, (ltvec-1), numts);
this->gradFGdata(gradfdata, gradgdata);
// initialize cubes to store all states and adjoints,
// at all internal time points (spi-1),
// for each pair of time series points (ltvec-1),
// and at all spatial grid points (ylen)
arma::cube dtqcube = arma::zeros(ylen,(ltvec-1),(spi-1));
arma::cube adjcube = arma::zeros(ylen,(ltvec-1),(spi-1));
// temporary matrix to store the initial state, phatinit
arma::mat phatinit = arma::zeros(ylen,(ltvec-1));
// cube to store the gradient of the initial state w.r.t. theta
arma::cube phatgrad = arma::zeros(ylen,(ltvec-1),curtheta.n_elem);
// build the big matrix of initial conditions
// and the gradients of those initial conditions!
this->phatinitgrad(phatinit, phatgrad, gradfdata, gradgdata);
dtqcube.slice(0) = phatinit;
// propagate states forward in time by (spi-2) steps
if (spi >= 3)
for (int i=1; i<=(spi-2); i++)
dtqcube.slice(i) = myk * prop * dtqcube.slice(i-1);
// now multiply on the left by the Gamma vectors
const arma::vec muvec = yvec + fy*myh;
const arma::vec sigvec = gy*sqrt(myh);
arma::cube allgamma = arma::zeros(ylen,numts,(ltvec-1));
for (int j=0; j<(ltvec-1); j++)
{
for (int l=0; l<numts; l++)
{
allgamma.slice(j).col(l) = myk*gausspdf((*odata)(j+1,l),muvec,sigvec);
loglikmat(j,l) = log(arma::dot(allgamma.slice(j).col(l),dtqcube.slice(spi-2).col(j)));
}
}
// std::cout << loglikmat << '\n';
// initialize the adjoint calculation
for (int j=0; j<(ltvec-1); j++)
for (int l=0; l<numts; l++)
adjcube.slice(spi-2).col(j) += allgamma.slice(j).col(l) / exp(loglikmat(j,l));
// propagate adjoints backward in time by (spi-2) steps
arma::sp_mat transprop = prop.t();
if (spi >= 3)
for (int i=(spi-2); i>=1; i--)
adjcube.slice(i-1) = myk * transprop * adjcube.slice(i);
// stuff that we need for a bunch of gradients
gradloglik = arma::zeros(curtheta.n_elem);
arma::vec gvecm1 = arma::pow(gy,-1);
arma::vec gvecm2 = arma::pow(gy,-2);
arma::vec gvecm3 = arma::pow(gy,-3);
// actual gradient calculation
// proceed element-wise through theta_i
for (int i=0; i<curtheta.n_elem; i++)
{
arma::vec temp1 = gvecm2 % gradfy.col(i);
arma::vec temp2 = gvecm1 % gradgy.col(i);
arma::vec temp3 = (1.0/myh)*gvecm3 % gradgy.col(i);
arma::sp_mat::const_iterator start = prop.begin();
arma::sp_mat::const_iterator end = prop.end();
arma::umat dkdtloc(2, prop.n_nonzero);
arma::vec dkdtval(prop.n_nonzero);
unsigned int dkdtc = 0;
for (arma::sp_mat::const_iterator it = start; it != end; ++it)
{
dkdtloc(0,dkdtc) = it.row();
dkdtloc(1,dkdtc) = it.col();
dkdtc++;
}
#pragma omp parallel for
for (unsigned int dkdtcount=0; dkdtcount < prop.n_nonzero; dkdtcount++)
{
unsigned int orow = dkdtloc(0,dkdtcount);
unsigned int ocol = dkdtloc(1,dkdtcount);
double comval = yvec(orow) - muvec(ocol);
dkdtval(dkdtcount) = myk*(prop.values[dkdtcount])*( comval*temp1(ocol) - temp2(ocol) + temp3(ocol)*comval*comval );
}
arma::sp_mat dkdtheta(dkdtloc, dkdtval, ylen, ylen, false, true);
// implement formula (22) from the DSAA paper
// need gradient of Gamma{F-1}
double tally = 0.0;
#pragma omp parallel for reduction(+:tally)
for (int j=0; j<(ltvec-1); j++)
{
tally += arma::dot(phatgrad.slice(i).col(j),adjcube.slice(0).col(j));
}
#pragma omp parallel for collapse(2) reduction(+:tally)
for (int j=0; j<(ltvec-1); j++)
for (int l=0; l<numts; l++)
{
double xi = (*odata)((j+1),l);
arma::vec gammagrad = (xi-muvec) % temp1;
gammagrad += arma::pow(xi-muvec,2) % temp3;
gammagrad -= temp2;
gammagrad = gammagrad % allgamma.slice(j).col(l);
tally += arma::dot(gammagrad,dtqcube.slice(spi-2).col(j)) / exp(loglikmat(j,l));
}
// we have tested and found that the dot product is better than the
// triple matrix product here, i.e., it is worth taking the transpose
// arma::mat dkdtheta = dkdthetatrans.t();
#pragma omp parallel for collapse(2) reduction(+:tally)
for (int j=0; j<(ltvec-1); j++)
for (int l=0; l<(spi-2); l++)
{
tally += arma::dot((dkdtheta*dtqcube.slice(l).col(j)),adjcube.slice(l+1).col(j));
}
gradloglik(i) = tally;
}
}
haveLoglik = true;
haveGradloglik = true;
return 0;
}