428 lines
14 KiB
C++
428 lines
14 KiB
C++
// only `RcppArmadillo.h` which includes `Rcpp.h`
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#include <RcppArmadillo.h>
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// through the depends attribute `Rcpp` is tolled to create
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// hooks for `RcppArmadillo` needed by the build process.
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//
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// [[Rcpp::depends(RcppArmadillo)]]
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// required for `R::qchisq()` used in `estimateBandwidth()`
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#include <Rmath.h>
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// for proper error handling
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#include <stdexcept>
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//' Estimated bandwidth for CVE.
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//'
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//' Estimates a propper bandwidth \code{h} according
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//' \deqn{h = \chi_{p-q}^{-1}\left(\frac{nObs - 1}{n-1}\right)\frac{2 tr(\Sigma)}{p}}{%
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//' h = qchisq( (nObs - 1)/(n - 1), p - q ) 2 tr(Sigma) / p}
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//'
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//' @param X data matrix of dimension (n x p) with n data points X_i of dimension
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//' q. Therefor each row represents a datapoint of dimension p.
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//' @param k Guess for rank(B).
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//' @param nObs Ether numeric of a function. If specified as numeric value
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//' its used in the computation of the bandwidth directly. If its a function
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//' `nObs` is evaluated as \code{nObs(nrow(x))}. The default behaviou if not
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//' supplied at all is to use \code{nObs <- nrow(x)^0.5}.
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//'
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//' @seealso [qchisq()]
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//'
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//' @export
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// [[Rcpp::export]]
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double estimateBandwidth(const arma::mat& X, arma::uword k, double nObs) {
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using namespace arma;
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uword n = X.n_rows; // nr samples
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uword p = X.n_cols; // dimension of rand. var. `X`
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// column mean
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mat M = mean(X);
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// center `X`
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mat C = X.each_row() - M;
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// trace of covariance matrix, `traceSigma = Tr(C' C)`
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double traceSigma = accu(C % C);
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// compute estimated bandwidth
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double qchi2 = R::qchisq((nObs - 1.0) / (n - 1), static_cast<double>(k), 1, 0);
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return 2.0 * qchi2 * traceSigma / (p * n);
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}
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//' Random element from Stiefel Manifold `S(p, q)`.
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//'
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//' Draws an element of \eqn{S(p, q)} which is the Stiefel Manifold.
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//' This is done by taking the Q-component of the QR decomposition
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//' from a `(p, q)` Matrix with independent standart normal entries.
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//' As a semi-orthogonal Matrix the result `V` satisfies \eqn{V'V = I_q}.
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//'
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//' @param p Row dimension
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//' @param q Column dimension
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//'
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//' @return Matrix of dim `(p, q)`.
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//'
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//' @seealso <https://en.wikipedia.org/wiki/Stiefel_manifold>
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//'
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//' @export
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// [[Rcpp::export]]
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arma::mat rStiefel(arma::uword p, arma::uword q) {
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arma::mat Q, R;
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arma::qr_econ(Q, R, arma::randn<arma::mat>(p, q));
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return Q;
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}
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//' Gradient computation of the loss `L_n(V)`.
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//'
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//' The loss is defined as
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//' \deqn{L_n(V) := \frac{1}{n}\sum_{j=1}^n y_2(V, X_j) - y_1(V, X_j)^2}{L_n(V) := 1/n sum_j( (y_2(V, X_j) - y_1(V, X_j)^2 )}
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//' with
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//' \deqn{y_l(s_0) := \sum_{i=1}^n w_i(V, s_0)Y_i^l}{y_l(s_0) := sum_i(w_i(V, s_0) Y_i^l)}
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//'
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//' @rdname optStiefel
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//' @keywords internal
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//' @name gradient
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double gradient(const arma::mat& X,
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const arma::mat& X_diff,
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const arma::mat& Y,
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const arma::mat& Y_rep,
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const arma::mat& V,
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const double h,
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arma::mat* G = 0 // out
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) {
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using namespace arma;
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uword n = X.n_rows;
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// orthogonal projection matrix `Q = I - VV'` for dist computation
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mat Q = -(V * V.t());
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Q.diag() += 1;
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// calc pairwise distances as `D` with `D(i, j) = d_i(V, X_j)`
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vec D_vec = sum(square(X_diff * Q), 1);
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mat D = reshape(D_vec, n, n);
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// calc weights as `W` with `W(i, j) = w_i(V, X_j)`
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mat W = exp(D / (-2.0 * h));
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// column-wise normalization via 1-norm
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W = normalise(W, 1);
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vec W_vec = vectorise(W);
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// weighted `Y` means (first and second order)
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vec y1 = W.t() * Y;
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vec y2 = W.t() * square(Y);
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// loss for each `X_i`, meaning `L(i) = L_n(V, X_i)`
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vec L = y2 - square(y1);
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// `loss = L_n(V)`
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double loss = mean(L);
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// check if gradient as output variable is set
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if (G != 0) {
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// `G = grad(L_n(V))` a.k.a. gradient of `L` with respect to `V`
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vec scale = (repelem(L, n, 1) - square(Y_rep - repelem(y1, n, 1))) % W_vec % D_vec;
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mat X_diff_scale = X_diff.each_col() % scale;
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(*G) = X_diff_scale.t() * X_diff * V;
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(*G) *= -2.0 / (h * h * n);
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}
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return loss;
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}
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//' Stiefel Optimization.
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//'
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//' Stiefel Optimization for \code{V} given a dataset \code{X} and responces
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//' \code{Y} for the model \deqn{Y\sim g(B'X) + \epsilon}{Y ~ g(B'X) + epsilon}
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//' to compute the matrix `B` such that \eqn{span{B} = span(V)^{\bot}}{%
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//' span(B) = orth(span(B))}.
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//'
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//' @param X data points
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//' @param Y response
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//' @param k assumed \eqn{rank(B)}
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//' @param nObs parameter for bandwidth estimation, typical value
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//' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
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//' @param tau Initial step size
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//' @param tol Tolerance for update error used for stopping criterion
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//' \eqn{|| V(j) V(j)' - V(j+1) V(j+1)' ||_2 < tol}{%
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//' \| V_j V_j' - V_{j+1} V_{j+1}' \|_2 < tol}.
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//' @param maxIter Upper bound of optimization iterations
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//'
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//' @return List containing the bandwidth \code{h}, optimization objective \code{V}
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//' and the matrix \code{B} estimated for the model as a orthogonal basis of the
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//' orthogonal space spaned by \code{V}.
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//'
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//' @rdname optStiefel
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//' @keywords internal
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//' @name optStiefel_simple
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double optStiefel_simple(
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const arma::mat& X,
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const arma::vec& Y,
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const int k,
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const double h,
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const double tauInitial,
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const double tol,
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const int maxIter,
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arma::mat& V, // out
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arma::vec& history // out
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) {
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using namespace arma;
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// get dimensions
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const uword n = X.n_rows; // nr samples
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const uword p = X.n_cols; // dim of random variable `X`
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const uword q = p - k; // rank(V) e.g. dim of ortho space of span{B}
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// all `X_i - X_j` differences, `X_diff.row(i * n + j) = X_i - X_j`
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mat X_diff(n * n, p);
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for (uword i = 0, k = 0; i < n; ++i) {
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for (uword j = 0; j < n; ++j) {
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X_diff.row(k++) = X.row(i) - X.row(j);
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}
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}
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const vec Y_rep = repmat(Y, n, 1);
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const mat I_p = eye<mat>(p, p);
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// initial start value for `V`
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V = rStiefel(p, q);
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// init optimization `loss`es, `error` and stepsize `tau`
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// double loss_next = datum::inf;
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double loss;
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double error = datum::inf;
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double tau = tauInitial;
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int iter;
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// main optimization loop
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for (iter = 0; iter < maxIter && error > tol; ++iter) {
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// calc gradient `G = grad_V(L)(V)`
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mat G;
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loss = gradient(X, X_diff, Y, Y_rep, V, h, &G);
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// matrix `A` for colescy-transform of the gradient
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mat A = tau * ((G * V.t()) - (V * G.t()));
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// next iteration step of `V`
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mat V_tau = inv(I_p + A) * (I_p - A) * V;
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// loss after step `L(V(tau))`
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double loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h);
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// store `loss` in `history` and increase `iter`
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history(iter) = loss;
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// validate if loss decreased
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if ((loss_tau - loss) > 0.0) {
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// ignore step, retry with half the step size
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tau = tau / 2.;
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error = datum::inf;
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} else {
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// compute step error (break condition)
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error = norm((V * V.t()) - (V_tau * V_tau.t()), 2) / (2 * q);
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// shift for next iteration
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V = V_tau;
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loss = loss_tau;
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}
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}
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// store final `loss`
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history(iter) = loss;
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return loss;
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}
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//' @rdname optStiefel
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//' @keywords internal
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//' @name optStiefel_linesearch
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double optStiefel_linesearch(
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const arma::mat& X,
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const arma::vec& Y,
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const int k,
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const double h,
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const double tauInitial,
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const double tol,
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const int maxIter,
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const double rho1,
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const double rho2,
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const int maxLineSearchIter,
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arma::mat& V, // out
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arma::vec& history // out
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) {
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using namespace arma;
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// get dimensions
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const uword n = X.n_rows; // nr samples
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const uword p = X.n_cols; // dim of random variable `X`
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const uword q = p - k; // rank(V) e.g. dim of ortho space of span{B}
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// all `X_i - X_j` differences, `X_diff.row(i * n + j) = X_i - X_j`
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mat X_diff(n * n, p);
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for (uword i = 0, k = 0; i < n; ++i) {
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for (uword j = 0; j < n; ++j) {
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X_diff.row(k++) = X.row(i) - X.row(j);
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}
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}
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const vec Y_rep = repmat(Y, n, 1);
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const mat I_p = eye<mat>(p, p);
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const mat I_2q = eye<mat>(2 * q, 2 * q);
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// initial start value for `V`
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V = rStiefel(p, q);
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// first gradient initialization
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mat G;
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double loss = gradient(X, X_diff, Y, Y_rep, V, h, &G);
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// set first `loss` in history
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history(0) = loss;
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// main curvilinear optimization loop
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double error = datum::inf;
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int iter = 0;
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while (iter++ < maxIter && error > tol) {
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// helper matrices `lU` (linesearch U), `lV` (linesearch V)
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// as describet in [Wen, Yin] Lemma 4.
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mat lU = join_rows(G, V); // linesearch "U"
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mat lV = join_rows(V, -1.0 * G); // linesearch "V"
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// `A = G V' - V G'`
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mat A = lU * lV.t();
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// set initial step size for curvilinear line search
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double tau = tauInitial, lower = 0., upper = datum::inf;
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// TODO: check if `tau` is valid for inverting
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// set line search internal gradient and loss to cycle for next iteration
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mat V_tau; // next position after a step of size `tau`, a.k.a. `V(tau)`
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mat G_tau; // gradient of `V` at `V(tau) = V_tau`
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double loss_tau; // loss (objective) at position `V(tau)`
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int lsIter = 0; // linesearch iter
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// start line search
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do {
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mat HV = inv(I_2q + (tau/2.) * lV.t() * lU) * lV.t();
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// next step `V`
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V_tau = V - tau * (lU * (HV * V));
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double LprimeV = -trace(G.t() * A * V);
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mat lB = I_p - (tau / 2.) * lU * HV;
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loss_tau = gradient(X, X_diff, Y, Y_rep, V_tau, h, &G_tau);
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double LprimeV_tau = -2. * trace(G_tau.t() * lB * A * (V + V_tau));
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// Armijo condition
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if (loss_tau > loss + (rho1 * tau * LprimeV)) {
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upper = tau;
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tau = (lower + upper) / 2.;
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// Wolfe condition
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} else if (LprimeV_tau < rho2 * LprimeV) {
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lower = tau;
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if (upper == datum::inf) {
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tau = 2. * lower;
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} else {
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tau = (lower + upper) / 2.;
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}
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} else {
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break;
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}
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} while (++lsIter < maxLineSearchIter);
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// compute error (break condition)
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// Note: `error` is the decrease of the objective `L_n(V)` and not the
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// norm of the gradient as proposed in [Wen, Yin] Algorithm 1.
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error = loss - loss_tau;
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// cycle `V`, `G` and `loss` for next iteration
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V = V_tau;
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loss = loss_tau;
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G = G_tau;
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// store final `loss`
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history(iter) = loss;
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}
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return loss;
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}
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//' Conditional Variance Estimation (CVE) method.
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//'
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//' This version uses a "simple" stiefel optimization schema.
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//'
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//' @param X data points
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//' @param Y response
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//' @param k assumed \eqn{rank(B)}
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//' @param nObs parameter for bandwidth estimation, typical value
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//' \code{nObs = nrow(X)^lambda} with \code{lambda} in the range [0.3, 0.8].
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//' @param h Bandwidth, if not specified \code{nObs} is used to compute a bandwidth.
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//' on the other hand if given (positive floating point number) \code{nObs} is ignored.
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//' @param tau Initial step size (default 1)
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//' @param tol Tolerance for update error used for stopping criterion (default 1e-5)
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//' @param slack Ratio of small negative error allowed in loss optimization (default -1e-10)
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//' @param maxIter Upper bound of optimization iterations (default 50)
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//' @param attempts Number of tryes with new random optimization starting points (default 10)
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//'
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//' @return List containing the bandwidth \code{h}, optimization objective \code{V}
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//' and the matrix \code{B} estimated for the model as a orthogonal basis of the
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//' orthogonal space spaned by \code{V}.
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//'
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//' @keywords internal
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// [[Rcpp::export]]
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Rcpp::List cve_cpp(
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const arma::mat& X,
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const arma::vec& Y,
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const std::string method,
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const int k,
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const double nObs,
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double h = -1., // default value to be overwritten
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const double tauInitial = 1.,
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const double rho1 = 0.1,
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const double rho2 = 0.9,
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const double tol = 1e-5,
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const int maxIter = 50,
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const int maxLineSearchIter = 10,
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const int attempts = 10
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) {
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using namespace arma;
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// tracker of current best results
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mat V_best;
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double loss_best = datum::inf;
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// estimate bandwidth
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if (h <= 0.0) {
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h = estimateBandwidth(X, k, nObs);
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}
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// loss history for each optimization attempt
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// each column contaions the iteration history for the loss
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mat history = mat(maxIter + 1, attempts);
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// multiple stiefel optimization attempts
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for (int i = 0; i < attempts; ++i) {
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// declare output variables
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mat V; // estimated `V` space
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vec hist = vec(history.n_rows, fill::zeros); // optimization history
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double loss;
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if (method == "simple") {
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loss = optStiefel_simple(
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X, Y, k, h,
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tauInitial, tol, maxIter,
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V, hist
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);
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} else if (method == "linesearch") {
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loss = optStiefel_linesearch(
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X, Y, k, h,
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tauInitial, tol, maxIter, rho1, rho2, maxLineSearchIter,
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V, hist
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);
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} else {
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throw std::invalid_argument("Unknown method.");
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}
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if (loss < loss_best) {
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loss_best = loss;
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V_best = V;
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}
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// add history to history collection
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history.col(i) = hist;
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}
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// get `B` as kernal of `V.t()`
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mat B = null(V_best.t());
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return Rcpp::List::create(
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Rcpp::Named("history") = history,
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Rcpp::Named("loss") = loss_best,
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Rcpp::Named("h") = h,
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Rcpp::Named("V") = V_best,
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Rcpp::Named("B") = B
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);
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}
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