intro_to_ml/19_nystroem_approximation_code.R

# compute the gaussian kernel between each row of X1 and each row of X2
# should be done more efficiently (C code, threads)
gausskernel.nakr <-
function(X1, X2, sigma2)
{
  n1 <- dim(X1)[1]
  n2 <- dim(X2)[1]
  K <- matrix(nrow = n1, ncol = n2)
  for(i in 1:n1)
    for(j in 1:n2)
      K[i,j] <- sum((X1[i,] - X2[j,])^2)
  K <- exp(-1*K/sigma2)
}

# Nystroem Approximation Kernel Ridge Regression
nakr <-
function(X, y, sigma2=NULL, lambda=1E-8, landmarks=NULL, nb.landmarks=NULL)
{
  X <- as.matrix(X)
  n <- nrow(X)
  p <- ncol(X)

  if(is.null(sigma2)) { sigma2 <- p }

  ldm <- landmarks.nakr(X, landmarks, nb.landmarks)

  X <- scale(X)
  y <- scale(y)

  C <- gausskernel.nakr(X, as.matrix(X[ldm$idx,]), sigma2)
  K11 <- C[ldm$idx,]

  svdK11 <- svd(K11)
  # K11 often ill-formed -> drop small sv
  ks <- which(svdK11$d < 1E-12)
  if (length(ks)>0) {k <- ks[1]} else {k <- length(svdK11$d)}

  US <- svdK11$u[,1:k] %*% diag(1 / sqrt(svdK11$d[1:k]))
  L <- C %*% US
  Ginv <- t(L) %*% L
  diag(Ginv) <- diag(Ginv) + lambda
  Ginv <- chol2inv(chol(Ginv))
  Ginv <- L %*% Ginv %*% t(L)
  Ginv <- - Ginv / lambda
  diag(Ginv) <- diag(Ginv) + (1/lambda)
  coef <- Ginv %*% y

  K11inv <- svdK11$v[,1:k] %*% diag(1/svdK11$d[1:k]) %*% t(svdK11$u[,1:k])
  beta <- K11inv %*% t(C) %*% coef

  r <- list(X=X,
            y=y,
            sigma2=sigma2,
            lambda=lambda,
            ldmidx=ldm$idx,
            coef=coef,
            beta=beta
           )
  class(r) <- "nakr"
  return(r)
}

landmarks.nakr <-
function(X, landmarks, nb.landmarks)
{
  n <- nrow(X)
  if(is.null(landmarks)) {
    if(is.null(nb.landmarks)) { nb.landmarks <- round(sqrt(n)) }
    ldmidx <- sample(1:n, nb.landmarks, replace = FALSE)
  } else {
    ldmidx <- which(rownames(X) %in% as.character(landmarks))
  }
  ldmidx <- sort(ldmidx)
  ldmnms <- as.numeric(rownames(X)[ldmidx])
  return(list(idx=ldmidx, nms=ldmnms))
}

predict.nakr <-
function(o, newdata)
{
  if(class(o) != "nakr") {
    warning("Object is not of class 'nakr'")
    UseMethod("predict")
    return(invisible(NULL))
  }
  newdata <- as.matrix(newdata)
  if(ncol(o$X)!=ncol(newdata)) {
    stop("Not the same number of variables btwn fitted nakr object and new data")
  }
  newdata <- scale(newdata,center=attr(o$X,"scaled:center"),
                   scale=attr(o$X,"scaled:scale"))
  Ktest <- gausskernel.nakr(newdata, as.matrix(o$X[o$ldmidx,]), o$sigma2)
  yh <- Ktest %*% o$beta
  yh <- (yh * attr(o$y,"scaled:scale")) + attr(o$y,"scaled:center")
}

kfold.nakr <-
function(X, y, K=5, lambdas=NULL, sigma2=NULL, landmarks=NULL, nb.landmarks=NULL)
{
  if(is.null(lambdas)) { lambdas <- 10^seq(-8, 2, by=1) }

  n <- nrow(X)
  folds <- rep_len(1:K, n)
  folds <- sample(folds, n)
  maes <- matrix(data = NA, nrow = K, ncol = length(lambdas))
  colnames(maes) <- lambdas
  lambda_idx <- 1
  ldm <- landmarks.nakr(X, landmarks, nb.landmarks)
  for(lambda in lambdas) {
    for(k in 1:K) {
      fold <- folds == k
      ldmnms2keep <- ldm$nms[! ldm$idx %in% which(fold)]
      nakrm <- nakr(X[!fold,], y[!fold], sigma2, lambda, landmarks=ldmnms2keep)
      pred <- predict(nakrm, X[fold,])
      maes[k,lambda_idx] <- mean(abs(pred - y[fold]))
      print(paste("lbd =", lambda, "; k =", k, "; mae =", maes[k,lambda_idx]))
    }
    lambda_idx <- lambda_idx + 1
  }
  mmaes <- colMeans(maes)
  minmmaes <- min(mmaes)
  bestlambda <- lambdas[which(mmaes == minmmaes)]
  nakrm <- nakr(X, y, sigma2, bestlambda, landmarks=ldm$nms)
}