# Nystroem Approximation Kernel Ridge Regression
nakr <-
function(X, y, sigma2=NULL, lambdas=NULL, nb.landmarks=NULL)
{
  set.seed(1123)
  X <- as.matrix(X)
  n <- nrow(X)
  p <- ncol(X)
  if(is.null(sigma2)) { sigma2 <- p }
  if(is.null(nb.landmarks)) { nb.landmarks <- 15*n^(1/3) }
  X <- scale(X)
  y <- scale(y)
  idx.landmarks <- sample(1:n, nb.landmarks, replace = FALSE)
  S <- X[idx.landmarks, ]
  K11 <- gausskernel.nakr(S, S, sigma2)
  C <- gausskernel.nakr(X, S, sigma2)
  svd.K11 <- svd(K11)
  ks <- which(svd.K11$d < 1E-12) # K11 often ill-formed -> drop small sv
  if (length(ks)>0) {k <- ks[1]} else {k <- length(svd.K11$d)}
  W <- svd.K11$u[,1:k] %*% diag(1/sqrt(svd.K11$d[1:k]))
  Phi <- C %*% W
  ridge <- ridge(Phi, y, lambdas)
  r <- list(center.X=attr(X,"scaled:center"),
            scale.X=attr(X,"scaled:scale"),
            center.y=attr(y,"scaled:center"),
            scale.y=attr(y,"scaled:scale"),
            S=S,
            sigma2=sigma2,
            W=W,
            ridge=ridge
           )
  class(r) <- "nakr"
  return(r)
}

predict.nakr <-
function(o, newdata)
{
  if(class(o) != "nakr") {
    warning("Object is not of class 'nakr'")
    UseMethod("predict")
    return(invisible(NULL))
  }
  test <- as.matrix(newdata)
  test <- scale(test,center=o$center.X,scale=o$scale.X)
  K.test <- gausskernel.nakr(test, o$S, o$sigma2)
  Phi.test <- K.test %*% o$W
  yh <- predict(o$ridge, Phi.test)
  yh <- yh * o$scale.y + o$center.y
}

# 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)
{
  if(is(X1,"vector"))
    X1 <- as.matrix(X1)
  if(is(X2,"vector"))
    X2 <- as.matrix(X2)
  if (!(dim(X1)[2]==dim(X2)[2]))
    stop("X1 and X2 must have the same number of columns")
  n1 <- dim(X1)[1]
  n2 <- dim(X2)[1]
  dotX1 <- rowSums(X1*X1)
  dotX2 <- rowSums(X2*X2)
  res <- X1%*%t(X2)
  for(i in 1:n2) res[,i] <- exp((2*res[,i] - dotX1 - rep(dotX2[i],n1))/sigma2)
  return(res)
}