137 lines
4.1 KiB
R
137 lines
4.1 KiB
R
#' Simple implementation of the CVE method. 'Simple' means that this method is
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#' a classic GD method unsing no further tricks.
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#'
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#' @keywords internal
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#' @export
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cve_simple <- function(X, Y, k,
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nObs = sqrt(nrow(X)),
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h = NULL,
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tau = 1.0,
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tol = 1e-3,
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slack = 0,
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epochs = 50L,
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attempts = 10L
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) {
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# Set `grad` functions environment to enable if to find this environments
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# local variabels, needed to enable the manipulation of this local variables
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# from within `grad`.
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environment(grad) <- environment()
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# Setup histories.
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loss.history <- matrix(NA, epochs, attempts)
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error.history <- matrix(NA, epochs, attempts)
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# Get dimensions.
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n <- nrow(X)
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p <- ncol(X)
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q <- p - k
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# Save initial learning rate `tau`.
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tau.init <- tau
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# Addapt tolearance for break condition.
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tol <- sqrt(2 * q) * tol
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# Estaimate bandwidth if not given.
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if (missing(h) | !is.numeric(h)) {
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h <- estimate.bandwidth(X, k, nObs)
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}
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# Compute persistent data.
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# Compute lookup indexes for symmetrie, lower/upper
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# triangular parts and vectorization.
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pair.index <- elem.pairs(seq(n))
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i <- pair.index[, 1] # `i` indices of `(i, j)` pairs
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j <- pair.index[, 2] # `j` indices of `(i, j)` pairs
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# Matrix of vectorized indices. (vec(index) -> seq)
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index <- matrix(seq(n * n), n, n)
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lower <- index[lower.tri(index)]
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upper <- t(index)[lower]
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# Create all pairewise differences of rows of `X`.
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X_diff <- X[i, , drop = F] - X[j, , drop = F]
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# Identity matrix.
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I_p <- diag(1, p)
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# Init tracking of current best (according multiple attempts).
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V.best <- NULL
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loss.best <- Inf
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# Start loop for multiple attempts.
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for (attempt in 1:attempts) {
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# reset step width `tau`
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tau <- tau.init
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# Sample a `(p, q)` dimensional matrix from the stiefel manifold as
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# optimization start value.
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V <- rStiefl(p, q)
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# Initial loss and gradient.
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loss <- Inf
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G <- grad(X, Y, V, h, loss.out = TRUE, persistent = TRUE)
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# Set last loss (aka, loss after applying the step).
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loss.last <- loss
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# Cayley transform matrix `A`
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A <- (G %*% t(V)) - (V %*% t(G))
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## Start optimization loop.
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for (epoch in 1:epochs) {
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# Apply learning rate `tau`.
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A.tau <- tau * A
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# Parallet transport (on Stiefl manifold) into direction of `G`.
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V.tau <- solve(I_p + A.tau) %*% ((I_p - A.tau) %*% V)
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# Loss at position after a step.
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loss <- grad(X, Y, V.tau, h, loss.only = TRUE, persistent = TRUE)
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# Check if step is appropriate
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if ((loss - loss.last) > slack * loss.last) {
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tau <- tau / 2
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next() # Keep position and try with smaller `tau`.
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}
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# Compute error.
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error <- norm(V %*% t(V) - V.tau %*% t(V.tau), type = "F")
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# Write history.
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loss.history[epoch, attempt] <- loss
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error.history[epoch, attempt] <- error
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# Check break condition (epoch check to skip ignored gradient calc).
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# Note: the devision by `sqrt(2 * k)` is included in `tol`.
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if (error < tol | epoch >= epochs) {
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# take last step and stop optimization.
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V <- V.tau
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break()
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}
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# Perform the step and remember previous loss.
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V <- V.tau
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loss.last <- loss
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# Compute gradient at new position.
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G <- grad(X, Y, V, h, persistent = TRUE)
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# Cayley transform matrix `A`
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A <- (G %*% t(V)) - (V %*% t(G))
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}
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# Check if current attempt improved previous ones
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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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}
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return(list(
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loss.history = loss.history,
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error.history = error.history,
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loss = loss.best,
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V = V.best,
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B = null(V.best),
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h = h
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))
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}
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