235 lines
8.2 KiB
R
235 lines
8.2 KiB
R
#' Using unbiased (but not MLE) estimates for the Kronecker decomposed
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#' covariance matrices Delta_1, Delta_2 for approximating the log-likelihood
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#' giving a closed form solution for the gradient.
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#'
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#' Delta_1 = n^-1 sum_i R_i' R_i,
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#' Delta_2 = n^-1 sum_i R_i R_i'.
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#'
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#' @export
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kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
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max.iter = 500L, max.line.iter = 50L, step.size = 1e-3,
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nesterov.scaling = function(a, t) 0.5 * (1 + sqrt(1 + (2 * a)^2)),
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eps = .Machine$double.eps,
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logger = NULL
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) {
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# Check if X and Fy have same number of observations
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stopifnot(nrow(X) == NROW(Fy))
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n <- nrow(X) # Number of observations
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# Get and check predictor dimensions
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if (length(dim(X)) == 2L) {
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stopifnot(!missing(shape))
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stopifnot(ncol(X) == prod(shape[1:2]))
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p <- as.integer(shape[1]) # Predictor "height"
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q <- as.integer(shape[2]) # Predictor "width"
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} else if (length(dim(X)) == 3L) {
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p <- dim(X)[2]
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q <- dim(X)[3]
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dim(X) <- c(n, p * q)
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} else {
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stop("'X' must be a matrix or 3-tensor")
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}
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# Get and check response dimensions
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if (!is.array(Fy)) {
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Fy <- as.array(Fy)
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}
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if (length(dim(Fy)) == 1L) {
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k <- r <- 1L
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dim(Fy) <- c(n, 1L)
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} else if (length(dim(Fy)) == 2L) {
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stopifnot(!missing(shape))
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stopifnot(ncol(Fy) == prod(shape[3:4]))
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k <- as.integer(shape[3]) # Response functional "height"
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r <- as.integer(shape[4]) # Response functional "width"
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} else if (length(dim(Fy)) == 3L) {
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k <- dim(Fy)[2]
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r <- dim(Fy)[3]
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dim(Fy) <- c(n, k * r)
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} else {
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stop("'Fy' must be a vector, matrix or 3-tensor")
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}
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### Step 1: (Approx) Least Squares solution for `X = Fy B' + epsilon`
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# Vectorize
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dim(Fy) <- c(n, k * r)
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dim(X) <- c(n, p * q)
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# Solve
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cpFy <- crossprod(Fy) # TODO: Check/Test and/or replace
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if (n <= k * r || qr(cpFy)$rank < k * r) {
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# In case of under-determined system replace the inverse in the normal
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# equation by the Moore-Penrose Pseudo Inverse
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B <- t(matpow(cpFy, -1) %*% crossprod(Fy, X))
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} else {
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# Compute OLS estimate by the Normal Equation
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B <- t(solve(cpFy, crossprod(Fy, X)))
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}
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# De-Vectroize (from now on tensor arithmetics)
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dim(Fy) <- c(n, k, r)
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dim(X) <- c(n, p, q)
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# Decompose `B = alpha x beta` into `alpha` and `beta`
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c(alpha0, beta0) %<-% approx.kronecker(B, c(q, r), c(p, k))
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# Compute residuals
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R <- X - (Fy %x_3% alpha0 %x_2% beta0)
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# Covariance estimates and scaling factor
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Delta.1 <- tcrossprod(mat(R, 3)) / n
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Delta.2 <- tcrossprod(mat(R, 2)) / n
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s <- mean(diag(Delta.1))
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# Inverse Covariances
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Delta.1.inv <- solve(Delta.1)
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Delta.2.inv <- solve(Delta.2)
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# cross dependent covariance estimates
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S.1 <- tcrossprod(mat(R, 3), mat(R %x_2% Delta.2.inv, 3)) / n
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S.2 <- tcrossprod(mat(R, 2), mat(R %x_3% Delta.1.inv, 2)) / n
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# Evaluate negative log-likelihood (2 pi term dropped)
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loss <- -0.5 * (n * (p * q * log(s) - p * log(det(Delta.1)) -
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q * log(det(Delta.2))) - s * sum(S.1 * Delta.1.inv))
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# Call history callback (logger) before the first iteration
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if (is.function(logger)) {
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logger(0L, loss, alpha0, beta0, Delta.1, Delta.2, NA)
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}
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### Step 2: MLE estimate with LS solution as starting value
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a0 <- 0
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a1 <- 1
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alpha1 <- alpha0
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beta1 <- beta0
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# main descent loop
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no.nesterov <- TRUE
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break.reason <- NA
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for (iter in seq_len(max.iter)) {
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if (no.nesterov) {
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# without extrapolation as fallback
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alpha.moment <- alpha1
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beta.moment <- beta1
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} else {
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# extrapolation using previous direction
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alpha.moment <- alpha1 + ((a0 - 1) / a1) * (alpha1 - alpha0)
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beta.moment <- beta1 + ((a0 - 1) / a1) * ( beta1 - beta0)
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}
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# Extrapolated residuals
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R <- X - (Fy %x_3% alpha.moment %x_2% beta.moment)
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# Recompute Covariance Estimates and scaling factor
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Delta.1 <- tcrossprod(mat(R, 3)) / n
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Delta.2 <- tcrossprod(mat(R, 2)) / n
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s <- mean(diag(Delta.1))
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# Inverse Covariances
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Delta.1.inv <- solve(Delta.1)
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Delta.2.inv <- solve(Delta.2)
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# cross dependent covariance estimates
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S.1 <- tcrossprod(mat(R, 3), mat(R %x_2% Delta.2.inv, 3)) / n
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S.2 <- tcrossprod(mat(R, 2), mat(R %x_3% Delta.1.inv, 2)) / n
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# Gradient "generating" tensor
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G <- (sum(S.1 * Delta.1.inv) - p * q / s) * R
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G <- G + R %x_2% ((diag(q, p) - s * (Delta.2.inv %*% S.2)) %*% Delta.2.inv)
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G <- G + R %x_3% ((diag(p, q) - s * (Delta.1.inv %*% S.1)) %*% Delta.1.inv)
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G <- G + s * (R %x_2% Delta.2.inv %x_3% Delta.1.inv)
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# Calculate Gradients
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grad.alpha <- tcrossprod(mat(G, 3), mat(Fy %x_2% beta.moment, 3))
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grad.beta <- tcrossprod(mat(G, 2), mat(Fy %x_3% alpha.moment, 2))
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# Backtracking line search (Armijo type)
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# The `inner.prod` is used in the Armijo break condition but does not
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# depend on the step size.
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inner.prod <- sum(grad.alpha^2) + sum(grad.beta^2)
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# backtracking loop
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for (delta in step.size * 0.618034^seq.int(0L, len = max.line.iter)) {
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# Update `alpha` and `beta` (note: add(+), the gradients are already
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# pointing into the negative slope direction of the loss cause they are
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# the gradients of the log-likelihood [NOT the negative log-likelihood])
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alpha.temp <- alpha.moment + delta * grad.alpha
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beta.temp <- beta.moment + delta * grad.beta
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# Update Residuals, Covariances, ...
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R <- X - (Fy %x_3% alpha.temp %x_2% beta.temp)
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Delta.1 <- tcrossprod(mat(R, 3)) / n
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Delta.2 <- tcrossprod(mat(R, 2)) / n
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s <- mean(diag(Delta.1))
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Delta.1.inv <- solve(Delta.1)
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Delta.2.inv <- solve(Delta.2)
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S.1 <- tcrossprod(mat(R, 3), mat(R %x_2% Delta.2.inv, 3)) / n
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# S.2 not needed
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# Re-evaluate negative log-likelihood
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loss.temp <- -0.5 * (n * (p * q * log(s) - p * log(det(Delta.1)) -
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q * log(det(Delta.2))) - s * sum(S.1 * Delta.1.inv))
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# Armijo line search break condition
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if (loss.temp <= loss - 0.1 * delta * inner.prod) {
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break
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}
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}
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# Call logger (invoke history callback)
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if (is.function(logger)) {
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logger(iter, loss.temp, alpha.temp, beta.temp, Delta.1, Delta.2, delta)
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}
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# Enforce descent
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if (loss.temp < loss) {
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alpha0 <- alpha1
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alpha1 <- alpha.temp
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beta0 <- beta1
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beta1 <- beta.temp
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# check break conditions
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if (mean(abs(alpha1)) + mean(abs(beta1)) < eps) {
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break.reason <- "alpha, beta numerically zero"
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break # estimates are basically zero -> stop
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}
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if (inner.prod < eps * (p * q + r * k)) {
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break.reason <- "mean squared gradient is smaller than epsilon"
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break # mean squared gradient is smaller than epsilon -> stop
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}
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if (abs(loss.temp - loss) < eps) {
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break.reason <- "decrease is too small (slow)"
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break # decrease is too small (slow) -> stop
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}
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loss <- loss.temp
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no.nesterov <- FALSE # always reset
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} else if (!no.nesterov) {
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no.nesterov <- TRUE # retry without momentum
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next
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} else {
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break.reason <- "failed even without momentum"
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break # failed even without momentum -> stop
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}
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# update momentum scaling
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a0 <- a1
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a1 <- nesterov.scaling(a1, iter)
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# Set next iter starting step.size to line searched step size
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# (while allowing it to encrease)
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step.size <- 1.618034 * delta
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}
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list(
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loss = loss,
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alpha = alpha1, beta = beta1,
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Delta.1 = Delta.1, Delta.2 = Delta.2, tr.Delta = s,
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break.reason = break.reason
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)
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}
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