#' Using unbiased (but not MLE) estimates for the Kronecker decomposed
#' covariance matrices Delta_1, Delta_2 for approximating the log-likelihood
#' giving a closed form solution for the gradient.
#'
#'      Delta_1 = n^-1 sum_i R_i' R_i,
#'      Delta_2 = n^-1 sum_i R_i R_i'.
#'
#' @export
kpir.approx <- function(X, Fy, shape = c(dim(X)[-1], dim(Fy[-1])),
    max.iter = 500L, max.line.iter = 50L, step.size = 1e-3,
    nesterov.scaling = function(a, t) 0.5 * (1 + sqrt(1 + (2 * a)^2)),
    max.init.iter = 20L, init.method = c("ls", "vlp"),
    eps = .Machine$double.eps,
    logger = NULL
) {

    # Check if X and Fy have same number of observations
    stopifnot(nrow(X) == NROW(Fy))
    n <- nrow(X)                        # Number of observations

    # Check predictor dimensions
    if (length(dim(X)) == 2L) {
        stopifnot(!missing(shape))
        stopifnot(ncol(X) == prod(shape[1:2]))
        p <- as.integer(shape[1])       # Predictor "height"
        q <- as.integer(shape[2])       # Predictor "width"
    } else if (length(dim(X)) == 3L) {
        p <- dim(X)[2]
        q <- dim(X)[3]
    } else {
        stop("'X' must be a matrix or 3-tensor")
    }

    # Check response dimensions
    if (!is.array(Fy)) {
        Fy <- as.array(Fy)
    }
    if (length(dim(Fy)) == 1L) {
        k <- r <- 1L
    } else if (length(dim(Fy)) == 2L) {
        stopifnot(!missing(shape))
        stopifnot(ncol(Fy) == prod(shape[3:4]))
        k <- as.integer(shape[3])       # Response functional "height"
        r <- as.integer(shape[4])       # Response functional "width"
    } else if (length(dim(Fy)) == 3L) {
        k <- dim(Fy)[2]
        r <- dim(Fy)[3]
    } else {
        stop("'Fy' must be a vector, matrix or 3-tensor")
    }


    ### Step 1: (Approx) Least Squares initial estimate
    init.method <- match.arg(init.method)
    if (init.method == "ls") {
        # De-Vectroize (from now on tensor arithmetics)
        dim(Fy) <- c(n, k, r)
        dim(X)  <- c(n, p, q)

        ls <- kpir.ls(X, Fy, max.iter = max.init.iter, sample.axis = 1L, eps = eps)
        c(beta0, alpha0) %<-% ls$alphas
    } else { # Van Loan and Pitsianis
        # Vectorize
        dim(Fy) <- c(n, k * r)
        dim(X)  <- c(n, p * q)

        # solution for `X = Fy B' + epsilon`
        cpFy <- crossprod(Fy)               # TODO: Check/Test and/or replace
        if (n <= k * r || qr(cpFy)$rank < k * r) {
            # In case of under-determined system replace the inverse in the normal
            # equation by the Moore-Penrose Pseudo Inverse
            B <- t(matpow(cpFy, -1) %*% crossprod(Fy, X))
        } else {
            # Compute OLS estimate by the Normal Equation
            B <- t(solve(cpFy, crossprod(Fy, X)))
        }

        # De-Vectroize (from now on tensor arithmetics)
        dim(Fy) <- c(n, k, r)
        dim(X)  <- c(n, p, q)

        # Decompose `B = alpha x beta` into `alpha` and `beta`
        c(alpha0, beta0) %<-% approx.kronecker(B, c(q, r), c(p, k))
    }

    # Compute residuals
    R <- X - (Fy %x_3% alpha0 %x_2% beta0)

    # Covariance estimates and scaling factor
    Delta.1 <- tcrossprod(mat(R, 3)) / n
    Delta.2 <- tcrossprod(mat(R, 2)) / n
    s <- mean(diag(Delta.1))

    # Inverse Covariances
    Delta.1.inv <- solve(Delta.1)
    Delta.2.inv <- solve(Delta.2)

    # cross dependent covariance estimates
    S.1 <- tcrossprod(mat(R, 3), mat(R %x_2% Delta.2.inv, 3)) / n
    S.2 <- tcrossprod(mat(R, 2), mat(R %x_3% Delta.1.inv, 2)) / n

    # Evaluate negative log-likelihood (2 pi term dropped)
    loss <- -0.5 * (n * (p * q * log(s) - p * log(det(Delta.1)) -
        q * log(det(Delta.2))) - s * sum(S.1 * Delta.1.inv))

    # Call history callback (logger) before the first iteration
    if (is.function(logger)) {
        logger(0L, loss, alpha0, beta0, Delta.1, Delta.2, NA)
    }


    ### Step 2: MLE estimate with LS solution as starting value
    a0 <- 0
    a1 <- 1
    alpha1 <- alpha0
    beta1 <- beta0

    # main descent loop
    no.nesterov <- TRUE
    break.reason <- NA
    for (iter in seq_len(max.iter)) {
        if (no.nesterov) {
            # without extrapolation as fallback
            alpha.moment <- alpha1
            beta.moment  <- beta1
        } else {
            # extrapolation using previous direction
            alpha.moment <- alpha1 + ((a0 - 1) / a1) * (alpha1 - alpha0)
            beta.moment  <-  beta1 + ((a0 - 1) / a1) * ( beta1 -  beta0)
        }

        # Extrapolated residuals
        R <- X - (Fy %x_3% alpha.moment %x_2% beta.moment)

        # Recompute Covariance Estimates and scaling factor
        Delta.1 <- tcrossprod(mat(R, 3)) / n
        Delta.2 <- tcrossprod(mat(R, 2)) / n
        s <- mean(diag(Delta.1))

        # Inverse Covariances
        Delta.1.inv <- solve(Delta.1)
        Delta.2.inv <- solve(Delta.2)

        # cross dependent covariance estimates
        S.1 <- tcrossprod(mat(R, 3), mat(R %x_2% Delta.2.inv, 3)) / n
        S.2 <- tcrossprod(mat(R, 2), mat(R %x_3% Delta.1.inv, 2)) / n

        # Gradient "generating" tensor
        G <- (sum(S.1 * Delta.1.inv) - p * q / s) * R
        G <- G + R %x_2% ((diag(q, p) - s * (Delta.2.inv %*% S.2)) %*% Delta.2.inv)
        G <- G + R %x_3% ((diag(p, q) - s * (Delta.1.inv %*% S.1)) %*% Delta.1.inv)
        G <- G + s * (R %x_2% Delta.2.inv %x_3% Delta.1.inv)

        # Calculate Gradients
        grad.alpha <- tcrossprod(mat(G, 3), mat(Fy %x_2% beta.moment, 3))
        grad.beta  <- tcrossprod(mat(G, 2), mat(Fy %x_3% alpha.moment, 2))

        # Backtracking line search (Armijo type)
        # The `inner.prod` is used in the Armijo break condition but does not
        # depend on the step size.
        inner.prod <- sum(grad.alpha^2) + sum(grad.beta^2)

        # backtracking loop
        for (delta in step.size * 0.618034^seq.int(0L, length = max.line.iter)) {
            # Update `alpha` and `beta` (note: add(+), the gradients are already
            # pointing into the negative slope direction of the loss cause they are
            # the gradients of the log-likelihood [NOT the negative log-likelihood])
            alpha.temp <- alpha.moment + delta * grad.alpha
            beta.temp  <- beta.moment  + delta * grad.beta

            # Update Residuals, Covariances, ...
            R <- X - (Fy %x_3% alpha.temp %x_2% beta.temp)
            Delta.1 <- tcrossprod(mat(R, 3)) / n
            Delta.2 <- tcrossprod(mat(R, 2)) / n
            s <- mean(diag(Delta.1))
            Delta.1.inv <- solve(Delta.1)
            Delta.2.inv <- solve(Delta.2)
            S.1 <- tcrossprod(mat(R, 3), mat(R %x_2% Delta.2.inv, 3)) / n
            # S.2 not needed

            # Re-evaluate negative log-likelihood
            loss.temp <- -0.5 * (n * (p * q * log(s) - p * log(det(Delta.1)) -
                q * log(det(Delta.2))) - s * sum(S.1 * Delta.1.inv))

            # Armijo line search break condition
            if (loss.temp <= loss - 0.1 * delta * inner.prod) {
                break
            }
        }

        # Call logger (invoke history callback)
        if (is.function(logger)) {
            logger(iter, loss.temp, alpha.temp, beta.temp, Delta.1, Delta.2, delta)
        }

        # Enforce descent
        if (loss.temp < loss) {
            alpha0 <- alpha1
            alpha1 <- alpha.temp
            beta0 <- beta1
            beta1 <- beta.temp

            # check break conditions
            if (mean(abs(alpha1)) + mean(abs(beta1)) < eps) {
                break.reason <- "alpha, beta numerically zero"
                break   # estimates are basically zero -> stop
            }
            if (inner.prod < eps * (p * q + r * k)) {
                break.reason <- "mean squared gradient is smaller than epsilon"
                break   # mean squared gradient is smaller than epsilon -> stop
            }
            if (abs(loss.temp - loss) < eps) {
                break.reason <- "decrease is too small (slow)"
                break   # decrease is too small (slow) -> stop
            }

            loss  <- loss.temp
            no.nesterov <- FALSE    # always reset
        } else if (!no.nesterov) {
            no.nesterov <- TRUE     # retry without momentum
            next
        } else {
            break.reason <- "failed even without momentum"
            break       # failed even without momentum -> stop
        }

        # update momentum scaling
        a0 <- a1
        a1 <- nesterov.scaling(a1, iter)

        # Set next iter starting step.size to line searched step size
        # (while allowing it to encrease)
        step.size <- 1.618034 * delta

    }

    list(
        loss = loss,
        alpha = alpha1, beta = beta1,
        Delta.1 = Delta.1, Delta.2 = Delta.2, tr.Delta = s,
        break.reason = break.reason
    )
}