#' Per mode (axis) alternating least squares estimate
#'
#' @param sample.axis index of the sample mode, a.k.a. observation axis index
#'
#' @export
kpir.ls <- function(X, Fy, max.iter = 20L, sample.axis = 1L,
    eps = .Machine$double.eps, logger = NULL
) {
    # Check if X and Fy have same number of observations
    if (!is.array(Fy)) {
        # scalar response case (add new axis of size 1)
        dim(Fy) <- local({
            dims <- rep(1, length(dim(X)))
            dims[sample.axis] <- length(Fy)
            dims
        })
    } else {
        stopifnot(dim(X)[sample.axis] == dim(Fy)[sample.axis])
    }
    # Check dimensions
    stopifnot(length(dim(X)) == length(dim(Fy)))
    stopifnot(dim(X)[sample.axis] == dim(Fy)[sample.axis])
    # and model constraints
    stopifnot(all(dim(Fy) <= dim(X)))

    # mode index sequence (exclude sample mode, a.k.a. observation axis)
    modes <- seq_along(dim(X))[-sample.axis]


    ### Step 1: initial per mode estimates
    alphas <- Map(function(mode, ncol) {
        La.svd(mcrossprod(X, mode = mode), ncol)$u
    }, modes, dim(Fy)[modes])

    # # Scaling of alpha, such that `tr(alpha_i' alpha_i) = tr(alpha_j' alpha_j)``
    # # for `i, j = 1, ..., r`.
    # traces <- unlist(Map(function(alpha) sum(alpha^2)))
    # alphas <- Map(`*`, prod(traces)^(1 / length(alphas)) / traces, alphas)

    # Call history callback (logger) before the first iteration
    if (is.function(logger)) { do.call(logger, c(0L, NA, rev(alphas))) }


    ### Step 2: iterate per mode (axis) least squares estimates
    for (iter in seq_len(max.iter)) {
        # cyclic iterate over modes
        for (j in seq_along(modes)) {
            # least squares solution for `alpha_j | alpha_i, i != j`
            Z <- mlm(Fy, alphas[-j], modes = modes[-j])
            alphas[[j]] <- t(solve(mcrossprod(Z, mode = modes[j]),
                tcrossprod(mat(Z, modes[j]), mat(X, modes[j]))))
            # TODO: alphas[[j]] <- t(solve(mcrossprod(Z, j), mcrossprod(Z, X, j)))
        }

        # # Scaling of alpha, such that `tr(alpha_i' alpha_i) = tr(alpha_j' alpha_j)``
        # # for `i, j = 1, ..., r`.
        # traces <- unlist(Map(function(alpha) sum(alpha^2)))
        # alphas <- Map(`*`, prod(traces)^(1 / length(alphas)) / traces, alphas)

        # Call logger (invoke history callback)
        if (is.function(logger)) { do.call(logger, c(iter, NA, rev(alphas))) }

        # TODO: add some kind of break condition
    }

    ### Step 3: Moment estimates for `Delta_i`
    # Residuals
    R <- X - mlm(Fy, alphas, modes = modes)
    # Moment estimates for `Delta_i`s
    Deltas <- Map(mcrossprod, list(R), mode = modes)
    Deltas <- Map(`*`, 1 / dim(X)[sample.axis], Deltas)

    list(
        alphas = structure(alphas, names = as.character(modes)),
        Deltas = structure(Deltas, names = as.character(modes))
    )
}