#' Fitting Generalized Multi-Linear Models
#'
#' @export
GMLM <- function(...) {
    stop("Not Implemented")
}

#' @export
GMLM.default <- function(X, Fy, sample.axis = 1L,
    family = "normal",
    ...,
    eps = sqrt(.Machine$double.eps),
    logger = NULL
) {
    stopifnot(exprs = {
        length(sample.axis) == 1L
        (1L <= sample.axis) && (sample.axis <= length(dim(X)))
        (dim(X) == dim(Fy))[sample.axis]
    })

    # rearrange `X`, `Fy` such that the last axis enumerates observations
    axis.perm <- c(seq_along(dim(X))[-sample.axis], sample.axis)
    X  <- aperm(X,  axis.perm)
    Fy <- aperm(Fy, axis.perm)

    # setup family specific GLM (pseudo) "inverse" link
    family <- make.gmlm.family(family)

    # wrap logger in callback for NAGD
    callback <- if (is.function(logger)) {
        function(iter, params) {
            do.call(logger, c(list(iter), params))
        }
    }

    # optimize likelihood using Nesterov Excelerated Gradient Descent
    params.fit <- NAGD(
        fun.loss = function(params) {
            # scaled negative lig-likelihood
            #                            eta1         alphas       Omegas
            family$log.likelihood(X, Fy, params[[1]], params[[2]], params[[3]])
        },
        fun.grad = function(params) {
            # gradient of the scaled negative lig-likelihood
            #                  eta1         alphas       Omegas
            family$grad(X, Fy, params[[1]], params[[2]], params[[3]])
        },
        params = family$initialize(X, Fy), # initial parameter estimates
        fun.lincomb = function(a, lhs, b, rhs) {
            list(
                a * lhs[[1]] + b * rhs[[1]],
                Map(function(l, r) a * l + b * r, lhs[[2]], rhs[[2]]),
                Map(function(l, r) a * l + b * r, lhs[[3]], rhs[[3]])
            )
        },
        fun.norm2 = function(params) {
            sum(unlist(params)^2)
        },
        callback = callback,
        ...
    )

    structure(params.fit, names = c("eta1", "alphas", "Omegas"))
}