#' Tensor Sliced Inverse Regression
#'
#' @export
TSIR <- function(X, y, d, sample.axis = 1L,
    max.iter = 50L,
    eps = sqrt(.Machine$double.eps)
) {

    if (!(is.factor(y) || is.integer(y))) { # TODO: Implement continuous case!
        stop("Only factor and integer response implemented!")
    }

    stopifnot(exprs = {
        dim(X)[sample.axis] == length(y)
        length(d) + 1L == length(dim(X))
    })

    # 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)

    # get dimensions
    n <- tail(dim(X), 1)
    p <- head(dim(X), -1)
    modes <- seq_along(p)

    # reinterpretation of `y` as factor index as number of different slices
    y <- as.factor(y)
    nr.slices <- length(levels(y))
    slice.sizes <- table(y)
    y <- as.integer(y)

    # center `X`
    X <- X - c(rowMeans(X, dim = length(modes)))

    # Slice `X` into slices governed by `y`
    slices <- Map(function(s) {
        X_s <- X[rep(s == y, each = prod(p))]
        dim(X_s) <- c(p, slice.sizes[s])
        X_s
    }, seq_len(nr.slices))

    # For each slice we get the slice means
    slice.means <- Map(rowMeans, slices, MoreArgs = list(dim = length(modes)))

    # Initial `Gamma_k` estimates as dominent eigenvalues of `Cov_c(E[X_(k) | Y])`
    Sigmas <- Map(function(k) {
        Reduce(`+`, Map(function(X_s, mu_s, n_s) {
            mcrossprod(rowMeans(X_s - c(mu_s), dims = length(modes)), mode = k)
        }, slices, slice.means, slice.sizes)) / nr.slices
    }, modes)
    Gammas <- Map(`[[`, Map(La.svd, Sigmas, nu = d, nv = 0), list("u"))

    # setup projections as `Proj_k = Gamma_k Gamma_k'`
    projections <- Map(tcrossprod, Gammas)

    # compute initial loss for the break condition
    loss.last <- n^-1 * Reduce(`+`, Map(function(X_s, n_s) {
        n_s * sum((X_s - mlm(X_s, projections))^2)
    }, slice.means, slice.sizes))

    # Iterate till convergence
    for (iter in seq_len(max.iter)) {
        # For each mode assume mode reduction `Gamma`s fixed and update
        # current mode reduction `Gamma_k`
        for (k in modes) {
            # Other mode projection of slice means
            slice.proj <- Map(mlm, slice.means, MoreArgs = list(
                Gammas[-k], modes[-k], transposed = TRUE
            ))

            # Update mode `k` slice mean covariances
            Sigmas[[k]] <- n^-1 * Reduce(`+`, Map(function(PX_s, n_s) {
                n_s * mcrossprod(PX_s, mode = k)
            }, slice.proj, slice.sizes))

            # Recompute mode `k` basis `Gamma_k`
            Gammas[[k]] <- La.svd(Sigmas[[k]], nu = d[k], nv = 0)$u

            # update mode `k` projection matrix onto the new span of `Gamma_k`
            projections[[k]] <- tcrossprod(Gammas[[k]])

            # compute loss for break condition
            loss <- n^-1 * Reduce(`+`, Map(function(X_s, n_s) {
                n_s * sum((X_s - mlm(X_s, projections))^2)
            }, slice.means, slice.sizes))
        }

        # check break condition
        if (abs(loss - loss.last) < eps * loss) {
            break
        }
        loss.last <- loss
    }

    # mode sample covariance matrices
    Omegas <- Map(function(k) n^-1 * mcrossprod(X, mode = k), modes)

    # reductions matrices `Omega_k^-1 Gamma_k` where there (reverse) kronecker
    # product spans the central tensor subspace (CTS) estimate
    Map(solve, Omegas, Gammas)
}