#' Compute get gradient of `L(V)` given a dataset `X`.
#'
#' @param X Data matrix.
#' @param Y Responce.
#' @param V Position to compute the gradient at, aka point on Stiefl manifold.
#' @param h Bandwidth
#' @param loss.only Boolean to only compute the loss, of \code{TRUE} a single
#'  value loss is returned and \code{envir} is ignored.
#' @keywords internal
#' @export
grad <- function(X, Y, V, h, loss.out = FALSE, loss.log = FALSE, loss.only = FALSE) {
    # Get number of samples and dimension.
    n <- nrow(X)
    p <- ncol(X)

    # Compute lookup indexes for symmetrie, lower/upper
    # triangular parts and vectorization.
    pair.index <- elem.pairs(seq(n))
    i <- pair.index[, 1] # `i` indices of `(i, j)` pairs
    j <- pair.index[, 2] # `j` indices of `(i, j)` pairs
    # Matrix of vectorized indices. (vec(index) -> seq)
    index <- matrix(seq(n * n), n, n)
    lower <- index[lower.tri(index)]
    upper <- t(index)[lower]

    # Create all pairewise differences of rows of `X`.
    X_diff <- X[i, , drop = F] - X[j, , drop = F]

    # Projection matrix onto `span(V)`
    Q <- diag(1, p) - (V %*% t(V))

    # Vectorized distance matrix `D`.
    vecD <- rowSums((X_diff %*% Q)^2)

    # Weight matrix `W` (dnorm ... gaussean density function)
    W <- matrix(dnorm(0), n, n)
    W[lower] <- dnorm(vecD / h) # Set lower tri. part
    W[upper] <- t(W)[upper] # Mirror lower tri. to upper
    W <- sweep(W, 2, colSums(W), FUN = `/`) # Col-Normalize

    # Weighted `Y` momentums
    y1 <- Y %*% W # Result is 1D -> transposition irrelevant
    y2 <- Y^2 %*% W

    # Per example loss `L(V, X_i)`
    L <- y2 - y1^2
    if (loss.out | loss.log | loss.only) {
        meanL <- mean(L)
        if (loss.out) {
            # Bubble environments up and write to loss variable, aka out param.
            loss <<- meanL
        }
        if (loss.log) {
            loss.history[epoch, attempt] <<- meanL
        }
        if (loss.only) {
            # Mean for total loss `L(V)`.
            return(meanL)
        }
    }

    # Vectorized Weights with forced symmetry
    vecS <- (L[i] - (Y[j] - y1[i])^2) * W[lower]
    vecS <- vecS + ((L[j] - (Y[i] - y1[j])^2) * W[upper])
    # Compute scaling of `X` row differences
    vecS <- vecS * vecD

    # The gradient.
    G <- t(X_diff) %*% sweep(X_diff %*% V, 1, vecS, `*`)
    G <- (-2 / (n * h^2)) * G
    return(G)
}