#' Regularized Matrix Regression
#'
#' Solved the regularized problem
#'      \deqn{min h(B) = l(B) + J(B)}
#' for a matrix \eqn{B}.
#' where \eqn{l} is a loss function; for the GLM, we use the negative
#' log-likelihood as the loss. \eqn{J(B) = f(\sigma(B))}, where \eqn{f} is a
#' function of the singular values of \eqn{B}.
#'
#' The default parameterization is a nuclear norm penalized least squares regression.
#'
#' The least squares loss combined with \eqn{f(s) = \lambda \sum_i |s_i|}
#' corresponds to the nuclear norm regularization problem.
#'
#' In case of \code{lambda = Inf} the maximum penalty \eqn{\lambda} is computed.
#' In this case the return value is only estimate as a single value.
#'
#' @param X the singnal data ether as a 3D tensor or a 2D matrix. In case of a
#'  3D tensor the axis are assumed to be \eqn{n\times p\times q} meaning the
#'  first dimension are the observations while the second and third are the
#'  `image' dimensions. When the data is provided as a matix it's assumed to be
#'  of shape \eqn{n\times p q} where each observation is the vectorid `image'.
#' @param Z additional covariate vector (can be \code{NULL} if not required.
#'  For regression with intercept set \code{Z = rep(1, n)})
#' @param y univariate response vector
#' @param lambda penalty term, if set to \code{Inf} 
#' @param loss loss function, part of the objective function
#' @param penalty penalty function with a vector of the singular values if the
#'  current iterate as arguments. The default function
#'  \code{function(sigma) sum(sigma)} is the nuclear norm penalty.
#' @param grad.loss gradient with respect to \eqn{B} of the loss function
#'  (required, there is no support for numerical gradients)
#' @param shape Shape of the matrix valued predictors. Required iff the
#'  predictors \code{X} are provided in vectorized form, e.g. as a 2D matrix.
#' @param step.size max. stepsize for gradient updates
#' @param alpha iterative Nesterov momentum scaling values
#' @param B0 initial value for optimization. Matrix of dimensions \eqn{p\times q}
#' @param beta initial value of additional covatiates coefficient for \eqn{Z}
#' @param max.iter maximum number of gadient updates
#' @param max.line.iter maximum number of line search iterations
#' @param logger logging callback invoced after every line search before break
#'  condition checks. The expected function signature is of the form
#'  \code{logger(iter, loss, penalty, grad, B, beta, step.size)}.
#'
#' @export
RMReg <- function(X, Z, y, lambda = 0,
    loss = function(B, beta, X, Z, y) 0.5 * sum((y - Z %*% beta - X %*% c(B))^2),
    penalty = function(sigma) sum(sigma),
    grad.loss = function(B, beta, X, Z, y) crossprod(X %*% c(B) + Z %*% beta - y, X),
    shape = dim(X)[-1],
    step.size = 1e-3,
    alpha = function(a, t) { (1 + sqrt(1 + (2 * a)^2)) / 2 },
    B0 = array(0, dim = shape),
    beta = rep(0, NCOL(Z)),
    max.iter = 500,
    max.line.iter = ceiling(log(step.size / sqrt(.Machine$double.eps), 2)),
    logger = NULL
) {
    ### Check (prepair) params
    stopifnot(nrow(X) == length(y))
    if (!missing(shape)) {
        stopifnot(ncol(X) == prod(shape))
    } else {
        stopifnot(length(dim(X)) == 3)
        dim(X) <- c(nrow(X), prod(shape))
    }
    if (missing(Z) || is.null(Z)) {
        Z <- matrix(0, nrow(X), 1)
        ZZiZ <- NULL
    } else {
        # Compute (Z' Z)^{-1} Z used to solve for beta. This is constant
        # throughout and the variable name stands for "((Z' Z) Inverse) Z"
        ZZiZ <- solve(crossprod(Z, Z), t(Z))
    }

    ### Set initial values
    # Note: Naming convention; a name ending with 1 is the current iterate while
    # names ending with 0 are the previous iterate value.
    # Init singular values of B1 (require only current point, not previous B0)
    if (missing(B0)) {
        b1 <- rep(0, min(shape))
    } else {
        b1 <- La.svd(B0, 0, 0)$d
    }
    # Init current to previous (start position)
    B1 <- B0
    a0 <- 0
    a1 <- 1
    loss1 <- loss(B1, beta, X, Z, y)

    # Start without, the nesterov momentum is zero anyway
    no.nesterov <- TRUE
    ### Repeat untill convergence
    for (iter in 1:max.iter) {
        # Extrapolation with Nesterov Momentum
        if (no.nesterov) {
            S <- B1
        } else {
            S <- B1 + ((a0 - 1) / a1) * (B1 - B0)
        }

        # Solve for beta at extrapolation point
        if (!is.null(ZZiZ)) {
            beta <- ZZiZ %*% (y - X %*% c(S))
        }

        # Compute Nesterov Gradient of the Loss
        grad <- array(grad.loss(S, beta, X, Z, y), dim = shape)

        # Line Search (executed at least once)
        for (delta in step.size * 0.5^seq(0, max.line.iter - 1)) {
            # (potential) next step with delta as stepsize for gradient update
            A <- S - delta * grad

            if (lambda == Inf) {
                # Application of Corollary 1 (only nuclear norm supported) to
                # estimate maximum lambda. In this case (first time this line is
                # hit when lambda set to Inf, then B is zero (ignore B0 param))
                lambda.max <- max(La.svd(A, 0, 0)$d) / delta
                return(lambda.max)
            } else if (lambda > 0) {
                # SVD of (potential) next step
                svdA <- La.svd(A)

                # Get (potential) next penalized iterate (nuclear norm version only)
                b.temp <- pmax(0, svdA$d - delta * lambda)  # Singular values of B.temp
                B.temp <- svdA$u %*% (b.temp * svdA$vt)
            } else {
                # in case of no penalization (pure least squares solution)
                b.temp <- La.svd(A, 0, 0)$d
                B.temp <- A
            }

            # Solve for beta at (potential) next step
            if (!is.null(ZZiZ)) {
                beta <- ZZiZ %*% (y - X %*% c(B.temp))
            }

            # Check line search break condition
            #   h(B.temp) <= g(B.temp | S, delta)
            #  \_ left _/    \_____ right _____/
            # where g(B.temp | S, delta) is the first order approx. of the loss
            #   l(S) + <grad l(S), B - S> + | B - S |_F^2 / 2 delta + J(B)
            left <- loss(B.temp, beta, X, Z, y) # + penalty(b.temp)
            right <- loss(S, beta, X, Z, y) + sum(grad * (B1 - S)) +
                norm(B1 - S, 'F')^2 / (2 * delta) # + penalty(b.temp)
            if (left <= right) {
                break
            }
        }

        # Evaluate loss at (potential) new parameters
        loss.temp <- loss(B.temp, beta, X, Z, y)

        # logging callback
        if (is.function(logger)) {
            logger(iter, loss.temp, penalty(b.temp), grad, B1, beta, delta)
        }

        # After gradient update enforce descent (stop if not decreasing)
        if (loss.temp + penalty(b.temp) <= loss1 + penalty(b1)) {
            no.nesterov <- FALSE
            loss1 <- loss.temp
            B0 <- B1
            B1 <- B.temp
            b1 <- b.temp
        } else if (!no.nesterov) {
            # Retry without Nesterov extrapolation
            no.nesterov <- TRUE
            next
        } else {
            break
        }

        # If estimate is zero, stop algorithm
        if (all(b.temp < .Machine$double.eps)) {
            loss1 <- loss.temp
            B1 <- array(0, dim = shape)
            b1 <- rep(0, min(shape))
            break
        }

        # Update momentum scaling
        a0 <- a1
        a1 <- alpha(a1, iter)
    }

    ### Degrees of Freedom estimate (TODO: this is like in `matrix_sparsereg.m`)
    sigma <- c(La.svd(A, 0, 0)$d, rep(0, max(shape) - min(shape)))
    df <- if (!is.null(ZZiZ)) { ncol(Z) } else { 0 }
    for (i in seq_len(sum(b1 > 0))) {
        df <- df + 1 + sigma[i] * (sigma[i] - delta * lambda) * (
            sum(ifelse((1:shape[1]) != i, 1 / (sigma[i]^2 - sigma[1:shape[1]]^2), 0)) +
            sum(ifelse((1:shape[2]) != i, 1 / (sigma[i]^2 - sigma[1:shape[2]]^2), 0)))
    }

    # return estimates and some additional stats
    list(
        B = B1,
        beta = if(is.null(ZZiZ)) { NULL } else { beta },
        singular.values = b1,
        iter = iter,
        df = df,
        loss = loss1,
        lambda = lambda,
        AIC = loss1 / var(y) + 2 * df,
        BIC = loss1 / var(y) + log(nrow(X)) * df,
        call = match.call() # invocing function call, collects params like lambda
    )
}