R/instantiation.R

In nnR: Neural Networks Made Algebraic

source("R/aux_fun.R")
source("R/is_nn.R")

#' @title inst
#' @description The function that instantiates a neural network as created
#' by create_nn().
#'
#'
#' @param neural_network An ordered list of lists, of the type generated by
#' create_nn() where each element in the
#' list of lists is a pair \eqn{(W,b)} representing the weights and biases of
#' that layer.
#'
#' \emph{NOTE:} We will call istantiation what Grohs et. al. call "realization".
#'
#' @references  Grohs, P., Hornung, F., Jentzen, A. et al. Space-time error estimates for deep
#' neural network approximations for differential equations. (2019).
#' \url{https://arxiv.org/abs/1908.03833}.
#'
#' Definition 1.3.4. Jentzen, A., Kuckuck, B., and von Wurstemberger, P. (2023).
#' Mathematical introduction to deep learning: Methods, implementations,
#' and theory. \url{https://arxiv.org/abs/2310.20360}
#'
#' Very precisely we will use the definition in:
#'
#' @references Definition 2.3 in Rafi S., Padgett, J.L., Nakarmi, U. (2024) Towards an Algebraic Framework For
#' Approximating Functions Using Neural Network Polynomials
#' \url{https://arxiv.org/abs/2402.01058}
#'
#' @param activation_function A continuous function applied to the output of each layer. For now we only
#' have ReLU, Sigmoid, and Tanh. Note, all proofs are only valid for ReLU activation.
#' @param x our input to the continuous function formed from activation. Our input will
#' be an element in \eqn{\mathbb{R}^d} for some appropriate \eqn{d}.
#'
#' @return The output of the continuous function that is the instantiation of the given
#' neural network with the given activation function at the given \eqn{x}. Where \eqn{x}
#' is of vector size equal to the input layer of the neural network.
#'
#'
#' @examples
#' create_nn(c(1, 3, 5, 6)) |> inst(ReLU, 5)
#' create_nn(c(3, 3, 5, 6)) |> inst(ReLU, c(4, 4, 4))
#'
#' @export
#'

inst <- function(neural_network, activation_function, x) {
  if (neural_network |> is_nn() == FALSE) {
    stop("Only neural networks can be instantiated")
  } else if (neural_network |> inn() != x |>
    matrix() |>
    nrow()) {
    stop("x does not match input size required by neural network")
  } else {
    if (dep(neural_network) == 1) {
      neural_network[[1]]$W %*% x + neural_network[[1]]$b -> output
      return(output)
    }

    x |> matrix() -> output

    for (i in 1:(length(neural_network) - 1)) {
      neural_network[[i]]$W %*% output + neural_network[[i]]$b -> linear_transform
      apply(linear_transform,
        MARGIN = 1,
        FUN = activation_function
      ) -> output
    }
    neural_network[[length(neural_network)]]$W %*% output +
      neural_network[[length(neural_network)]]$b -> output
    return(output)
  }
}