add first sketch for estimators

R/estimators.R

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+# load local files
+source(here::here("R", "singular_values.R"))
+source(here::here("R", "graphon_distribution.R"))
+source(here::here("R","singular_value_plot.R"))
+source(here::here("R", "build_network.R"))
+
+# Helper functions -------------------------------------------------------------
+# helper function for wrapping the parameters of the Q_a creation funciton
+# TODO rename this function
+make_matrix_creation <- function(seed, n, K, sample_X_fn, fv, Fv, guard) {
+  function(a) {
+    compute_matrix(seed, a, n, K, sample_X_fn, fv, Fv, guard)
+  }
+}
+
+# estimators -------------------------------------------------------------------
+
+## Moore-Penrose Inverse -------------------------------------------------------
+#' Moore‑Penrose pseudoinverse of a matrix
+#'
+#' Computes the Moore‑Penrose generalized inverse of a numeric matrix using a
+#' singular‑value decomposition (SVD). Singular values smaller than the
+#' tolerance are treated as zero to improve numerical stability.
+#'
+#' @param A A numeric matrix (or an object coercible to a matrix) whose
+#'   pseudoinverse is required.
+#' @param tol Tolerance for treating singular values as zero.  By default it
+#'   is set to `max(dim(A)) * max(svd(A)$d) * .Machine$double.eps`, which
+#'   scales with the size of the matrix and machine precision.
+#' @return A matrix representing the Moore‑Penrose inverse of `A`.  The
+#'   dimensions of the result are `ncol(A) × nrow(A)`.
+#' @examples
+#' set.seed(123)
+#' A <- matrix(rnorm(12), nrow = 3)
+#' A_pinv <- pinv(A)
+#' # Verify the Moore‑Penrose properties (A %*% A_pinv %*% A ≈ A)
+#' all.equal(A %*% A_pinv %*% A, A)
+#' @importFrom stats svd
+#' @export
+pinv <- function(A, tol = NULL) {
+  # Coerce to matrix and check type
+  if (!is.matrix(A)) {
+    A <- as.matrix(A)
+  }
+  if (!is.numeric(A)) {
+    stop("`A` must be a numeric matrix.", call. = FALSE)
+  }
+  
+  # Singular value decomposition
+  s <- svd(A)
+  
+  # Determine tolerance if not supplied
+  if (is.null(tol)) {
+    tol <- max(dim(A)) * max(s$d) * .Machine$double.eps
+  }
+  
+  # Invert non‑zero singular values
+  d_inv <- ifelse(s$d > tol, 1 / s$d, 0)
+  
+  # Construct diagonal matrix of inverted singular values
+  D_plus <- diag(d_inv, nrow = length(d_inv))
+  
+  # Moore‑Penrose inverse: V %*% D⁺ %*% t(U)
+  s$v %*% D_plus %*% t(s$u)
+}
+
+
+
+## Estimate Matrix B -----------------------------------------------------------
+#' Estimate the matrix \$B\$ for a graphon model
+#'
+#' For a given graphon scaling parameter `rho_n`, a square matrix `Q_a`, and an
+#' adjacency matrix `A`, this function computes  
+
+#' $ B = \\rho_n \, Q_a^{+\\,T} \, A \, Q_a^{+} $
+
+#' where `Q_a^{+}` denotes the Moore‑Penrose pseudoinverse of `Q_a`. 
+#'
+#' @param rho_n Numeric scalar.  The graphon scaling parameter.
+#' @param Q_a   Square numeric matrix. TODO: write description of the Matrix
+#' @param A     Square numeric adjacency matrix (same dimension as `Q_a`).
+#' @return A numeric matrix of the same dimension as `Q_a` representing the
+#'   estimated \$B\$.
+#' @examples
+#' set.seed(42)
+#' n <- 5
+#' Q_a <- diag(n)               # simple identity basis
+#' A   <- matrix(rbinom(n^2, 1, 0.3), n, n)
+#' rho_n <- 0.5
+#' B_est <- estimate_B_matrix(rho_n, Q_a, A)
+#' str(B_est)
+#' @importFrom stats svd
+#' @export
+estimate_B_matrix <- function(rho_n, Q_a, A) {
+  # ---- Input checks ---------------------------------------------------------
+  if (!is.numeric(rho_n) || length(rho_n) != 1) {
+    stop("`rho_n` must be a single numeric value.", call. = FALSE)
+  }
+  if (!is.matrix(Q_a) || !is.numeric(Q_a)) {
+    stop("`Q_a` must be a numeric matrix.", call. = FALSE)
+  }
+  if (!is.matrix(A) || !is.numeric(A)) {
+    stop("`A` must be a numeric matrix.", call. = FALSE)
+  }
+  if (nrow(Q_a) != ncol(Q_a)) {
+    stop("`Q_a` must be square.", call. = FALSE)
+  }
+  if (nrow(A) != ncol(A)) {
+    stop("`A` must be square.", call. = FALSE)
+  }
+  if (nrow(Q_a) != nrow(A)) {
+    stop("Dimensions of `Q_a` and `A` must agree for matrix multiplication.", call. = FALSE)
+  }
+  
+  # ---- Compute pseudoinverse ------------------------------------------------
+  pinv_Qa <- pinv(Q_a)   # assumes `pinv()` is available in the namespace
+  
+  # ---- Estimate B -----------------------------------------------------------
+  B <- rho_n * (t(pinv_Qa) %*% A %*% pinv_Qa)
+  
+  B
+}
+
+# TODO rename this function
+# TODO test the convergence of the function estimate_B
+# with given graphon (block function, Hölder continuous functions)
+# and given K with growing N
+# and other options
+# plot the loss function with respect to a
+estimate_a <- function(A, # adjacency matrix
+                       a0, # start value
+                       n,
+                       K,
+                       sample_X_fn,
+                       fv,
+                       Fv, 
+                       guard
+) {
+  calc_Q_a <- make_matrix_creation(seed, n, K, sample_X_fn, fv, Fv, guard)
+  
+  loss_func <- function(a) {
+    Q_a <- calc_Q_a(a)
+    pinv_Qa <- pinv(Q_a)
+    
+    norm(pinv_Qa %*% Q_a %*% A %*% pinv_Qa %*% Q_a - A)^2
+  }
+  
+  optim(a0, loss_func)
+}
+
+# test the estimator routines
+seed <- 1L
+set.seed(seed)
+X  <- matrix(seq(-1, 1, length.out = 5), ncol = 1)
+a  <- 2
+n <- 2
+K <- 2
+sample_X_fn <- function(n) {matrix(rnorm(n), ncol = 1L)}
+fv <- function(x) {dnorm(x, mean=0, sd=1)}
+Fv <- function(x) {pnorm(x, mean=0, sd=1)}
+guard <- 1e-12
+v  <- seq(0, 0.8, length.out = 5)
+phi_fun <- function(x, y) x * y          # multiplicative kernel
+adj <- compute_adj_matrix(
+   X_matrix = X,
+   v        = v,
+   a        = a,
+   phi      = phi_fun,
+   rho_n    = 0.5,
+   Fv       = Fv
+ )
+adj
+
+# Q_a matrix
+Qa <- compute_matrix(seed, a, n, K, sample_X_fn, fv, Fv, guard)
+
+estimate_B()