Empirical Analysis of Iterated Correlation Matrices: A Geometric Framework for Stability and Reproducibility in AI Systems

This presentation reports a large-scale empirical study of the iterative correlation operator—a procedure widely used in psychometrics, machine learning, and signal processing to uncover structure in multivariate data. Starting from a square matrix, each iteration centers the matrix, replaces it by its correlation, and repeats the process. The practice, originating with the CONCOR algorithm and Chen’s Generalized Association Plots (GAP), has been used for decades although its convergence behavior remains largely unproved. Using custom C and R implementations, we performed random uniform initializations for matrix sizes from 3 × 3 to 2000 × 2000 with 1,000 trials per size. At each step we measured Frobenius norms and maximum element-wise differences between consecutive iterates, declaring convergence at a fixed tolerance. Three universal empirical laws emerge: (1) a sharp first-step contraction across all dimensions with high probability; (2) a nearly monotone decay of stepwise differences with small, diminishing oscillations; and (3) a convergence time essentially independent of dimension. These findings consolidate long-standing practice and establish a reproducible empirical baseline for the theoretical analysis of the iterative correlation operator—work that will be extended in forthcoming research using Lyapunov-function and manifold-geometry frameworks.