Empirical Laws for Iterated Correlation Matrices

TL;DR

This paper investigates the iterative behavior of correlation matrices, revealing empirical laws that describe its stabilization patterns and decay properties across various dimensions, despite the lack of a complete theoretical convergence proof.

Contribution

It introduces a geometric framework to analyze the nonlinear iteration and uncovers four universal empirical laws governing its behavior across dimensions.

Findings

Iteration stabilizes at a block pattern

Displays finite total variation

Rapid decay near fixed points

Abstract

We study the discrete dynamical system obtained by repeatedly applying the Pearson correlation operator to a real matrix. Each step centers every row, normalizes each centered row to unit Euclidean norm, and forms the Gram matrix of the resulting rows. This produces a nonlinear map that underlies the classical CONCOR and GAP procedures. Despite its simple formulation and long history, the global behavior of this iteration has remained analytically unresolved. We present a geometric formulation that separates directions associated with changes in row means and row norms from directions that preserve them. This formulation clarifies why local analysis does not extend to a global convergence theorem: the iteration is nonlinear, the structure of its fixed-point set is not fully characterized, and standard uniform contractive or Fejer-type techniques do not directly apply. Empirically, the iteration stabilizes at a block plus or minus one pattern, exhibits finite total variation, and displays rapid decay once trajectories enter a neighborhood of a fixed pattern. We develop a dimension-uniform experimental framework and perform a large-scale numerical study over dimensions from 3 to 2000 with thousands of random initializations. Using the Frobenius step size, the entrywise step size, and the one-step ratio, we identify four universal empirical laws that persist uniformly across all tested dimensions. These observations provide a quantitative, dimension-uniform description of the iteration and formulate a precise target for future global analysis.