Empirical Laws for Iterated Correlation Matrices

TL;DR

This work studies the global dynamics of repeatedly applying the Pearson correlation operator by decomposing the update into row centering/normalization and Gram construction, yielding a nonlinear map $f=\Phi\circ\Psi$ that preserves the elliptope $\mathcal{E}_n$. Through a dimension-uniform, large-scale numerical study across $n\in[3,2000]$, the authors uncover four universal empirical laws: (i) a sharp first-step contraction, (ii) nearly monotone decay with bounded overshoots and finite total variation, (iii) a dimension-independent $\Delta_k$–$\rho_k$ contraction profile with a characteristic V-shape, and (iv) uniformly bounded iteration counts. They provide a geometric framework based on row-sphericalization and Gram mapping, clarifying the anisotropic nature of the dynamics and offering concrete benchmarks for any future global convergence analysis. Together, these results yield a dimension-stable, quantitative portrait of iterated correlation and identify the structural features any analytic theory must reproduce.

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.

Figures (17)

• Figure 1: Frobenius step-size decay across iterations. Each curve corresponds to a fixed matrix dimension $n$. All dimensions exhibit a pronounced contraction between $k=0$ and $k=1$, followed by a near-isometric regime. This universal pattern is the content of Law I.
• Figure 2: Per iteration view of trajectories per matrix size. A detailed view of iterations for multiple sizes $n$. The Frobenius step size undergoes a large reduction at $k=1$ for every dimension, illustrating that the sharp first-step contraction is uniform in $n$.
• Figure 3: Representative trajectories by matrix size. Each trajectory shows the Frobenius step-size decay for a representative size from each dimension group. All four groups exhibit an immediate and sharp first-step contraction, confirming the dimension-independent nature of the phenomenon. Larger dimensions display a sharper drop and a quantitatively smaller ratio $\rho_0$, but the qualitative behavior is identical across sizes.
• Figure 4: Median first-step contraction $\rho_0$ versus dimension $n$ (logarithmic horizontal axis). Contraction occurs for every dimension and every trial, while its magnitude strengthens monotonically with $n$. The shaded region denotes the interquartile range, which decreases sharply for large $n$, indicating increasing stability of contraction strength.
• Figure 5: Frobenius step decay at $n=3$ over $N=1000$ trials. Colored curves show individual trajectories $\{\Delta_k^{(t)}\}$. The black curve is the mean $\bar{\Delta}_k$ over trials that are still active at iteration $k$. A universal sharp drop is visible at $k=1$, followed by near-monotone decay with shrinking oscillations. The mean curve ends at $k=14$, indicating that all trajectories have converged by this iteration.