Finite-Step Bounds for Iterated Correlation Matrices

Abstract: Iterated Pearson correlation matrices arise from repeatedly applying the row–row correlation operator and appear in relational clustering, network analysis, and related normalization procedures. Although these dynamics are often convergent, their finite-step behavior remains poorly understood: even along convergent trajectories, successive update ratios may temporarily exceed one, producing finite-step expansion phenomena that are not captured by local asymptotic analyses.

In this talk, I present a probabilistic framework for studying finite-step behavior in iterated correlation dynamics. The approach constructs explicit state-dependent upper bounds for successive contraction ratios conditioned on the current normalized step size. These bounds are obtained through nonparametric conditional quantile estimation and are validated on large ensembles of randomly initialized trajectories across matrix dimensions ranging from 3 to 2000.

The results reveal a remarkable empirical dimension-uniformity in the dynamics and yield practical finite-step guarantees. In particular, the analysis identifies a nearly universal contraction threshold: once the normalized step size falls below approximately 0.03, the next iteration is contractive with probability at least 0.95 across all tested dimensions. More broadly, the work provides the first quantitative finite-step probabilistic control of iterated Pearson correlation dynamics and suggests new directions toward an analytical theory of nonlinear correlation-matrix iterations.

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.