Random Frame Decompositions from Weighted Residual Flows

TL;DR

This paper investigates the dynamics of positive operators under weighted residual maps, revealing convergence properties and the emergence of random Parseval frames through a probabilistic tree-based framework.

Contribution

It introduces a novel framework for analyzing weighted residual flows, demonstrating convergence and frame generation under specific coverage conditions.

Findings

Residuals converge strongly to zero along almost every branch.

Dissipated energy admits a rank-one decomposition reconstructing the initial operator.

In special cases, generates random Parseval frames intrinsically.

Abstract

We study the evolution of a positive operator under weighted residual maps determined by a finite family of orthogonal projections. Iterating these maps along the rooted tree of multi-indices produces a "weighted residual energy tree", together with natural path measures obtained by normalizing the dissipated energy or trace at each step. Under a quantitative coverage condition on the projections, we show that along almost every branch the residuals converge strongly to zero and the dissipated pieces admit a rank-one decomposition that reconstructs the initial operator. In the special case where the initial operator is the identity on a subspace, this yields almost surely a random Parseval frame generated intrinsically by the weighted residual dynamics.