Convergence of a data-driven time-frequency analysis method

TL;DR

This paper proves the convergence and exact recovery capabilities of a data-driven time-frequency analysis method based on nonlinear matching pursuit, applicable to nonlinear, non-stationary signals with sparse representations.

Contribution

It provides a rigorous proof of convergence for the nonlinear matching pursuit method and demonstrates exact signal recovery in noise-free conditions.

Findings

Convergence of the nonlinear matching pursuit method is established.

Exact recovery of the original signal is achieved without noise.

Applicable to both well-resolved and sparsely sampled signals.

Abstract

In a recent paper, Hou and Shi introduced a new adaptive data analysis method to analyze nonlinear and non-stationary data. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form $\{a(t) \cos(\theta(t))\}$, where $a \in V(\theta)$, $V(\theta)$ consists of the functions smoother than $\cos(\theta(t))$ and $\theta'\ge 0$. This problem was formulated as a nonlinear $L^0$ optimization problem and an iterative nonlinear matching pursuit method was proposed to solve this nonlinear optimization problem. In this paper, we prove the convergence of this nonlinear matching pursuit method under some sparsity assumption on the signal. We consider both well-resolved and sparse sampled signals. In the case without noise, we prove that our method gives exact recovery of the original signal.