Convergence of a data-driven time–frequency analysis method

In a recent paper [11], 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(θ(t))}{a(t)cos(θ(t))}, where a∈V(θ)a∈V(θ), V(θ)V(θ) consists of the functions that are less oscillatory than cos(θ(t))cos(θ(t)) and θ′⩾0θ′⩾0. This problem was formulated as a nonlinear L0L0 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 scale separation assumptions on the signal. We consider both well-resolved and poorly sampled signals, as well as signals with noise. In the case without noise, we prove that our method gives exact recovery of the original signal.