In this paper, we introduce a new adaptive data analysis method to study trend and
instantaneous frequency of nonlinear and nonstationary data. This method is inspired by
the Empirical Mode Decomposition method (EMD) and the recently developed compressed
(compressive) sensing theory. 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))}, where a ∈ V (θ), V (θ) consists of the functions smoother than
cos(θ(t)) and θ
0. This problem can be formulated as a nonlinear l0 optimization
problem. In order to solve this optimization problem, we propose a nonlinear matching
pursuit method by generalizing the classical matching pursuit for the l0 optimization
problem. One important advantage of this nonlinear matching pursuit method is it can
be implemented very efficiently and is very stable to noise. Further, we provide an
error analysis of our nonlinear matching pursuit method under certain scale separation
assumptions. Extensive numerical examples will be given to demonstrate the robustness
of our method and comparison will be made with the state-of-the-art methods. We also
apply our method to study data without scale separation, and data with incomplete or
under-sampled data.