  In this paper, we introduce a new adaptive data analysis method to study
trend and instantaneous frequency of nonlinear and non-stationary 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 $\{a(t)
\cos(\theta(t))\}$, where $a \in V(\theta)$, $V(\theta)$ consists of the
functions smoother than $\cos(\theta(t))$ and $\theta'\ge 0$. This problem can
be formulated as a nonlinear $L^0$ optimization problem. In order to solve this
optimization problem, we propose a nonlinear matching pursuit method by
generalizing the classical matching pursuit for the $L^0$ 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 a
convergence 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
EMD/EEMD method. We also apply our method to study data without scale
separation, data with intra-wave frequency modulation, and data with incomplete
or under-sampled data.
