This paper develops new theory and algorithms for 1D general mode decompositions. First, we introduce the 1D synchrosqueezed wave packet transform and prove that it is able to estimate the instantaneous information of well-separated modes from their superposition accurately. The synchrosqueezed wave packet transform has a better resolution than the synchrosqueezed wavelet transform in the time-frequency domain for separating high frequency modes. Second, we present a new approach based on diffeomorphisms for the spectral analysis of general shape functions. These two methods lead to a framework for general mode decompositions under a weak well-separation condition and a well different condition. Numerical examples of synthetic and real data are provided to demonstrate the fruitful applications of these methods.
By extracting the components one-by-one from the most oscillatory one, Hou and Shi proposed a nonlinear optimization scheme to decompose signals. The first model in @cite_32 is based on nonlinear @math minimization, which is computationally costly. To deal with this problem, the second paper @cite_15 proposed a nonlinear matching pursuit model based on sparse representations of signals in a data-driven time-frequency dictionary, which has a fast algorithm for periodic data. Under some sparsity assumptions, the analysis of convergence for the latter scheme has been recently studied in @cite_29 .
@cite_15 @cite_29 @cite_32
2963917297 2156455340 2952801764 1980454827
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)) , where a∈V(θ),V(θ) consists of the functions that are less oscillatory than cos(θ(t)) and θ′⩾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 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. 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 @math , where @math , @math consists of the functions smoother than @math and @math . This problem was formulated as a nonlinear @math 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. 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 @math , where @math , @math consists of the functions smoother than @math and @math . This problem can be formulated as a nonlinear @math optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the @math 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. Sparse signal expansions represent or approximate a signal using a small number of elements from a large collection of elementary waveforms. Finding the optimal sparse expansion is known to be NP hard in general and non-optimal strategies such as Matching Pursuit, Orthogonal Matching Pursuit, Basis Pursuit and Basis Pursuit De-noising are often called upon. These methods show good performance in practical situations, however, they do not operate on the l0 penalised cost functions that are often at the heart of the problem. In this paper we study two iterative algorithms that are minimising the cost functions of interest. Furthermore, each iteration of these strategies has computational complexity similar to a Matching Pursuit iteration, making the methods applicable to many real world problems. However, the optimisation problem is non-convex and the strategies are only guaranteed to find local solutions, so good initialisation becomes paramount. We here study two approaches. The first approach uses the proposed algorithms to refine the solutions found with other methods, replacing the typically used conjugate gradient solver. The second strategy adapts the algorithms and we show on one example that this adaptation can be used to achieve results that lie between those obtained with Matching Pursuit and those found with Orthogonal Matching Pursuit, while retaining the computational complexity of the Matching Pursuit algorithm.