Robustness Analysis of Synchrosqueezed Transforms

Identifying and extracting principle wave-like components underlying a complex physical phenomenon are of great importance in modern data science. It is difficult to estimate all the wave-like components simultaneously from their superposition in order to reduce the influence of a sifting bias, which is crucial to many scientific problems. The newly developed synchrosqueezed transform has been proved a good option for this simultaneous analysis. Although its mathematical background is clear and is well-developed in a noiseless model, there is relatively little study on its robustness under noise. This paper is concerned with several fundamental robustness properties of synchrosqueezed transforms. We prove that it is possible to develop compactly supported synchrosqueezed transforms for oscillatory component analysis and give the conditions for accurate and robust estimation. Considering a generalized Gaussian random noise, we address the multiscale robustness problem of a wide range of existing synchrosqueezed transforms in one and two dimensions. It is shown that their multiscale robustness can be improved by tuning their corresponding multiscale geometry in the frequency domain. This dependence is clarified by quantitative probability analysis. As a supplement, new insights and numerical implementations are introduced for estimates with better accuracy and robustness. A software package together with several heavily noisy examples is provided to demonstrate these proposed properties.