PiPs: A kernel-based optimization scheme for analyzing non-stationary 1D signals

This paper proposes a novel kernel-based optimization scheme to handle tasks in the analysis, e.g., signal spectral estimation and single-channel source separation of 1D non-stationary oscillatory data. The key insight of our optimization scheme for reconstructing the time-frequency information is that when a nonparametric regression is applied on some input values, the output regressed points would lie near the oscillatory pattern of the oscillatory 1D signal only if these input values are a good approximation of the ground-truth phase function. In this work, Gaussian Process (GP) is chosen to conduct this nonparametric regression: the oscillatory pattern is encoded as the Pattern-inducing Points (PiPs) which act as the training data points in the GP regression; while the targeted phase function is fed in to compute the correlation kernels, acting as the testing input. Better approximated phase function generates more precise kernels, thus resulting in smaller optimization loss error when comparing the kernel-based regression output with the original signals. To the best of our knowledge, this is the first algorithm that can satisfactorily handle fully non-stationary oscillatory data, close and crossover frequencies, and general oscillatory patterns. Even in the example of a signal produced by slow variation in the parameters of a trigonometric expansion, we show that PiPs admits competitive or better performance in terms of accuracy and robustness than existing state-of-the-art algorithms.