Celebratio Mathematica — Daubechies

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J. Xu, H. Yang, and I. Daubechies: “Recursive diffeomorphism-based regression for shape functions,” SIAM J. Math. Anal. 50 : 1 (2018), pp. 5–32. MR 3740323 Zbl 1381.65104 ArXiv 1610.03819 article

This paper proposes a recursive diffeomorphism based regression method for one-dimensional generalized mode decomposition problem that aims at extracting generalized modes \( \alpha_k(t) \) \( s_k(2\pi N_k\phi_k(t)) \) from their superposition \[ \sum_{k=1}^K \alpha_k(t)\,s_k(2\pi N_k\phi_k(t)) .\] First, a one-dimensional synchrosqueezed transform is applied to estimate instantaneous information, e.g., \( \alpha_k(t) \) and \( N_k\phi_k(t) \). Second, a novel approach based on diffeomorphisms and nonparametric regression is proposed to estimate wave shape functions \( s_k(t) \). These two methods lead to a framework for the generalized mode decomposition problem under a weak well-separation condition. Numerical examples of synthetic and real data are provided to demonstrate the fruitful applications of these methods.

Jieren Xu	Related
Haizhao Yang	Related

J. Xu, Y. Li, D. Dunson, I. Daubechies, and H. Yang: Non-oscillatory pattern learning for non-stationary signals. Preprint, May 2018. ArXiv 1805.08102 techreport

This paper proposes a novel non-oscillatory pattern (NOP) learning scheme for several oscillatory data analysis problems including signal decomposition, super-resolution, and signal sub-sampling. To the best of our knowledge, the proposed NOP is the first algorithm for these problems with fully non-stationary oscillatory data with close and crossover frequencies, and general oscillatory patterns. NOP is capable of handling complicated situations while existing algorithms fail; even in simple cases, e.g., stationary cases with trigonometric patterns, numerical examples show that NOP admits competitive or better performance in terms of accuracy and robustness than several state-of-the-art algorithms.

David Brian Dunson	Related
Jieren Xu	Related
Haizhao Yang	Related
Yitong Li	Related