Speaker: Professor Thomas Y. Hou (California Institute of Technology)
Title: Data-Driven Time-Frequency Analysis via Nonlinear Optimization
Time: 2013-01-03 (Thu.)  15:00 - 16:00
Abstract: Developing effective data analysis methods is an important path through which we can understand the underlying processes of natural phenomena. So far, most data analysis methods use a predetermined basis to process data. Applications of these methods to nonlinear and non-stationary data tend to give many unphysical harmonic modes. To better understand the physical mechanisms hidden in the data, one needs to develop truly adaptive data analysis methods whose basis is derived from the data rather than determined a priori. We introduce to develop a data-driven time-frequency analysis method to study nonlinear and non-stationary data. The key idea is to look for the sparsest time-frequency representation of a signal over the largest possible dictionary using nonlinear optimization. This method can be used to extract physically meaningful information of the signal such as instantaneous frequency and trend. Applications of our method to some real world data from geoscience and biomedical applications have led to some new discoveries. We have carried out rigorous convergence analysis for this nonlinear L1 optimization problem. Under some nonlinear sparsity assumption, we can prove that our adaptive data analysis gives exact recovery of the data. In this sense, our method can be considered as a nonlinear version of compressed sensing.
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