MTH 994 (Sect. 003) - Computational Harmonic Analysis and Data Science - SS18

Time & Location: SS18, Arranged

Instructor: Matthew J. Hirn Room 2507F 428 S. Shaw Ln. (Engineering Bldg)

Description:

This course will cover aspects of modern computational harmonic analysis at the interface of signal processing and data science. A central theme of the course is to find “good” representations of functional data (e.g., time series, images, etc), where the quality of the representation is measured through notions of sparsity, characterization of certain functional classes, and eventually empirical data driven measures.

The prologue of the course will cover the rudiments of Fourier analysis and discrete signal processing. The shortcomings of the Fourier transform will motivate us to study localized time-frequency representations of functions, which will introduce the windowed Fourier transform as well as the continuous and dyadic wavelet transforms. Unlike the Fourier transform, which characterizes only the global regularity of a function, wavelet transforms characterize the local regularity of functions, and we will prove fundamental results along these lines. Windowed Fourier and wavelet transforms will be placed in a more general mathematical context via the study of redundant dictionaries and frame theory. Motivated in part by the sparsity of wavelet transforms, we will then aim to understand how to find sparse representations in general dictionaries, and at the conclusion of this part of the course look at recent methods that learn, in a data driven fashion, dictionaries of functions that yield sparse representations. Dictionary learning in turn leads to the study of more complicated learning models; convolutional neural networks are a natural place to turn. In the final part of the course we will study these networks, as well as mathematically tractable models for them (i.e. ones in which we can prove theorems) based upon nonlinear cascades of semi-discrete frame operators.

The primary textbook for the course will be “A Wavelet Tour of Signal Processing: The Sparse Way,” 3rd edition, by Stephane Mallat. The course may also draw a bit of material from “Wavelets and Operators,” by Yves Meyer. The final part of the course on convolutional neural networks will be based on current papers in the field. All parts of the course (minus the prologue) will highlight current research and papers.

The course will assume knowledge of real analysis (Lebesgue integration, L^p spaces, Banach and Hilbert spaces).