## PIMS CRG in High Dimensional Data Analysis: Short Course (CANCELLED)

- Start Date: 04/07/2020
- End Date: 04/16/2020

Simon Fraser University

Short course on Sparse Fourier Transforms for Approximating Functions of Many Variables

Compressive sensing has generated tremendous amounts of interest since first being proposed by Emmanuel Candes, David Donoho, Terry Tao, and others a bit more than a decade ago. This mathematical framework has its origins in (i) the observation that traditional signal processing applications, such as MRI imaging problems, often deal with the acquisition of signals which are known a priori to be sparse in some basis, as well as (ii) the subsequent realization that this knowledge could in fact be used to help streamline the signal acquisition process in the first place (by taking the bare minimum of signal measurements necessary in order to discover and then reconstruct the important basis coefficients only). The resulting mathematical theory has since led to dramatic reductions in measurement needs over traditional approaches in many situations where one would previously have reconstructed a fuller set of a given signal's basis coefficients only to later discard most of them as insignificant.

Though extremely successful at reducing the number of measurements needed in order to reconstruct a given signal, most standard compressive sensing recovery algorithms still individually represent every basis function during the signal's numerical reconstruction. This leads one to ask a computationally oriented variant of the original question which led to the development of compressive sensing in the first place: why should one consider all possible basis coefficients individually during the numerical reconstruction of a given signal when one knows in advance that only a few of them will end up being significant? In fact, it turns out that one often does not have to explicitly consider each basis function individually during the reconstruction process, and so can reduce both the measurement needs *and* computational complexity of signal reconstruction to depend on the bare minimum of signal measurements necessary in order to reconstruct the important basis coefficients in many settings. This series of lectures will discuss a class of sublinear-time numerical methods which do exactly this for functions that are compressible in the one-dimensional Fourier basis, as well as the extension of such techniques to produce new fast methods for approximating functions of many variables. If time permits, the further extension of similar techniques to tackle functions which exhibit approximate sparsity in other orthonormal polynomial bases will also be discussed.

**Suggested Prerequisites**:

-- a strong background in linear algebra

-- A working familiarity with Fourier series and trigonometric polynomials

-- basic knowledge of introductory number theory (the Chinese remainder thm in particular)

-- basic knowledge of finite fields (first lecture only)

**Schedule**:

Tue Apr 7, 3:00 - 4:30

Lecture 1: Combinatorial Compressive Sensing Using Binary Low Coherence Matrices

Thu Apr 9, 3:00 - 4:30

Lecture 2: Sublinear-time Compressive Sensing Using Lowe Coherence Matrices

Tue Apr 14, 3:00 - 4:30

Lecture 3: Sparse Fourier Transforms for Periodic Functions of One Variable

Thu Apr 16, 3:00 - 4:30

Lecture 4: Sparse Fourier Transforms for approximating Functions of Many Variables

**Registration**:

Registration for this event is not required.

This Short Course is part of the PIMS CRG in High Dimensional Data Analysis.

**Location**: Simon Fraser University

Room K9509