Local analysis of redundant frame expansions

One of the most successful techniques in analysis is the one of decomposing function spaces into elementary building blocks. This is commonly known as a frame expansion. A general function is presented as a weighted superimposition of more basic ones, often called atoms. Diverse operations on functions can be understood by first considering their action on the basic building blocks. Time-scale and time-frequency analysis provide examples of such techniques. In the first case, the dictionary of building blocks consists of translated and dilated copies of a basic shape. In the second, it consists of time and frequency translations of a fundamental atom. Several subtle properties of functions (e.g. smoothness, decay, directional regularity) are encoded in the coefficients (weights) involved in such representations. The usefulness of frame expansions, both in theory and applications, comes from the possibility of designing dictionaries of atoms with prescribed properties. The counterpart of this flexibility is redundancy. In order to construct dictionaries with certain properties one often has to give up perfect independence between different atoms, thus yielding a redundant expansion. The aim of this project is to study local operations on redundant frame expansions. The challenge in this is that a perfect manipulation of the coefficients is no longer possible, since changing a coefficient associated with an atom can have an effect on the other ones. The main insight we will apply to overcome this difficulty is to replace the role of strict independence by the more flexible notion of low-correlation.

A very powerful technique to analyze data consists in representing it as a superimposition of basic building blocks with a simple structure (often called atoms). The choice of the building blocks is tailored to a specific problem or need. For example, in audio processing, the basic blocks consist of brief oscillatory signals, whose oscillation frequencies correspond to the sound's pitch. Hence, the representation of auditory signals by these blocks resembles a musical score.If the signals in question are images, then atoms are basic geometric shapes with various sizes and orientations. Analyzing an image in this way amounts to decomposing it into more basic elements like lines and circles. Such representations are very useful in practice. Once data is effectively represented as superimposition of basic pieces, it can be compressed by simply discarding those parts having a small influence, thus retaining only the essential features.Other signals of interest include measurements of natural phenomena (temperatures, pollutions, etc.). In each case a crucial task is designing good building blocks. It is quite often the case that, in order to do that, one has to allow for some redundancy. This means that data can be decomposed into atoms in several different ways.The extra freedom brought by redundancy can be greatly exploited in applications. Redundancy also entails, however, a mathematical challenge. Redundancy means that there is not a perfect correspondence between data and the description of the data in terms of basic building block.This project addressed questions of the form: which kind of building blocks provide an optimal description for a certain problem? If the problem slightly changes or evolves, do we need a completely new representation or can we just tweak what we have already computed? How many building blocks are necessary to effectively represent a certain phenomenon? Is the corresponding data description numerically stable? I.e., can it be effectively processed with a computer?