Subspace approximation on the continuum

Abstract

Many signal processing problems---such as analysis, compression, reconstruction, and denoising---can be facilitated by exploiting the underlying models the signals and data sets obey. A model often deals with the notion of conciseness and suggests a signal has few degrees of freedom relative to its ambient dimensionality. For instance, the Shannon-Nyquist sampling theorem works on bandlimited signals obeying{\em subspace model}. As an another example, the power of sparse signal processing often relies on the assumption that the signals live in some union of subspaces. In many cases, signals have concise representations which are often obtained by ($i$) constructing a{\em dictionary} of elements drawn from the signal space, and then ($ii$) expressing the signal of interest as a linear combination of a small number of atoms drawn from the dictionary. Such representations serve as an efficient way to describe the conciseness of the signals and enable effective signal processing methods. For example, the sparse representation forms the core of compressive sensing (CS), an emerging research area that aims to break through the Shannon-Nyquist limit for sampling analog signals. However, despite its recent success, there are many important applications in signal processing that do not naturally fall into the subspace models and sparse recovery framework. As a classical example, a finite-length vector obtained by sampling a bandlimited signal is not sparse using the discrete Fourier transform (DFT), the natural tool for frequency analysis on finite-dimensional space. In other words, the DFT cannot excavate the concise structure within the sampled bandlimited signals. These signals obey a so-called{\em parameterized subspace model} in which the signals of interest are inherently low-dimensional and live in a union of subspaces, but the choice of subspace is controlled by a small number of continuous-valued parameters (the parameter controlling sampled bandlimited signals is the frequency). This continuous-valued parameterized subspace model appears in many problems including spectral estimation, mitigation of narrowband interference, feature extraction, and steerable filters for rotation-invariant image recognition. The purpose of this thesis is to 1) construct a subspace---whose dimension matches the effective number of local degrees of freedom---for approximating (almost) all the signals controlled by a small number of continuous-valued parameters ranging within some certain intervals; 2) develop rigorous, theoretically-backed techniques for computing projections onto and orthogonal to these subspaces. By developing an appropriate basis to economically represent the signals of interest, one can apply effective tools developed for the subspace model and sparse recovery framework for signal processing. In the process of building local subspace fits, we will also obtain the effective dimensionality of such signals. Our key contributions include (i) new non-asymptotic results on the eigenvalue distribution of (periodic) discrete time-frequency localization operators and fast constructions for computing approximate projections onto the{\em discrete prolate spheroidal sequences} (DPSS's) subspace; (ii) an orthogonal approximate Slepian transform that has computational complexity comparable to the fast Fourier transform (FFT); (iii) results on the spectrum of combined time- and multiband-limiting operations in the discrete-time domain and analysis for a dictionary formed by concatenating a collection of modulated DPSS's; (iv) analysis for the dimensionality of wall and target return subspaces in through-the-wall radar imaging and algorithms for mitigating wall clutter and identifying non-point targets; (v) asymptotic performance guarantee of the individual eigenvalue estimates for Toeplitz matrices by circulant matrices; and (vi) analysis of the eigenvalue distribution of time-frequency limiting operators on locally compact abelian groups.