Research Overview

The field of Digital Signal Processing (DSP) has been tremendously successful over the past 50 years. This success stems in part from the fact that most operations can be carried out much more precisely by a digital processor (or computer) than by analog hardware. In a sense, DSP has become a victim of its own success. As more and more processing is pushed into the digital domain, the demands on DSP hardware and algorithms - to be faster and more efficient - continues to grow. In some areas we have begun to reach a limit, and we need new approaches to some classical problems.

Compressive sensing

What does it really mean to "sample" a signal? And how can we ensure that our samples contain enough information to accurately reconstruct the original signal? compressive sensing is an emerging field that extends our notion of sampling - we are no longer interested in sampling such that we can recover any bandlimited signal, because in many scenarios our signal of interest exhibits a great deal of structure. Specifically, many signals of interest can be modeled as being sparse or compressible in some basis or dictionary, or obeying a low-dimensional parametric or manifold model. Using this a priori information, we are able to "sample" a signal (not necessarily bandlimited) in totally new ways and at a dramatically lower rate, and still perfectly reconstruct the signal. This gives us a way to "sample" signals that would otherwise be impossible acquire. It is also an extremely computationally efficient compression strategy, combining compression and sensing into a single process.

See: single-pixel camera -- random matrices for compressive sensing -- compressive detection and estimation -- smashed filtering -- joint manifolds for ensembles of data

Cost-sensitive learning with support vector machines

In addition to the challenges we face in acquiring, storing, and transmitting very large amounts of data, we also frequently desire to "learn" from the data in a number of senses. Example applications include search engines, medical diagnosis, detecting credit card fraud, stock market analysis, speech and handwriting recognition, object recognition in computer vision, and spam filtering. Support vector machines (SVMs) are a particularly effective technique for both supervised and unsupervised machine learning problems - achieving high accuracy while simultaneously scaling well to large or high-dimensional data sets. Traditionally, research on machine learning in general, and SVMs in particular, has focused on minimizing the probability of making an error. However, in both supervised and unsupervised learning there are two types of errors we can make (associated with the two classes - in the unsupervised case the second class is implicitly assumed), and in many practical settings the two types of errors have very different costs. For example, in tumor classification the probability of error is not the most appropriate criterion. The Neyman-Pearson (NP) approach seeks instead to minimize false negatives while constraining false positives to be below a certain significance level. This approach is useful in both the supervised and unsupervised settings. We show that SVMs can be adapted to this setting in a natural manner.

See: Neyman-Pearson and minimax SVMs -- regression level-set estimation.