## Learning Low-dimensional Signal Models -- A Bayesian approach based on incomplete measurements (2011)

### BibTeX

@MISC{Carin11learninglow-dimensional,

author = {Lawrence Carin and Richard G. Baraniuk and Volkan Cevher and David Dunson and Michael I. Jordan and Guillermo Sapiro and Michael B. Wakin},

title = { Learning Low-dimensional Signal Models -- A Bayesian approach based on incomplete measurements},

year = {2011}

}

### OpenURL

### Abstract

Sampling, coding, and streaming even the most essential data, e.g., in medical imaging and weather-monitoring applications, produce a data deluge that severely stresses the available analog-to-digital converter, communication bandwidth, and digital-storage resources. Surprisingly, while the ambient data dimension is large in many problems, the relevant information in the data can reside in a much lower dimensional space. This observation has led to several important theoretical © DIGITAL STOCK & LUSPHIX and algorithmic developments under different low-dimensional modeling frameworks, such as compressive sensing (CS) [1], [2], matrix completion [3], [4], and general factor-model representations [5], [6]. These approaches have enabled new measurement systems, tools, and methods for information extraction from dimensionality-reduced or incomplete data. A key aspect of maximizing the potential of such techniques is to develop appropriate data models. In this article, we investigate this challenge from the perspective of nonparametric Bayesian analysis. Before detailing the Bayesian modeling techniques, we review the form of measurements. Specifically, we consider measurement systems based on dimensionality reduction, where we linearly project the signal of interest into a lower-dimensional space via y Ux þ d: (1) The signal is x 2 R d, the measurements are y 2 R d0, U is a d0 3 d matrix with d05 d,anddaccounts for noise. Such a projection process loses signal information in general, since U has a nontrivial null space. Hence, there has been significant interest over the last few decades in finding dimensionality reductions that preserve as much information as possible in the incomplete measurements y about certain signals x. One way to preserve information is for U to provide a stable embedding that approximately preserves pairwise distances between all signals in some set of interest. In some cases, this property allows the recovery of x from its measurement y.