@article{dmnw-tsp-2016,
author = {M. A. Davenport and A. K. Massimino and D. Needell and T. Woolf},
journal = {IEEE Trans. Signal Processing},
title = {Constrained Adaptive Sensing},
year = {2016},
volume = {64},
number = {20},
pages = {5437-5449},
keywords = {Algorithm design and analysis;Atmospheric measurements;Compressed
sensing;Noise measurement;Particle measurements;Sensors;Signal to noise
ratio;Sparsity;adaptive sensing;compressive sensing;constrained
sensing;information bounds;optimal experimental design},
abstract = {
Suppose that we wish to estimate a vector $\x \in \C^n$ from a small
number of noisy linear measurements of the form $\y = \A \x + \z$,
where $\z$ represents measurement noise. When the vector $\x$ is
sparse, meaning that it has only $s$ nonzeros with $s \ll n$, one can
obtain a significantly more accurate estimate of $\x$ by adaptively
selecting the rows of $\A$ based on the previous measurements provided
that the signal-to-noise ratio (SNR) is sufficiently large. In this
paper we consider the case where we wish to realize the potential of
adaptivity but where the rows of $\A$ are subject to physical
constraints. In particular, we examine the case where the rows of
$\A$ are constrained to belong to a finite set of allowable
measurement vectors. We demonstrate both the limitations and
advantages of adaptive sensing in this constrained setting. We prove
that for certain measurement ensembles, the benefits offered by
adaptive designs fall far short of the improvements that are possible
in the unconstrained adaptive setting. On the other hand, we also
provide both theoretical and empirical evidence that in some scenarios
adaptivity does still result in substantial improvements even in the
constrained setting. To illustrate these potential gains, we propose
practical algorithms for constrained adaptive sensing by exploiting
connections to the theory of optimal experimental design and show that
these algorithms exhibit promising performance in some representative
applications.
},
doi = {10.1109/TSP.2016.2597130},
ISSN = {1053-587X},
month = oct,
note = {arXiv:1506.05889}
}