Papers โบ Constrained adaptive sensing
Constrained adaptive sensing
Mark A. Davenport, Andrew K. Massimino, Deanna Needell, Tina Woolf
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Suppose that we wish to estimate a vector ๐ฑ โโโฟ from a small number of noisy linear measurements of the form ๐ฒ = ๐ ๐ฑ + ๐ณ, where ๐ณ represents measurement noise. When the vector ๐ฑ is sparse, meaning that it has only s nonzeros with s โชn, one can obtain a significantly more accurate estimate of ๐ฑ by adaptively selecting the rows of ๐ 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 ๐ are subject to physical constraints. In particular, we examine the case where the rows of ๐ 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.
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