Adaptive search for sparse targets with informative priors

Abstract

This works considers the problem of efficient energy allocation of resources in a continuous fashion in determining the location of targets in a sparse environment. We extend the work of Bashan [1] to analyze the use of non-uniform prior knowledge for the location of targets. We show that in the best-case scenario (i.e., when the known prior knowledge is also the underlying prior), then we can get significant gains (several dB) by using a two-level piecewise uniform prior over using the uniform prior that is assumed in [1]. Moreover, even when we have uncertainty in our prior knowledge, we show that we can always do at least as well as the uniform alternative in terms of worst-case and expected gains. In future work, we plan to extend our analysis to general piecewise uniform priors in order to develop multistage (i.e., greater than 2) adaptive energy allocation policies. II. INTRODUCTIONIn many situations, it might be desirable to allocate a limited amount of energy to a small region of interest (ROI) within a larger environment by using adaptive sampling techniques. For example, consider the problem of minimizing communication costs when tracking a target in a distributed sensor network.Clearly, when a node in our sensor network is far from our previous estimate of the target, we would like to reduce the communication from that node in order to preserve its battery life. On the other hand, we would like all sensors within a region near the previous target estimate to be used to estimate the target's position at the next time step. For this purpose, then, we can use adaptive sampling to provide both smart sensor management and improved tracking performance.In another application, we may be interested in locating and estimating the content of tumors in early cancer detection. In this situation, the ROI consists of the tumor location, which in early cancer detection is assumed to be much smaller in area than the entire image. Moreover, the total amount of energy used

Multistage adaptive estimation of sparse signals

“…x(t + 1) is specified by x(t), λ(t), and y(t + 1) through (8). Thus the choice of λ(t) depends on Y(t) only through the state x(t), which is a property of dynamic programs [24].…”

“…An optimal two-stage allocation policy was developed in [3] for a cost function related to bounds on estimation and detection performance. Subsequent developments stemming from [3] include a modification to handle non-uniform signal priors [8], a simplification based on Lagrangian constraint relaxation [9], and a multiscale approach that uses linear combinations in the first stage to reduce the number of measurements [4]. Based on a similar model but in a different direction, a method known as distilled sensing [10] was proposed for signal support identification and was shown to be asymptotically reliable (as the ambient dimension increases) at SNR levels significantly lower than non-adaptive limits.…”

On the Fundamental Limits of Recovering Tree Sparse Vectors From Noisy Linear Measurements

“…The results of Figure 3 allow us to make one additional comparison with the behavior identified by the sufficient condition (7). Namely, note that for fixed m, β, and σ 2 , we expect from (7) that the minimum signal amplitude µ above which exact support recovery is achieved (with high probability) by the tree sensing procedure should increase in proportion to √ k log k. Now, the results of Figure 3…”

“…Our second experimental evaluation is designed to investigate the scaling behavior predicted by the theoretical guarantees we provide in Corollary I.1 -namely, that accurate support estimation is achievable provided the nonzero signal components satisfy the condition given in (7). To that end, we provide a phase transition plot for our approach that depicts, for a measurement budget m max , whether the tree sensing procedure results in accurate support recovery of k tree-sparse signals whose nonzero amplitudes each have amplitude µ (for varying parameters k and µ).…”

Adaptive sensing for sparse recovery

“…Two-stage sequential sampling techniques have also been examined recently in the signal processing literature. In [32], two-stage target detection procedures were examined, and a follow-on work examined a Bayesian approach for incorporating prior information into such two-step detection procedures [33]. The problem of target detection and localization from sequential compressive measurements was recently examined in [34].…”