Active Search for Sparse Signals with Region Sensing
This addresses the challenge of efficient signal localization for autonomous systems like aerial robots, offering a novel approach that bridges gaps between existing methods, but it is incremental as it builds on prior work in active sensing.
The paper tackles the problem of searching for sparse signals using noisy average measurements of rectangular regions, proposing an algorithm based on greedy information gain maximization. It shows that the algorithm requires O~(n/μ^2 + k^2) measurements to recover k signal locations with small Bayes error in 1D and demonstrates empirical performance on satellite image data and high-dimensional problems.
Autonomous systems can be used to search for sparse signals in a large space; e.g., aerial robots can be deployed to localize threats, detect gas leaks, or respond to distress calls. Intuitively, search algorithms may increase efficiency by collecting aggregate measurements summarizing large contiguous regions. However, most existing search methods either ignore the possibility of such region observations (e.g., Bayesian optimization and multi-armed bandits) or make strong assumptions about the sensing mechanism that allow each measurement to arbitrarily encode all signals in the entire environment (e.g., compressive sensing). We propose an algorithm that actively collects data to search for sparse signals using only noisy measurements of the average values on rectangular regions (including single points), based on the greedy maximization of information gain. We analyze our algorithm in 1d and show that it requires $\tilde{O}(\frac{n}{μ^2}+k^2)$ measurements to recover all of $k$ signal locations with small Bayes error, where $μ$ and $n$ are the signal strength and the size of the search space, respectively. We also show that active designs can be fundamentally more efficient than passive designs with region sensing, contrasting with the results of Arias-Castro, Candes, and Davenport (2013). We demonstrate the empirical performance of our algorithm on a search problem using satellite image data and in high dimensions.