Information-Theoretic Bounds for Adaptive Sparse Recovery
This work addresses theoretical limits in sparse recovery for researchers in signal processing and machine learning, providing a unified analysis that is incremental in extending existing non-adaptive bounds to adaptive settings.
The paper tackles the problem of determining sample complexity lower bounds for adaptive sparse recovery across various models, showing that adaptivity cannot reduce sample complexity in group testing, 1-bit compressive sensing, and compressive sensing with linear sparsity, but might offer mild gains in sublinear regimes.
We derive an information-theoretic lower bound for sample complexity in sparse recovery problems where inputs can be chosen sequentially and adaptively. This lower bound is in terms of a simple mutual information expression and unifies many different linear and nonlinear observation models. Using this formula we derive bounds for adaptive compressive sensing (CS), group testing and 1-bit CS problems. We show that adaptivity cannot decrease sample complexity in group testing, 1-bit CS and CS with linear sparsity. In contrast, we show there might be mild performance gains for CS in the sublinear regime. Our unified analysis also allows characterization of gains due to adaptivity from a wider perspective on sparse problems.