🤖 AI Summary
This work addresses the recovery of $k$-sparse binary signals of length $n$ under extreme undersampling conditions where the number of measurements $m$ is less than the sparsity level $k$, a regime in which conventional compressive sensing methods fail. The authors propose ISDP-MVSIC, a novel approach that integrates randomized semidefinite programming (SDP) sampling, majority voting (MV), and successive interference cancellation (SIC), enhanced by a residual-driven retry mechanism for staged signal reconstruction. This method achieves, for the first time, exact recovery with high probability even when $m < k$, while offering a tunable trade-off between computational complexity and reconstruction performance. Experimental results demonstrate empirically perfect recovery for $n = 100$ and $144$ across a wide range of sparsity ratios, specifically for $m/k \in [0.4, 5.0]$, at the cost of modestly increased computational overhead.
📝 Abstract
Binary compressive sensing (BCS) seeks to recover a $k$-sparse binary vector of length $n$ from $m$ linear measurements. Classical CS guarantees break down for $m < k$ and convex/greedy BCS algorithms with random Gaussian sensing matrices perform poorly. We introduce ISDP-MVSIC, which combines randomized semidefinite programming (SDP) sampling, majority voting (MV) and successive interference cancellation (SIC) across $L \ll n$ stages, wrapped in a residual-cost driven retry loop. The method exposes a tunable complexity--performance trade-off: for $n=100, 144$, raising the worst-case complexity $\mathcal{C}_{max}$ from $7.9 \times 10^9$ to $2.0 \times 10^{10}$ enables empirical exact recovery over $m/k \in [0.4,5.0]$ as the sparsity ratio $s=k/n$ decreases from $0.5$ to $0.1$, by practically targeting the undersampled regime.