Support Recovery in One-bit Compressed Sensing with Near-Optimal Measurements and Sublinear Time

Abstract

One-bit compressed sensing (1bCS) addresses the recovery of sparse signals from highly quantized measurements, retaining only the sign of each linear measurement. From a data compression perspective, the one-bit measurements form a compact binary representation of sparse signals. The support recovery problem seeks to recover the support of an unknown signal $x\in {\mathbb{R}}^n$, $\mathrm{supp}(x)$, from $y=\mathrm{sgn}(Ax)$, where $A\in {\mathbb{R}}^{m\times n}$ is the measurement matrix and $|\mathrm{supp}(x)|\le k\ll n$. Existing methods seek to minimize the number of measurements but often incur $\Omega (n)$ decoding complexity, limiting their applicability to large-scale problems. We propose new 1bCS schemes that achieve sublinear decoding complexity while maintaining near-optimal measurement bounds. For universal support recovery, our framework provides: (i) exact recovery with $m=O(k^2\log (n/k)\log n)$ measurements and decoding complexity $D=O(km)$, and (ii) $\epsilon$-approximate recovery with $m=O(k{\epsilon }^{-1}\log (n/k)\log n)$ and $D=O({\epsilon }^{-1}m)$. For probabilistic exact recovery, we design a scheme with $m=O(k\log k\log n)$ and $D=O(m)$, achieving vanishing error probability. Our schemes leverage ideas from group testing to achieve near-optimal support compression with substantially reduced decoding complexity.