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

TL;DR

This paper introduces new one-bit compressed sensing methods that recover sparse signals efficiently with near-optimal measurements and sublinear decoding time, improving scalability for large signals.

Contribution

The authors develop schemes achieving near-optimal measurement bounds with sublinear decoding complexity, combining group testing ideas with sparse recovery techniques.

Findings

Universal support recovery with $O(k^2 \, \log(n/k) \, \log n)$ measurements

Probabilistic exact recovery with $O(k \, \frac{\log k}{\log\log k} \, \log n)$ measurements

Decoding complexity is sublinear, e.g., $O(km)$ or $O(m)$ depending on the scheme.

Abstract

One-bit compressed sensing (1bCS) addresses the recovery of sparse signals from highly quantized measurements, retaining only the sign of each linear measurement. In the support recovery setting, the goal is to identify $\text{supp}(x)$, the nonzero coordinates of an unknown signal $x \in \mathbb{R}^n$ from $y = \text{sign}(Ax)$, where $A \in \mathbb{R}^{m \times n}$ and $|\text{supp}(x)| \le k \ll n$. Existing methods minimize the number of measurements but often incur $\Omega(n)$ decoding complexity, limiting large-scale applicability. 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 $m = O\big(k \frac{\log k}{\log\log k} \log n\big)$ and $D=O(m)$, achieving vanishing error probability. Our approach leverages ideas from group testing to bridge classical sparse recovery techniques with modern algorithmic efficiency considerations, highlighting a new trade-off between compression efficiency and computational complexity.