Near-Optimal Lower Bounds on One-Bit Compressed Sensing of Approximately Sparse Signals

2026-07-07

AI Summary

This work investigates fundamental lower bounds on the reconstruction error for approximately sparse signals—lying within a scaled ℓ₁ ball—in the setting of one-bit compressed sensing. By constructing pairs of indistinguishable signals embedded in small Euclidean balls and leveraging lifting maps, minimax analysis, information-theoretic arguments, and recent results on sub-Gaussian and sub-Weibull random matrices, the authors establish the first near-optimal lower bounds applicable to ℓ_q-sparse signals for all q ∈ [0,1]. The resulting Euclidean error bound scales as Ω̃((k/m)^{(2−q)/(2+q)}), which matches the known upper bound Õ((k/m)^{1/3}) when q = 1. The analysis unifies standard and uniformly dithered one-bit quantization models and extends to adversarial bit flips and matrix recovery, revealing intrinsic trade-offs between sparsity and the number of measurements.

Abstract

This paper provides the first near-optimal lower bounds for one-bit compressed sensing of approximately sparse signals lying in a scaled $\ell_1$ ball, which is a commonly adopted relaxation of the exactly $k$-sparse assumption. In prior works, the best known upper bounds on uniform Euclidean error are of order $\widetilde{O}((k/m)^{1/3})$, where $m$ is the number of measurements. Under sub-Gaussian matrices, we establish nearly matching lower bounds for both the canonical one-bit compressed sensing model and the uniformly dithered model. Our argument is to first embed a small Euclidean ball into the signal set, which is straightforward for the dithered model but relies on a lifting map for the canonical model, and then construct two signals in this small ball that are separated in Euclidean distance by at least $(k/m)^{1/3}$ (up to logarithmic factor) but are indistinguishable from the binary measurements. Moreover, our argument extends to approximately sparse signals that live in a properly scaled $\ell_q$ ball $(q\in [0,1])$, yielding a lower bound $\widetildeΩ((k/m)^{\frac{2-q}{2+q}})$ that smoothly bridges the cases of exact sparsity ($q=0$) and $\ell_1$ sparsity ($q=1$). Finally, we discuss the extensions of our lower bounds to sub-Weibull matrices, adversarial bit flipping, matrix recovery, and characterize the transition to the non-sparse case.

Problem

• one-bit compressed sensing
• approximately sparse signals
• lower bounds
• ℓ_q ball
• Euclidean error

Innovation

• one-bit compressed sensing
• near-optimal lower bounds
• approximately sparse signals
• lifting map
• ℓ_q sparsity