We study Toeplitz covariance estimation when fixed-threshold one-bit quantization is combined with deterministic sparse-ruler sampling, so that each observed bit is reused across many lag products. At a nonzero threshold the signs have nonzero mean, and this reuse gives raw sign products a coherent one-vertex variance component governed by weighted row sums; centering removes it and leaves a degenerate sparse-pair statistic. We prove a Gaussian variance contraction theorem for hollow quadratic forms of bounded coordinate transforms, including hard threshold signs: the variance is bounded by the squared correlation operator norm times the squared Frobenius norm of the edge weights, with constants independent of dimension, support size and maximum degree. For the oracle centered sparse-ruler estimator, the leading operator-norm term is \(γ_0L_1κ_{\rm obs}\sqrt{\varphi(Ω)\log d/n}\), where \(\varphi(Ω)=\sum_{s=1}^{d-1}q_s^{-1}\) is the coverage coefficient of the ruler; pooled marginal calibration from the \(n|Ω|\) observed bits adds a plug-in term. A spectral-packing lower bound in a known-scale identity-neighborhood submodel shows that this dependence is intrinsic under balanced coverage geometry; in the non-saturated regime where the coverage term dominates, the oracle estimator is minimax rate optimal over this submodel.

翻译：我们研究当固定阈值单比特量化与确定性稀疏尺采样相结合时的Toeplitz协方差估计问题，使得每个观测比特在多个时滞乘积中被重复使用。在非零阈值下，符号具有非零均值，这种重复使用使得原始符号乘积产生由加权行和支配的相干单顶点方差分量；中心化处理可消除该分量，并留下一个退化的稀疏对统计量。我们证明了有界坐标变换下空心二次型的高斯方差压缩定理，包括硬阈值符号：该方差受平方相关算子范数与边权平方Frobenius范数的乘积上界约束，且常数与维度、支撑集大小及最大度无关。对于最优中心化稀疏尺估计量，主导算子范数项为\(γ_0L_1κ_{\rm obs}\sqrt{\varphi(Ω)\log d/n}\)，其中\(\varphi(Ω)=\sum_{s=1}^{d-1}q_s^{-1}\)是尺的覆盖系数；基于\(n|Ω|\)个观测比特的汇合边际校准引入了一个插件项。在已知尺度恒等邻域子模型下的谱包络下界表明，在平衡覆盖几何条件下该依赖关系是本质的；在覆盖项主导的非饱和区域中，最优估计量在该子模型上达到极小极大最优速率。