We study Toeplitz covariance estimation when fixed-threshold one-bit quantization is combined with deterministic sparse-ruler sampling. Each observed bit can enter many lag products. At a nonzero threshold the signs have nonzero mean, and this deterministic vertex reuse gives raw sign products a coherent one-vertex component. This component changes the variance geometry. Raw nonzero-threshold products are governed by weighted-degree row sums rather than by lag coverage or edge Frobenius geometry. Centering the signs removes the vertex component and leaves a degenerate sparse-pair statistic. We then prove a dimension-free Gaussian variance contraction theorem for hollow quadratic forms of bounded coordinate transforms. The theorem applies to hard threshold signs and controls arbitrary deterministic sparse supports by the Frobenius norm of the edge weights, with no dependence on dimension, support size or maximum degree. For operator-norm estimation, we construct centered sparse-ruler Toeplitz estimators with pooled marginal calibration. The leading oracle term is \[ γ_0 L_1κ_{\rm obs} \sqrt{\frac{\varphi(Ω)\log d}{n}}, \qquad \varphi(Ω)=\sum_{s=1}^{d-1}q_s^{-1}, \] while the plug-in term is governed by the marginal bit budget \(n|Ω|\). A real spectral-packing lower bound in a known-scale identity-neighborhood submodel shows that the \(\sqrt{\varphi(Ω)\log d/n}\) dependence is intrinsic under balanced coverage geometry. In the non-saturated regime where this coverage term dominates, the oracle estimator is therefore minimax rate optimal over the submodel; the optimal dependence on the conditioning, curvature and plug-in calibration constants is left open.

翻译：本文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。每个观测比特可参与多个延迟乘积。在非零阈值下，符号具有非零均值，且这种确定性顶点复用使得原始符号乘积产生一个相干单顶点分量，该分量改变了方差几何结构。原始非零阈值乘积由加权度行和主导，而非延迟覆盖或边Frobenius几何结构。对符号进行中心化处理可移除顶点分量，留下退化稀疏对统计量。随后，我们证明了一个关于有界坐标变换的空心二次型的无维高斯方差收缩定理。该定理适用于硬阈值符号，并通过边权重的Frobenius范数控制任意确定性稀疏支撑集，且与维度、支撑集大小或最大度数均无关。对于算子范数估计，我们构建了具有池化边缘标定的中心化稀疏尺Toeplitz估计器。其最优项为 \[ γ_0 L_1κ_{\rm obs} \sqrt{\frac{\varphi(Ω)\log d}{n}}, \qquad \varphi(Ω)=\sum_{s=1}^{d-1}q_s^{-1}, \] 而内插项则由边际比特预算 \(n|Ω|\) 决定。在已知尺度恒等邻域子模型中，实谱填充下界表明，在均衡覆盖几何结构下，\(\sqrt{\varphi(Ω)\log d/n}\) 依赖性是本质性的。在覆盖项占主导的非饱和机制下，该最优估计器在子模型上达到极小极大速率最优；而关于条件数、曲率及内插标定常数的最优依赖性仍有待研究。