Fixed-Threshold One-Bit Toeplitz Covariance Estimation under Sparse-Ruler Sampling

2026-06-09

AI Summary

This study addresses the problem of efficiently estimating Toeplitz covariance matrices under one-bit quantization with a fixed threshold and sparse ruler sampling. The authors propose a centered sparse ruler Toeplitz estimator that mitigates vertex bias through sign centering, controls edge weights via the Frobenius norm, and leverages a dimension-free Gaussian variance shrinkage theorem to show that, under non-zero thresholds, the variance of sign products is dominated by weighted row sums. The method achieves the theoretically optimal convergence rate of $\sqrt{\phi(\Omega) \log d / n}$ in the non-saturated regime and attains minimax optimality within specific submodels, while maintaining marginal calibration and computational efficiency.

Abstract

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.

Problem

• Toeplitz covariance estimation
• one-bit quantization
• sparse-ruler sampling
• fixed-threshold
• covariance matrix

Innovation

• one-bit quantization
• sparse-ruler sampling
• Toeplitz covariance estimation
• dimension-free variance contraction
• minimax optimality