Statistical Robust Chinese Remainder Theorem for Multiple Numbers

TL;DR

This paper introduces a statistical framework for the Chinese Remainder Theorem applied to multiple numbers, enhancing robustness against noise through MAP estimation and Gaussian mixture models, with superior performance demonstrated in experiments.

Contribution

It provides the first rigorous statistical analysis of CRT-based multiple parameter estimation and proposes two novel robust approaches incorporating residue error correction.

Findings

Statistical schemes outperform deterministic methods in noisy environments

MAP estimation improves residue clustering accuracy

Gaussian mixture models enhance robustness in heavy-noise scenarios

Abstract

Generalized Chinese Remainder Theorem (CRT) is a well-known approach to solve ambiguity resolution related problems. In this paper, we study the robust CRT reconstruction for multiple numbers from a view of statistics. To the best of our knowledge, it is the first rigorous analysis on the underlying statistical model of CRT-based multiple parameter estimation. To address the problem, two novel approaches are established. One is to directly calculate a conditional maximum a posteriori probability (MAP) estimation of the residue clustering, and the other is based on a generalized wrapped Gaussian mixture model to iteratively search for MAP of both estimands and clustering. Residue error correcting codes are introduced to improve the robustness further. Experimental results show that the statistical schemes achieve much stronger robustness compared to state-of-the-art deterministic schemes, especially in heavy-noise scenarios.