Polynomial Linear System Solving with Random Errors: new bounds and early termination technique

TL;DR

This paper improves bounds on the number of evaluations needed for polynomial linear system solving with errors, introducing an early termination technique to optimize the process when errors are randomly distributed.

Contribution

It presents a new, tighter bound on evaluations and an early termination method to reduce computational effort in error-prone polynomial linear system solving.

Findings

Lowered the evaluation bound based on system parameters.

Introduced an early termination strategy for efficiency.

Enhanced robustness against random errors.

Abstract

This paper deals with the polynomial linear system solving with errors (PLSwE) problem. Specifically, we focus on the evaluation-interpolation technique for solving polynomial linear systems and we assume that errors can occur in the evaluation step. In this framework, the number of evaluations needed to recover the solution of the linear system is crucial since it affects the number of computations. It depends on the parameters of the linear system (degrees, size) and on a bound on the number of errors. Our work is part of a series of papers about PLSwE aiming to reduce this number of evaluations. We proved in [Guerrini et al., Proc. ISIT'19] that if errors are randomly distributed, the bound of the number of evaluations can be lowered for large error rate. In this paper, following the approach of [Kaltofen et al., Proc. ISSAC'17], we improve the results of [Guerrini et al., Proc. ISIT'19] in two directions. First, we propose a new bound of the number of evaluations, lowering the dependency on the parameters of the linear system, based on work of [Cabay, Proc. SYMSAC'71]. Second, we introduce an early termination strategy in order to handle the unnecessary increase of the number of evaluations due to overestimation of the parameters of the system and on the bound on the number of errors.