Multiple-Error-Correcting Codes for Analog Computing on Resistive Crossbars

TL;DR

This work develops error-correcting codes over the real field to robustly locate multiple outlying errors in analog crossbar computations, addressing vector-matrix products implemented with resistive memory. It introduces two complementary code families: a spherical-code-based construction that preserves $\Gamma_{\lambda}(\mathcal{C})=O(n/\sqrt{r})$ for various $\lambda$ with $r=O(\log n)$, and a disjunct-matrix-based construction yielding $\Gamma_{\lambda}(\mathcal{C})\le 2\rho$ with efficient decoders. The paper also provides efficient decoding algorithms for both generic and weight-constrained disjunct designs and offers new lower/upper bounds on maximum row weights for disjunct matrices. By leveraging restricted isometry properties and coherence, the results establish scalable, multi-error localization performance and connect combinatorial designs to practical analog ECCs for resistive crossbars. The findings advance reliable analog computing by enabling precise error localization with compact redundancy and efficient decoding.

Abstract

Error-correcting codes over the real field are studied which can locate outlying computational errors when performing approximate computing of real vector--matrix multiplication on resistive crossbars. Prior work has concentrated on locating a single outlying error and, in this work, several classes of codes are presented which can handle multiple errors. It is first shown that one of the known constructions, which is based on spherical codes, can in fact handle multiple outlying errors. A second family of codes is then presented with $\zeroone$~parity-check matrices which are sparse and disjunct; such matrices have been used in other applications as well, especially in combinatorial group testing. In addition, a certain class of the codes that are obtained through this construction is shown to be efficiently decodable. As part of the study of sparse disjunct matrices, this work also contains improved lower and upper bounds on the maximum Hamming weight of the rows in such matrices.