# Random Khatri-Rao-Product Codes for Numerically-Stable Distributed Matrix Multiplication

## TL;DR

This paper introduces random Khatri-Rao-Product (RKRP) codes for distributed matrix multiplication, offering improved numerical stability and lower decoding complexity compared to existing codes, especially in the presence of stragglers.

## Contribution

The paper proposes RKRP codes that are maximally distance separable with high probability and demonstrate superior numerical stability and decoding efficiency over prior codes.

## Key findings

- RKRP codes are maximum distance separable with probability 1.
- RKRP codes have lower average decoding complexity than OrthoPoly codes.
- Numerical results show RKRP codes have substantially better reconstruction error.

## Abstract

We propose a class of codes called random Khatri-Rao-Product (RKRP) codes for distributed matrix multiplication in the presence of stragglers. The main advantage of the proposed codes is that decoding of RKRP codes is highly numerically stable in comparison to decoding of Polynomial codes and decoding of the recently proposed OrthoPoly codes. We show that RKRP codes are maximum distance separable with probability 1. The communication cost and encoding complexity for RKRP codes are identical to that of OrthoPoly codes and Polynomial codes and the average decoding complexity of RKRP codes is lower than that of OrthoPoly codes. Numerical results show that the average relative $L_2$-norm of the reconstruction error for RKRP codes is substantially better than that of OrthoPoly codes.