# Numerically Stable Polynomially Coded Computing

## TL;DR

This paper introduces orthogonal polynomial-based codes for matrix multiplication that enhance numerical stability and fault tolerance, with theoretical bounds and empirical validation showing reduced errors compared to traditional Vandermonde-based methods.

## Contribution

It develops new orthogonal polynomial codes, especially using Chebyshev polynomials, with proven bounds on condition numbers, leading to more numerically stable coded computing techniques.

## Key findings

- Orthogonal polynomial codes achieve similar fault tolerance as previous codes.
- Chebyshev-Vandermonde matrices have polynomially bounded condition numbers.
- Empirical results show significantly lower numerical errors with the new methods.

## Abstract

We study the numerical stability of polynomial based encoding methods, which has emerged to be a powerful class of techniques for providing straggler and fault tolerance in the area of coded computing. Our contributions are as follows: 1) We construct new codes for matrix multiplication that achieve the same fault/straggler tolerance as the previously constructed MatDot Codes and Polynomial Codes. Unlike previous codes that use polynomials expanded in a monomial basis, our codes uses a basis of orthogonal polynomials. 2) We show that the condition number of every $m \times m$ sub-matrix of an $m \times n, n \geq m$ Chebyshev-Vandermonde matrix, evaluated on the $n$-point Chebyshev grid, grows as $O(n^{2(n-m)})$ for $n > m$. An implication of this result is that, when Chebyshev-Vandermonde matrices are used for coded computing, for a fixed number of redundant nodes $s=n-m,$ the condition number grows at most polynomially in the number of nodes $n$. 3) By specializing our orthogonal polynomial based constructions to Chebyshev polynomials, and using our condition number bound for Chebyshev-Vandermonde matrices, we construct new numerically stable techniques for coded matrix multiplication. We empirically demonstrate that our constructions have significantly lower numerical errors compared to previous approaches which involve inversion of Vandermonde matrices. We generalize our constructions to explore the trade-off between computation/communication and fault-tolerance. 4) We propose a numerically stable specialization of Lagrange coded computing. Motivated by our condition number bound, our approach involves the choice of evaluation points and a suitable decoding procedure that involves inversion of an appropriate Chebyshev-Vandermonde matrix. Our approach is demonstrated empirically to have lower numerical errors as compared to standard methods.