Computing minimal interpolation bases

TL;DR

This paper introduces a fast, deterministic divide-and-conquer algorithm for computing minimal interpolation bases of polynomial matrices, improving efficiency over previous iterative methods especially in decoding applications.

Contribution

It presents a novel divide-and-conquer algorithm leveraging fast matrix computations, reducing complexity for minimal basis computation in polynomial interpolation problems.

Findings

Achieves $O(m^{ ext{w}-1} (\sigma + |s|))$ complexity for shifted minimal bases

Improves efficiency for bivariate interpolation in soft decoding

Matches the fastest existing algorithms for Hermite-Padé approximation

Abstract

We consider the problem of computing univariate polynomial matrices over a field that represent minimal solution bases for a general interpolation problem, some forms of which are the vector M-Pad\'e approximation problem in [Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rational interpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22, 2000]. Particular instances of this problem include the bivariate interpolation steps of Guruswami-Sudan hard-decision and K\"otter-Vardy soft-decision decodings of Reed-Solomon codes, the multivariate interpolation step of list-decoding of folded Reed-Solomon codes, and Hermite-Pad\'e approximation. In the mentioned references, the problem is solved using iterative algorithms based on recurrence relations. Here, we discuss a fast, divide-and-conquer version of this recurrence, taking advantage of fast matrix computations over the scalars and over the polynomials. This new algorithm is deterministic, and for computing shifted minimal bases of relations between $m$ vectors of size $\sigma$ it uses $O~( m^{\omega-1} (\sigma + |s|) )$ field operations, where $\omega$ is the exponent of matrix multiplication, and $|s|$ is the sum of the entries of the input shift $s$, with $\min(s) = 0$. This complexity bound improves in particular on earlier algorithms in the case of bivariate interpolation for soft decoding, while matching fastest existing algorithms for simultaneous Hermite-Pad\'e approximation.