Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations

• Published in: IEEE transactions on information theory Vol. 61; no. 5; pp. 2370 - 2387
• Main Authors: Chowdhury, Muhammad F. I., Jeannerod, Claude-Pierre, Neiger, Vincent, Schost, Eric, Villard, Gilles
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-05-2015; The Institute of Electrical and Electronics Engineers, Inc. (IEEE); Institute of Electrical and Electronics Engineers
• Subjects: Algorithms; Computer Science; Context; Information Theory; Interpolation; Lattice theory; Linear algebra; Linear systems; list decoding; Mathematical problems; Mathematics; multivariate polynomial interpolation; Polynomials; Reed-Solomon codes; structured matrix; Symbolic Computation; Reed-Solomon codes; structured matrix; Multivariate polynomial interpolation; polynomial approximation; list decoding; Multivariate interpolation; List-decoding; Structured matrices; Fast algorithms
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2015.2416068

The interpolation step in the Guruswami-Sudan algorithm is a bivariate interpolation problem with multiplicities commonly solved in the literature using either structured linear algebra or basis reduction of polynomial lattices. This problem has been extended to three or more variables; for this generalization, all fast algorithms proposed so far rely on the lattice approach. In this paper, we reduce this multivariate interpolation problem to a problem of simultaneous polynomial approximations, which we solve using fast structured linear algebra. This improves the best known complexity bounds for the interpolation step of the list-decoding of Reed-Solomon codes, Parvaresh-Vardy codes, and folded Reed-Solomon codes. In particular, for Reed-Solomon list-decoding with re-encoding, our approach has complexity O ~ (ℓ ω-1 m 2 (n - k)), where ℓ, m, n, and k are the list size, the multiplicity, the number of sample points, and the dimension of the code, and ω is the exponent of linear algebra; this accelerates the previously fastest known algorithm by a factor of ℓ/m.