Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations

Chowdhury, Muhammad F. I.; Jeannerod, Claude-Pierre; Neiger, Vincent; Schost, Éric; Villard, Gilles

doi:10.1109/tit.2015.2416068

Cited by 29 publications

(43 citation statements)

References 48 publications

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“…(i) L r δ (A) is left-unimodularly equivalent to [ A 0 B I ] for some B ∈ K[X] ( m−m)×m[16, Theorem 10 (i)]. Then, let R be the remainder of B modulo P, that is, the unique matrix in K[X] ( m−m)×m which has column degree bounded by the column degree of P componentwise and such that R = B + QP for some matrix Q (see for example[22, Theorem 6 [3][4][5][6][7][8][9][10][11][12][13][14][15],. noting that P is 0-column reduced).Let W denote the unimodular matrix such that P = WA.Then, [ W 0 QW I ][ A 0 B I ] = [ P 0 R I ] is left-unimodularly equivalent to L r δ (A).…”

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confidence: 99%

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

Neiger

2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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We give a Las Vegas algorithm which computes the shifted Popov form of an m × m nonsingular polynomial matrix of degree d in expected O(m ω d) field operations, where ω is the exponent of matrix multiplication and O(·) indicates that logarithmic factors are omitted. This is the first algorithm in O(m ω d) for shifted row reduction with arbitrary shifts. Using partial linearization, we reduce the problem to the case d ⌈σ/m⌉ where σ is the generic determinant bound, with σ/m bounded from above by both the average row degree and the average column degree of the matrix. The cost above becomes O(m ω ⌈σ/m⌉), improving upon the cost of the fastest previously known algorithm for row reduction, which is deterministic.Our algorithm first builds a system of modular equations whose solution set is the row space of the input matrix, and then finds the basis in shifted Popov form of this set. We give a deterministic algorithm for this second step supporting arbitrary moduli in O(m ω−1 σ) field operations, where m is the number of unknowns and σ is the sum of the degrees of the moduli. This extends previous results with the same cost bound in the specific cases of order basis computation and M-Padé approximation, in which the moduli are products of known linear factors.

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confidence: 99%

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

Neiger

2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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“…Guruswami [16] (Section 4.4.1) showed that any [n, r] q -RS code is also an n r − 1, O n 4 r 2 -listrecoverable code. To efficiently decode RS code, Chowdhury et al [17] proposed an efficient scheme, which they summarized in Table 1 of their paper with ω < 2.38, as follows:…”

Section: Reed-solomon Codesmentioning

confidence: 99%

Efficient (nonrandom) Construction and Decoding for Non-adaptive Group Testing

Bui

Kuribayashi²,

Kojima

et al. 2019

Journal of Information Processing

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The task of non-adaptive group testing is to identify up to d defective items from N items, where a test is positive if it contains at least one defective item, and negative otherwise. If there are t tests, they can be represented as a t × N measurement matrix. We have answered the question of whether there exists a scheme such that a larger measurement matrix, built from a given t × N measurement matrix, can be used to identify up to d defective items in time O(t log 2 N ). In the meantime, a t × N nonrandom measurement matrix with t = O arXiv:1804.03819v4 [cs.IT]

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“…. . , x σ are pairwise distinct, the best known cost bound for computing a minimal basis is O˜(m ω σ) (Bernstein, 2011;Cohn and Heninger, 2015;Nielsen, 2014); the cost bound O˜(m ω−1 σ) was achieved in (Chowdhury et al, 2015) with a probabilistic algorithm which outputs only one interpolant satisfying the degree constraints.…”

Section: Introductionmentioning

confidence: 99%

Computing minimal interpolation bases

Jeannerod¹,

Neiger²,

Schost³

et al. 2017

Journal of Symbolic Computation

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International audienceWe consider the problem of computing univariate polynomial matrices over afield that represent minimal solution bases for a general interpolationproblem, some forms of which are the vector M-Pad\'e approximation problem in[Van Barel and Bultheel, Numerical Algorithms 3, 1992] and the rationalinterpolation problem in [Beckermann and Labahn, SIAM J. Matrix Anal. Appl. 22,2000]. Particular instances of this problem include the bivariate interpolationsteps of Guruswami-Sudan hard-decision and K\"otter-Vardy soft-decisiondecodings of Reed-Solomon codes, the multivariate interpolation step oflist-decoding of folded Reed-Solomon codes, and Hermite-Pad\'e approximation. In the mentioned references, the problem is solved using iterative algorithmsbased on recurrence relations. Here, we discuss a fast, divide-and-conquerversion of this recurrence, taking advantage of fast matrix computations overthe scalars and over the polynomials. This new algorithm is deterministic, andfor 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 theentries of the input shift $s$, with $\min(s) = 0$. This complexity boundimproves in particular on earlier algorithms in the case of bivariateinterpolation for soft decoding, while matching fastest existing algorithms forsimultaneous Hermite-Pad\'e approximation

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Faster Algorithms for Multivariate Interpolation With Multiplicities and Simultaneous Polynomial Approximations

Cited by 29 publications

References 48 publications

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

Fast Computation of Shifted Popov Forms of Polynomial Matrices via Systems of Modular Polynomial Equations

Efficient (nonrandom) Construction and Decoding for Non-adaptive Group Testing

Computing minimal interpolation bases

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