Faster Sparse Interpolation of Straight-Line Programs

Arnold, Andrew; Giesbrecht, Mark; Roche, Daniel S.

doi:10.1007/978-3-319-02297-0_5

Cited by 19 publications

(53 citation statements)

References 12 publications

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“…Type Dense LD n Deterministic Garg & Schost [11] Ln 2 T 4 log 2 D Deterministic Randomized G & S [12] Ln 2 T 3 log 2 D Las Vegas Arnold, Giesbrecht & Roche [4] Ln 3 T log 3 D Monte Carlo This paper (Thm 5.9) Ln 2 T 2 log 2 D + LnT log 3 D Deterministic This paper (Thm 6.8)…”

Section: Algorithms Total Costmentioning

confidence: 99%

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Faster interpolation algorithms for sparse multivariate polynomials given by straight-line programs

Huang

Gao

2020

Journal of Symbolic Computation

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In this paper, we propose new deterministic and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let f be an n-variate polynomial given by a straight-line program, which has a degree bound D and a term bound T . Our deterministic algorithm is quadratic in n, T and cubic in log D in the Soft-Oh sense, which has better complexities than existing deterministic interpolation algorithms in most cases. Our Monte Carlo interpolation algorithms have better complexities than existing Monte Carlo interpolation algorithms and are the first algorithms whose complexities are linear in nT in the Soft-Oh sense. Since nT is a factor of the size of f , our Monte Carlo algorithms are optimal in n and T in the Soft-Oh sense.

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Section: Algorithms Total Costmentioning

confidence: 99%

“…Noting that a dense polynomial with degree d has d d + n terms, our algorithm works better in most cases. For probabilistic algorithms, our method is the only one whose complexity is linear in nT and is better than that of the Monte Carlo method given in [4].…”

Section: Algorithms Total Costmentioning

confidence: 99%

Faster interpolation algorithms for sparse multivariate polynomials given by straight-line programs

Huang

Gao

2020

Journal of Symbolic Computation

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“…A Las Vegas randomized algorithm that uses fewer black box evaluations was presented in [13] for this remainder black box. Arnold, Giesbrecht and Roche [2] devised a Monte Carlo algorithm for interpolating polynomials represented by straight-line programs. The algorithm in [2] is more efficient than the ones presented in [10] and [13] .…”

Section: Introductionmentioning

confidence: 99%

“…Arnold, Giesbrecht and Roche [2] devised a Monte Carlo algorithm for interpolating polynomials represented by straight-line programs. The algorithm in [2] is more efficient than the ones presented in [10] and [13] . Arnold, Giesbrecht and Roche [3] devised a randomized algorithm for interpolating sparse polynomials represented by straight-line programs over finite fields, which is faster than previous known algorithms.…”

Section: Introductionmentioning

confidence: 99%

A new deterministic algorithm for sparse multivariate polynomial interpolation

Bläser

Jindal

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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We present a deterministic algorithm to interpolate an msparse n-variate polynomial which uses poly(n, m, log H, log d) bit operations. Our algorithm works over the integers. Here H is a bound on the magnitude of the coefficient values of the given polynomial. The degree of given polynomial is bounded by d and m is upper bound on number of monomials. This running time is polynomial in the output size. Our algorithm only requires modular black box access to the given polynomial, as introduced in [12]. As an easy consequence, we obtain an algorithm to interpolate polynomials represented by arithmetic circuits.

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“…Algorithms for sparse multivariate rational functions are in [7,18,6]. Finally, large degrees are dealt with in [8,11,14,1]. Univariate algorithms for imprecise inputs go back to French revolution times [21], but came to life with early termination [9].…”

Section: Introductionmentioning

confidence: 99%

Sparse multivariate function recovery with a high error rate in the evaluations

Kaltofen

Yang

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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In [Kaltofen and Yang, Proc. ISSAC 2013] we have generalized algebraic error-correcting decoding to multivariate sparse rational function interpolation from evaluations that can be numerically inaccurate and where several evaluations can have severe errors ("outliers"). Here we present a different algorithm that can interpolate a sparse multivariate rational function from evaluations where the error rate is 1/q for any q > 2, which our ISSAC 2013 algorithm could not handle. When implemented as a numerical algorithm we can, for instance, reconstruct a fraction of trinomials of degree 15 in 50 variables with non-outlier evaluations of relative noise as large as 10 −7 and where as much as 1/4 of the 14717 evaluations are outliers with relative error as small as 0.01 (large outliers are easily located by our method).For the algorithm with exact arithmetic and exact values at non-erroneous points, we provide a proof that for random evaluations one can avoid quadratic oversampling. Our argument already applies to our original 2007 sparse rational function interpolation algorithm [Kaltofen, Yang and Zhi, Proc. SNC 2007], where we have experimentally observed that for T unknown non-zero coefficients in a sparse candidate ansatz one only needs T + O(1) evaluations rather than the proven O(T 2 ) (cf. Candès and Tao sparse sensing). Here we finally can give the probabilistic analysis for this fact.

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Faster Sparse Interpolation of Straight-Line Programs

Cited by 19 publications

References 12 publications

Faster interpolation algorithms for sparse multivariate polynomials given by straight-line programs

Faster interpolation algorithms for sparse multivariate polynomials given by straight-line programs

A new deterministic algorithm for sparse multivariate polynomial interpolation

Sparse multivariate function recovery with a high error rate in the evaluations

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