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Noisy Interpolation of Sparse Polynomials, and Applications

Shubhangi Saraf, Sergey Yekhanin
2011 2011 IEEE 26th Annual Conference on Computational Complexity  
In this paper we establish a similar result for sparse polynomials.  ...  We show that a k-sparse polynomial f ∈ F q [x] of degree d ≤ q/2 can be recovered from its values at O(k) randomly chosen points, even if a small fraction of the values of f are adversarially corrupted  ...  . • Our results assert the existence of O(k)-sized noisy interpolating sets for k-sparse polynomials over finite fields. It would be very interesting to find such sets explicitly.  ... 
doi:10.1109/ccc.2011.38 dblp:conf/coco/SarafY11 fatcat:ljv6wrwe5fe6jl2jsuvshjgnum

Page 2736 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
The class of finite fields is constructed with irreducible AOPs (all one polynomials) and ESPs (equally spaced polynomials).  ...  Summary: “New implementations of bit-parallel multipliers for a class of finite fields are proposed.  ... 

Sparse interpolation of multivariate rational functions

Annie Cuyt, Wen-shin Lee
2011 Theoretical Computer Science  
In general, the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation.  ...  The latter is illustrated with several examples, running from exact finite field arithmetic to noisy floating point evaluations.  ...  Acknowledgements We thank Erich Kaltofen and Zhengfeng Yang for valuable remarks and providing their sparse rational interpolation codes and benchmarks.  ... 
doi:10.1016/j.tcs.2010.11.050 fatcat:pmnhbkkj3zethawvfmovzdnohq

Guruswami--Sudan List Decoding for Complex Reed--Solomon Codes [article]

Mostafa H. Mohamed and Sven Puchinger and Martin Bossert
2016 arXiv   pre-print
We analyze the Guruswami--Sudan list decoding algorithm for Reed--Solomon codes over the complex field for sparse recovery in Compressed Sensing.  ...  We propose methods of stabilizing both the interpolation and the root-finding steps against numerical instabilities, where the latter is the most sensitive.  ...  ACKNOWLEDGMENTS The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.  ... 
arXiv:1611.07811v1 fatcat:zqmnlrqzebafnobbnfdwhqwkfy

Sparse polynomial interpolation and Berlekamp/Massey algorithms that correct outlier errors in input values

Matthew T. Comer, Erich L. Kaltofen, Clément Pernet
2012 Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12  
The algorithms are applied to sparse interpolation algorithms with numeric noise, into which we now can bring outlier errors in the values.  ...  First, we present an algorithm that can recover a t-sparse polynomial f from a sequence of values, where some of the values are wrong, spoiled by either random or misleading errors.  ...  Let K be the finite field Fq, set n = q − 1 and let ξ be a primitive n-th root of unity in K.  ... 
doi:10.1145/2442829.2442852 dblp:conf/issac/ComerKP12 fatcat:dnxp6kcrsjd3lljcgekaklcvf4

What Can (and Can't) we Do with Sparse Polynomials?

Daniel S. Roche
2018 Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18  
In this tutorial we examine the state of the art for sparse polynomial algorithms in three areas: arithmetic, interpolation, and factorization.  ...  Simply put, a sparse polynomial is one whose zero coefficients are not explicitly stored.  ...  This work was supported in part by the National Science Foundation under grants 1319994 (https://www.nsf.gov/awardsearch/ showAward?  ... 
doi:10.1145/3208976.3209027 dblp:conf/issac/Roche18 fatcat:fygzzsxjwrdk7kfxw4z4g7zsjq

Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization

A. S. Bandeira, K. Scheinberg, L. N. Vicente
2012 Mathematical programming  
We suggest an approach for building sparse quadratic polynomial interpolation models by minimizing the 1 -norm of the entries of the model Hessian subject to the interpolation conditions.  ...  The sparse recovery theory developed recently in the field of compressed sensing characterizes conditions under which a sparse vector can be accurately recovered from few random measurements.  ...  with the testing environment of Section 5.3.  ... 
doi:10.1007/s10107-012-0578-z fatcat:qgmkvkpmcnb63j47ke5vms6kfm

A matrix-free approach to geostatistical filtering [article]

Mike Pereira, Nicolas Desassis, Cédric Magneron, Nathan Palmer
2020 arXiv   pre-print
The approach is based on a finite element approximation of Gaussian random fields expressed as an expansion of the eigenfunctions of a Laplace--Beltrami operator defined to account for local anisotropies  ...  The numerical approximation of the resulting random fields using a finite element approach is then leveraged to solve the scalability issue through a matrix-free approach.  ...  Then, the precision matrix in (19) , which corresponds to the precision matrix of Z at the triangulation nodes, is a matrix polynomial of a sparse matrix.  ... 
arXiv:2004.02799v1 fatcat:5lzyhej2x5ae5n5ordoo4fq2na

On exact and approximate interpolation of sparse rational functions

Erich Kaltofen, Zhengfeng Yang
2007 Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07  
The black box algorithm for separating the numerator from the denominator of a multivariate rational function can be combined with sparse multivariate polynomial interpolation algorithms to interpolate  ...  Finally, one can deploy the sparse rational function interpolation algorithm in the hybrid symbolic-numeric setting when the black box for the rational function returns real and complex values with noise  ...  Acknowledgement: We thank Wen-shin Lee for providing her numeric sparse interpolation code to us, Arne Storjohann for sending us [22] and discussions on rational vector recovery, and Lihong Zhi for discussions  ... 
doi:10.1145/1277548.1277577 dblp:conf/issac/KaltofenY07 fatcat:vumng65q5zdyvof4qg3z3gu3xu

Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization [article]

Afonso S. Bandeira and Katya Scheinberg and Luis Nunes Vicente
2013 arXiv   pre-print
We suggest an approach for building sparse quadratic polynomial interpolation models by minimizing the l1-norm of the entries of the model Hessian subject to the interpolation conditions.  ...  The sparse recovery theory developed recently in the field of compressed sensing characterizes conditions under which a sparse vector can be accurately recovered from few random measurements.  ...  us assistance with the testing environment of Section 5.3.  ... 
arXiv:1306.5729v1 fatcat:mparcuu7ubgclfejfhsji2w6z4

Reconstructing Algebraic Functions from Mixed Data

Sigal Ar, Richard J. Lipton, Ronitt Rubinfeld, Madhu Sudan
1998 SIAM journal on computing (Print)  
Our model and techniques can be applied in the areas of computer vision, machine learning, curve fitting and polynomial approximation, self-correcting programs and bivariate polynomial factorization. 1  ...  Our methods are robust in the presence of errors in the black box.  ...  We thank Ronen Basri, Oded Goldreich and Mike Kearns for their comments on the writeup of this paper.  ... 
doi:10.1137/s0097539796297577 fatcat:m7uy5bemdjcaxmsvu5rsafvzte

Reconstructing algebraic functions from mixed data

S. Ar, R.J. Lipton, R. Rubinfeld, M. Sudan
1992 Proceedings., 33rd Annual Symposium on Foundations of Computer Science  
Our model and techniques can be applied in the areas of computer vision, machine learning, curve fitting and polynomial approximation, self-correcting programs and bivariate polynomial factorization. 1  ...  Our methods are robust in the presence of errors in the black box.  ...  We thank Ronen Basri, Oded Goldreich and Mike Kearns for their comments on the writeup of this paper.  ... 
doi:10.1109/sfcs.1992.267801 dblp:conf/focs/ArLRS92 fatcat:nffsykz3ozh4lllxamsp2lbixm

Error correction in fast matrix multiplication and inverse [article]

Daniel S. Roche
2018 arXiv   pre-print
We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field.  ...  These algorithms build on the recent result of Gasieniec et al (2017) on correcting matrix products, as well as existing work on verification algorithms, sparse low-rank linear algebra, and sparse polynomial  ...  Batched low-degree sparse interpolation Our algorithms use techniques from sparse polynomial interpolation to find the locations and values of erroneous entries in a matrix product or inverse.  ... 
arXiv:1802.02270v1 fatcat:3mzf5dobhbgutb5quaprzwe3wy

Noisy polynomial interpolation modulo prime powers [article]

Marek Karpinski, Igor Shparlinski
2020 arXiv   pre-print
We consider the noisy polynomial interpolation problem of recovering an unknown s-sparse polynomial f(X) over the ring ℤ_p^k of residues modulo p^k, where p is a small prime and k is a large integer parameter  ...  We give a deterministic polynomial time algorithm, which for almost given more than a half bits of f(t) for sufficiently many randomly chosen points t ∈ℤ_p^k^*, recovers f(X).  ...  During the preparation of this work the first author was supported in part by the Deutsche Forschungsgemeinschaft and the second author by the Australian Research Council.  ... 
arXiv:2006.05685v2 fatcat:3rbxgiajifhivf2lkm4fiyvjlu

Sparse multivariate function recovery from values with noise and outlier errors

Erich L. Kaltofen, Zhengfeng Yang
2013 Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13  
Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D.  ...  Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.  ...  For the exact problem for multivariate polynomials, say with K a finite field, we also mention [22] , where the minimum number of points is studied for unique recovery.  ... 
doi:10.1145/2465506.2465524 dblp:conf/issac/KaltofenY13 fatcat:ccdgfxh7cjckvj5lneljnwblue
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