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Noisy Interpolation of Sparse Polynomials, and Applications

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. ...

##
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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

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. ...

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

2016
*
arXiv
*
pre-print

We analyze the Guruswami--Sudan list decoding algorithm for Reed--Solomon codes over the complex

arXiv:1611.07811v1
fatcat:zqmnlrqzebafnobbnfdwhqwkfy
*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. ...##
###
Sparse polynomial interpolation and Berlekamp/Massey algorithms that correct outlier errors in input values

2012
*
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12
*

The algorithms are applied to

doi:10.1145/2442829.2442852
dblp:conf/issac/ComerKP12
fatcat:dnxp6kcrsjd3lljcgekaklcvf4
*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. ...##
###
What Can (and Can't) we Do with Sparse Polynomials?

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? ...

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

2012
*
Mathematical programming
*

We suggest an approach for building

doi:10.1007/s10107-012-0578-z
fatcat:qgmkvkpmcnb63j47ke5vms6kfm
*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. ...##
###
A matrix-free approach to geostatistical filtering
[article]

2020
*
arXiv
*
pre-print

The approach is based on a

arXiv:2004.02799v1
fatcat:5lzyhej2x5ae5n5ordoo4fq2na
*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. ...##
###
On exact and approximate interpolation of sparse rational functions

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

doi:10.1145/1277548.1277577
dblp:conf/issac/KaltofenY07
fatcat:vumng65q5zdyvof4qg3z3gu3xu
*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 ...##
###
Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization
[article]

2013
*
arXiv
*
pre-print

We suggest an approach for building

arXiv:1306.5729v1
fatcat:mparcuu7ubgclfejfhsji2w6z4
*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. ...##
###
Reconstructing Algebraic Functions from Mixed Data

1998
*
SIAM journal on computing (Print)
*

Our model and techniques can be applied

doi:10.1137/s0097539796297577
fatcat:m7uy5bemdjcaxmsvu5rsafvzte
*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. ...##
###
Reconstructing algebraic functions from mixed data

1992
*
Proceedings., 33rd Annual Symposium on Foundations of Computer Science
*

Our model and techniques can be applied

doi:10.1109/sfcs.1992.267801
dblp:conf/focs/ArLRS92
fatcat:nffsykz3ozh4lllxamsp2lbixm
*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. ...##
###
Error correction in fast matrix multiplication and inverse
[article]

2018
*
arXiv
*
pre-print

We present new algorithms to detect and correct errors

arXiv:1802.02270v1
fatcat:3mzf5dobhbgutb5quaprzwe3wy
*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. ...##
###
Noisy polynomial interpolation modulo prime powers
[article]

2020
*
arXiv
*
pre-print

We consider the

arXiv:2006.05685v2
fatcat:3rbxgiajifhivf2lkm4fiyvjlu
*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. ...##
###
Sparse multivariate function recovery from values with noise and outlier errors

2013
*
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13
*

Our multivariate

doi:10.1145/2465506.2465524
dblp:conf/issac/KaltofenY13
fatcat:ccdgfxh7cjckvj5lneljnwblue
*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. ...
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