Greatest common divisors of polynomials given by straight-line programs
Journal of the ACM (JACM)
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Interpolating polynomials from their values
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Algorithms for computing sparsest shifts of polynomials in power, Chebyshev, and Pochhammer bases
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Algorithms for the non-monic case of the sparse modular GCD algorithm
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
A displacement approach to decoding algebraic codes
Contemporary mathematics
On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms
Proceedings of the 2007 international workshop on Symbolic-numeric computation
On exact and approximate interpolation of sparse rational functions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Expressing a fraction of two determinants as a determinant
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Symbolic-numeric sparse interpolation of multivariate polynomials
Journal of Symbolic Computation
A universal Reed-Solomon decoder
IBM Journal of Research and Development
Sparse interpolation of multivariate rational functions
Theoretical Computer Science
Supersparse black box rational function interpolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Noisy Interpolation of Sparse Polynomials, and Applications
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Decoding by linear programming
IEEE Transactions on Information Theory
Fast estimates of Hankel matrix condition numbers and numeric sparse interpolation
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Error-correcting decoding is generalized to multivariate sparse rational function recovery from evaluations that can be numerically inaccurate and where several evaluations can have severe errors ("outliers"). The generalization of the Berlekamp-Welch decoder to exact Cauchy interpolation of univariate rational functions from values with faults is by Kaltofen and Pernet in 2012. We give a different univariate solution based on structured linear algebra that yields a stable decoder with floating point arithmetic. Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers ("cleans up data") through techniques from error correcting codes. Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.