### Reconstructing Algebraic Functions from Mixed Data

We consider a variant of the traditional task of explicitly reconstructing algebraic functions from black box representations. In the traditional setting for such problems, one is given access to an unknown function f that is represented by a black box, or an oracle, which can be queried for the value of f at any input. Given a guarantee that this unknown function f is some nice algebraic function, say a polynomial in its input of degree bound d, the goal of the reconstruction problem is to explicitly determine the coefficients of the unknown polynomial. All work on polynomial interpolation, especially sparse ones, are or may be presented in such a setting. The work of Kaltofen and Trager [Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators, in Proc. 29th Ann. IEEE Symp. on Foundations of Computer Science, 1988, pp. 296--305], for instance, highlights the utility of this setting, by performing numerous manipulations on polynomials presented as black boxes. The variant considered in this paper differs from the traditional setting in that our black boxes represent several algebraic functions f1,...,fk, where at each input x, the box arbitrarily chooses a subset of f1(x),...,fk(x) to output and we do not know which subset it outputs. We show how to reconstruct the functions f1,...,fk from the black box, provided the black box outputs according to these functions "often." This allows us to group the sample points into sets, such that for each set, all outputs to points in the set are from the same algebraic function. Our methods are robust in the presence of a small fraction of arbitrary errors in the black box. 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]  David Haussler,et al.  Decision Theoretic Generalizations of the PAC Model for Neural Net and Other Learning Applications , 1992, Information and Computation.

[2]  Richard Zippel Interpolating Polynomials from Their Values , 1990, J. Symb. Comput..

[4]  Avrim Blum,et al.  Learning switching concepts , 1992, COLT '92.

[5]  J. Gathen Algebraic complexity theory , 1988 .

[6]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[7]  Madhu Sudan,et al.  Highly Resilient Correctors for Polynomials , 1992, Inf. Process. Lett..

[8]  Marek Karpinski,et al.  Interpolation of sparse rational functions without knowing bounds on exponents , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[9]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[10]  R. J. Walker Algebraic curves , 1950 .

[11]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, STOC '84.

[12]  Ronitt Rubinfeld,et al.  A new modular interpolation algorithm for factoring multivariate polynominals , 1994, ANTS.

[13]  Rajeev Motwani,et al.  Randomized algorithms , 1996, CSUR.

[14]  Jin-Yi Cai,et al.  A Note on Enumarative Counting , 1991, Inf. Process. Lett..

[15]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[16]  Madhu Sudan Decoding of Reed Solomon Codes beyond the Error-Correction Bound , 1997, J. Complex..

[17]  Michael Ben-Or,et al.  Probabilistic algorithms in finite fields , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[18]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[19]  E. Berlekamp Factoring polynomials over large finite fields , 1970 .

[20]  Ronitt Rubinfeld,et al.  Learning polynomials with queries: The highly noisy case , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[21]  Erich Kaltofen,et al.  Computing with polynomials given by black boxes for their evaluations: greatest common divisors, factorization, separation of numerators and denominators , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[23]  Richard J. Lipton,et al.  New Directions In Testing , 1989, Distributed Computing And Cryptography.

[24]  David Haussler,et al.  Occam's Razor , 1987, Inf. Process. Lett..

[25]  Elwyn R. Berlekamp Bounded distance+1 soft-decision Reed-Solomon decoding , 1996, IEEE Trans. Inf. Theory.

[26]  Marek Karpinski,et al.  Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields , 1988, SIAM J. Comput..

[27]  Michael Ben-Or,et al.  A deterministic algorithm for sparse multivariate polynomial interpolation , 1988, STOC '88.

[28]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[29]  Eyal Kushilevitz,et al.  Learning decision trees using the Fourier spectrum , 1991, STOC '91.