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.

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