Reconstructing algebraic functions from mixed data

The authors consider the task of reconstructing algebraic functions given by black boxes. Unlike traditional settings, they are interested in black boxes which represent several algebraic functions-f/sub 1/, . . ., f/sub k/, where at each input x, the box arbitarrily chooses a subset of f/sub 1/(x), . . ., f/sub k/(x) to output. They show how to reconstruct the functions f/sub 1/,. . ., f/sub k/ from the black box. This allows them to group the same points into sets, such that for each set, all outputs to points in the set are from the same algebraic function. The methods are robust in the presence of errors in the black box. The 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.<<ETX>>

[1]  Marek Karpinski,et al.  Algorithms for sparse rational interpolation , 1991, ISSAC '91.

[2]  Ronald L. Rivest,et al.  Learning decision lists , 2004, Machine Learning.

[3]  Ronitt Rubinfeld,et al.  Self-testing/correcting for polynomials and for approximate functions , 1991, STOC '91.

[4]  D. Grigor'ev,et al.  Factorization of polynomials over a finite field and the solution of systems of algebraic equations , 1986 .

[5]  D. Garling,et al.  Algebra, Volume 1 , 1969, Mathematical Gazette.

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

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

[8]  Erich Kaltofen.  A Polynomial-Time Reduction from Bivariate to Univariate Integral Polynomial Factorization , 1982, FOCS.

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

[10]  R. Tennant.  Algebra , 1941, Nature.

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

[12]  Richard Zippel,et al.  Interpolating Polynomials from Their Values , 1990, J. Symb. Comput..

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

[14]  Michael O. Rabin,et al.  Probabilistic Algorithms in Finite Fields , 1980, SIAM J. Comput..

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

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

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

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

[19]  H. Niederreiter,et al.  Introduction to finite fields and their applications: Factorization of Polynomials , 1994 .

[20]  Joan Feigenbaum,et al.  Hiding Instances in Multioracle Queries , 1990, STACS.

[21]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1971, SYMSAC '71.

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

[23]  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.

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

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

[26]  J. Canny.  Finding Edges and Lines in Images , 1983 .

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

[28]  Leonid A. Levin,et al.  A hard-core predicate for all one-way functions , 1989, STOC '89.

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

[30]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[31]  Erich Kaltofen,et al.  A polynomial-time reduction from bivariate to univariate integral polynomial factorization , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[32]  Erich Kaltofen,et al.  Computing with Polynomials Given By Black Boxes for Their Evaluations: Greatest Common Divisors, Factorization, Separation of Numerators and Denominators , 1990, J. Symb. Comput..

[33]  C. Hoffmann.  Algebraic curves , 1988 .

[34]  David Haussler,et al.  Decision Theoretic Generalizations of the PAC Model for Neural Net and Other Learning Applications , 1992, Inf. Comput..