An introduction to computational learning theory
An introduction to computational learning theory
Representing Boolean functions as polynomials modulo composite numbers
Computational Complexity - Special issue on circuit complexity
Detecting perfect powers in essentially linear time
Mathematics of Computation
A lower bound on the MOD 6 degree of the or function
Computational Complexity
Zero testing of p-adic and modular polynomials
Theoretical Computer Science
Learning Polynomials with Queries: The Highly Noisy Case
SIAM Journal on Discrete Mathematics
Set systems with restricted intersections modulo prime powers
Journal of Combinatorial Theory Series A
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Learning Matrix Functions over Rings
EuroCOLT '97 Proceedings of the Third European Conference on Computational Learning Theory
On-line Algorithms in Machine Learning
Developments from a June 1996 seminar on Online algorithms: the state of the art
Primality and Identity Testing via Chinese Remaindering
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Lower Bounds for Approximations by Low Degree Polynomials Over Z_m
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Symmetric Polynomials over \mathbb{Z}_m and Simultaneous Communication Protocols
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Factoring into coprimes in essentially linear time
Journal of Algorithms
| Hi-index | 0.00 |
The problem of polynomial interpolation is to reconstruct a polynomial based on its valuations on a set of inputs I. We consider the problem over Zm when m is composite. We ask the question: Given I ⊆ Zm, how many evaluations of a polynomial at points in I are required to compute its value at every point in I? Surprisingly for composite m, this number can vary exponentially between log[I] and [I] in contrast to the prime case where [I] evaluations are necessary. While this minimization problem is NP-complete, we give an efficient algorithm of query complexity within a factor t of the optimum where t is the number of prime factors of m. We use our interpolation algorithm to design algorithms for zero-testing and distributional learning of polynomials over Zm. In some cases, we get an exponential improvement over known algorithms in query complexity and running time. Our main technical contribution is the notion of an interpolating set for I which is a subset S of I such that a polynomial which is 0 over S must be 0 at every point in I. Any interpolation algorithm needs to query an interpolating set for I. Our query-efficient algorithms are obtained by constructing interpolating sets whose size is close to optimal.