Learning matrix functions over rings

Abstract

Let R be a commutative Artinian ring with identity and let X be a finite subset of R. We present an exact learning algorithm with a polynomial query complexity for the class of functions representable as

$$f(x) = \prod\limits_{i = 1}^n {A_i (x_i )}$$

where for each 1≤i≤n, A _i is a matrix-valued mapping A _i: X→ R ^mixmi+1 and m ₁=m _n+1=1. These functions are referred to as matrix functions. Our algorithm uses a decision tree based hypothesis class called decision programs that takes advantage of linear dependencies. We also show that the class of matrix functions is equivalent to the class of decision programs.

Our learning algorithm implies the following results.

• 1. Multivariate polynomials over a finite commutative ring with identity are learnable using equivalence and substitution queries.

• 2. Bounded degree multivariate polynomials over Z _n can be interpolated using substitution queries.

This paper generalizes the learning algorithm for automata over fields given in [4].

Work done partly while the author was a student at the Dept. Computer Science, University of Calgary.