theory of computing and complexity

This paper takes a computational learning theory approach to a problem of linear systems identification. It is assumed that input signals have only a finite number k of frequency components, and systems to be identified have dimension no greater than n. The main result establishes that the sample complexity needed for identification scales polynomially with n and logarithmically with k.

Recent work by H.T. Siegelmann and E.D. Sontag (1992) has demonstrated that polynomial time on linear saturated recurrent neural networks equals polynomial time on standard computational models: Turing machines if the weights of the net are rationals, and nonuniform circuits if the weights are real. Here, further connections between the languages recognized by such neural nets and other complexity classes are developed. Connections to space-bounded classes, simulation of parallel computational models such as Vector Machines, and a discussion of the characterizations of various nonuniform classes in terms of Kolmogorov complexity are presented.

This paper deals with analog circuits. It establishes the finiteness of VC dimension, teaching dimension, and several other measures of sample complexity which arise in learning theory. It also shows that the equivalence of behaviors, and the loading problem, are effectively decidable, modulo a widely believed conjecture in number theory. The results, the first ones that are independent of weight size, apply when the gate function is the "standard sigmoid" commonly used in neural networks research. The proofs rely on very recent developments in the elementary theory of real numbers with exponentiation. (Some weaker conclusions are also given for more general analytic gate functions.) Applications to learnability of sparse polynomials are also mentioned.

Some results are given in the theory of rational power series over a broad class of semirings. In particular, it is shown that for unambiguous sets the notion of rationality is independent of the semiring over which representations are defined. The undecidability of the rationality of probabilistic word functions is also established.