Optimal Spectral-Norm Approximate Minimization of Weighted Finite Automata

TL;DR

This paper introduces a novel method for optimally approximating weighted finite automata with bounded states using spectral norm minimization, grounded in advanced Hankel operator theory, with proven guarantees and an efficient algorithm.

Contribution

It adapts Adamyan-Arov-Krein approximation theory to weighted automata, providing the first optimal spectral-norm approximation algorithm with theoretical bounds.

Findings

Provides theoretical guarantees for approximation quality.

Develops an algorithm for optimal spectral-norm approximation.

Demonstrates effectiveness through mathematical bounds.

Abstract

We address the approximate minimization problem for weighted finite automata (WFAs) with weights in $\mathbb{R}$, over a one-letter alphabet: to compute the best possible approximation of a WFA given a bound on the number of states. This work is grounded in Adamyan-Arov-Krein Approximation theory, a remarkable collection of results on the approximation of Hankel operators. In addition to its intrinsic mathematical relevance, this theory has proven to be very effective for model reduction. We adapt these results to the framework of weighted automata over a one-letter alphabet. We provide theoretical guarantees and bounds on the quality of the approximation in the spectral and $\ell^2$ norm. We develop an algorithm that, based on the properties of Hankel operators, returns the optimal approximation in the spectral norm.