Towards an AAK Theory Approach to Approximate Minimization in the Multi-Letter Case

TL;DR

This paper explores extending AAK theory-based approximate minimization of weighted finite automata from one-letter to multi-letter alphabets, reformulating the problem using noncommutative operator theory.

Contribution

The paper advances the theoretical framework by reformulating the approximation problem in terms of noncommutative Hankel operators, addressing the multi-letter case.

Findings

Reformulated the approximation problem using noncommutative Hankel operators.

Successfully addressed the first step towards multi-letter AAK theory extension.

The second step for a constructive solution remains open.

Abstract

We study the approximate minimization problem of weighted finite automata (WFAs): given a WFA, we want to compute its optimal approximation when restricted to a given size. We reformulate the problem as a rank-minimization task in the spectral norm, and propose a framework to apply Adamyan-Arov-Krein (AAK) theory to the approximation problem. This approach has already been successfully applied to the case of WFAs and language modelling black boxes over one-letter alphabets \citep{AAK-WFA,AAK-RNN}. Extending the result to multi-letter alphabets requires solving the following two steps. First, we need to reformulate the approximation problem in terms of noncommutative Hankel operators and noncommutative functions, in order to apply results from multivariable operator theory. Secondly, to obtain the optimal approximation we need a version of noncommutative AAK theory that is constructive. In this paper, we successfully tackle the first step, while the second challenge remains open.