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Optimal Approximate Minimization of One-Letter Weighted Finite Automata

Clara Lacroce, Borja Balle, Prakash Panangaden, Guillaume Rabusseau

TL;DR

The paper addresses the problem of optimally approximating a weighted finite automaton (WFA) under a fixed state bound by reframing the task as a low-rank Hankel operator approximation and applying Adamyan-Arov-Krein (AAK) theory to achieve a spectral-norm optimum. It confines to irredundant WFAs with real weights over a one-letter alphabet, deriving a closed-form symbol, an algorithm, and a rigorous error analysis, all executable in polynomial time. The core contributions include a detailed linkage between WFA parameters and Hankel symbols, a constructive procedure to obtain the size-$k$ WFA that minimizes the spectral norm error with bound $\ig|\mathbf H - \mathbf G\big| = \sigma_k(\mathbf H)$, and an explicit example demonstrating the method. This framework supports principled model compression and learning, with extensions toward removing key assumptions and extending to broader alphabets and model classes in future work.

Abstract

In this paper, we study the approximate minimization problem of weighted finite automata (WFAs): to compute the best possible approximation of a WFA given a bound on the number of states. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. We solve the optimal spectral-norm approximate minimization problem for irredundant WFAs with real weights, defined over a one-letter alphabet. We present a theoretical analysis based on AAK theory, and bounds on the quality of the approximation in the spectral norm and $\ell^2$ norm. Moreover, we provide a closed-form solution, and an algorithm, to compute the optimal approximation of a given size in polynomial time.

Optimal Approximate Minimization of One-Letter Weighted Finite Automata

TL;DR

The paper addresses the problem of optimally approximating a weighted finite automaton (WFA) under a fixed state bound by reframing the task as a low-rank Hankel operator approximation and applying Adamyan-Arov-Krein (AAK) theory to achieve a spectral-norm optimum. It confines to irredundant WFAs with real weights over a one-letter alphabet, deriving a closed-form symbol, an algorithm, and a rigorous error analysis, all executable in polynomial time. The core contributions include a detailed linkage between WFA parameters and Hankel symbols, a constructive procedure to obtain the size- WFA that minimizes the spectral norm error with bound , and an explicit example demonstrating the method. This framework supports principled model compression and learning, with extensions toward removing key assumptions and extending to broader alphabets and model classes in future work.

Abstract

In this paper, we study the approximate minimization problem of weighted finite automata (WFAs): to compute the best possible approximation of a WFA given a bound on the number of states. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. We solve the optimal spectral-norm approximate minimization problem for irredundant WFAs with real weights, defined over a one-letter alphabet. We present a theoretical analysis based on AAK theory, and bounds on the quality of the approximation in the spectral norm and norm. Moreover, we provide a closed-form solution, and an algorithm, to compute the optimal approximation of a given size in polynomial time.
Paper Structure (26 sections, 18 theorems, 73 equations, 1 figure, 2 algorithms)

This paper contains 26 sections, 18 theorems, 73 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Hankel matrix and Weighted Automata
  5. AAK Theory
  6. AAK Theory and Approximate Minimization
  7. Defining a Hankel Operator: the one-letter assumption
  8. Solving the Approximate Minimization Problem
  9. Assumptions
  10. Problem Formulation
  11. Finding the Optimal Approximation
  12. Finding a Symbol for the WFA
  13. Finding the Optimal Symbol
  14. Extracting the Rational Component
  15. Solving the Approximation Problem
  16. ...and 11 more sections

Key Result

Theorem 2

A function $f:\Sigma^*\rightarrow \mathbb{R}$ is realized by a WFA $A$ if and only if $\mathbf{H}_f$ has finite rank. In that case, the rank of $\mathbf{H}_f$ corresponds to the minimal number of states of any automaton realizing $f$.

Figures (1)

  • Figure 1: Graphical representation of the generative probabilistic automaton described in Example \ref{['example1']}.

Theorems & Definitions (33)

  • Definition 1
  • Theorem 2: CP71Fli
  • Definition 3
  • Definition 4
  • Theorem 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 9: Nehari
  • Definition 10
  • ...and 23 more