This paper studies the following approximate minimization problem: given a minimal weighted tree automaton A with n states recognizing a weighted tree language f, can we construct a smaller automaton $\hat{A}$ with $\hat{n}<n$ states recognizing a language $\hat{f}$ that is a good approximation of f? The corresponding problem for weighted automata on strings was recently studied by Balle et al. [16], [15], where the authors introduced a new canonical form for weighted automata called singular value automata inspired by spectral methods, and showed that truncating such canonical form yields a solution for the problem satisfying a certain approximation criteria. In this paper we take a similar approach and show that in the tree case one can obtain an analogous canonical form that we call singular value tree automata, and use it to study the approximate minimization problem for weighted tree automata. We first establish the existence of this canonical form for weighted tree automata and then provide bounds on the quality of the resulting approximation method based on truncation. We also study the problem of computing the canonical form given a minimal weighted tree automaton and show that in the tree case this task is considerably more complicated than in the string case. In particular, computing the canonical form reduces to solving a system of polynomial equations. By further reducing this problem to the computation of generalized partition functions for weighted tree automata, we propose and analyze two methods for computing the canonical form based on iterative methods: fixed point iteration and Newton's method. Our analysis of Newton's method unveils a connection between iterative methods and sequences of sets of trees satisfying a certain technical condition that might be of independent interest.

本文研究了以下近似最小化问题：给定一个最小加权树自动机A，有n个状态识别加权树语言f，我们能否构造一个更小的自动机$\widehat{\mathrm{一个}}$ 和 $\widehat{n}<n$ 承认语言的国家 $\widehat{F}$这是f 的一个很好的近似值？Balle 等人最近研究了字符串上加权自动机的相应问题。[16]、[15]，作者在其中引入了一种新的加权自动机规范形式，称为奇异值自动机，其灵感来自谱方法，并表明截断这种规范形式可以为满足某个近似标准的问题提供解决方案。在本文中，我们采用了类似的方法，并表明在树的情况下可以得到一种类似的规范形式，我们称之为奇异值树自动机，并用它来研究加权树自动机的近似最小化问题。我们首先建立加权树自动机的这种规范形式的存在，然后提供基于截断的结果近似方法的质量的界限。我们还研究了在给定最小加权树自动机的情况下计算规范形式的问题，并表明在树情况下此任务比在字符串情况下复杂得多。特别是，计算规范形式简化为求解多项式方程组。通过将这个问题进一步简化为广义分区函数的计算对于加权树自动机，我们提出并分析了两种基于迭代法计算规范形式的方法：不动点迭代法和牛顿法。我们对牛顿方法的分析揭示了迭代方法与满足可能具有独立意义的特定技术条件的树集序列之间的联系。