论文标题
采用AAK理论方法，以最小化在多字母的情况下
Towards an AAK Theory Approach to Approximate Minimization in the Multi-Letter Case
论文作者
论文摘要
我们研究加权有限自动机（WFA）的近似最小化问题：鉴于WFA，我们希望计算其最佳近似值时，仅限于给定尺寸。我们将问题重新制定为光谱规范中的排名最小化任务，并提出了一个将Adamyan-Arov-Krein（AAK）理论应用于近似问题的框架。这种方法已经成功地应用于WFA和语言对一个字母字母\ citep {aak-wfa，aak-rnn}上的黑匣子建模的情况。将结果扩展到多字母字母需要求解以下两个步骤。首先，我们需要根据非交通性的汉克尔操作员和非交通功能来重新重新估算问题，以应用多变量操作员理论的结果。其次，要获得最佳近似，我们需要一个具有建设性的非交流性AAK理论的版本。在本文中，我们成功解决了第一步，而第二个挑战仍然开放。
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.
