We address the approximate minimization problem for weighted finite automata (WFAs) 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.

中文翻译：

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加权有限自动机的最优频谱范数近似最小化

我们解决一个字母上的加权有限自动机（WFA）的近似最小化问题：在给定状态数的范围内，计算WFA的最佳近似值。这项工作基于Adamyan-Arov-Krein逼近理论，这是有关Hankel算子逼近的显着结果。除了其内在的数学意义外，该理论还被证明对模型简化非常有效。我们将这些结果调整为一个字母组成的加权自动机的框架。我们为频谱和$ \ ell ^ 2 $范数中的逼近质量提供了理论上的保证和界限。我们开发了一种算法，该算法基于Hankel算子的属性，返回频谱范数中的最佳近似值。