Model complexity plays an essential role in its selection, namely, by choosing a model that fits the data and is also succinct. Two-part codes and the minimum description length have been successful in delivering procedures to single out the best models, avoiding overfitting. In this work, we pursue this approach and complement it by performing further assumptions in the parameter space. Concretely, we assume that the parameter space is a smooth manifold, and by using tools of Riemannian geometry, we derive a sharper expression than the standard one given by the stochastic complexity, where the scalar curvature of the Fisher information metric plays a dominant role. Furthermore, we compute a sharper approximation to the capacity for exponential families and apply our results to derive optimal dimensional reduction in the context of principal component analysis.

中文翻译：

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统计流形中的模型复杂性：曲率的作用

模型复杂度在其选择中起着至关重要的作用，即通过选择适合数据且简洁的模型。两部分代码和最小描述长度已成功交付程序以挑选出最佳模型，避免过度拟合。在这项工作中，我们采用这种方法并通过在参数空间中执行进一步的假设来补充它。具体来说，我们假设参数空间是一个光滑的流形，并且通过使用黎曼几何工具，我们得到了比随机复杂度给出的标准表达式更清晰的表达式，其中 Fisher 信息度量的标量曲率起主导作用。此外，