论文标题
关于统计流形的Minmax遗憾：曲率的作用
On the minmax regret for statistical manifolds: the role of curvature
论文作者
论文摘要
模型复杂性在其选择中起着至关重要的作用，即选择适合数据并且简洁的模型。两部分的代码和最小描述长度已成功地提供了挑选最佳型号的程序，以避免过度拟合。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进行补充。具体而言，我们假设参数空间是一个平滑的歧管，并且通过使用Riemannian几何形状的工具，我们比随机复杂性给出的标准表达式得出一个更清晰的表达式，其中Fisher Information指标的标态曲率起着主导作用。此外，我们为一般统计流形引发了Minmax的遗憾，并将结果应用于主成分分析的背景下得出最佳的维度减少。
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 derive the minmax regret for general statistical manifolds and apply our results to derive optimal dimensional reduction in the context of principal component analysis.
