Geometric methods in statistical learning theory and applications

该项目旨在发展统计学习理论中的微分几何方法，特别是在有效估计器的几何方面。在研究大量数据时，找到一个表示数据结构的密度函数是至关重要的。这是通过基于数据的某些特征函数给出所谓的估计器来实现的。这种估计量的效率是根据估计量与实际密度的偏差来定义的。近年来，人们发展了微分几何方法来构造有效的估计量，本项目的目的是改进这些方法。特别是，我们希望研究指数模型和自然梯度流的几何，并将它们应用于机器学习。

This project aims at developing differential geometric methods in statistical learning theory, in particular in the geometry of efficient estimators. When investigating large amounts of data, it is essential to find a density function representing the structure of the data. This is done by giving a so called estimator, based on some feature function of the data. The efficiency of this estimator is then defined in terms of the deviation of the estimated from the actual density.In recent years, differential geometric methods were developed for constructing efficient estimators, and it is the aim of the present project to refine these methods. In particular, we wish to investigate exponential models and the geometry of the natural gradient flow, and apply them to machine learning.