模型的固有复杂度和泛化能力与几何曲率的关系

The Relation Between Intrinsic Complexity and Generalization of a Model and the Geometric Curvature

从微分几何角度考察与参数化形式无关的统计模型流形的固有复杂度,指出模型流形的Gauss-Kroneker曲率可以完全刻画模型流形在一点处的全部性质,进而分析了曲率与体积的关系;给出了基于参数估计量邻域附近的解轨迹方法的曲率计算方法;证明了用于衡量泛化能力的未来残差可以用模型的曲率来表示,由此给出一种新的以曲率度量模型复杂度的模型选择准则GKCIC;对几何方法和统计学习理论进行了分析比较．在人工数据集和真实数据集上的比较实验结果表明了文中提出的方法的有效性．

The paper uses the conception of curvature from the point of view of differential geometry to explore the intrinsic model complexity that is free of reparametrization; and then through theoretical analysis, shows that the Gauss-Kroneker curvature can describe the whole properties of the statistical manifold, thus gives the relation between curvature and the volume of the manifold. An algorithm is proposed based on study of the solution locus in the neighborhood of the expectation of parameters to calculate the curvature of the model. This paper proves that the future residual that is qualified to measure the generalizability can be expressed by using the intrinsic curvature array of model, from which a new model selection criterion GKCIC is given. It not only considers the factors such as the number of parameters, sample size and functional form, but also with very clear and intuitive geometric understanding of model selection. The geometrical method of the statistical manifold is compared with the statistical learning theory, in particular, the VC dimension versus the Gauss-Kroneker curvature. By running the algorithm on synthetic and real datasets, the author argue that the GKCIC work efficiently.