论文标题
皮尔森相关与黎曼歧管的概括
A Generalization of the Pearson Correlation to Riemannian Manifolds
论文作者
论文摘要
深度学习的越来越多的应用伴随着向高度非线性统计模型的转变。就其几何形状而言，自然可以用riemannian流形识别这些模型。因此，对统计模型的进一步分析提出了相关度量的问题，即切线的切割平面等于相应的皮尔逊相关性，并扩展到相关度量，该相关度量相对于基础歧管进行了归一化。在这个目的中，文章重构了皮尔森相关性的基本特性，以连续得出多个维度的线性概括，并因此由Riemann-Pearson相关性给出了对主歧管的非线性概括。
The increasing application of deep-learning is accompanied by a shift towards highly non-linear statistical models. In terms of their geometry it is natural to identify these models with Riemannian manifolds. The further analysis of the statistical models therefore raises the issue of a correlation measure, that in the cutting planes of the tangent spaces equals the respective Pearson correlation and extends to a correlation measure that is normalized with respect to the underlying manifold. In this purpose the article reconstitutes elementary properties of the Pearson correlation to successively derive a linear generalization to multiple dimensions and thereupon a nonlinear generalization to principal manifolds, given by the Riemann-Pearson Correlation.
