奇异叶结构与ReLU网络中学习流形的几何
Geometry of Singular Foliations and Learning Manifolds in ReLU Networks via the Data Information Matrix
AI总结：
本文提出通过ReLU网络中的数据信息矩阵揭示数据在高维空间中的奇异叶结构，并通过谱分析实现跨数据集的知识迁移。
摘要：
理解高维空间中真实数据的分布是许多机器学习任务的关键。我们希望通过ReLU神经网络训练为分类器，在数据空间上提供一种自然的几何结构。通过数据信息矩阵（DIM），一种Fisher信息矩阵的变种，模型将能够在数据空间上 discern 一种奇异叶结构。我们证明此类叶结构的奇点包含在测度零集合中，并且几乎处处存在局部正则叶结构。实验显示数据与此类叶结构的叶相关联。此外，我们通过分析DIM的谱来测量数据集之间的距离，展示了该方法在知识迁移方面的潜力。
英文摘要：
Understanding how real data is distributed in high dimensional spaces is the key to many tasks in machine learning. We want to provide a natural geometric structure on the space of data employing a ReLU neural network trained as a classifier. Through the Data Information Matrix (DIM), a variation of the Fisher information matrix, the model will discern a singular foliation structure on the space of data. We show that the singular points of such foliation are contained in a measure zero set, and that a local regular foliation exists almost everywhere. Experiments show that the data is correlated with leaves of such foliation. Moreover we show the potential of our approach for knowledge transfer by analyzing the spectrum of the DIM to measure distances between datasets.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
