Manifold Learning via Foliations and Knowledge Transfer
This work addresses the challenge of geometric data analysis for machine learning practitioners, but it appears incremental as it builds on existing concepts like foliations and information matrices.
The paper tackled the problem of understanding high-dimensional data distributions by using a deep ReLU neural network classifier to induce a geometric structure called a singular foliation via a data information matrix, showing that singular points are measure zero and data correlates with leaves, and demonstrated knowledge transfer by measuring dataset distances through the matrix's spectrum.
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 deep 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.