Topological Approaches to Deep Learning

TL;DR

This paper uses topological data analysis to understand and modify deep neural networks, leading to faster computations and better generalization, and introduces a new geometric methodology for constructing CNN analogues for various data structures.

Contribution

It introduces a topological approach to analyze and enhance deep learning models, including a new geometric framework for CNNs on different data structures.

Findings

Topological analysis reveals geometric structures in neural network states.

Modified computations improve speed and generalization.

Develops a methodology for CNN analogues on various geometries.

Abstract

We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed up computations and (b) improve generalization from one data set of digits to another. One byproduct of the analysis is the production of a geometry on new sets of features on data sets of images, and use this observation to develop a methodology for constructing analogues of CNN's for many other geometries, including the graph structures constructed by topological data analysis.