Topological Learning in Multi-Class Data Sets

TL;DR

This paper applies topological data analysis to characterize the complexity of multi-class datasets and explores its impact on the learning ability of deep neural networks, introducing a topological classifier based on simplicial complexes.

Contribution

It introduces a novel topological classification method for multi-class data and investigates the relationship between topological complexity and neural network learning performance.

Findings

Topological complexity negatively correlates with neural network classification accuracy.

The topological classifier effectively characterizes dataset complexity.

Experimental validation on multiple datasets supports the hypotheses.

Abstract

We specialize techniques from topological data analysis to the problem of characterizing the topological complexity (as defined in the body of the paper) of a multi-class data set. As a by-product, a topological classifier is defined that uses an open sub-covering of the data set. This sub-covering can be used to construct a simplicial complex whose topological features (e.g., Betti numbers) provide information about the classification problem. We use these topological constructs to study the impact of topological complexity on learning in feedforward deep neural networks (DNNs). We hypothesize that topological complexity is negatively correlated with the ability of a fully connected feedforward deep neural network to learn to classify data correctly. We evaluate our topological classification algorithm on multiple constructed and open source data sets. We also validate our hypothesis regarding the relationship between topological complexity and learning in DNN's on multiple data sets.