Topological comparison of some dimension reduction methods using persistent homology on EEG data

TL;DR

This paper uses topological data analysis to compare various dimension reduction methods on EEG data, revealing significant qualitative differences in their topological features across homologies.

Contribution

It introduces a topological framework to compare dimension reduction methods, applying persistent homology to EEG data to quantify differences.

Findings

Wasserstein and Bottleneck distances effectively distinguish methods

Topological features vary significantly across methods

Qualitative differences are evident in persistent homology analysis

Abstract

In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used dimension reduction such as Isometric Feature Mapping, Laplacian Eigenmaps, Fast Independent Component Analysis, Kernel Ridge Regression, t-distributed Stochastic Neighbor Embedding. We then give a brief overview of some topological notions used in topological data analysis, such as, barcodes, persistent homology, and Wasserstein distance. Theoretically, these methods applied on a data set can be interpreted differently. From EEG data embedded into a manifold of high dimension, we apply these methods and we compare them across persistent homologies of dimension 0, 1, and 2, that is, across connected components, tunnels and holes, shells around voids or cavities. We find that from three dimension clouds of points, it is not clear how distinct from each other the methods are, but Wasserstein and Bottleneck distances, topological tests of hypothesis, and various methods show that the methods qualitatively and significantly differ across homologies.