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

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