# Visualization of AE's Training on Credit Card Transactions with Persistent Homology

## TL;DR

This paper introduces PHom-WAE, a novel topological method using persistent homology to evaluate and improve the data distribution quality of Wasserstein Auto-Encoders, especially on complex real-world credit card transaction data.

## Contribution

It proposes a new topological distance measure, the bottleneck distance, for Wasserstein Auto-Encoders to better assess latent space quality compared to traditional methods.

## Key findings

- Persistent homology effectively captures topological features of data manifolds.
- PHom-WAE outperforms traditional distance measures on real-world datasets.
- The methodology provides a new way to evaluate generative models' data distribution quality.

## Abstract

Auto-encoders are among the most popular neural network architecture for dimension reduction. They are composed of two parts: the encoder which maps the model distribution to a latent manifold and the decoder which maps the latent manifold to a reconstructed distribution. However, auto-encoders are known to provoke chaotically scattered data distribution in the latent manifold resulting in an incomplete reconstructed distribution. Current distance measures fail to detect this problem because they are not able to acknowledge the shape of the data manifolds, i.e. their topological features, and the scale at which the manifolds should be analyzed. We propose Persistent Homology for Wasserstein Auto-Encoders, called PHom-WAE, a new methodology to assess and measure the data distribution of a generative model. PHom-WAE minimizes the Wasserstein distance between the true distribution and the reconstructed distribution and uses persistent homology, the study of the topological features of a space at different spatial resolutions, to compare the nature of the latent manifold and the reconstructed distribution. Our experiments underline the potential of persistent homology for Wasserstein Auto-Encoders in comparison to Variational Auto-Encoders, another type of generative model. The experiments are conducted on a real-world data set particularly challenging for traditional distance measures and auto-encoders. PHom-WAE is the first methodology to propose a topological distance measure, the bottleneck distance, for Wasserstein Auto-Encoders used to compare decoded samples of high quality in the context of credit card transactions.