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.Comment: arXiv admin note: substantial text overlap with arXiv:1905.0989
