InvMap and Witness Simplicial Variational Auto-Encoders

Abstract

Variational Auto-Encoders (VAEs) are deep generative models used for unsupervised learning, however their standard version is not topology-aware in practice since the data topology may not be taken into consideration. In this paper, we propose two different approaches with the aim to preserve the topological structure between the input space and the latent representation of a VAE. Firstly, we introduce InvMap-VAE as a way to turn any dimensionality reduction technique, given an embedding it produces, into a generative model within a VAE framework providing an inverse mapping into original space. Secondly, we propose the Witness Simplicial VAE as an extension of the Simplicial Auto-Encoder to the variational setup using a witness complex for computing the simplicial regularization, and we motivate this method theoretically using tools from algebraic topology. The Witness Simplicial VAE is independent of any dimensionality reduction technique and together with its extension, Isolandmarks Witness Simplicial VAE, preserves the persistent Betti numbers of a data set better than a standard VAE.

InvMap and Witness Simplicial Variational Auto-Encoders

“…In this section, we present briefly some notions of Computational Topology that are relevant to our work. For a complete introduction, we refer the reader to any book of computational topology, such as the one by Edelsbrunner and Harer [7], or to the theoretical background of our master thesis [4] which is self-contained. Computational topology aims to compute and develop algorithms in order to analyse topological structures, that is the shapes of considered objects.…”

“…In addition to its ability to generate new data, the VAE can, thus, be used for many applications, especially for dimensionality reduction which is useful for signal compression, high dimensional data visualisation, classification tasks, or clustering in a lower dimensional space, etc. This paper presents in a more concise way our main work developed during Medbouhi's master thesis [4] and provides an extension of the Witness Simplicial VAE method. We investigate here the use of TDA in order to modify the VAE with the hope that it will lead to an improvement of its performance.…”