# Topological Autoencoders

## TL;DR

This paper introduces Topological Autoencoders, which incorporate topological data analysis to preserve the multi-scale connectivity of input data in latent representations, improving structure retention without sacrificing reconstruction quality.

## Contribution

It presents a differentiable topological loss based on persistent homology, enabling autoencoders to better preserve topological features in latent space.

## Key findings

- Effective preservation of topological structures in latent space
- Favorable latent representations on synthetic and real-world data
- Maintains low reconstruction errors

## Abstract

We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of both the input and latent space to derive a topological loss term. Under weak theoretical assumptions, we construct this loss in a differentiable manner, such that the encoding learns to retain multi-scale connectivity information. We show that our approach is theoretically well-founded and that it exhibits favourable latent representations on a synthetic manifold as well as on real-world image data sets, while preserving low reconstruction errors.