Why top🍩logy?

$👕 = 👖 + 👖$

🤨

✨ 🤯 ✨

img source

I like to think of it as space bending

(Abstract) simplicial complex:

A finite set of simplices $\{\sigma_1, \dots, \sigma_i, \dots, \sigma_n\}$ is a simplicial complex $K$ if two properties hold:

• if $\sigma_i \in K$, then every face of $\sigma_i$ is an element of $K$,
• for $\sigma_i, \sigma_j \in K$, the non-empty intersection $\sigma_i \cap \sigma_j$ is a face of both $\sigma_i$ and $\sigma_j$.

$\emptyset \subseteq K_0 \subseteq K_1 \subseteq \dots \subseteq K_n$

Acknowledgments

Max Horn

@ExpectationMax

Bastian Rieck

@Pseudomanifold

Karsten Borgwardt

@kmborgwardt

Motivation

• We are given samples that lie on high-dimensional manifolds.
• Can we recover the global data structure at hand?

Standard autoencoder ($2D$ latent space):

Topology-aware autoencoder:

Method overview

Results

💡 Insights & Takeaways

• TopoAE was uniquely able to capture global topological features of the data manifolds.
• The topological features can be approximated on the level of mini-batches.
• Our resulting loss term is differentiable and generic (PCA, AE, VAE, $\dots$).