Pavel Snopov

Mathilde Papillon, Mustafa Hajij, Helen Jenne et al.

This paper presents the computational challenge on topological deep learning that was hosted within the ICML 2023 Workshop on Topology and Geometry in Machine Learning. The competition asked participants to provide open-source implementations of topological neural networks from the literature by contributing to the python packages TopoNetX (data processing) and TopoModelX (deep learning). The challenge attracted twenty-eight qualifying submissions in its two-month duration. This paper describes the design of the challenge and summarizes its main findings.

Guillermo Bernárdez, Lev Telyatnikov, Marco Montagna et al.

This paper describes the 2nd edition of the ICML Topological Deep Learning Challenge that was hosted within the ICML 2024 ELLIS Workshop on Geometry-grounded Representation Learning and Generative Modeling (GRaM). The challenge focused on the problem of representing data in different discrete topological domains in order to bridge the gap between Topological Deep Learning (TDL) and other types of structured datasets (e.g. point clouds, graphs). Specifically, participants were asked to design and implement topological liftings, i.e. mappings between different data structures and topological domains --like hypergraphs, or simplicial/cell/combinatorial complexes. The challenge received 52 submissions satisfying all the requirements. This paper introduces the main scope of the challenge, and summarizes the main results and findings.

Pavel Snopov, Oleg R. Musin

This study explores novel activation functions that enhance the ability of neural networks to manipulate data topology during training. Building on the limitations of traditional activation functions like $\mathrm{ReLU}$, we propose $\mathrm{SmoothSplit}$ and $\mathrm{ParametricSplit}$, which introduce topology "cutting" capabilities. These functions enable networks to transform complex data manifolds effectively, improving performance in scenarios with low-dimensional layers. Through experiments on synthetic and real-world datasets, we demonstrate that $\mathrm{ParametricSplit}$ outperforms traditional activations in low-dimensional settings while maintaining competitive performance in higher-dimensional ones. Our findings highlight the potential of topology-aware activation functions in advancing neural network architectures. The code is available via https://github.com/Snopoff/Topology-Aware-Activations.

Lev Telyatnikov, Guillermo Bernardez, Marco Montagna et al.

This work introduces TopoBench, an open-source library designed to standardize benchmarking and accelerate research in topological deep learning (TDL). TopoBench decomposes TDL into a sequence of independent modules for data generation, loading, transforming and processing, as well as model training, optimization and evaluation. This modular organization provides flexibility for modifications and facilitates the adaptation and optimization of various TDL pipelines. A key feature of TopoBench is its support for transformations and lifting across topological domains. Mapping the topology and features of a graph to higher-order topological domains, such as simplicial and cell complexes, enables richer data representations and more fine-grained analyses. The applicability of TopoBench is demonstrated by benchmarking several TDL architectures across diverse tasks and datasets.

Maxim Beketov, Pavel Snopov

We generalise the reparameterization trick applied in variational autoencoders (VAEs) letting these have latent spaces of non-trivial topology - i.e. that of base manifolds covered with other ones, on which some technique for RT is available. That is possible since covering maps are measurable - moreover, in case of particular measure preservation property holding for the covering, one can establish an inequality on KL-divergence between pushforward (PF) densities on the base latent manifold, making the KL-term of VAE's ELBO analytically tractable, despite the topological non-triviality of the supporting latent manifold. Our development follows a route close but somewhat alternative to reparameterization on Lie groups, the latest proposal for which is to reparameterize PFs of normal densities from the Lie algebra - "through" the exponential map, seen by us as sometimes a particular case of what we propose to call reparameterization through a covering. Covering maps need not be global diffeomorphisms (although Lie-exp maps, in general, need not either, but, to date only smooth ones were considered in this context, to the best of our knowledge), which makes many non-trivial topologies tamable to our proposed technique, that we detail on a particular such example. We demonstrate the working of our approach by constructing a VAE with the latent space of Klein bottle (not a Lie group) topology, which we call KleinVAE, successfully learning an appropriate artificial dataset. We discuss potential applicability of such topology-informed generative models as weight priors in Bayesian learning, particularly for convolutional vision models, where said manifold was peculiarly shown to have some relevance.