Olga Zaghen

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.

Jinwoo Kim, Olga Zaghen, Ayhan Suleymanzade et al.

We revisit a simple model class for machine learning on graphs, where a random walk on a graph produces a machine-readable record, and this record is processed by a deep neural network to directly make vertex-level or graph-level predictions. We call these stochastic machines random walk neural networks (RWNNs), and through principled analysis, show that we can design them to be isomorphism invariant while capable of universal approximation of graph functions in probability. A useful finding is that almost any kind of record of random walks guarantees probabilistic invariance as long as the vertices are anonymized. This enables us, for example, to record random walks in plain text and adopt a language model to read these text records to solve graph tasks. We further establish a parallelism to message passing neural networks using tools from Markov chain theory, and show that over-smoothing in message passing is alleviated by construction in RWNNs, while over-squashing manifests as probabilistic under-reaching. We empirically demonstrate RWNNs on a range of problems, verifying our theoretical analysis and demonstrating the use of language models for separating strongly regular graphs where 3-WL test fails, and transductive classification on arXiv citation network. Code is available at https://github.com/jw9730/random-walk.

Lev Telyatnikov, Maria Sofia Bucarelli, Guillermo Bernardez et al.

Most of the current hypergraph learning methodologies and benchmarking datasets in the hypergraph realm are obtained by lifting procedures from their graph analogs, leading to overshadowing specific characteristics of hypergraphs. This paper attempts to confront some pending questions in that regard: Q1 Can the concept of homophily play a crucial role in Hypergraph Neural Networks (HNNs)? Q2 Is there room for improving current HNN architectures by carefully addressing specific characteristics of higher-order networks? Q3 Do existing datasets provide a meaningful benchmark for HNNs? To address them, we first introduce a novel conceptualization of homophily in higher-order networks based on a Message Passing (MP) scheme, unifying both the analytical examination and the modeling of higher-order networks. Further, we investigate some natural, yet mostly unexplored, strategies for processing higher-order structures within HNNs such as keeping hyperedge-dependent node representations, or performing node/hyperedge stochastic samplings, leading us to the most general MP formulation up to date -MultiSet-, as well as to an original architecture design, MultiSetMixer. Finally, we conduct an extensive set of experiments that contextualize our proposals and successfully provide insights about our inquiries.

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.

Olga Zaghen, Maksim Zhdanov, Dario Coscia et al.

Machine Learning Interatomic Potentials (MLIPs) achieve near ab initio accuracy at a fraction of the cost of quantum-mechanical simulations, yet they remain prone to silent failures on out-of-distribution configurations, making principled uncertainty quantification (UQ) essential for error-aware simulations and active learning. Existing non-ensemble UQ methods for MLIPs rely either on variational inference or on parametric distributional assumptions, both of which add architectural complexity and hyper-parameters that must be tuned per task. Inspired by recent advances in probabilistic weather forecasting, we propose a simpler alternative: turn a deterministic MLIP into a probabilistic one through learned functional perturbations and finetune it end-to-end with the Continuous Ranked Probability Score (CRPS), a proper scoring rule. We validate the approach with an equivariant GNN (P-EGNN) trained from scratch and by finetuning the foundation model the Orb-v3 for silica. On the N-body charged particle benchmark, P-EGNN improves CRPS over the state-of-the-art Bayesian MLIP method BLIP by 19-32% across all training sizes; on silica, P-Orb raises the Spearman correlation between predicted uncertainty and actual error from 0.75 (BLIP-Orb) to 0.84.

Martin Carrasco, Olga Zaghen, Erik Bekkers et al.

For far too long, expressivity of graph neural networks has been measured \emph{only} in terms of combinatorial properties. In this work we stray away from this tradition and provide a principled way to measure similarity between vertex attributed graphs. We denote this measure as the \emph{graph homomorphism distortion}. We show it can \emph{completely characterize} graphs and thus is also a \emph{complete graph embedding}. However, somewhere along the road, we run into the graph canonization problem. To circumvent this obstacle, we devise to efficiently compute this measure via sampling, which in expectation ensures \emph{completeness}. Additionally, we also discovered that we can obtain a metric from this measure. We validate our claims empirically and find that the \emph{graph homomorphism distortion}: (1.) fully distinguishes the \texttt{BREC} dataset with up to $4$-WL non-distinguishable graphs, and (2.) \emph{outperforms} previous methods inspired in homomorphisms under the \texttt{ZINC-12k} dataset. These theoretical results, (and their empirical validation), pave the way for future characterization of graphs, extending the graph theoretic tradition to new frontiers.

Mustafa Hajij, Mathilde Papillon, Florian Frantzen et al.

We introduce TopoX, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes. TopoX consists of three packages: TopoNetX facilitates constructing and computing on these domains, including working with nodes, edges and higher-order cells; TopoEmbedX provides methods to embed topological domains into vector spaces, akin to popular graph-based embedding algorithms such as node2vec; TopoModelX is built on top of PyTorch and offers a comprehensive toolbox of higher-order message passing functions for neural networks on topological domains. The extensively documented and unit-tested source code of TopoX is available under MIT license at https://pyt-team.github.io/}{https://pyt-team.github.io/.

Olga Zaghen, Floor Eijkelboom, Alison Pouplin et al.

We present Riemannian Gaussian Variational Flow Matching (RG-VFM), a geometric extension of Variational Flow Matching (VFM) for generative modeling on manifolds. In Euclidean space, predicting endpoints (VFM), velocities (FM), or noise (diffusion) are largely equivalent due to affine interpolations. On curved manifolds this equivalence breaks down, and we hypothesize that endpoint prediction provides a stronger learning signal by directly minimizing geodesic distances. Building on this insight, we derive a variational flow matching objective based on Riemannian Gaussian distributions, applicable to manifolds with closed-form geodesics. We formally analyze its relationship to Riemannian Flow Matching (RFM), exposing that the RFM objective lacks a curvature-dependent penalty - encoded via Jacobi fields - that is naturally present in RG-VFM. Experiments on synthetic spherical and hyperbolic benchmarks, as well as real-world tasks in material and protein generation, demonstrate that RG-VFM more effectively captures manifold structure and improves downstream performance over Euclidean and velocity-based baselines.

Olga Zaghen

This work focuses on exploring the potential benefits of introducing a nonlinear Laplacian in Sheaf Neural Networks for graph-related tasks. The primary aim is to understand the impact of such nonlinearity on diffusion dynamics, signal propagation, and performance of neural network architectures in discrete-time settings. The study primarily emphasizes experimental analysis, using real-world and synthetic datasets to validate the practical effectiveness of different versions of the model. This approach shifts the focus from an initial theoretical exploration to demonstrating the practical utility of the proposed model.