arXiv subjects

Ernst Röell

Publications and source records attributed to Ernst Röell.

6 recordsLinked to original sources

arXiv2026

NervePool: A Simplicial Pooling Layer

For deep learning problems on graph-structured data, pooling layers are important for down sampling, reducing computational cost, and to minimize overfitting. We define a pooling layer, nervePool, for data structured as simplicial complexes, which are generalizations of graphs that include higher-dimensional simplices beyond vertices and edges; this structure allows for greater flexibility in modeling higher-order relationships. The proposed simplicial coarsening scheme is built upon partitions of vertices, which allow us to generate hierarchical representations of simplicial complexes, collapsing information in a learned fashion. NervePool builds on the learned vertex cluster assignments and extends to coarsening of higher dimensional simplices in a deterministic fashion. While in practice the pooling operations are computed via a series of matrix operations, the topological motivation is a set-theoretic construction based on unions of stars of simplices and the nerve complex.

cs.CG↗

arXiv2026

LEAP: Local ECT-Based Learnable Positional Encodings for Graphs

Graph neural networks (GNNs) largely rely on the message-passing paradigm, where nodes iteratively aggregate information from their neighbors. Yet, standard message passing neural networks (MPNNs) face well-documented theoretical and practical limitations. Graph positional encoding (PE) has emerged as a promising direction to address these limitations. The Euler Characteristic Transform (ECT) is an efficiently computable geometric-topological invariant that characterizes shapes and graphs. In this work, we combine the differentiable approximation of the ECT (DECT) and its local variant ($\ell$-ECT) to propose LEAP, a new end-to-end trainable local structural PE for graphs. We evaluate our approach on multiple real-world datasets as well as on a synthetic task designed to test its ability to extract topological features. Our results underline the potential of LEAP-based encodings as a powerful component for graph representation learning pipelines.

cs.LG↗

arXiv2026

No Triangulation Without Representation: Generalization in Topological Deep Learning

Despite an ever-increasing interest in topological deep learning models that target higher-order datasets, there is no consensus on how to evaluate such models. This is exacerbated by the fact that topological objects permit operations, such as structural refinements, that are not appropriate for graph data. In this work, we extend MANTRA, a benchmark dataset containing manifold triangulations, to a larger class of manifolds with more diverse homeomorphism types. We show that, unlike prior claims, both graph neural networks (GNNs) and higher-order message passing (HOMP) methods can saturate the benchmark. However, we find that this is contingent on the right representation and feature assignment, emphasizing their importance in baseline models. We thus provide a novel evaluation protocol based on representational diversity and triangulation refinement. Surprisingly, we find no indication that existing models are capable of generalizing beyond the combinatorial structure of the data. This points towards a research gap in developing models that understand topological structure independent of scale. Our work thus provides the necessary scaffolding to evaluate future models and enable the development of topology-aware inductive biases.

cs.LG↗

arXiv2026

Point Cloud Synthesis Using Inner Product Transforms

Point cloud synthesis, i.e. the generation of novel point clouds from an input distribution, remains a challenging task, for which numerous complex machine learning models have been devised. We develop a novel method that encodes geometrical-topological characteristics of point clouds using inner products, leading to a highly-efficient point cloud representation with provable expressivity properties. Integrated into deep learning models, our encoding exhibits high quality in typical tasks like reconstruction, generation, and interpolation, with inference times orders of magnitude faster than existing methods.

cs.CV↗

arXiv2025

MANTRA: The Manifold Triangulations Assemblage

The rising interest in leveraging higher-order interactions present in complex systems has led to a surge in more expressive models exploiting higher-order structures in the data, especially in topological deep learning (TDL), which designs neural networks on higher-order domains such as simplicial complexes. However, progress in this field is hindered by the scarcity of datasets for benchmarking these architectures. To address this gap, we introduce MANTRA, the first large-scale, diverse, and intrinsically higher-order dataset for benchmarking higher-order models, comprising over 43,000 and 250,000 triangulations of surfaces and three-dimensional manifolds, respectively. With MANTRA, we assess several graph- and simplicial complex-based models on three topological classification tasks. We demonstrate that while simplicial complex-based neural networks generally outperform their graph-based counterparts in capturing simple topological invariants, they also struggle, suggesting a rethink of TDL. Thus, MANTRA serves as a benchmark for assessing and advancing topological methods, leading the way for more effective higher-order models.

cs.LG↗

arXiv2023

A Diffusion Model Predicts 3D Shapes from 2D Microscopy Images

Diffusion models are a special type of generative model, capable of synthesising new data from a learnt distribution. We introduce DISPR, a diffusion-based model for solving the inverse problem of three-dimensional (3D) cell shape prediction from two-dimensional (2D) single cell microscopy images. Using the 2D microscopy image as a prior, DISPR is conditioned to predict realistic 3D shape reconstructions. To showcase the applicability of DISPR as a data augmentation tool in a feature-based single cell classification task, we extract morphological features from the red blood cells grouped into six highly imbalanced classes. Adding features from the DISPR predictions to the three minority classes improved the macro F1 score from $F1_\text{macro} = 55.2 \pm 4.6\%$ to $F1_\text{macro} = 72.2 \pm 4.9\%$. We thus demonstrate that diffusion models can be successfully applied to inverse biomedical problems, and that they learn to reconstruct 3D shapes with realistic morphological features from 2D microscopy images.

cs.CV↗