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Stefan Huber

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4 papers

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NeurIPS Conference 2025 Conference Paper

The Flood Complex: Large-Scale Persistent Homology on Millions of Points

Florian Graf
Paolo Pellizzoni
Martin Uray
Stefan Huber
Roland Kwitt

We consider the problem of computing persistent homology (PH) for large-scale Euclidean point cloud data, aimed at downstream machine learning tasks, where the exponential growth of the most widely-used Vietoris-Rips complex imposes serious computational limitations. Although more scalable alternatives such as the Alpha complex or sparse Rips approximations exist, they often still result in a prohibitively large number of simplices. This poses challenges in the complex construction and in the subsequent PH computation, prohibiting their use on large-scale point clouds. To mitigate these issues, we introduce the Flood complex, inspired by the advantages of the Alpha and Witness complex constructions. Informally, at a given filtration value $r\geq 0$, the Flood complex contains all simplices from a Delaunay triangulation of a small subset of a point cloud $X$ that are fully covered by the union of balls of radius $r$ emanating from $X$, a process we call flooding. Our construction allows for efficient PH computation, possesses several desirable theoretical properties, and is amenable to GPU parallelization. Scaling experiments on 3D point cloud data show that we can compute PH of up to dimension 2 on several millions of points. Importantly, when evaluating object classification performance on real-world and synthetic data, we provide evidence that this scaling capability is needed, especially if objects are geometrically or topologically complex, yielding performance superior to other PH-based methods and neural networks for point cloud data. Source code and datasets are available on GitHub: https: //github. com/plus-rkwitt/flooder

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NeurIPS Conference 2024 Conference Paper

Neural Persistence Dynamics

Sebastian Zeng
Florian Graf
Martin Uray
Stefan Huber
Roland Kwitt

We consider the problem of learning the dynamics in the topology of time-evolving point clouds, the prevalent spatiotemporal model for systems exhibiting collective behavior, such as swarms of insects and birds or particles in physics. In such systems, patterns emerge from (local) interactions among self-propelled entities. While several well-understood governing equations for motion and interaction exist, they are notoriously difficult to fit to data, as most prior work requires knowledge about individual motion trajectories, i. e. , a requirement that is challenging to satisfy with an increasing number of entities. To evade such confounding factors, we investigate collective behavior from a topological perspective, but instead of summarizing entire observation sequences (as done previously), we propose learning a latent dynamical model from topological features per time point. The latter is then used to formulate a downstream regression task to predict the parametrization of some a priori specified governing equation. We implement this idea based on a latent ODE learned from vectorized (static) persistence diagrams and show that a combination of recent stability results for persistent homology justifies this modeling choice. Various (ablation) experiments not only demonstrate the relevance of each model component but provide compelling empirical evidence that our proposed model -- Neural Persistence Dynamics -- substantially outperforms the state-of-the-art across a diverse set of parameter regression tasks.

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YNIMG Journal 2018 Journal Article

Neuro-cognitive mechanisms of global Gestalt perception in visual quantification

Johannes Bloechle
Stefan Huber
Elise Klein
Julia Bahnmueller
Korbinian Moeller
Johannes Rennig

Recent neuroimaging studies identified posterior regions in the temporal and parietal lobes as neuro-functional correlates of subitizing and global Gestalt perception. Beyond notable overlap on a neuronal level both mechanisms are remarkably similar on a behavioral level representing both a specific form of visual top-down processing where single elements are integrated into a superordinate entity. In the present study, we investigated whether subitizing draws on principles of global Gestalt perception enabling rapid top-down processes of visual quantification. We designed two functional neuroimaging experiments: a task identifying voxels responding to global Gestalt stimuli in posterior temporo-parietal brain regions and a visual quantification task on dot patterns with magnitudes within and outside the subitizing range. We hypothesized that voxels activated in global Gestalt perception should respond stronger to dot patterns within than those outside the subitizing range. The results confirmed this prediction for left-hemispheric posterior temporo-parietal brain areas. Additionally, we trained a classifier with response patterns from global Gestalt perception to predict neural responses of visual quantification. With this approach we were able to classify from TPJ Gestalt ROIs of both hemispheres whether a trial requiring subitizing was processed. The present study demonstrates that mechanisms of subitizing seem to build on processes of high-level visual perception.

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NeurIPS Conference 2015 Conference Paper

Statistical Topological Data Analysis - A Kernel Perspective

Roland Kwitt
Stefan Huber
Marc Niethammer
Weili Lin
Ulrich Bauer

We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel which would enable a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in two-sample hypothesis testing on synthetic as well as real-world data.

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