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PhD Student

Computational Network Science
RWTH Aachen University
hoppe@cs.rwth-aachen.de

           

Bio

I am currently a PhD Student in Michael T. Schaub’s Group at RWTH Aachen University.

Previously, I also obtained my Bachelor’s and Master’s in Computer Science at RWTH University. My master’s thesis led to my first (joint) publication on passenger prediction in public transport, but my focus has since shifted toward graph signal processing, algebraic topology, and higher-order networks. In that area, I am proud to report that our paper Representing Edge Flows on Graphs via Sparse Cell Complexes received the Best Paper award at the 2023 Learning on Graphs Conference.

In connection with my research, I have also published or contributed to the following software packages:

In my free time, I am active in the german-language debating scene, more specifically, the debating club Aachen. Recently, I organized the Western German Debating Championship (WDM 2024).

Recent Talks & Poster Presentations

Publications

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Don't be Afraid of Cell Complexes! An Introduction from an Applied Perspective
In arXiv Preprints, 2025.
Cell complexes (CCs) are a higher-order network model deeply rooted in algebraic topology that has gained interest in signal processing and network science recently. However, while the processing of signals supported on CCs can be described in terms of easily-accessible algebraic or combinatorial notions, the commonly presented definition of CCs is grounded in abstract concepts from topology and remains disconnected from the signal processing methods developed for CCs. In this paper, we aim to bridge this gap by providing a simplified definition of CCs that is accessible to a wider audience and can be used in practical applications. Specifically, we first introduce a simplified notion of abstract regular cell complexes (ARCCs). These ARCCs only rely on notions from algebra and can be shown to be equivalent to regular cell complexes for most practical applications. Second, using this new definition we provide an accessible introduction to (abstract) cell complexes from a perspective of network science and signal processing. Furthermore, as many practical applications work with CCs of dimension 2 and below, we provide an even simpler definition for this case that significantly simplifies understanding and working with CCs in practice.
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Random Abstract Cell Complexes
In arXiv Preprints, 2024.
We define a model for random (abstract) cell complexes (CCs), similiar to the well-known Erdős-Rényi model for graphs and its extensions for simplicial complexes. To build a random cell complex, we first draw from an Erdős-Rényi graph, and consecutively augment the graph with cells for each dimension with a specified probability. As the number of possible cells increases combinatorially -- e.g., 2-cells can be represented as cycles, or permutations -- we derive an approximate sampling algorithm for this model limited to two-dimensional abstract cell complexes. As a basis for this algorithm, we first introduce a spanning-tree-based method that samples simple cycles and allows the efficient approximation of various properties, most notably the probability of occurence of a given cycle. This approximation is of independent interest as it enables the approximation of a wide variety of cycle-related graph statistics using importance sampling. We use this to approximate the number of cycles of a given length on a graph, allowing us to calculate the sampling probability to arrive at a desired expected number of sampled 2-cells. The probability approximation also trivially leads to a sampling algorithm for 2-cells with a desired sampling probability. We provide some initial analysis into the properties of random CCs drawn from this model. We further showcase practical applications for our random CCs as null models, and in the context of (random) liftings of graphs to cell complexes. The cycle sampling, cycle count estimation, and combined cell sampling algorithms are available in the package `py-raccoon` on PyPI.
PDFGithubPyPIPoster
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Faster Inference of Cell Complexes from Flows via Matrix Factorization
In EUSIPCO, 2025 (in Press).
We consider the following inference problem: Given a set of edge-flow signals observed on a graph, lift the graph to a cell complex, such that the observed edge-flow signals can be represented as a sparse combination of gradient and curl flows on the cell complex. Specifically, we aim to augment the observed graph by a set of 2-cells (polygons encircled by closed, non-intersecting paths), such that the eigenvectors of the Hodge Laplacian of the associated cell complex provide a sparse, interpretable representation of the observed edge flows on the graph. As it has been shown that the general problem is NP-hard in prior work, we here develop a novel matrix-factorization-based heuristic to solve the problem. Using computational experiments, we demonstrate that our new approach is significantly less computationally expensive than prior heuristics, while achieving only marginally worse performance in most settings. In fact, we find that for specifically noisy settings, our new approach outperforms the previous state of the art in both solution quality and computational speed.
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Representing Edge Flows on Graphs via Sparse Cell Complexes
In The Second Learning on Graphs Conference, 2023.
Obtaining sparse, interpretable representations of observable data is crucial in many machine learning and signal processing tasks. For data representing flows along the edges of a graph, an intuitively interpretable way to obtain such representations is to lift the graph structure to a simplicial complex: The eigenvectors of the associated Hodge-Laplacian, respectively the incidence matrices of the corresponding simplicial complex then induce a Hodge decomposition, which can be used to represent the observed data in terms of gradient, curl, and harmonic flows. In this paper, we generalize this approach to cellular complexes and introduce the cell inference optimization problem, i.e., the problem of augmenting the observed graph by a set of cells, such that the eigenvectors of the associated Hodge Laplacian provide a sparse, interpretable representation of the observed edge flows on the graph. We show that this problem is NP-hard and introduce an efficient approximation algorithm for its solution. Experiments on real-world and synthetic data demonstrate that our algorithm outperforms current state-of-the-art methods while being computationally efficient.
PDFDOIGithubPosterSlidesTalk at LoG 2023arXivPyPI
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Topological Trajectory Classification and Landmark Inference on Simplicial Complexes
In Asilomar Conference on Signals, Systems, and Computers (in press), 2024.
We consider the problem of classifying trajectories on a discrete or discretised 2-dimensional manifold modelled by a simplicial complex. Previous works have proposed to project the trajectories into the harmonic eigenspace of the Hodge Laplacian, and then cluster the resulting embeddings. However, if the considered space has vanishing homology (i.e., no "holes"), then the harmonic space of the 1-Hodge Laplacian is trivial and thus the approach fails. Here we propose to view this issue akin to a sensor placement problem and present an algorithm that aims to learn "optimal holes" to distinguish a set of given trajectory classes. Specifically, given a set of labelled trajectories, which we interpret as edge-flows on the underlying simplicial complex, we search for 2-simplices whose deletion results in an optimal separation of the trajectory labels according to the corresponding spectral embedding of the trajectories into the harmonic space. Finally, we generalise this approach to the unsupervised setting.
PDFDOIGit
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Improving the Prediction of Passenger Numbers in Public Transit Networks by Combining Short-Term Forecasts With Real-Time Occupancy Data
In IEEE Open Journal of Intelligent Transportation Systems, 2023.
Good public transport occupancy predictions are important for both passengers (to avoid crowded vehicles) and operating companies (to intervene with dispositive actions, hereby increasing the service quality). In this paper, we present a novel approach to improve the real-time prediction of passenger numbers. It combines a day-ahead prediction made using SARIMA with a real-time detection of characteristic deviation profiles. In our experiments with data from Germany, the proposed model outperforms artificial neural networks when only little data is available or the prediction happens with larger lead times than 90 minutes. It also has the inherent advantage of being faster to train and explainable in all steps.
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