Josef Hoppe

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. Since there is a large variance in the number of simple cycles on graphs drawn from the same configuration of ER, we also provide an efficient method to approximate that number, which is of independent interest. Moreover, it enables us to specify the expected number of 2-cells of each boundary length we want to sample. 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.

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.

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.