TY - CHAP A1 - Maignant, Elodie A1 - Conrad, Tim A1 - von Tycowicz, Christoph T1 - Tree inference with varifold distances T2 - Geometric Science of Information. GSI 2025 N2 - In this paper, we consider a tree inference problem motivated by the critical problem in single-cell genomics of reconstructing dynamic cellular processes from sequencing data. In particular, given a population of cells sampled from such a process, we are interested in the problem of ordering the cells according to their progression in the process. This is known as trajectory inference. If the process is differentiation, this amounts to reconstructing the corresponding differentiation tree. One way of doing this in practice is to estimate the shortest-path distance between nodes based on cell similarities observed in sequencing data. Recent sequencing techniques make it possible to measure two types of data: gene expression levels, and RNA velocity, a vector that predicts changes in gene expression. The data then consist of a discrete vector field on a (subset of a) Euclidean space of dimension equal to the number of genes under consideration. By integrating this velocity field, we trace the evolution of gene expression levels in each single cell from some initial stage to its current stage. Eventually, we assume that we have a faithful embedding of the differentiation tree in a Euclidean space, but which we only observe through the curves representing the paths from the root to the nodes. Using varifold distances between such curves, we define a similarity measure between nodes which we prove approximates the shortest-path distance in a tree that is isomorphic to the target tree. Y1 - 2025 U6 - https://doi.org/10.1007/978-3-032-03921-7_30 VL - 16034 ER - TY - CHAP A1 - Schade, Johannes A1 - von Tycowicz, Christoph A1 - Hanik, Martin T1 - Bi-invariant Geodesic Regression with Data from the Osteoarthritis Initiative T2 - Information Processing in Medical Imaging N2 - Many phenomena are naturally characterized by measuring continuous transformations such as shape changes in medicine or articulated systems in robotics. Modeling the variability in such datasets requires performing statistics on Lie groups, that is, manifolds carrying an additional group structure. As the Lie group captures the symmetries in the data, it is essential from a theoretical and practical perspective to ask for statistical methods that respect these symmetries; this way they are insensitive to confounding effects, e.g., due to the choice of reference coordinate systems. In this work, we investigate geodesic regression---a generalization of linear regression originally derived for Riemannian manifolds. While Lie groups can be endowed with Riemannian metrics, these are generally incompatible with the group structure. We develop a non-metric estimator using an affine connection setting. It captures geodesic relationships respecting the symmetries given by left and right translations. For its computation, we propose an efficient fixed point algorithm requiring simple differential expressions that can be calculated through automatic differentiation. We perform experiments on a synthetic example and evaluate our method on an open-access, clinical dataset studying knee joint configurations under the progression of osteoarthritis. Y1 - 2025 U6 - https://doi.org/10.1007/978-3-031-96628-6_4 SP - 49 EP - 63 PB - Springer CY - Lecture Notes in Computer Science ER - TY - CHAP A1 - Stokke, Jo Andersson A1 - Bergmann, Ronny A1 - Hanik, Martin A1 - von Tycowicz, Christoph T1 - p-Laplacians for Manifold-valued Hypergraphs T2 - Geometric Science of Information. GSI 2025 N2 - Hypergraphs extend traditional graphs by enabling the representation of N-ary relationships through higher-order edges. Akin to a common approach of deriving graph Laplacians, we define function spaces and corresponding symmetric products on the nodes and edges to derive hypergraph Laplacians. While this has been done before for Euclidean features, this work generalizes previous hypergraph Laplacian approaches to accommodate manifold-valued hypergraphs for many commonly encountered manifolds. Y1 - 2025 U6 - https://doi.org/10.1007/978-3-032-03924-8_17 VL - 16035 ER -