Stefan Kober

Christoph Hertrich, Stefan Kober, Georg Loho

We prove that there exist uniform $(+,\times,/)$-circuits of size $O(n^3)$ to compute the basis generating polynomial of regular matroids on $n$ elements. By tropicalization, this implies that there exist uniform $(\max,+,-)$-circuits and ReLU neural networks of the same size for weighted basis maximization of regular matroids. As a consequence in linear programming theory, we obtain a first example where taking the difference of two extended formulations can be more efficient than the best known individual extended formulation of size $O(n^6)$ by Aprile and Fiorini. Such differences have recently been introduced as virtual extended formulations. The proof of our main result relies on a fine-tuned version of Seymour's decomposition of regular matroids which allows us to identify and maintain graphic substructures to which we can apply a local version of the star-mesh transformation.

Stefan Kober

Traditional k-means clustering underperforms on non-convex shapes and requires the number of clusters k to be specified in advance. We propose a simple geometric enhancement: after standard k-means, each cluster center is assigned a radius (the distance to its farthest assigned point), and clusters whose radii overlap are merged. This post-processing step loosens the requirement for exact k: as long as k is overestimated (but not excessively), the method can often reconstruct non-convex shapes through meaningful merges. We also show that this approach supports recursive partitioning: clustering can be performed independently on tiled regions of the feature space, then globally merged, making the method scalable and suitable for distributed systems. Implemented as a lightweight post-processing step atop scikit-learn's k-means, the algorithm performs well on benchmark datasets, achieving high accuracy with minimal additional computation.