Searcharxiv⌕ Search

arXiv subjects

Maximilian Penka

Publications and source records attributed to Maximilian Penka.

3 recordsLinked to original sources
arXiv2024

Genetic Column Generation for Computing Lower Bounds for Adversarial Classification

Recent theoretical results on adversarial multi-class classification showed a similarity to the multi-marginal formulation of Wasserstein-barycenter in optimal transport. Unfortunately, both problems suffer from the curse of dimension, making it hard to exploit the nice linear program structure of the problems for numerical calculations. We investigate how ideas from Genetic Column Generation for multi-marginal optimal transport can be used to overcome the curse of dimension in computing the minimal adversarial risk in multi-class classification.

math.NA↗
arXiv2023

Convergence proof for the GenCol algorithm in the case of two-marginal optimal transport

The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor $β$. Here we prove that for $β\geq 2$ and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from $O(\ell^2)$ to $O(\ell)$ without any loss in accuracy, where $\ell$ is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.

math.NA↗