Raphael Hauser

Giancarlo Antonino Antonucci, Raphael Andreas Hauser, Debasmita Samaddar et al.

Time-parallel algorithms, such as Parareal, are well-understood for linear problems, but their convergence analysis for nonlinear, chaotic systems remains limited. This paper introduces a new theoretical framework for analysing time-decomposition methods as contraction mappings that converge in a finite number of iterations. We derive a finite-time guarantee linking the initial error, convergence rate, and iteration count, defined via a geometric outer--inner-ball condition. We apply this framework to Parareal, deriving explicit estimates for the convergence factor $β$ on nonlinear problems and showing it scales as $\mathcal{O}(h^2)$ when the macroscopic time grid is uniformly refined. Further, we address the failure of standard convergence criteria in chaotic regimes by introducing a proximity function. This chaos-aware criterion weighs solution discontinuities by the system's Lyapunov exponent (or the solver's Lipschitz constant), allowing the algorithm to converge to the correct statistical attractor without enforcing futile pointwise accuracy on divergent trajectories. Numerical experiments on the Logistic, Lorenz, and Lorenz-96 systems demonstrate that this approach decouples the iteration count from the total simulation time. By isolating the intrinsic mathematical bounds from hardware-dependent overheads, we establish that the method is strictly algorithmically scalable.

Reka A. Kovacs, Oktay Gunluk, Raphael A. Hauser

Binary matrix factorisation is an essential tool for identifying discrete patterns in binary data. In this paper we consider the rank-k binary matrix factorisation problem (k-BMF) under Boolean arithmetic: we are given an n x m binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension n x k and k x m respectively, which minimise the distance between X and the Boolean product of A and B in the squared Frobenius distance. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak LP relaxation, while the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times the zero is erroneously covered in a rank-k factorisation. For one of the exponential size IPs we describe a computational approach based on column generation. Experimental results on synthetic and real word datasets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorisations.

Reka A. Kovacs, Oktay Gunluk, Raphael A. Hauser

Identifying discrete patterns in binary data is an important dimensionality reduction tool in machine learning and data mining. In this paper, we consider the problem of low-rank binary matrix factorisation (BMF) under Boolean arithmetic. Due to the hardness of this problem, most previous attempts rely on heuristic techniques. We formulate the problem as a mixed integer linear program and use a large scale optimisation technique of column generation to solve it without the need of heuristic pattern mining. Our approach focuses on accuracy and on the provision of optimality guarantees. Experimental results on real world datasets demonstrate that our proposed method is effective at producing highly accurate factorisations and improves on the previously available best known results for 15 out of 24 problem instances.

Reka Kovacs, Oktay Gunluk, Raphael Hauser

Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding low-rank approximations to Boolean matrices. In this paper we give what is to the best of our knowledge the first integer programming formulation that relies on only polynomially many variables and constraints, we discuss how to solve it computationally and report numerical tests on synthetic and real-world data.

Felipe Cucker, Raphael Hauser, Martin Lotz

The purpose of this note is to extend the results on uniform smoothed analysis of condition numbers from \cite{BuCuLo:07} to the case where the perturbation follows a radially symmetric probability distribution. In particular, we will show that the bounds derived in \cite{BuCuLo:07} still hold in the case of distributions whose density has a singularity at the center of the perturbation, which we call {\em adversarial}.

Raphael A Hauser, Yongdo Lim

Self-scaled barrier functions are fundamental objects in the theory of interior-point methods for linear optimization over symmetric cones, of which linear and semidefinite programming are special cases. We are classifying all self-scaled barriers over irreducible symmetric cones and show that these functions are merely homothetic transformations of the universal barrier function. Together with a decomposition theorem for self-scaled barriers this concludes the algebraic classification theory of these functions. After introducing the reader to the concepts relevant to the problem and tracing the history of the subject, we start by deriving our result from first principles in the important special case of semidefinite programming. We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras.