arXiv subjects

Rasmus Jensen

Publications and source records attributed to Rasmus Jensen.

6 recordsLinked to original sources

arXiv2026

Conditioning and interpolation error bounds for second-order Stiefel retractions with closed-form inverses

Retractions provide a computationally efficient alternative to the Riemannian exponential and logarithm maps for practical data-processing tasks on manifolds. In particular, second-order retractions with closed-form inverse are well-suited for interpolation problems on manifolds. On the Stiefel manifold of orthogonal frames, there are only two retractions of this type: the Cayley retraction, which is second-order accurate under the canonical metric, and the recently proposed polar-light retraction, which is second-order accurate under the Euclidean metric. In this paper, we study the properties of these maps in the context of interpolation on the Stiefel manifold. To obtain explicit interpolation error bounds, we examine the conditioning of the retraction maps and their inverses. We show that the retractions are well-conditioned, and we derive interpolation error bounds similar to those of classical Euclidean interpolation. The inverse retractions are not well-conditioned in general, and we discuss how data can be mapped via an isometric group action to ensure stable computations. As with all retractions on compact manifolds, the inverse canonical Cayley retraction and the inverse polar-light retraction exist only locally, and we construct normal neighborhoods around any point in which either the inverse Cayley retraction or the invese polar-light retraction are guaranteed to be computable. As an application of the retraction maps, we consider Hermite interpolation, where the objective is to reproduce both sampled function values and derivative information. A numerical example demonstrates that retraction-based interpolation is competitive with classical methods based on Riemannian normal coordinates.

math.NA↗

arXiv2026

Scalable Neighborhood-Based Multi-Agent Actor-Critic

We propose MADDPG-K, a scalable extension to Multi-Agent Deep Deterministic Policy Gradient (MADDPG) that addresses the computational limitations of centralized critic approaches. Centralized critics, which condition on the observations and actions of all agents, have demonstrated significant performance gains in cooperative and competitive multi-agent settings. However, their critic networks grow linearly in input size with the number of agents, making them increasingly expensive to train at scale. MADDPG-K mitigates this by restricting each agent's critic to the $k$ closest agents under a chosen metric which in our case is Euclidean distance. This ensures a constant-size critic input regardless of the total agent count. We analyze the complexity of this approach, showing that the quadratic cost it retains arises from cheap scalar distance computations rather than the expensive neural network matrix multiplications that bottleneck standard MADDPG. We validate our method empirically across cooperative and adversarial environments from the Multi-Particle Environment suite, demonstrating competitive or superior performance compared to MADDPG, faster convergence in cooperative settings, and better runtime scaling as the number of agents grows. Our code is available at https://github.com/TimGop/MADDPG-K .

cs.LG↗

arXiv2026

An new polar factor retraction on the Stiefel manifold with closed-form inverse

Retractions are the workhorses in Riemannian computing applications, where computational efficiency is of the essence. This work introduces a new retraction on the compact Stiefel manifold of orthogonal frames. The retraction is second-order accurate under the Euclidean metric and features a closed-form inverse that can be efficiently computed. A variety of retractions is known on the Stiefel manifold, including the Riemannian exponential map, the polar factor retraction, the QR-retraction, quasi--geodesics and the Cayley retraction. The Cayley retraction is second--order accurate under the canonical metric and features a closed-form inverse. The new retraction is the first one with the corresponding features under the Euclidean metric. We present numerical experiments which illustrates the properties of the new retraction, as well as compare it to numerous of the currently available alternatives. In addition, we examine the performance of the retraction when used for interpolation and for computing a Riemannian barycenter.

math.NA↗

arXiv2025

Maximum volume coordinates for Grassmann interpolation: Lagrange, Hermite, and errors

We present a novel approach to Riemannian interpolation on the Grassmann manifold. Instead of relying on the Riemannian normal coordinates, i.e. the Riemannian exponential and logarithm maps, we approach the interpolation problem with an alternative set of local coordinates and corresponding parameterizations. A special property of these coordinates is that their calculation does not require any matrix decompositions. This is a numerical advantage over Riemann normal coordinates and many other retractions on the Grassmann manifold, especially when derivative data are to be treated. To estimate the interpolation error, we examine the conditioning of these mappings and state explicit bounds. It turns out that the parameterizations are well-conditioned, but the coordinate mappings are generally not. As a remedy, we introduce maximum-volume coordinates that are based on a search for subblocks of column-orthogonal matrices of large absolute determinant. We show that the order of magnitude of the asymptotic interpolation error on $\Gr(n,p)$ is the same as in the Euclidean space. Two numerical experiments are conducted. The first is an academic one, where we interpolate a parametric orthogonal projector $QQ^T$, where the $Q$--factor stems from a parametric compact QR--decomposition. The second experiment is in the context of parametric model reduction of dynamical systems, where we interpolate reduced subspaces that are obtained by proper orthogonal decomposition.

math.NA↗

arXiv2024

Riemannian optimization on the symplectic Stiefel manifold using second-order information

Riemannian optimization is concerned with problems, where the independent variable lies on a smooth manifold. There is a number of problems from numerical linear algebra that fall into this category, where the manifold is usually specified by special matrix structures, such as orthogonality or definiteness. Following this line of research, we investigate tools for Riemannian optimization on the symplectic Stiefel manifold. We complement the existing set of numerical optimization algorithms with a Riemannian trust region method tailored to the symplectic Stiefel manifold. To this end, we derive a matrix formula for the Riemannian Hessian under a right-invariant metric. Moreover, we propose a novel retraction for approximating the Riemannian geodesics. Finally, we conduct a comparative study in which we juxtapose the performance of the Riemannian variants of the steepest descent, conjugate gradients, and trust region methods on selected matrix optimization problems that feature symplectic constraints.

math.NA↗

arXiv2023

Fighting Money Laundering with Statistics and Machine Learning

Money laundering is a profound global problem. Nonetheless, there is little scientific literature on statistical and machine learning methods for anti-money laundering. In this paper, we focus on anti-money laundering in banks and provide an introduction and review of the literature. We propose a unifying terminology with two central elements: (i) client risk profiling and (ii) suspicious behavior flagging. We find that client risk profiling is characterized by diagnostics, i.e., efforts to find and explain risk factors. On the other hand, suspicious behavior flagging is characterized by non-disclosed features and hand-crafted risk indices. Finally, we discuss directions for future research. One major challenge is the need for more public data sets. This may potentially be addressed by synthetic data generation. Other possible research directions include semi-supervised and deep learning, interpretability, and fairness of the results.

stat.ML↗