Aktuelles & Events

Graphical continuous Lyapunov models offer a novel framework for the statistical modeling of correlated multivariate data. These models define the covariance matrix through a continuous Lyapunov equation, parameterized by the drift matrix of the underlying dynamic process. In this talk, I will discuss key results on the defining equations of these models and explore the challenge of structural identifiability. Specifically, I will present conditions under which models derived from different directed acyclic graphs (DAGs) are equivalent and provide a transformational characterization of such equivalences. This is based on ongoing work with Carlos Amendola, Tobias Boege, and Ben Hollering.

We study opportunistic traders that try to detect and exploit the order flow of dealers hedging their net exposure to the FX fix. We also discuss how dealers can take this into account to balance not only risk and trading costs but also information leakage in an appropriate manner. It turns out that information leakage significantly expands the set of scenarios where both dealers and the clients whose orders they execute benefit from hedging part of the exposure before the fixing window itself. (Joint work in progress with Roel Oomen (Deutsche Bank) and Mateo Rodriguez Polo (ETH Zurich))

The exact treatment of Markovian models on complex networks requires knowledge of probability distributions expo- nentially large in the number of nodes n. Mean-field approximations provide an effective reduction in complexity of the models, requiring only a number of phase space variables polynomial in system size. However, this comes at the cost of losing accuracy close to critical points in the systems dynamics and an inability to capture correlations in the system. In this talk, we introduce a tunable approximation scheme for Markovian spreading models on networks based on matrix product states (MPSs). By controlling the bond dimensions of the MPS, we can investigate the effective dimensional- ity needed to accurately represent the exact 2n dimensional steady-state distribution. We introduce the entanglement entropy as a measure of the compressibility of the system and find that it peaks just after the phase transition on the disordered side, in line with the intuition that more complex states are at the ’edge of chaos’. The MPS provides a systematic way to tune the accuracy of the approximation by reducing the dimensionality of the systems state vector, leading to an improvement over second-order mean-field approximations for sufficiently large bond dimensions.

In this talk, I will discuss two of our recent results on three-dimensional (3D) packing problems: the 3D Knapsack problem and the 3D Bin Packing problem. In both settings, we are given a collection of axis-aligned cuboids. In the Knapsack problem, each cuboid is associated with a profit, and the objective is to pack a subset of cuboids non-overlappingly into a unit cube to maximize total profit. In contrast, the Bin Packing problem seeks to pack all the cuboids using the minimum number of unit cubes (bins). Both problems are NP-hard and unlike their two-dimensional counterparts that have been extensively studied, the 3D variants have received much less attention. The previously best-known approximation ratios for 3D Knapsack and 3D Bin Packing are 7 + ε and (T_∞)^2 + ε ≈ 2.86, respectively for any constant ε > 0, where T_∞ ≈ 1.691 is the well-known Harmonic constant in Bin Packing. We provide improved approximation ratios of 139/29 + ε ≈ 4.794, and 3T_∞/2 + ε ≈ 2.54, for 3D Knapsack and 3D Bin Packing, respectively. Our key technical contribution is container packing -- a structured packing in 3D wherein all items are assigned into a constant number of containers, and each container is packed using a specific strategy based on its type. I shall also discuss few extensions of our techniques to related 3D packing problems.