Triangle inequalities of quantum Wasserstein distances on noncommutative tori

In 2022, Golse and Paul defined a pseudometric for quantum optimal transport that extends the classical Wasserstein distance. They proved that the pseudometric satisfies the triangle inequality in certain cases. This thesis reviews their proof in the case where the middle point is a classical density. Motivated by that proof, we formulate the...

Subgraph Matching via Fused Gromov-Wasserstein Distance

Subgraph matching is a fundamental problem in various fields such as machine learning, computer vision, image processing, and bioinformatics, where detecting specific substructures within an object is often crucial. In these domains, not only structure plays an essential role, but also the feature information on nodes should be incorporated,...

BECLR: Batch Enhanced Contrastive Unsupervised Few-Shot Learning

There exists a fundamental gap between human and artificial intelligence. Deep learning models are exceedingly data hungry for learning even the simplest of tasks, whereas humans can easily adapt to new tasks with just a handful of samples. Unsupervised few-shot learning (U-FSL) aspires to bridge this gap, without relying on costly annotations....

Hamilton–Jacobi equations for controlled gradient flows: The comparison principle

Motivated by recent developments in the fields of large deviations for interacting particle systems and mean field control, we establish a comparison principle for the Hamilton–Jacobi equation corresponding to linearly controlled gradient flows of an energy function E defined on a metric space (E,d). Our analysis is based on a systematic use...

Modelling a Simulation Environment to analyse Race Strategy in Formula E

In this report, I investigate strategic decision making in the Formula E racing series for Porsche. Formula E is an electric car circuit racing series, where the main tasks of race strategy are allocating energy consumption across the race and timing mandatory "attack mode" activations (similar to small-scale pitstops). I work on a flawed...

Self-Supervised Few Shot Learning: Prototypical Contrastive Learning with Graphs

A primary trait of humans is the ability to learn rich representations and relationships between entities from just a handful of examples without much guidance. Unsupervised few-shot learning is an undertaking aimed at reducing this fundamental gap between smart human adaptability and machines. We present a contrastive learning scheme for...

Marginal and Dependence Uncertainty: Bounds, Optimal Transport, And Sharpness

Motivated by applications in model-free finance and quantitative risk management, we consider Frechet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in the marginals is also possible. We derive optimal transport duality results for these Frechet classes that...

Quantifying the quality of coastal morphological predictions

This thesis investigates the behaviour of the often used point-wise skill score, the MSESSini a.k.a. BSS, and develops new error metrics that, as opposed to point-wise metrics, take the spatial structure of morphological patterns into account. The...

Optimal sediment transport for morphodynamic model validation

Although commonly used for the validation of morphological predictions, point-wise accuracy metrics, such as the root-mean-squared error (RMSE), are not well suited to demonstrate the quality of a high-variability prediction; in the presence of (often inevitable) location errors, the comparison of depth values per grid point tends to favour...

Comparison and unification of material-point and optimal transportation meshfree methods

Both the material-point method (MPM) and optimal transportation meshfree (OTM) method have been developed to efficiently solve partial differential equations that are based on the conservation laws from continuum mechanics. However, the methods are derived in a different fashion and have been studied independently of one another. In this...

Algorithmic improvements of the material-point method and Taylor least-squares function reconstruction

The material-point method (MPM) is a continuum-based numerical tool to simulate problems that involve large deformations. Within MPM, a continuum is discretized by defining a set of Lagrangian particles, called material points, which store all relevant material properties. Themethod adopts an Eulerian background grid, where the equations of...

Comparison between Material Point Method and meshfree schemes derived from optimal transportation theory

Both the Material Point Method (MPM) and meshfree schemes based on optimal transport theory have been developed for efficient and robust integration of the weak form equations originating from computational mechanics. Although the methods are derived in a different fashion, their algorithms share many similarities. In this paper, we outline the...

Regularization via mass transportation

The goal of regression and classification methods in supervised learning is to minimize the empirical risk, that is, the expectation of some loss function quantifying the prediction error under the empirical distribution. When facing scarce training data, overfitting is typically mitigated by adding regularization terms to the objective that...