The problem of hedging derivatives represents a central challenge in financial markets. Under Markovian models, the theory is very well developed, specifically for European-style derivatives. However, significant challenges arise when considering path-dependent options where the payoff depends on the asset's entire price path, or further still, when the underlying asset has non-Markovian dynamics, where conventional parametrized hedging approaches tend to be too restrictive or untractable. In response to these challenges, non-parametric approaches have gained a lot of popularity, and more specifically with the improvement of machine learning software and hardware, deep hedging approaches for their versatility, ease of train and ability to capture nonlinearities, see for instance B uhler et al. (2018).

We propose Path Signatures Logistic Regression (PSLR), a semi-parametric framework for classifying vector-valued functional data with scalar covariates. Classical functional logistic regression models rely on linear assumptions and fixed basis expansions, which limit flexibility and degrade performance under irregular sampling. PSLR overcomes these issues by leveraging truncated path signatures to construct a finite-dimensional, basis-free representation that captures nonlinear and cross-channel dependencies. By embedding trajectories as time-augmented paths, PSLR extracts stable, geometry-aware features that are robust to sampling irregularity without requiring a common time grid, while still preserving subject-specific timing patterns. We establish theoretical guarantees for the existence and consistent estimation of the optimal truncation order, along with non-asymptotic risk bounds. Experiments on synthetic and real-world datasets show that PSLR outperforms traditional functional classifiers in accuracy, robustness, and interpretability, particularly under non-uniform sampling schemes. Our results highlight the practical and theoretical benefits of integrating rough path theory into modern functional data analysis.

The paper starts by situating the problem and motivating their approach - essentially, enabling embeddings for weighted graphs by extending the original implicit embedding performed in Lovasz "Shannon capacity of a graph" paper, which operated on binary graphs. The paper then presents the connection between kernel machines and the Lovasz number in the unweighted case, and extends previous work for vertex-weighted, edge-weighted, and LS-labelled graphs. Section 3 provides practical details for computation, and section 4 motivates the use of their approach for the clustering problem - setting the number of clusters by using their \vartheta 1 bound, and initialising the clusters by starting by vertices with large alpha_i values. Finally, they show results on max-cut, clustering, overlapping clustering, and summarization tasks. The paper ties together very different work to propose a coherent approach to graph embedding. The contributions are clearly laid out, and the references to previous work is well established and used.