Topological Data Analysis and Graph-Theoretic Approaches for Tennis Match Prediction

We present two approaches for predicting tennis match outcomes using topological data analysis and graph theory on ATP singles matches from 2000-2025. The first method applies lower-star filtration to player competitive networks, extracting topological features through persistent homology using four summary methods (VAB, HNAV, HWNAV, OW-HNPV) combined with Modified Band Depth analysis. Algorithmic optimizations including ego graph approximations and triangle elimination enable analysis of about 66k matches. Our Random Forest model achieves 66.2% accuracy (AUC = 0.719) using topological, graph-theoretic, and ranking features. Feature importance analysis reveals that rankings contribute 36.3%, centralities 25.5%, and TDA features 24.0%, with topological features providing complementary signal. When rankings are unavailable, the topology-only model maintains 63.56% accuracy, demonstrating that network-derived features alone capture meaningful competitive structure. The second method uses a modified Katz similarity index with temporal edge weighting, achieving 62.48% accuracy on held-out test data. This work represents the first application of lower-star filtration to tennis prediction, provides systematic comparison of four topological summary methods in sports analytics, and demonstrates that TDA can achieve above-chance prediction using network topology alone while providing additional value when combined with traditional features.

A Universal Dense Football Event Representation Based on TabTransformer

Football event data constitute a rich spatiotemporal source for quantitative analysis of player actions in team sports. These datasets contain heterogeneous features, combining continuous location coordinates with categorical variables such as action type, action outcome, and body part. Such data have been applied in sports analytics for match outcome forecasting, player evaluation, and tactical pattern recognition. However, existing approaches predominantly encode categorical features using one-hot or ordinal embedding representations, overlooking the intrinsic semantics of action descriptors. The Transformer is a deep neural network architecture based on self-attention that captures dependencies between input features at arbitrary positions. We propose and implement a Transformer-based model to learn latent dependencies among categorical event features and produce dense representations of football events. By encoding categorical features as learned embedding vectors, sport-specific action semantics are captured during pretraining, enabling the representations to support downstream tasks such as action value estimation and play style recognition. Empirical evaluation shows that the embedding representations yield superior probability calibration over task-specific baselines on the downstream prediction tasks, as measured by Brier score.