Michael Horton

Michael Horton, Patrick Lucey

Football (soccer) is a sport that is characterised by complex game play, where players perform a variety of actions, such as passes, shots, tackles, fouls, in order to score goals, and ultimately win matches. Accurately forecasting the total number of each action that each player will complete during a match is desirable for a variety of applications, including tactical decision-making, sports betting, and for television broadcast commentary and analysis. Such predictions must consider the game state, the ability and skill of the players in both teams, the interactions between the players, and the temporal dynamics of the game as it develops. In this paper, we present a transformer-based neural network that jointly and recurrently predicts the expected totals for thirteen individual actions at multiple time-steps during the match, and where predictions are made for each individual player, each team and at the game-level. The neural network is based on an \emph{axial transformer} that efficiently captures the temporal dynamics as the game progresses, and the interactions between the players at each time-step. We present a novel axial transformer design that we show is equivalent to a regular sequential transformer, and the design performs well experimentally. We show empirically that the model can make consistent and reliable predictions, and efficiently makes $\sim$75,000 live predictions at low latency for each game.

Lintao Wang, Shiwen Xu, Michael Horton et al.

Predicting soccer match outcomes is a challenging task due to the inherently unpredictable nature of the game and the numerous dynamic factors influencing results. While it conventionally relies on meticulous feature engineering, deep learning techniques have recently shown a great promise in learning effective player and team representations directly for soccer outcome prediction. However, existing methods often overlook the heterogeneous nature of interactions among players and teams, which is crucial for accurately modeling match dynamics. To address this gap, we propose HIGFormer (Heterogeneous Interaction Graph Transformer), a novel graph-augmented transformer-based deep learning model for soccer outcome prediction. HIGFormer introduces a multi-level interaction framework that captures both fine-grained player dynamics and high-level team interactions. Specifically, it comprises (1) a Player Interaction Network, which encodes player performance through heterogeneous interaction graphs, combining local graph convolutions with a global graph-augmented transformer; (2) a Team Interaction Network, which constructs interaction graphs from a team-to-team perspective to model historical match relationships; and (3) a Match Comparison Transformer, which jointly analyzes both team and player-level information to predict match outcomes. Extensive experiments on the WyScout Open Access Dataset, a large-scale real-world soccer dataset, demonstrate that HIGFormer significantly outperforms existing methods in prediction accuracy. Furthermore, we provide valuable insights into leveraging our model for player performance evaluation, offering a new perspective on talent scouting and team strategy analysis.

Joachim Gudmundsson, Michael Horton, John Pfeifer et al.

We present a scalable approach for range and $k$ nearest neighbor queries under computationally expensive metrics, like the continuous Fréchet distance on trajectory data. Based on clustering for metric indexes, we obtain a dynamic tree structure whose size is linear in the number of trajectories, regardless of the trajectory's individual sizes or the spatial dimension, which allows one to exploit low `intrinsic dimensionality' of data sets for effective search space pruning. Since the distance computation is expensive, generic metric indexing methods are rendered impractical. We present strategies that (i) improve on known upper and lower bound computations, (ii) build cluster trees without any or very few distance calls, and (iii) search using bounds for metric pruning, interval orderings for reduction, and randomized pivoting for reporting the final results. We analyze the efficiency and effectiveness of our methods with extensive experiments on diverse synthetic and real-world data sets. The results show improvement over state-of-the-art methods for exact queries, and even further speed-ups are achieved for queries that may return approximate results. Surprisingly, the majority of exact nearest-neighbor queries on real data sets are answered without any distance computations.

Kevin Buchin, Anne Driemel, Joachim Gudmundsson et al.

The Euclidean $k$-center problem is a classical problem that has been extensively studied in computer science. Given a set $\mathcal{G}$ of $n$ points in Euclidean space, the problem is to determine a set $\mathcal{C}$ of $k$ centers (not necessarily part of $\mathcal{G}$) such that the maximum distance between a point in $\mathcal{G}$ and its nearest neighbor in $\mathcal{C}$ is minimized. In this paper we study the corresponding $(k,\ell)$-center problem for polygonal curves under the Fréchet distance, that is, given a set $\mathcal{G}$ of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity $m$, determine a set $\mathcal{C}$ of $k$ polygonal curves in $\mathbb{R}^d$, each of complexity $\ell$, such that the maximum Fréchet distance of a curve in $\mathcal{G}$ to its closest curve in $\mathcal{C}$ is minimized. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension $2$ and higher. We show that, if $\ell$ is part of the input, then there is no polynomial-time approximation scheme unless $\mathsf{P}=\mathsf{NP}$. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to $2.598$. This result also holds when $k=1$, and the $\mathsf{NP}$-hardness extends to the case that $\ell=\infty$, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a $3$-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.

Michael Horton, Joachim Gudmundsson, Sanjay Chawla et al.

A knowledgeable observer of a game of football (soccer) can make a subjective evaluation of the quality of passes made between players during the game. We investigate the problem of producing an automated system to make the same evaluation of passes. We present a model that constructs numerical predictor variables from spatiotemporal match data using feature functions based on methods from computational geometry, and then learns a classification function from labelled examples of the predictor variables. Furthermore, the learned classifiers are analysed to determine if there is a relationship between the complexity of the algorithm that computed the predictor variable and the importance of the variable to the classifier. Experimental results show that we are able to produce a classifier with 85.8% accuracy on classifying passes as Good, OK or Bad, and that the predictor variables computed using complex methods from computational geometry are of moderate importance to the learned classifiers. Finally, we show that the inter-rater agreement on pass classification between the machine classifier and a human observer is of similar magnitude to the agreement between two observers.