An optimal transport based embedding to quantify the distance between playing styles in collective sports

A Distance between teams in the NBA

Similarly to Section 3.3, we apply the optimal transport framework to tracking data from the first half of the 2015–2016 NBA season. For each team, the tracking data was subsampled by retaining one out of every 25 frames. We set L = 6 to obtain an embedding in R 5 × 6 by projecting along the grid of directions defined for l = 1,…, L by

Figure 9 illustrates an example of the ten clusters identified from the collection of frames across all Golden State Warriors games, and Table 5 provides the percentage of frames in each cluster. We observe that the clustering algorithm effectively distinguishes different phases of play in a consistent manner. Specifically, we identify three distinct defensive settings in clusters 6, 2, and 10, where the defensive block transitions from very deep positions to more advanced ones. Additionally, clusters 3, 5, 8, and 9 represent transitions between defense and offense, characterized by players being spread across the court. The dynamics observed in these clusters differ significantly from those in football. In particular, clusters where players are positioned in the center of the court account for a total of 18.62 % of the frames, unlike in football, where central clusters are more prevalent. This difference is expected because basketball rules prohibit the ball from moving backward from the offensive half of the court. Consequently, basketball is essentially a transition-oriented game where teams alternate between defense and offense.

Figure 9:

Example of 10 clusters observed in the frames from the games of the Golden State Warriors. For each cluster, the closest frame to the barycenter and a random example from the cluster are shown. The shaded areas represent the total density of player locations taken from a random sample of 100 frames from each cluster. Clusters 2, 6 and 10 represent defensive phases. Clusters 1 and 4 represent attacking phases while clusters 3, 5, 7, 8 and 9 represent transitions between attack and defense.

Finally, Figure 10 illustrates the similarity in playing styles between NBA teams. This similarity is computed by considering the collection of frames from all available games in the first half of the NBA season, subsampled at a rate of one frame every 25 frames. We observe that the Golden State Warriors exhibit the largest deviation from the other franchises. This is not surprising, as they are credited with introducing a new style of play heavily reliant on 3-pointers. In fact, this season marked the record for the most points scored in the regular season, which naturally leads to a different spatial distribution of play. Interestingly, the Utah Jazz display the second largest deviation from the rest of the teams and its distance to the Golden State Warriors is the greatest. Figure 11 presents the similarity metric after centering the frames prior to embedding. We observe in this case that the distance metrics become more uniform, and the significant deviations of the Utah Jazz and Golden State Warriors diminish. This suggests that these effects are primarily due to the average locations both teams occupy during their games rather than relative placement of players.

Figure 10:

Distance matrix between the collections of frames in the games of each team in the NBA.

Figure 11:

Distance matrix between the collections of frames in the games of each team in the NBA after centering the data.

A.1 Predicting team identity

Similarly to Section 3.3.3, we demonstrate that our representation of collections of frames as distributions in the embedding space effectively captures the stylistic identity of teams in the NBA. Specifically, we show that it is possible to predict a team’s identity from an out-of-sample collection of frames with high accuracy, which provides strong evidence that our embedding preserves spatial information while differentiating between unique playing styles.

Using all frames in each test fold yields an average Top-1 accuracy of 99.33 % and a Top-2 accuracy of 100 %. This performance suggests that the distribution of frames in the embedding perfectly captures the style of play of a team with a large enough test sample of frames. Figure 12 shows how accuracy varies as a function of the number of frames in each subset. Similarly to the football data, larger subsets naturally improve accuracy, as they offer a more representative idea of a team’s positioning style. Compared to football, we need more frames to achieve an accuracy of 70 %, but the ceiling of precision is higher. In fact, we achieve 93.8 % Top-1 accuracy with 2000 frames. Overall, the results further confirm that our embedding captures team-specific spatial configurations, enabling accurate and fast team identification.

Figure 12:

Accuracy of NBA team identity prediction as a function of test sample size. For each sample size, 100 randomly selected subsets of the test fold are used to compute the accuracy and its standard deviation.

B Proof of Proposition 1

First, we prove that Proj _θ is injective. We proceed by induction on n

(H _n ): For any choice of grid θ ₁, θ ₂,…, θ _L such that L ≥ n + 1 and θ _l , θ _k are noncollinear for all l, k ≤ L, Proj _θ is injective.

If n = 1, take μ = δ _x and ν = δ _y and θ ₁, θ ₂,…, θ _L such that L ≥ 2 and Proj _θ (μ) = Proj _θ (ν). Then we have δ ⟨ θ 1 , x ⟩ = δ ⟨ θ 1 , y ⟩ and δ ⟨ θ 2 , x ⟩ = δ ⟨ θ 2 , y ⟩ . Equivalently, ⟨θ ₁, x⟩ = ⟨θ ₁, y⟩ and ⟨θ ₂, x⟩ = ⟨θ ₂, y⟩ which implies x = y since θ ₁ and θ ₂ are non collinear. Thus, μ = ν.

Let n ≥ 2 and assume (H _n−1) is true. Take μ and ν in P n u ( R 2 ) such that Proj _θ (μ) = Proj _θ (ν). We can write μ = ∑ i = 1 n 1 n δ x i and ν = ∑ i = 1 n 1 n δ y i and we have for all l in {1, …, L}:

Therefore, for all l in {1, …, L}, there exists i _l in {1, …, n} such that

Since L ≥ n + 1 and using the pigeon-hole principle, there must exist l, k such that i _l = i _k . Thus

we can deduce that x n = y i l . And we have

We conclude using (H _n−1) on μ ′ = ∑ i = 1 n − 1 1 n − 1 δ x h , i and ν ′ = ∑ i ≠ i l 1 n − 1 δ y h , i .

Finally, for μ and ν in P n u ( R 2 ) we have

This justifies that S W ̂ p ( . , . , ( θ l ) l ≤ L ) is a distance over P n u ( R 2 ) .

C Justification of the normalisation term

The following result allows to determine a suitable normalisation factor for the sliced-Wasserstein distance.

To prove Equation (10), consider μ = 1 n ∑ i = 1 n δ x i and ν = 1 n ∑ i = 1 n δ y i and let σ be a permutation such that

Then, for all l = 1, …, L, we have

Thus,

where Θ = ∑ l = 1 L θ l θ l T is a positive semi-definite matrix. Let ρ(Θ) be its spectral radius, we have

For the choice of grid in Equation (7), we have θ l θ l T = cos 2 ( π ( l − 1 ) 2 L ) cos ( π ( l − 1 ) 2 L ) sin ( π ( l − 1 ) 2 L ) cos ( π ( l − 1 ) 2 L ) sin ( π ( l − 1 ) 2 L ) sin 2 ( π ( l − 1 ) 2 L ) and hence

This yields ρ ( Θ ) = L 2 ( 1 + 1 L ⁡ sin ( π 2 L ) ) and we deduce that

For the second bound, we consider the case where ν = δ _y is concentrated in one location. Similar calculations yield

where γ(Θ) is the smallest eigenvalue of Θ and is given, for the grid in Equation (7), by γ ( Θ ) = L 2 ( 1 − 1 L ⁡ sin ( π 2 L ) ) . Therefore, we obtain