Which metric on the space of collider events?

Which is the best metric for the space of collider events? Motivated by the success of the energy mover’s distance in characterizing collider events, we explore the larger space of unbalanced optimal transport distances, of which the energy mover’s distance is a particular case. Geometric and computational considerations favor an unbalanced optimal transport distance known as the Hellinger-Kantorovich distance, which possesses a Riemannian structure that lends itself to efficient linearization. We develop the particle linearized unbalanced optimal transport framework for collider events based on the linearized Hellinger-Kantorovich distance and demonstrate its efficacy in boosted jet tagging. This provides a flexible and computationally efficient optimal transport framework ideally suited for collider physics applications.

Figure 1

Optimal transport plans from uniform reference measures to jets. In the top row, the reference measure (blue) consists of $8\times 8$ particles and is transported to an artificial jet (green), composed of a single particle at the origin. In the second row, the uniform reference measure (blue) consists of $15\times 15$ particles and is transported to a simulated $W$ jet (red). The columns correspond to different choices of OT metric: $W_2$ and ${\mathrm{HK}}^{\kappa }$ with $\kappa =100$, 10, 1, 0.1 (from left to right). The size of the filled dots indicates the amount of $p_T$ at that point. The darkness of the lines indicates how much $p_T$ is moved from one particle to another. For ${\mathrm{HK}}^{\kappa }$, the thickness of the circles around the points represents how much $p_T$ is destroyed for that particular particle. Also shown at the bottom of the plots are the total OT distances between the jets, which are similar for $\kappa =+\infty ,100$, 10, the transport regime.

Figure 2

AUC scores for classifying $W$ vs QCD jets using kNN and SVM models coupled to linear $W_2$ and ${\mathrm{HK}}^{\kappa }$ embedding with $\kappa \in [0.01,+\infty ]$. Jet $p_T$ ranges from 500 to 550 (1500) GeV in the first (second, third) column. The datasets for column 1 and column 2 (column 3) have 10 000 (200 000) jets. Solid (dashed) blue lines show the results calculated on normalized (unnormalized) jets; horizontal gray solid (dashed) lines use the EMD metrics on normalized (unnormalized) jets; and gray dash-dotted lines give the performance using ${\tau }_{21}$ as the discriminator.

Figure 3

Distribution of the LOT norm, i.e., the distance from each jet’s LOT coordinate to the origin, for the HK distance with $\kappa =100$, 1, 0.5, 0.1, 0.05, 0.01. The uniform reference measures used include uniref15 with $15\times 15$ particles, uniref30, and uniref60. The y coordinate of the rightmost plot follows the scale on the right, while the other plots follow the scale on the left.

Figure 4

AUC scores for classifying 10 000 W vs QCD jets using different reference measures, with uniref4, uniref8, uniref15, uniref30, and uniref60 (from left to right). The machine learning model used here is kNN. Jet $p_T$ is in between 500 and 550 (1500) GeV in the first (second) row. Solid (dashed) blue lines show the results calculated on normalized (unnormalized) jets for $W_2$ and HK distance; horizontal gray solid (dashed) lines use the EMD metrics on normalized (unnormalized) jets; and gray dash-dotted lines give the tagging performance of ${\tau }_{21}$.

Figure 5

AUC scores for using kNN to classify 10 000 W vs QCD jets with different amounts of pileup where the average numbers of pileup particles in each event are $⟨N_{PU}⟩=20$ (red), 80 (blue), and 140 (green). From left to right, the reference measures used are the $15\times 15$ uniform reference, the $30\times 30$ uniform reference, and a jet drawn from the pileup template corresponding to each $N_{PU}$. As usual, solid (dashed) lines show the AUC scores using the linear HK and $W_2$ distances on normalized (unnormalized) jets, and solid horizontal lines give the tagging performance of ${\tau }_{21}$ on unpruned jets, whereas the dash-dotted lines are the results using ${\tau }_{21}$ on pruned jets (denoted by ${\tau }_{21}^{pr}$).