# Learning the Geometry of Collider Events with Metric-Aware Deep Sets

arXiv: https://arxiv.org/abs/2609.12024 · HTML: https://mlsanity.com/paper/2609.12024/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 36 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 11 of 11)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Energy Flow Networks: Deep Sets for Particle Jets](https://mlsanity.com/paper/1810.05165/) (2018)
- [Sinkhorn Divergences for Unbalanced Optimal Transport](https://mlsanity.com/paper/1910.12958/) (2019)
- [Linearized Optimal Transport for Collider Events](https://mlsanity.com/paper/2008.08604/) (2020)
- [Particle Graph Autoencoders and Differentiable, Learned Energy Mover's Distance](https://mlsanity.com/paper/2111.12849/) (2021)
- [Universal Representation of Permutation-Invariant Functions on Vectors and Tensors](https://mlsanity.com/paper/2310.13829/) (2023)
- [PINE: Universal Deep Embedding for Graph Nodes via Partial Permutation Invariant Set Functions](https://mlsanity.com/paper/1909.12903/) (2019)
- [GPU coprocessors as a service for deep learning inference in high energy physics](https://mlsanity.com/paper/2007.10359/) (2020)
- [Neural Embedding: Learning the Embedding of the Manifold of Physics Data](https://mlsanity.com/paper/2208.05484/) (2022)
- [Finding NEEMo: Geometric Fitting using Neural Estimation of the Energy Mover's Distance](https://mlsanity.com/paper/2209.15624/) (2022)

## Abstract (arXiv)

Optimal transport gives structured data a geometry, but exact evaluation is costly in large pairwise analyses that exploit relationships among distances. Learned surrogates are faster, but need not preserve this metric structure. We develop a Deep Sets surrogate for OT between variable-size weighted point clouds that enforces non-negativity, exchange symmetry, and zero self-distance, leaving the triangle inequality unconstrained. Applied to the Energy Mover's Distance between collider events in a particle physics application, the Metric-Aware Particle Flow Network achieves percent-level mean absolute percentage error while significantly improving inference throughput over other exact and approximate methods surveyed. The architectural constraints are found to improve properties that are not explicitly enforced: across 106 held-out event triplets, triangle-inequality violations fall from 199 for a matched unconstrained network to 2, and the maximum from 149.5 to 5.8 GeV. These results demonstrate that targeted inductive biases can yield fast neural surrogates with substantially improved geometric fidelity.