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  • Open Access

Optimal transport event representation for anomaly detection

Tianji Cai1,2,*, Aditya Bhargava3, and Benjamin Nachman4,5

  • *Contact author: tianji_cai@tongji.edu.cn
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Phys. Rev. D 114, 016011 – Published 10 July, 2026

DOI: https://doi.org/10.1103/wwzk-25sy

Abstract

We introduce optimal transport (OT) as a physics-based intermediate event representation for weakly supervised anomaly detection. With only 0.5% injection of resonant signals in the LHC Olympics benchmark datasets, the OT-augmented feature set achieves nearly twice the significance improvement of the standard high-level observables using an idealized setup, while end-to-end deep learning on low-level four-momenta is less effective in this low-signal regime. The observed gains persist across signal types and classifiers considered in this study, suggesting that structured, physics-informed representations can provide a useful complement to existing approaches for anomaly detection.

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References (59)

  1. G. Kasieczka et al., The LHC Olympics 2020 a community challenge for anomaly detection in high energy physics, Rep. Prog. Phys. 84, 124201 (2021).
  2. T. Aarrestad et al., The Dark machines anomaly score challenge: Benchmark data and model independent event classification for the large hadron collider, SciPost Phys. 12, 043 (2022).
  3. G. Karagiorgi, G. Kasieczka, S. Kravitz, B. Nachman, and D. Shih, Machine learning in the search for new fundamental physics, arXiv:2112.03769.
  4. V. Belis, P. Odagiu, and T. K. Aarrestad, Machine learning for anomaly detection in particle physics, Rev. Phys. 12, 100091 (2024).
  5. J. Hernández-González, I. Inza, and J. A. Lozano, Weak supervision and other non-standard classification problems: A taxonomy, Pattern Recogn. Lett. 69, 49 (2016).
  6. E. M. Metodiev, B. Nachman, and J. Thaler, Classification without labels: Learning from mixed samples in high energy physics, J. High Energy Phys. 10 (2017) 174.
  7. J. Collins, K. Howe, and B. Nachman, Anomaly detection for resonant new physics with machine learning, Phys. Rev. Lett. 121, 241803 (2018).
  8. R. T. D’Agnolo and A. Wulzer, Learning new physics from a machine, Phys. Rev. D 99, 015014 (2019).
  9. G. Aad et al. (ATLAS Collaboration), Dijet resonance search with weak supervision using s=13  TeV pp collisions in the atlas detector, Phys. Rev. Lett. 125, 131801 (2020).
  10. G. Aad et al. (ATLAS Collaboration), Weakly supervised anomaly detection for resonant new physics in the dijet final state using proton-proton collisions at s=13  TeV with the atlas detector, Phys. Rev. D 112, 072009 (2025).
  11. V. Chekhovsky et al. (The CMS Collaboration), Model-agnostic search for dijet resonances with anomalous jet substructure in proton–proton collisions at s=13  TeV, Rep. Prog. Phys. 88, 067802 (2025).
  12. R. Gambhir, R. Mastandrea, B. Nachman, and J. Thaler, Isolating unisolated upsilons with anomaly detection in cms open data, Phys. Rev. Lett. 135, 021902 (2025).
  13. D. Shih, M. R. Buckley, L. Necib, and J. Tamanas, via machinae: Searching for stellar streams using unsupervised machine learning, Mon. Not. R. Astron. Soc. 509, 5992 (2021).
  14. D. Shih, M. R. Buckley, and L. Necib, Via machinae 2.0: Full-sky, model-agnostic search for stellar streams in Gaia DR2, Mon. Not. R. Astron. Soc. 529, 4745 (2024).
  15. A. Hallin, D. Shih, C. Krause, and M. R. Buckley, Via machinae 3.0: A search for stellar streams in Gaia with the CATHODE algorithm, arXiv:2509.08064.
  16. M. Pettee, S. Thanvantri, B. Nachman, D. Shih, M. R. Buckley, and J. H. Collins, Weakly supervised anomaly detection in the milky way, Mon. Not. R. Astron. Soc. 527, 8459 (2023).
  17. D. Sengupta, S. Mulligan, D. Shih, J. A. Raine, and T. Golling, skycurtainss: Model-agnostic search for stellar streams with Gaia data, Mon. Not. R. Astron. Soc. 536, 1104 (2024).
  18. A. Hallin, J. Isaacson, G. Kasieczka, C. Krause, B. Nachman, T. Quadfasel, M. Schlaffer, D. Shih, and M. Sommerhalder, Classifying anomalies through outer density estimation, Phys. Rev. D 106, 055006 (2022).
  19. K. Benkendorfer, L. Le Pottier, and B. Nachman, Simulation-assisted decorrelation for resonant anomaly detection, Phys. Rev. D 104, 035003 (2021).
  20. E. Buhmann, C. Ewen, G. Kasieczka, V. Mikuni, B. Nachman, and D. Shih, Full phase space resonant anomaly detection, Phys. Rev. D 109, 055015 (2024).
  21. B. Nachman and D. Shih, Anomaly detection with density estimation, Phys. Rev. D 101, 075042 (2020).
  22. G. Stein, U. Seljak, and B. Dai, Unsupervised in-distribution anomaly detection of new physics through conditional density estimation, arXiv:2012.11638.
  23. C. L. Cheng, G. Singh, and B. Nachman, Incorporating physical priors into weakly supervised anomaly detection, Phys. Rev. Lett. 135, 021801 (2025).
  24. V. Mikuni and B. Nachman, Method to simultaneously facilitate all jet physics tasks, Phys. Rev. D 111, L054015 (2025).
  25. G. Peyré and M. Cuturi, Computational optimal transport, arXiv:1803.00567.
  26. P. T. Komiske, E. M. Metodiev, and J. Thaler, Metric space of collider events, Phys. Rev. Lett. 123, 041801 (2019).
  27. T. Cai, Optimal transport for high energy physics, Ph.D. thesis, UC, Santa Barbara (main), 2023.
  28. T. Cai, J. Cheng, N. Craig, and K. Craig, Linearized optimal transport for collider events, Phys. Rev. D 102, 116019 (2020).
  29. T. Cai, J. Cheng, K. Craig, and N. Craig, Which metric on the space of collider events?, Phys. Rev. D 105, 076003 (2022).
  30. T. Cai, J. Cheng, N. Craig, G. Koszegi, and A. J. Larkoski, The phase space distance between collider events, J. High Energy Phys. 09 (2024) 054.
  31. P. T. Komiske, E. M. Metodiev, and J. Thaler, The hidden geometry of particle collisions, J. High Energy Phys. 07 (2020) 006.
  32. A. J. Larkoski and J. Thaler, A spectral metric for collider geometry, J. High Energy Phys. 12 (2023) 219.
  33. R. Gambhir, A. J. Larkoski, and J. Thaler, SPECTER: Efficient evaluation of the spectral EMD, J. High Energy Phys. 12 (2025) 219.
  34. P. Onyisi, D. Shen, and J. Thaler, Comparing point cloud strategies for collider event classification, Phys. Rev. D 108, 012001 (2023).
  35. T. Cai, N. Craig, K. Craig, and X. Lin, Multiscale optimal transport for complete collider events, Phys. Rev. D 112, 036021 (2025).
  36. G. Aad et al. (ATLAS Collaboration), A continuous calibration of the ATLAS flavour-tagging classifiers via optimal transportation maps, Eur. Phys. J. C 85, 1272 (2025).
  37. N. Craig, J. N. Howard, and H. Li, Exploring optimal transport for event-level anomaly detection at the large hadron collider, arXiv:2401.15542.
  38. L. Brennan, T. A. Vami, O. Amram, S. Sekhar, Y. Takahashi, L. Moureaux, M. Sommerhalder, P. Maksimovic, T. Cai, and N. Craig, Weakly supervised anomaly detection with event-level variables, Phys. Rev. D 112, 055040 (2025).
  39. R. T. D’Agnolo, A. Glioti, G. Rigo, and A. Valenti, The intrinsic dimension of collider events and model-independent searches in 100 dimensions, arXiv:2511.20760.
  40. G. Kasieczka, B. Nachman, and D. Shih, R&D Dataset for LHC Olympics 2020 Anomaly Detection Challenge, 10.5281/zenodo.6466204 (2022).
  41. D. Shih, Additional QCD Background Events for LHCO2020 R&D (signal region only). 10.5281/zenodo.8370758 (2023).
  42. T. Sjostrand, S. Mrenna, and P. Z. Skands, pythia 6.4 Physics and Manual, J. High Energy Phys. 05 (2006) 026.
  43. T. Sjostrand, S. Mrenna, and P. Z. Skands, A brief introduction to pythia 8.1, Comput. Phys. Commun. 178, 852 (2008).
  44. J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, and M. Selvaggi (DELPHES 3 Collaboration), delphes 3, A modular framework for fast simulation of a generic collider experiment, J. High Energy Phys. 02 (2014) 057.
  45. M. Cacciari, G. P. Salam, and G. Soyez, fastjet user manual, Eur. Phys. J. C 72, 1896 (2012).
  46. M. Cacciari, G. P. Salam, and G. Soyez, The anti-kt jet clustering algorithm, J. High Energy Phys. 04 (2008) 063.
  47. J. Neyman, E. S. Pearson, and K. Pearson, Ix. on the problem of the most efficient tests of statistical hypotheses, Phil. Trans. R. Soc. A 231, 289 (1933).
  48. A. Andreassen, B. Nachman, and D. Shih, Simulation assisted likelihood-free anomaly detection, Phys. Rev. D 101, 095004 (2020).
  49. J. A. Raine, S. Klein, D. Sengupta, and T. Golling, CURTAINs for your sliding window: Constructing unobserved regions by transforming adjacent intervals, Front. Big Data 6, 899345 (2023).
  50. T. Finke, M. Hein, G. Kasieczka, M. Krämer, A. Mück, P. Prangchaikul, T. Quadfasel, D. Shih, and M. Sommerhalder, Tree-based algorithms for weakly supervised anomaly detection, Phys. Rev. D 109, 034033 (2024).
  51. M. Freytsis, M. Perelstein, and Y. C. San, Anomaly detection in the presence of irrelevant features, J. High Energy Phys. (2024) 220.
  52. J. Thaler and K. Van Tilburg, Identifying boosted objects with N-subjettiness, J. High Energy Phys. 03 (2010) 015.
  53. J. Thaler and K. Van Tilburg, Maximizing boosted top identification by minimizing N-subjettiness, J. High Energy Phys. 02 (2011) 093.
  54. T. Cai, J. Cheng, B. Schmitzer, and M. Thorpe, The linearized Hellinger–Kantorovich distance, SIAM J. Imag. Sci. 15, 45 (2022).
  55. C. Cesarotti and J. Thaler, A robust measure of event isotropy at colliders, J. High Energy Phys. 08 (2020) 084.
  56. M. J. Strassler and K. M. Zurek, Echoes of a hidden valley at hadron colliders, Phys. Lett. B 651, 374 (2007).
  57. T. Han, Z. Si, K. M. Zurek, and M. J. Strassler, Phenomenology of hidden valleys at hadron colliders, J. High Energy Phys. 07 (2008) 008.
  58. K. Craig, N. G. Trillos, and D. Nikolić, Vector valued optimal transport: From dynamic to static formulations, arXiv:2505.03670.
  59. https://github.com/TianjiCai/ADwithOT.

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Phys. Rev. D 114, 016011– Published 10 July, 2026
  • Received 24 March 2026
  • Accepted 24 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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