## Abstract

The identification of interesting substructures within jets is an important tool for searching for new physics and probing the Standard Model at colliders. Many of these substructure tools have previously been shown to take the form of optimal transport problems, in particular the Energy Mover’s Distance (EMD). In this work, we show that the EMD is in fact *the* natural structure for comparing collider events, which accounts for its recent success in understanding event and jet substructure. We then present a Shape Hunting Algorithm using Parameterized Energy Reconstruction (Shaper), which is a general framework for defining and computing shape-based observables. Shaper generalizes *N*-jettiness from point clusters to any extended, parametrizable shape. This is accomplished by efficiently minimizing the EMD between events and parameterized manifolds of energy flows representing idealized shapes, implemented using the dual-potential Sinkhorn approximation of the Wasserstein metric. We show how the geometric language of observables as manifolds can be used to define novel observables with built-in infrared-and-collinear safety. We demonstrate the efficacy of the Shaper framework by performing empirical jet substructure studies using several examples of new shape-based observables.

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## References

E. Farhi,

*A QCD test for jets*,*Phys. Rev. Lett.***39**(1977) 1587 [INSPIRE].D.P. Barber et al.,

*Tests of quantum chromodynamics and a direct measurement of the strong coupling constant α*_{S}*at*\( \sqrt{s} \) = 30*GeV*,*Phys. Lett. B***89**(1979) 139 [INSPIRE].M. Dasgupta and G.P. Salam,

*Event shapes in e*^{+}*e*^{−}*annihilation and deep inelastic scattering*,*J. Phys. G***30**(2004) R143 [hep-ph/0312283] [INSPIRE].G. Dissertori et al.,

*e*^{+}*e*^{−}→ 3*jets and event shapes at NNLO*,*Nucl. Phys. B Proc. Suppl.***183**(2008) 2 [arXiv:0806.4601] [INSPIRE].L.G. Almeida et al.,

*Substructure of high-p*_{T}*jets at the LHC*,*Phys. Rev. D***79**(2009) 074017 [arXiv:0807.0234] [INSPIRE].G. Gur-Ari, M. Papucci and G. Perez,

*Classification of energy flow observables in narrow jets*, arXiv:1101.2905 [INSPIRE].S. Brandt, C. Peyrou, R. Sosnowski and A. Wroblewski,

*The principal axis of jets. An attempt to analyze high-energy collisions as two-body processes*,*Phys. Lett.***12**(1964) 57 [INSPIRE].A. De Rujula, J.R. Ellis, E.G. Floratos and M.K. Gaillard,

*QCD predictions for hadronic final states in e*^{+}*e*^{−}*annihilation*,*Nucl. Phys. B***138**(1978) 387 [INSPIRE].C.F. Berger, T. Kucs and G.F. Sterman,

*Event shape/energy flow correlations*,*Phys. Rev. D***68**(2003) 014012 [hep-ph/0303051] [INSPIRE].C.F. Berger and L. Magnea,

*Scaling of power corrections for angularities from dressed gluon exponentiation*,*Phys. Rev. D***70**(2004) 094010 [hep-ph/0407024] [INSPIRE].T. Becher and M.D. Schwartz,

*A precise determination of α*_{s}*from LEP thrust data using effective field theory*,*JHEP***07**(2008) 034 [arXiv:0803.0342] [INSPIRE].A. Banfi, H. McAslan, P.F. Monni and G. Zanderighi,

*A general method for the resummation of event-shape distributions in e*^{+}*e*^{−}*annihilation*,*JHEP***05**(2015) 102 [arXiv:1412.2126] [INSPIRE].TASSO collaboration,

*Jet production and fragmentation in e*^{+}*e*^{−}*annihilation at*12*GeV to*43*GeV*,*Z. Phys. C***22**(1984) 307 [INSPIRE].MARK-II collaboration,

*First measurements of hadronic decays of the Z boson*,*Phys. Rev. Lett.***63**(1989) 1558 [INSPIRE].AMY collaboration,

*Multi-hadron event properties in e*^{+}*e*^{−}*annihilation at*\( \sqrt{s} \) = 52*GeV to*57*GeV*,*Phys. Rev. D***41**(1990) 2675 [INSPIRE].ALEPH collaboration,

*Measurement of α*_{s}*from scaling violations in fragmentation functions in e*^{+}*e*^{−}*annihilation*,*Phys. Lett. B***357**(1995) 487 [INSPIRE].L3 collaboration,

*Determination of α*_{s}*from hadronic event shapes measured on the Z*^{0}*resonance*,*Phys. Lett. B***284**(1992) 471 [INSPIRE].TASSO collaboration,

*Global jet properties at*14*GeV to*44*GeV center-of-mass energy in e*^{+}*e*^{−}*annihilation*,*Z. Phys. C***47**(1990) 187 [INSPIRE].SLD collaboration,

*Measurement of α*_{s}(\( {M}_Z^2 \))*from hadronic event observables at the Z*^{0}*resonance*,*Phys. Rev. D***51**(1995) 962 [hep-ex/9501003] [INSPIRE].ALEPH collaboration,

*Studies of QCD at e*^{+}*e*^{−}*centre-of-mass energies between*91*GeV and*209*GeV*,*Eur. Phys. J. C***35**(2004) 457 [INSPIRE].DELPHI collaboration,

*A study of the energy evolution of event shape distributions and their means with the DELPHI detector at LEP*,*Eur. Phys. J. C***29**(2003) 285 [hep-ex/0307048] [INSPIRE].L3 collaboration,

*Studies of hadronic event structure in e*^{+}*e*^{−}*annihilation from*30*GeV to*209*GeV with the L*3*detector*,*Phys. Rept.***399**(2004) 71 [hep-ex/0406049] [INSPIRE].OPAL collaboration,

*Measurement of event shape distributions and moments in e*^{+}*e*^{−}→*hadrons at*91*–*209*GeV and a determination of α*_{s},*Eur. Phys. J. C***40**(2005) 287 [hep-ex/0503051] [INSPIRE].A. Abdesselam et al.,

*Boosted objects: a probe of beyond the standard model physics*,*Eur. Phys. J. C***71**(2011) 1661 [arXiv:1012.5412] [INSPIRE].P.T. Komiske, E.M. Metodiev and J. Thaler,

*The hidden geometry of particle collisions*,*JHEP***07**(2020) 006 [arXiv:2004.04159] [INSPIRE].P.T. Komiske, E.M. Metodiev and J. Thaler,

*Metric space of collider events*,*Phys. Rev. Lett.***123**(2019) 041801 [arXiv:1902.02346] [INSPIRE].S. Peleg, M. Werman and H. Rom,

*A unified approach to the change of resolution: space and gray-level*,*IEEE Trans. Pattern Anal. Machine Intell.***11**(1989) 739.Y. Rubner, C. Tomasi and L.J. Guibas,

*A metric for distributions with applications to image databases*, in*Proceedings of the sixth international conference on computer vision, ICCV*′98, IEEE Computer Society, U.S.A. (1998), p. 59.Y. Rubner, C. Tomasi and L.J. Guibas,

*The earth mover’s distance as a metric for image retrieval*,*Int. J. Comput. Vision***40**(2004) 99.O. Pele and M. Werman,

*A linear time histogram metric for improved SIFT matching*, in the proceedings of the*ECCV*, (2008) [https://doi.org/10.1007/978-3-540-88690-7_37].O. Pele and B. Taskar,

*The tangent earth mover’s distance*, in*Lecture notes in computer science*, Springer, Berlin, Germany (2013), p. 397 [https://doi.org/10.1007/978-3-642-40020-9_43].L.N. Wasserstein,

*Markov processes over denumerable products of spaces describing large systems of automata*,*Probl. Info. Transm.***5**(1969) 47.R.L. Dobrushin,

*Prescribing a system of random variables by conditional distributions*,*Theor. Probabil. Appl.***15**(1970) 458.P.T. Komiske et al.,

*Exploring the space of jets with CMS open data*,*Phys. Rev. D***101**(2020) 034009 [arXiv:1908.08542] [INSPIRE].J.H. Collins,

*An exploration of learnt representations of W jets*, arXiv:2109.10919 [INSPIRE].S.E. Park, P. Harris and B. Ostdiek,

*Neural embedding: learning the embedding of the manifold of physics data*, arXiv:2208.05484 [INSPIRE].M. Crispim Romão et al.,

*Use of a generalized energy Mover’s distance in the search for rare phenomena at colliders*,*Eur. Phys. J. C***81**(2021) 192 [arXiv:2004.09360] [INSPIRE].T. Cai, J. Cheng, N. Craig and K. Craig,

*Linearized optimal transport for collider events*,*Phys. Rev. D***102**(2020) 116019 [arXiv:2008.08604] [INSPIRE].T. Cai, J. Cheng, K. Craig and N. Craig,

*Which metric on the space of collider events?*,*Phys. Rev. D***105**(2022) 076003 [arXiv:2111.03670] [INSPIRE].C. Cesarotti and J. Thaler,

*A robust measure of event isotropy at colliders*,*JHEP***08**(2020) 084 [arXiv:2004.06125] [INSPIRE].C. Cesarotti, M. Reece and M.J. Strassler,

*The efficacy of event isotropy as an event shape observable*,*JHEP***07**(2021) 215 [arXiv:2011.06599] [INSPIRE].ATLAS collaboration,

*Measurements of multijet event isotropies using optimal transport with the ATLAS detector*, arXiv:2305.16930 [INSPIRE].R. Sinkhorn,

*A relationship between arbitrary positive matrices and stochastic matrices*,*Can. J. Math.***18**(1966) 303.M. Cuturi,

*Sinkhorn distances: lightspeed computation of optimal transport*,*Adv. Neural Inf. Process. Syst.***26**(2013) 2292.C. Clason, D.A. Lorenz, H. Mahler and B. Wirth,

*Entropic regularization of continuous optimal transport problems*,*J. Math. Anal. Appl.***494**(2021) 124432.J. Feydy et al.,

*Interpolating between optimal transport and MMD using Sinkhorn divergences*, in the proceedings of the 22^{nd}*international conference on artificial intelligence and statistics*, (2019), p. 2681.O. Kitouni, N. Nolte and M. Williams,

*Finding NEEMo: geometric fitting using Neural Estimation of the Energy Mover’s distance*, arXiv:2209.15624 [INSPIRE].A. Butter, G. Kasieczka, T. Plehn and M. Russell,

*Deep-learned top tagging with a Lorentz layer*,*SciPost Phys.***5**(2018) 028 [arXiv:1707.08966] [INSPIRE].A. Butter et al.,

*The machine learning landscape of top taggers*,*SciPost Phys.***7**(2019) 014 [arXiv:1902.09914] [INSPIRE].D. Krohn, J. Thaler and L.-T. Wang,

*Jets with variable R*,*JHEP***06**(2009) 059 [arXiv:0903.0392] [INSPIRE].L. Mackey, B. Nachman, A. Schwartzman and C. Stansbury,

*Fuzzy jets*,*JHEP***06**(2016) 010 [arXiv:1509.02216] [INSPIRE].B. Mukhopadhyaya, T. Samui and R.K. Singh,

*Dynamic radius jet clustering algorithm*,*JHEP***04**(2023) 019 [arXiv:2301.13074] [INSPIRE].A.J. Larkoski, D. Rathjens, J. Veatch and J.W. Walker,

*Jet SIFT-ing: a new scale-invariant jet clustering algorithm for the substructure era*, arXiv:2302.08609 [INSPIRE].I.W. Stewart et al.,

*XCone: N-jettiness as an exclusive cone jet algorithm*,*JHEP***11**(2015) 072 [arXiv:1508.01516] [INSPIRE].H. Georgi and M. Machacek,

*A simple QCD prediction of jet structure in e*^{+}*e*^{−}*annihilation*,*Phys. Rev. Lett.***39**(1977) 1237 [INSPIRE].A.J. Larkoski, D. Neill and J. Thaler,

*Jet shapes with the broadening axis*,*JHEP***04**(2014) 017 [arXiv:1401.2158] [INSPIRE].I.W. Stewart, F.J. Tackmann and W.J. Waalewijn,

*N-jettiness: an inclusive event shape to veto jets*,*Phys. Rev. Lett.***105**(2010) 092002 [arXiv:1004.2489] [INSPIRE].A. Banfi, G.P. Salam and G. Zanderighi,

*Resummed event shapes at hadron-hadron colliders*,*JHEP***08**(2004) 062 [hep-ph/0407287] [INSPIRE].A. Banfi, G.P. Salam and G. Zanderighi,

*Phenomenology of event shapes at hadron colliders*,*JHEP***06**(2010) 038 [arXiv:1001.4082] [INSPIRE].S. Catani, Y.L. Dokshitzer, M.H. Seymour and B.R. Webber,

*Longitudinally invariant K*_{t}*clustering algorithms for hadron hadron collisions*,*Nucl. Phys. B***406**(1993) 187 [INSPIRE].S.D. Ellis and D.E. Soper,

*Successive combination jet algorithm for hadron collisions*,*Phys. Rev. D***48**(1993) 3160 [hep-ph/9305266] [INSPIRE].Y.L. Dokshitzer, G.D. Leder, S. Moretti and B.R. Webber,

*Better jet clustering algorithms*,*JHEP***08**(1997) 001 [hep-ph/9707323] [INSPIRE].M. Wobisch and T. Wengler,

*Hadronization corrections to jet cross-sections in deep inelastic scattering*, in the proceedings of the*Workshop on Monte Carlo generators for HERA physics (plenary starting meeting)*, (1998), p. 270 [hep-ph/9907280] [INSPIRE].M. Cacciari, G.P. Salam and G. Soyez,

*The anti-k*_{t}*jet clustering algorithm*,*JHEP***04**(2008) 063 [arXiv:0802.1189] [INSPIRE].S.D. Ellis, Z. Kunszt and D.E. Soper,

*Jets at hadron colliders at order*\( {\alpha}_s^3 \)*: a look inside*,*Phys. Rev. Lett.***69**(1992) 3615 [hep-ph/9208249] [INSPIRE].CDF collaboration,

*A measurement of jet shapes in*\( p\overline{p} \)*collisions at*\( \sqrt{s} \) = 1*.*8*TeV*,*Phys. Rev. Lett.***70**(1993) 713 [INSPIRE].S.D. Ellis et al.,

*Jet shapes and jet algorithms in SCET*,*JHEP***11**(2010) 101 [arXiv:1001.0014] [INSPIRE].J. Thaler and K. Van Tilburg,

*Identifying boosted objects with N-subjettiness*,*JHEP***03**(2011) 015 [arXiv:1011.2268] [INSPIRE].M. Dasgupta and G.P. Salam,

*Resummation of nonglobal QCD observables*,*Phys. Lett. B***512**(2001) 323 [hep-ph/0104277] [INSPIRE].A. Banfi, M. Dasgupta, K. Khelifa-Kerfa and S. Marzani,

*Non-global logarithms and jet algorithms in high-p*_{T}*jet shapes*,*JHEP***08**(2010) 064 [arXiv:1004.3483] [INSPIRE].F.V. Tkachov,

*Measuring multi-jet structure of hadronic energy flow or what is a jet?*,*Int. J. Mod. Phys. A***12**(1997) 5411 [hep-ph/9601308] [INSPIRE].N.A. Sveshnikov and F.V. Tkachov,

*Jets and quantum field theory*,*Phys. Lett. B***382**(1996) 403 [hep-ph/9512370] [INSPIRE].G.P. Korchemsky, G. Oderda and G.F. Sterman,

*Power corrections and nonlocal operators*,*AIP Conf. Proc.***407**(1997) 988 [hep-ph/9708346] [INSPIRE].C.L. Basham, L.S. Brown, S.D. Ellis and S.T. Love,

*Energy correlations in electron-positron annihilation in quantum chromodynamics: asymptotically free perturbation theory*,*Phys. Rev. D***19**(1979) 2018 [INSPIRE].P.S. Cherzor and N.A. Sveshnikov,

*Jet observables and energy momentum tensor*, in the proceedings of the 12^{th}*international workshop on high-energy physics and quantum field theory (QFTHEP*′97*)*, (1997), p. 402 [hep-ph/9710349] [INSPIRE].F.V. Tkachov,

*A theory of jet definition*,*Int. J. Mod. Phys. A***17**(2002) 2783 [hep-ph/9901444] [INSPIRE].G.P. Korchemsky and G.F. Sterman,

*Power corrections to event shapes and factorization*,*Nucl. Phys. B***555**(1999) 335 [hep-ph/9902341] [INSPIRE].A.V. Belitsky, G.P. Korchemsky and G.F. Sterman,

*Energy flow in QCD and event shape functions*,*Phys. Lett. B***515**(2001) 297 [hep-ph/0106308] [INSPIRE].C.F. Berger et al.,

*Snowmass*2001*: jet energy flow project*,*eConf***C010630**(2001) P512 [hep-ph/0202207] [INSPIRE].C.W. Bauer, S.P. Fleming, C. Lee and G.F. Sterman,

*Factorization of e*^{+}*e*^{−}*event shape distributions with hadronic final states in soft collinear effective theory*,*Phys. Rev. D***78**(2008) 034027 [arXiv:0801.4569] [INSPIRE].D.M. Hofman and J. Maldacena,

*Conformal collider physics: energy and charge correlations*,*JHEP***05**(2008) 012 [arXiv:0803.1467] [INSPIRE].V. Mateu, I.W. Stewart and J. Thaler,

*Power corrections to event shapes with mass-dependent operators*,*Phys. Rev. D***87**(2013) 014025 [arXiv:1209.3781] [INSPIRE].A.V. Belitsky et al.,

*From correlation functions to event shapes*,*Nucl. Phys. B***884**(2014) 305 [arXiv:1309.0769] [INSPIRE].A. Ramdas, N. García Trillos and M. Cuturi,

*On Wasserstein two-sample testing and related families of nonparametric tests*,*Entropy***19**(2017) 47.H.G. Barrow, J.M. Tenenbaum, R.C. Bolles and H.C. Wolf,

*Parametric correspondence and chamfer matching: two new techniques for image matching*, in*Proceedings of the*5^{th}*international joint conference on artificial intelligence — volume*2*, IJCAI*^{1}77, Morgan Kaufmann Publishers Inc., San Francisco, CA, U.S.A. (1977), p. 659.G. Soyez,

*Pileup mitigation at the LHC: a theorist’s view*,*Phys. Rept.***803**(2019) 1 [arXiv:1801.09721] [INSPIRE].CDF collaboration,

*Charged jet evolution and the underlying event in*\( p\overline{p} \)*collisions at*1*.*8*TeV*,*Phys. Rev. D***65**(2002) 092002 [INSPIRE].A.G. Agocs, G.G. Barnafoldi and P. Levai,

*Jets and underlying events at LHC energies*,*J. Phys. Conf. Ser.***270**(2011) 012017 [arXiv:1011.5363] [INSPIRE].D.P. Kingma, T. Salimans and M. Welling,

*Variational dropout and the local reparameterization trick*, [arXiv:1506.02557].Y.L. Dokshitzer, V.A. Khoze and S.I. Troian,

*Particle spectra in light and heavy quark jets*,*J. Phys. G***17**(1991) 1481 [INSPIRE].Y.L. Dokshitzer, V.A. Khoze and S.I. Troian,

*On specific QCD properties of heavy quark fragmentation (‘dead cone’)*,*J. Phys. G***17**(1991) 1602 [INSPIRE].B.A. Schumm, Y.L. Dokshitzer, V.A. Khoze and D.S. Koetke,

*MLLA and the average charged multiplicity of events containing heavy quarks in e*^{+}*e*^{−}*annihilation*,*Phys. Rev. Lett.***69**(1992) 3025 [INSPIRE].M. Cacciari and G.P. Salam,

*Pileup subtraction using jet areas*,*Phys. Lett. B***659**(2008) 119 [arXiv:0707.1378] [INSPIRE].M. Cacciari, G.P. Salam and G. Soyez,

*The catchment area of jets*,*JHEP***04**(2008) 005 [arXiv:0802.1188] [INSPIRE].G. Soyez et al.,

*Pileup subtraction for jet shapes*,*Phys. Rev. Lett.***110**(2013) 162001 [arXiv:1211.2811] [INSPIRE].A.J. Larkoski, S. Marzani, G. Soyez and J. Thaler,

*Soft drop*,*JHEP***05**(2014) 146 [arXiv:1402.2657] [INSPIRE].M. Dasgupta, A. Fregoso, S. Marzani and G.P. Salam,

*Towards an understanding of jet substructure*,*JHEP***09**(2013) 029 [arXiv:1307.0007] [INSPIRE].J. Altschuler, J. Niles-Weed and P. Rigollet,

*Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration*, in the proceedings of the*Advances in neural information processing systems*, (2017).P. Dvurechensky, A. Gasnikov and A. Kroshnin,

*Computational optimal transport: complexity by accelerated gradient descent is better than by Sinkhorn’s algorithm*, in*Proceedings of the*35^{th}*international conference on machine learning*, J. Dy and A. Krause eds.,*Proc. Machine Learn. Res.***80**(2018) 1367.T. Lin, N. Ho and M.I. Jordan,

*On the efficiency of the Sinkhorn and Greenkhorn algorithms and their acceleration for optimal transport*, in*International conference on machine learning*(2019).J. Altschuler, F. Bach, A. Rudi and J. Niles-Weed,

*Massively scalable Sinkhorn distances via the Nyström method*, in the proceedings of the*Advances in neural information processing systems*, (2019).M. Scetbon and M. Cuturi,

*Linear time Sinkhorn divergences using positive features*, in the proceedings of the*Advances in neural information processing systems*, (2020), p. 13468.K. Atasu and T. Mittelholzer,

*Linear-complexity data-parallel earth mover’s distance approximations*, in the proceedings of the 36^{th}*international conference on machine learning*, (2019), p. 364.A. Paszke et al.,

*PyTorch: an imperative style, high-performance deep learning library*, in*Advances in neural information processing systems*32, H. Wallach et al. eds., Curran Associates Inc. (2019), p. 8024.J. Matousek and B. Gartner,

*Understanding and using linear programming*, Springer, Berlin, Heidelberg, Germany (2006).C. Vilani,

*Topics in optimal transportation*, American Mathematical Society, U.S.A. (2003).J. Feydy,

*Geometric data analysis, beyond convolutions*, ENS Paris-Saclay, France (2020).R. Sinkhorn,

*Diagonal equivalence to matrices with prescribed row and column sums*,*Am. Math. Mon.***74**(1967) 402.R. Sinkhorn and P. Knopp,

*Concerning nonnegative matrices and doubly stochastic matrices*,*Pacific J. Math.***21**(1967) 343.J.J. Kosowsky and A.L. Yuille,

*The invisible hand algorithm: solving the assignment problem with statistical physics*,*Neural Networks***7**(1994) 477.D.P. Bertsekas,

*The auction algorithm: a distributed relaxation method for the assignment problem*,*Ann. Operat. Res.***14**(1988) 105.D.P. Kingma and J. Ba,

*Adam: a method for stochastic optimization*, arXiv:1412.6980 [INSPIRE].P. Tankala, A. Tasissa, J.M. Murphy and D. Ba,

*K-deep simplex: deep manifold learning via local dictionaries*, arXiv:2012.02134.M. Mueller, S. Aeron, J.M. Murphy and A. Tasissa,

*Geometric sparse coding in Wasserstein space*, arXiv:2210.12135.W. Wang and M. Carreira-Perpiñán,

*Projection onto the probability simplex: an efficient algorithm with a simple proof, and an application*, arXiv:1309.1541.T. Sjöstrand et al.,

*An introduction to PYTHIA*8*.*2,*Comput. Phys. Commun.***191**(2015) 159 [arXiv:1410.3012] [INSPIRE].DELPHES 3 collaboration,

*DELPHES*3*, a modular framework for fast simulation of a generic collider experiment*,*JHEP***02**(2014) 057 [arXiv:1307.6346] [INSPIRE].M. Cacciari, G.P. Salam and G. Soyez,

*FastJet user manual*,*Eur. Phys. J. C***72**(2012) 1896 [arXiv:1111.6097] [INSPIRE].R. Flamary et al.,

*POT: Python Optimal Transport*,*J. Mach. Learn. Res.***22**(2021) 1.CMS collaboration,

*Jet energy scale and resolution in the CMS experiment in pp collisions at*8*TeV*, 2017*JINST***12**P02014 [arXiv:1607.03663] [INSPIRE].ATLAS collaboration,

*Jet energy scale and resolution measured in proton-proton collisions at*\( \sqrt{s} \) = 13*TeV with the ATLAS detector*,*Eur. Phys. J. C***81**(2021) 689 [arXiv:2007.02645] [INSPIRE].E. Wigner,

*The unreasonable effectiveness of mathematics in the natural sciences*,*Commun. Pure Appl. Math.***13**(1960) 1 [INSPIRE].M. Kac,

*Can one hear the shape of a drum?*,*Am. Math. Mon.***73**(1966) 1 [INSPIRE].S. Kullback and R.A. Leibler,

*On information and sufficiency*,*Annals Math. Statist.***22**(1951) 79.B. Piccoli and F. Rossi,

*Generalized Wasserstein distance and its application to transport equations with source*,*Archive for Rational Mechanics and Analysis***211**(2013) 335.M. Liero, A. Mielke and G. Savaré,

*Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves*,*SIAM J. Math. Anal.***48**(2016) 2869.

## Acknowledgments

Special thanks goes to Samuel Alipour-fard for helping to come up with the acronym Shaper, and for useful discussions about pileup. We thank Cari Cesarotti and Matthew LeBlanc for useful discussions about event isotropy, and Ouail Kitouni, Niklas Nolte, and Mike Williams for useful discussions on EMD estimation with Kantorovich potentials. Finally, we would like to thank Eugene Wigner of ref. [122] for inspiring the title of section 2, and Mark Kac of ref. [123] for inspiring the title of this paper.

DB, ASD, RG, and JT are supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions). ASD’s research was also funded by the President’s PhD Scholarship at Imperial College London and supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1). RG and JT are additionally supported by the U.S. DOE Office of High Energy Physics under grant number DE-SC0012567. AT’s research is supported by NSF DMS 2208392.

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Ba, D., Dogra, A.S., Gambhir, R. *et al.* SHAPER: can you hear the shape of a jet?.
*J. High Energ. Phys.* **2023**, 195 (2023). https://doi.org/10.1007/JHEP06(2023)195

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DOI: https://doi.org/10.1007/JHEP06(2023)195