(machine) learning jet substructure Machine Learning for Jet Physics - PowerPoint PPT Presentation

A complete linear basis for (machine) learning jet substructure Machine Learning for Jet Physics Workshop, 2017 Eric M. Metodiev Center for Theoretical Physics Massachusetts Institute of Technology Based on work with Patrick T. Komiske and Jesse Thaler December 12, 2017 Eric M. Metodiev, MIT 1

Energy Flow Polynomials (EFPs) The Energy Flow Basis from IRC safety … Jet Mass ECFs Angularities Taming the (IRC-safe) Substructure Zoo Planar Flow ECFGs Spanning Substructure with Linear Regression Eric M. Metodiev, MIT 2

Anatomy of an Energy Flow Polynomial: Pairwise Angular Distance Energy Fraction 𝑨 𝑗 𝜄 𝑗𝑘 𝛾 𝜈 𝑞 𝑘𝜈 𝐹 𝑘 2𝑞 𝑗 2 𝑓 + 𝑓 − : 𝑨 𝑗 = σ 𝑙 𝐹 𝑙 , 𝜄 𝑗𝑘 = 𝑨 𝑘 𝐹 𝑗 𝐹 𝑘 𝛾 𝑞 𝑈𝑘 2 + Δ𝜚 𝑗𝑘 2 Hadronic: 𝑨 𝑗 = σ 𝑙 𝑞 𝑈𝑙 , 𝜄 𝑗𝑘 = Δ𝑧 𝑗𝑘 2 𝑁 𝑁 𝑁 In equations: EFP G = ෍ ෍ ⋯ ෍ 𝑨 𝑗 1 𝑨 𝑗 2 ⋯ 𝑨 𝑗 𝑂 ෑ 𝜄 𝑗 𝑙 𝑗 𝑚 𝑗 1 =1 𝑗 2 =1 𝑗 𝑂 =1 𝑙,𝑚 ∈G multigraph In words: Correlator Energies Angles of and Sum over all N -tuples of Product of the N One 𝜄 𝑗 𝑙 𝑗 𝑚 for each particle in the event energy fractions edge in 𝑙, 𝑚 ∈ 𝐻 Eric M. Metodiev, MIT 3

Anatomy of an Energy Flow Polynomial: Pairwise Angular Distance Energy Fraction 𝑨 𝑗 𝜄 𝑗𝑘 𝛾 𝜈 𝑞 𝑘𝜈 𝐹 𝑘 2𝑞 𝑗 2 𝑓 + 𝑓 − : 𝑨 𝑗 = σ 𝑙 𝐹 𝑙 , 𝜄 𝑗𝑘 = 𝑨 𝑘 𝐹 𝑗 𝐹 𝑘 𝛾 𝑞 𝑈𝑘 2 + Δ𝜚 𝑗𝑘 2 Hadronic: 𝑨 𝑗 = σ 𝑙 𝑞 𝑈𝑙 , 𝜄 𝑗𝑘 = Δ𝑧 𝑗𝑘 2 𝑁 𝑁 𝑁 In equations: EFP G = ෍ ෍ ⋯ ෍ 𝑨 𝑗 1 𝑨 𝑗 2 ⋯ 𝑨 𝑗 𝑂 ෑ 𝜄 𝑗 𝑙 𝑗 𝑚 𝑗 1 =1 𝑗 2 =1 𝑗 𝑂 =1 𝑙,𝑚 ∈G multigraph In words: Correlator Energies Angles of and Sum over all N -tuples of Product of the N One 𝜄 𝑗 𝑙 𝑗 𝑚 for each particle in the event energy fractions edge in 𝑙, 𝑚 ∈ 𝐻 In pictures: 𝑨 𝑗 𝑘 𝜄 𝑗 𝑙 𝑗 𝑚 𝑘 𝑙 𝑚 3 𝑁 𝑁 𝑁 𝑁 1 2 (e.g.) 2 = ෍ ෍ ෍ ෍ 𝑨 𝑗 1 𝑨 𝑗 2 𝑨 𝑗 3 𝑨 𝑗 4 𝜄 𝑗 1 𝑗 2 𝜄 𝑗 2 𝑗 3 𝜄 𝑗 3 𝑗 4 𝜄 𝑗 2 𝑗 4 𝑗 1 =1 𝑗 2 =1 𝑗 3 =1 𝑗 4 =1 4 (any index labelling works) Eric M. Metodiev, MIT 4

Multigraph/EFP Correspondence Multigraph EFP 𝑁 𝑁 𝑁 𝑁 2 = ෍ ෍ ෍ ෍ 𝑨 𝑗 1 𝑨 𝑗 2 𝑨 𝑗 3 𝑨 𝑗 4 𝜄 𝑗 1 𝑗 2 𝜄 𝑗 2 𝑗 3 𝜄 𝑗 3 𝑗 4 𝜄 𝑗 2 𝑗 4 𝑗 1 =1 𝑗 2 =1 𝑗 3 =1 𝑗 4 =1 𝑨 𝑗 𝑘 𝑘 𝜄 𝑗 𝑙 𝑗 𝑚 𝑙 𝑚 N Number of vertices N -particle correlator d Number of edges Degree of angular monomial e.g. Tree graph EFPs are 𝑃(𝑁 2 ) ! Treewidth + 1 Optimal VE Complexity 𝜓 Surprisingly efficient to compute. Stay tuned… See P. Komiske’s talk . Prime Connected Disconnected Composite ⋮ Eric M. Metodiev, MIT 5

Energy Flow Polynomials (EFPs) The Energy Flow Basis from IRC safety … Jet Mass ECFs Angularities Taming the (IRC-safe) Substructure Zoo Planar Flow ECFGs Spanning Substructure with Linear Regression Eric M. Metodiev, MIT 6

EFPs linearly span IRC-safe observables IRC-safe Observable Eric M. Metodiev, MIT 7

EFPs linearly span IRC-safe observables IRC-safe Observable Energy Expansion : Expand/approximate the observable in polynomials of the particle energies IR safety : Observable unchanged by addition of infinitesimally soft particle C safety : Observable unchanged by the collinear splitting of a particle Relabeling Symmetry : All ways of indexing particles are equivalent New, direct argument from IRC safety See also: F. Tkachov, hep-ph/9601308 N. Sveshnikov and F. Tkachov, hep-ph/9512370 Energy correlators linearly span IRC-safe observables Eric M. Metodiev, MIT 8

EFPs linearly span IRC-safe observables IRC-safe Observable Energy Expansion : Expand/approximate the observable in polynomials of the particle energies IR safety : Observable unchanged by addition of infinitesimally soft particle C safety : Observable unchanged by the collinear splitting of a particle Relabeling Symmetry : All ways of indexing particles are equivalent New, direct argument from IRC safety See also: F. Tkachov, hep-ph/9601308 N. Sveshnikov and F. Tkachov, hep-ph/9512370 Energy correlators linearly span IRC-safe observables Angular Expansion : Expansion/approximation of angular part of correlators in pairwise angular distances Analyze : Identify the unique analytic structures that emerge as non-isomorphic multigraphs/EFPs Similar expansions & emergent multigraphs in: M. Hogervorst et al . arXiv:1409.1581 B. Henning et al . arXiv:1706.08520 EFPs linearly span/approximate IRC-safe observables! Eric M. Metodiev, MIT 9

Organization of the basis EFPs are truncated by angular degree d, the order of the angular expansion. Finite number at each order in d ! All prime EFPs up to d=5 Online Encyclopedia of Integer Sequences (OEIS) A050535 # of multigraphs with d edges # of EFPs of degree d A076864 # of connected multigraphs with d edges # of prime EFPs of degree d Image files for all of the prime EFP multigraphs up to d = 7 are available here. Exactly 1000 EFPs up to degree d=7! Eric M. Metodiev, MIT 10

Energy Flow Polynomials (EFPs) The Energy Flow Basis from IRC safety … Jet Mass ECFs Angularities Taming the (IRC-safe) Substructure Zoo Planar Flow ECFGs Spanning Substructure with Linear Regression Eric M. Metodiev, MIT 11

Jet Observables with Energy Flow Dumbbell EFP 𝑁 𝑁 Jet Mass 2 𝑛 𝐾 𝑨 𝑗 1 𝑨 𝑗 2 (cosh Δ𝑧 𝑗 1 𝑗 2 − cos Δ𝜚 𝑗 1 𝑗 2 ) = 1 2 = ෍ ෍ + ⋯ 𝑞 𝑈𝐾 2 𝑗 1 =1 𝑗 2 =1 Can include these using a fully general measure Eric M. Metodiev, MIT 12

Jet Observables with Energy Flow Dumbbell EFP 𝑁 𝑁 Jet Mass 2 𝑛 𝐾 𝑨 𝑗 1 𝑨 𝑗 2 (cosh Δ𝑧 𝑗 1 𝑗 2 − cos Δ𝜚 𝑗 1 𝑗 2 ) = 1 2 = ෍ ෍ + ⋯ 𝑞 𝑈𝐾 2 𝑗 1 =1 𝑗 2 =1 Can include these using 𝑁 a fully general measure 𝜇 (𝛽) = ෍ 𝛽 𝑨 𝑗 𝜄 𝑗 𝑗 [C. Berger, T. Kucs, and G. Sterman, hep-ph/0303051] [L. Almeida, et al ., arXiv:0807.0234] Angularities Star Graph EFPs [S. Ellis, et al ., arXiv:10010014] − 3 𝜇 (4) = [A. Larkoski, J. Thaler, and W. Waalewijn, arXiv:1408.3122] 4 (and so on, for all even angularities) − 3 + 5 𝜇 (6) = 2 8 using pT -centroid axis Eric M. Metodiev, MIT 13

Jet Observables with Energy Flow Energy Correlation Functions 𝑁 𝑁 𝑁 (𝛾) = ෍ 𝛾 𝑓 𝑂 ෍ ⋯ ෍ 𝑨 𝑗 1 𝑨 𝑗 2 ⋯ 𝑨 𝑗 𝑂 ෑ 𝜄 𝑗 𝑙 𝑗 𝑚 𝑗 1 =1 𝑗 2 =1 𝑗 𝑂 =1 𝑙<𝑚∈{1,⋯,𝑂} [A. Larkoski, G. Salam, and J. Thaler, arXiv:1305.0007] Complete Graph EFPs (𝛾) = (𝛾) = (𝛾) = ⋯ 𝑓 3 𝑓 4 𝑓 2 with measure choice of 𝛾 Eric M. Metodiev, MIT 14

Jet Observables with Energy Flow Energy Correlation Functions 𝑁 𝑁 𝑁 (𝛾) = ෍ 𝛾 𝑓 𝑂 ෍ ⋯ ෍ 𝑨 𝑗 1 𝑨 𝑗 2 ⋯ 𝑨 𝑗 𝑂 ෑ 𝜄 𝑗 𝑙 𝑗 𝑚 𝑗 1 =1 𝑗 2 =1 𝑗 𝑂 =1 𝑙<𝑚∈{1,⋯,𝑂} [A. Larkoski, G. Salam, and J. Thaler, arXiv:1305.0007] Complete Graph EFPs (𝛾) = (𝛾) = (𝛾) = ⋯ 𝑓 3 𝑓 4 𝑓 2 with measure choice of 𝛾 𝑁 2 Δ𝑧 𝑗 Δ𝑧 𝑗 Δ𝜚 𝑗 Pf = 4 det 𝐃 Geometric Moments 𝐃 = ෍ 𝑨 𝑗 e. g. 2 tr 𝐃 2 Δ𝜚 𝑗 Δ𝑧 𝑗 Δ𝜚 𝑗 𝑗 1 =1 [L. Almeida, et al ., arXiv:0807.0234] Higher dumbbell EFPs [J, Thaler and L.-T. Wang, arXiv:0806.0023] [J. Gallicchio and M. Schwartz, arXiv:1211.7038] tr 𝐃 = 1 − 1 det 𝐃 = 2 2 using pT -centroid axis Eric M. Metodiev, MIT 15

Energy Flow Polynomials (EFPs) The Energy Flow Basis from IRC safety … Jet Mass ECFs Angularities Taming the (IRC-safe) Substructure Zoo Planar Flow ECFGs Spanning Substructure with Linear Regression Eric M. Metodiev, MIT 16

Linear Models and Energy Flow 𝑇 = ෍ 𝑥 𝐻 EFP G Linear methods: Machine learn these 𝐻 Utilize the linear completeness of the Energy Flow basis. Convex and few/no hyperparameters to tune. Achieve global optimum via closed-form solution or convergent iteration. Simple models with the minimum number of parameters/input. Rich in tools and applications: First few chapters of C. Bishop’s Pattern Recognition and Machine Learning : See P. Komiske’s talk . This talk. Eric M. Metodiev, MIT 17

Confirming Analytic Relationships with Regression 𝜇 (2) = 1 2 − 3 𝜇 (4) = 4 − 3 + 5 𝜇 (6) = 2 8 Eric M. Metodiev, MIT 18

Linear Regression and IRC-safety 𝑛 𝐾 𝑞 𝑈𝐾 : IRC safe. No Taylor expansion due to square root. 𝜇 (𝛽=1/2) : IRC safe. No simple analytic relationship. 𝜐 2 : IRC safe. Algorithmically defined. 𝜐 21 : Sudakov safe. Safe for 2-prong jets and higher. [A. Larkoski, S. Marzani, and J. Thaler, arXiv:1502.01719] 𝜐 32 : Sudakov safe. Safe for 3-prong jets and higher. Multiplicity : IRC unsafe. QCD Jets (1 prong) W Jets (2 prong) T op Jets (3 prong) Expected to be IRC safe = Solid. Eric M. Metodiev, MIT 19 Expected to be IRC unsafe = Dashed.