(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 Spanning Substructure with Linear Regression

Eric M. Metodiev, MIT 2 ECFs Angularities Planar Flow ECFGs Jet Mass …

Taming the (IRC-safe) Substructure Zoo

EFP

G = ෍ 𝑗1=1 𝑁

෍

𝑗2=1 𝑁

⋯ ෍

𝑗𝑂=1 𝑁

𝑨𝑗1𝑨𝑗2 ⋯ 𝑨𝑗𝑂 ෑ

𝑙,𝑚 ∈G

𝜄𝑗𝑙𝑗𝑚

Correlator

Sum over all N-tuples of particle in the event

Energies

Product of the N energy fractions

Angles

One 𝜄𝑗𝑙𝑗𝑚 for each edge in 𝑙, 𝑚 ∈ 𝐻

Anatomy of an Energy Flow Polynomial:

In equations: In words:

• f

and

Eric M. Metodiev, MIT 3

𝑓+𝑓−: 𝑨𝑗 =

𝐹𝑘 σ𝑙 𝐹𝑙,

𝜄𝑗𝑘 =

2𝑞𝑗

𝜈𝑞𝑘𝜈

𝐹𝑗𝐹𝑘

𝛾 2

Hadronic: 𝑨𝑗 =

𝑞𝑈𝑘 σ𝑙 𝑞𝑈𝑙, 𝜄𝑗𝑘 = Δ𝑧𝑗𝑘 2 + Δ𝜚𝑗𝑘 2

𝛾 2

Energy Fraction Pairwise Angular Distance

𝑨𝑗 𝑨𝑘 𝜄𝑗𝑘

multigraph

EFP

G = ෍ 𝑗1=1 𝑁

෍

𝑗2=1 𝑁

⋯ ෍

𝑗𝑂=1 𝑁

𝑨𝑗1𝑨𝑗2 ⋯ 𝑨𝑗𝑂 ෑ

𝑙,𝑚 ∈G

𝜄𝑗𝑙𝑗𝑚

Correlator

Sum over all N-tuples of particle in the event

Energies

Product of the N energy fractions

Angles

One 𝜄𝑗𝑙𝑗𝑚 for each edge in 𝑙, 𝑚 ∈ 𝐻

Anatomy of an Energy Flow Polynomial:

In equations: In words:

• f

and

In pictures:

𝑨𝑗𝑘 𝑘 𝜄𝑗𝑙𝑗𝑚 𝑙 𝑚 (e.g.) = ෍

𝑗1=1 𝑁

෍

𝑗2=1 𝑁

෍

𝑗3=1 𝑁

෍

𝑗4=1 𝑁

𝑨𝑗1𝑨𝑗2𝑨𝑗3𝑨𝑗4 𝜄𝑗1𝑗2𝜄𝑗2𝑗3𝜄𝑗3𝑗4𝜄𝑗2𝑗4

2

1 2 3 4

(any index labelling works)

Eric M. Metodiev, MIT 4

𝑓+𝑓−: 𝑨𝑗 =

𝐹𝑘 σ𝑙 𝐹𝑙,

𝜄𝑗𝑘 =

2𝑞𝑗

𝜈𝑞𝑘𝜈

𝐹𝑗𝐹𝑘

𝛾 2

Hadronic: 𝑨𝑗 =

𝑞𝑈𝑘 σ𝑙 𝑞𝑈𝑙, 𝜄𝑗𝑘 = Δ𝑧𝑗𝑘 2 + Δ𝜚𝑗𝑘 2

𝛾 2

Energy Fraction Pairwise Angular Distance

𝑨𝑗 𝑨𝑘 𝜄𝑗𝑘

multigraph

Multigraph/EFP Correspondence

𝑨𝑗𝑘 𝑘 𝜄𝑗𝑙𝑗𝑚 𝑙 𝑚 Number of vertices N-particle correlator Number of edges Degree of angular monomial Treewidth + 1 Optimal VE Complexity Connected Disconnected Prime Composite = ෍

𝑗1=1 𝑁

෍

𝑗2=1 𝑁

෍

𝑗3=1 𝑁

෍

𝑗4=1 𝑁

𝑨𝑗1𝑨𝑗2𝑨𝑗3𝑨𝑗4 𝜄𝑗1𝑗2𝜄𝑗2𝑗3𝜄𝑗3𝑗4𝜄𝑗2𝑗4

2

Multigraph EFP

⋮

e.g. Tree graph EFPs are 𝑃(𝑁2)! Surprisingly efficient to compute. Stay tuned… See P. Komiske’s talk.

Eric M. Metodiev, MIT 5

N d 𝜓

Energy Flow Polynomials (EFPs) The Energy Flow Basis from IRC safety Spanning Substructure with Linear Regression

Eric M. Metodiev, MIT 6 ECFs Angularities Planar Flow ECFGs Jet Mass …

Taming the (IRC-safe) Substructure Zoo

IRC-safe Observable

EFPs linearly span IRC-safe observables

Eric M. Metodiev, MIT 7

IRC-safe Observable

IR safety: Observable unchanged by addition of infinitesimally soft particle Relabeling Symmetry: All ways of indexing particles are equivalent C safety: Observable unchanged by the collinear splitting of a particle

Energy correlators linearly span IRC-safe observables

Energy Expansion: Expand/approximate the observable in polynomials of the particle energies

New, direct argument from IRC safety See also: F. Tkachov, hep-ph/9601308

• N. Sveshnikov and F. Tkachov, hep-ph/9512370

Eric M. Metodiev, MIT 8

EFPs linearly span IRC-safe observables

IRC-safe Observable

IR safety: Observable unchanged by addition of infinitesimally soft particle Relabeling Symmetry: All ways of indexing particles are equivalent C safety: Observable unchanged by the collinear splitting of a particle

Energy correlators linearly span IRC-safe observables

Energy Expansion: Expand/approximate the observable in polynomials of the particle energies

New, direct argument from IRC safety See also: F. Tkachov, hep-ph/9601308

• N. Sveshnikov and F. Tkachov, hep-ph/9512370

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

EFPs linearly span/approximate IRC-safe observables!

Similar expansions & emergent multigraphs in:

• M. Hogervorst et al. arXiv:1409.1581
• B. Henning et al. arXiv:1706.08520

Eric M. Metodiev, MIT 9

EFPs linearly span IRC-safe observables

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) # of multigraphs with d edges # of EFPs of degree d # of connected multigraphs with d edges # of prime EFPs of degree d A050535 A076864 Exactly 1000 EFPs up to degree d=7!

Eric M. Metodiev, MIT 10

Image files for all of the prime EFP multigraphs up to d = 7 are available here.

Energy Flow Polynomials (EFPs) The Energy Flow Basis from IRC safety Spanning Substructure with Linear Regression

Eric M. Metodiev, MIT 11 ECFs Angularities Planar Flow ECFGs Jet Mass …

Taming the (IRC-safe) Substructure Zoo

Jet Observables with Energy Flow

𝑛𝐾

2

𝑞𝑈𝐾

2 = ෍ 𝑗1=1 𝑁

෍

𝑗2=1 𝑁

𝑨𝑗1𝑨𝑗2(cosh Δ𝑧𝑗1𝑗2 − cos Δ𝜚𝑗1𝑗2) = 1 2 + ⋯

Can include these using a fully general measure

Jet Mass

Dumbbell EFP

Eric M. Metodiev, MIT 12

Jet Observables with Energy Flow

𝑛𝐾

2

𝑞𝑈𝐾

2 = ෍ 𝑗1=1 𝑁

෍

𝑗2=1 𝑁

𝑨𝑗1𝑨𝑗2(cosh Δ𝑧𝑗1𝑗2 − cos Δ𝜚𝑗1𝑗2) = 1 2 + ⋯ 𝜇(4) = − 3 4 𝜇(6) = − 3 2 + 5 8

Can include these using a fully general measure (and so on, for all even angularities)

Jet Mass

Dumbbell EFP

Angularities

Star Graph EFPs

𝜇(𝛽) = ෍

𝑗 𝑁

𝑨𝑗𝜄𝑗

𝛽

Eric M. Metodiev, MIT 13

using pT

• centroid axis

[C. Berger, T. Kucs, and G. Sterman, hep-ph/0303051] [S. Ellis, et al., arXiv:10010014] [L. Almeida, et al., arXiv:0807.0234] [A. Larkoski, J. Thaler, and W. Waalewijn, arXiv:1408.3122]

Jet Observables with Energy Flow

Energy Correlation Functions

Complete Graph EFPs

𝑓3

(𝛾) =

𝑓𝑂

(𝛾) = ෍ 𝑗1=1 𝑁

෍

𝑗2=1 𝑁

⋯ ෍

𝑗𝑂=1 𝑁

𝑨𝑗1𝑨𝑗2 ⋯ 𝑨𝑗𝑂 ෑ

𝑙<𝑚∈{1,⋯,𝑂}

𝜄𝑗𝑙𝑗𝑚

𝛾

𝑓4

(𝛾) =

⋯ 𝑓2

(𝛾) =

with measure choice of 𝛾

Eric M. Metodiev, MIT 14

[A. Larkoski, G. Salam, and J. Thaler, arXiv:1305.0007]

Jet Observables with Energy Flow

Energy Correlation Functions

Complete Graph EFPs

Geometric Moments

Higher dumbbell EFPs

𝑓3

(𝛾) =

𝑓𝑂

(𝛾) = ෍ 𝑗1=1 𝑁

෍

𝑗2=1 𝑁

⋯ ෍

𝑗𝑂=1 𝑁

𝑨𝑗1𝑨𝑗2 ⋯ 𝑨𝑗𝑂 ෑ

𝑙<𝑚∈{1,⋯,𝑂}

𝜄𝑗𝑙𝑗𝑚

𝛾

𝑓4

(𝛾) =

⋯ 𝑓2

(𝛾) =

with measure choice of 𝛾

𝐃 = ෍

𝑗1=1 𝑁

𝑨𝑗 Δ𝑧𝑗

2

Δ𝑧𝑗Δ𝜚𝑗 Δ𝜚𝑗Δ𝑧𝑗 Δ𝜚𝑗

2

tr 𝐃 = 1 2 det 𝐃 = − 1 2

• e. g.

Pf = 4 det 𝐃 tr 𝐃 2

Eric M. Metodiev, MIT

using pT

• centroid axis

15

[A. Larkoski, G. Salam, and J. Thaler, arXiv:1305.0007] [L. Almeida, et al., arXiv:0807.0234] [J. Gallicchio and M. Schwartz, arXiv:1211.7038] [J, Thaler and L.-T. Wang, arXiv:0806.0023]

Energy Flow Polynomials (EFPs) The Energy Flow Basis from IRC safety Spanning Substructure with Linear Regression

Eric M. Metodiev, MIT 16 ECFs Angularities Planar Flow ECFGs Jet Mass …

Taming the (IRC-safe) Substructure Zoo

Linear Models and Energy Flow

Linear methods:

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:

𝑇 = ෍

𝐻

𝑥𝐻 EFP

G

Machine learn these Eric M. Metodiev, MIT 17

See P. Komiske’s talk. This talk.

Confirming Analytic Relationships with Regression

𝜇(4) = − 3 4 𝜇(6) = − 3 2 + 5 8 𝜇(2) = 1 2

Eric M. Metodiev, MIT 18

Linear Regression and IRC-safety

𝑛𝐾 𝑞𝑈𝐾: IRC safe. No Taylor expansion due to square root.

𝜇(𝛽=1/2): IRC safe. No simple analytic relationship. 𝜐2: IRC safe. Algorithmically defined. 𝜐21: Sudakov safe. Safe for 2-prong jets and higher. 𝜐32: Sudakov safe. Safe for 3-prong jets and higher. Multiplicity: IRC unsafe.

Expected to be IRC safe = Solid. Expected to be IRC unsafe = Dashed. T

• p Jets (3 prong)

Eric M. Metodiev, MIT QCD Jets (1 prong) W Jets (2 prong) 19

[A. Larkoski, S. Marzani, and J. Thaler, arXiv:1502.01719]

Eric M. Metodiev, MIT 20

Conclusions EFPs form a complete, linear representation of the jet

• EFPs energy correlators with monomial angular structure
• Encompass many existing classes of expert variables
• Opens the door to using linear methods for jet substructure
• IRC-unsafe information? Combine!
• Use EFPs & linearity to reduce radiation pattern to a

single optimal observable

(Linear) Learning is easy

• Linear models are convex & even closed-form at times
• Few or no hyperparameters to tune at all
• Guaranteed global optima

Energy Flow Polynomials (EFPs) The Energy Flow Basis from IRC safety Spanning Substructure with Linear Regression

Eric M. Metodiev, MIT 21 ECFs Angularities Planar Flow ECFGs Jet Mass …

Taming the (IRC-safe) Substructure Zoo

The End