theory perspective on machine learning for jets · 2021. 6. 3. · jesse thaler (mit) — theory...

1

Jets and their Substructure from LHC Data, CERN-TH Virtual Workshop — June 3, 2021

Jesse Thaler (MIT) — Theory Perspective on Machine Learning for Jets

Jesse Thaler

Theory Perspective on

Machine Learning for Jets

[http://iaifi.org, MIT News Announcement]

The NSF AI Institute for Artificial Intelligenceand Fundamental Interactions (IAIFI)

Advance physics knowledge — from the smallest building blocks of nature

to the largest structures in the universe — and galvanize AI research innovation

“eye-phi”

2Jesse Thaler (MIT) — Theory Perspective on Machine Learning for Jets

http://iaifi.org

https://news.mit.edu/2020/nsf-announces-mit-led-institute-artificial-intelligence-fundamental-interactions-0826


Machine learning that incorporatesfirst principles, best practices, and domain knowledge

from fundamental physics

Symmetries, conservation laws, scaling relations, limiting behaviors, locality, causality,unitarity, gauge invariance, entropy, least action, factorization, unit tests,

exactness, systematic uncertainties, reproducibility, verifiability, …

AI2: Ab Initio Artificial Intelligence


Confronting the Black Box

How do we develop robust machine learning for jet analyses?


Likelihood Ratio TrickKey example of simulation-based inference

[see e.g. Cranmer, Pavez, Louppe, arXiv 2015; D’Agnolo, Wulzer, PRD 2019;simulation-based inference in Cranmer, Brehmer, Louppe, PNAS 2020;

relation to f-divergences in Nguyen, Wainwright, Jordan, AoS 2009; Nachman, JDT, arXiv 2021]

Learnable Function:

Training Data: Finite samples P and Q

Goal: Estimate p(x) / q(x)

f(x) parametrized by, e.g., neural networks

−minf(x)

L =

Zdx p(x) log

p(x)

q(x)<latexit sha1_base64="DHXBjqaYO+bfMooFpxGVp6lB6cI=">AAADDHicfVLdbtMwGHXC3ygwOuCOG0OFNFBXxaH05wKpAi64GGJIdJtUV5XjOpm1xAm2g1oZvwJPwx3ilnfgBXgOnHSTWIb4JEtH33fO+X6SqEi50kHwy/OvXL12/cbWzdat23e277Z37h2qvJSUTWme5vI4IoqlXLCp5jplx4VkJItSdhSdvq7qR5+ZVDwXH/W6YPOMJILHnBLtUov279YezrhYGINrs5lMorkJei8CNB6gbtAL6uieZ2y8u3pqLdyHLyHmQkP8CGLNVtos7QriLmz4OC1CGx80HFQ+4/EoRCNbVD4Qp3kCcSwJbQ7wf6Ft0sejQdgf1PQhCmtdOOw/79tPNX3R7pyvAi8DdAY6kwegjoPFjreNlzktMyY0TYlSMxQUem6I1JymzLZwqVhB6ClJ2MxBQTKm5qaeycInLrOEcS7dczeqs38rDMmUWmeRY2ZEn6hmrUr+qzYrdTyaGy6KUjNBN43iMoU6h9XXhUsuGdXp2gFCJXezQnpC3Hm1+wcudInZWmSF2+MNc/tJ9s71el8wSXQunxlMZJKRlXX7JrhbIXdB1LzXZXAY9lDQQx/6ncmrzSnBFngIHoNdgMAQTMBbcACmgHr7nvSM98X/6n/zv/s/NlTfO9PcBxfC//kHt6vozA==</latexit><latexit sha1_base64="DHXBjqaYO+bfMooFpxGVp6lB6cI=">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</latexit><latexit sha1_base64="DHXBjqaYO+bfMooFpxGVp6lB6cI=">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</latexit><latexit sha1_base64="TU7q4dQBpc+LJp2rM/pH2uS50uM=">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</latexit>

Loss Function(al):

Asymptotically:

Kullback–Leibler divergence

argminf(x)

L =p(x)

q(x)<latexit sha1_base64="C44EPfIX41lNtDowReWFHN5q4Ug=">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</latexit><latexit sha1_base64="C44EPfIX41lNtDowReWFHN5q4Ug=">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</latexit><latexit sha1_base64="C44EPfIX41lNtDowReWFHN5q4Ug=">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</latexit><latexit sha1_base64="aV3VBbPgU5blfbcq1YETF1NKiLY=">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</latexit>

Likelihood ratio

L = −

⌦

log f(x)↵

P+

⌦

f(x)− 1↵

Q<latexit sha1_base64="/BAofgnPH6B4QebhXUNq+8pNlqE=">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</latexit><latexit sha1_base64="/BAofgnPH6B4QebhXUNq+8pNlqE=">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</latexit><latexit sha1_base64="/BAofgnPH6B4QebhXUNq+8pNlqE=">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</latexit><latexit sha1_base64="kPIkJdH6omNFCZSl9SSaGcCTYJg=">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</latexit>

Many QCD/collider/jet problemscan be expressed in this form!

https://arxiv.org/abs/1506.02169



https://arxiv.org/abs/math/0510521


[see e.g. Cranmer, Pavez, Louppe, arXiv 2015; D’Agnolo, Wulzer, PRD 2019;simulation-based inference in Cranmer, Brehmer, Louppe, PNAS 2020;

relation to f-divergences in Nguyen, Wainwright, Jordan, AoS 2009; Nachman, JDT, arXiv 2021]

Many QCD/collider/jet problemscan be expressed in this form!Likelihood Ratio Trick

Key example of simulation-based inference


Asymptotically, same structure as Lagrangian mechanics!

Action:

Lagrangian:

Euler-Lagrange: Solution:

L =

ZdxL(x)

<latexit sha1_base64="8vxtnXiFvfGW0xbeZIJI9D9XEPA=">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</latexit><latexit sha1_base64="8vxtnXiFvfGW0xbeZIJI9D9XEPA=">AAACJHicbVDLattAFL1K+nDdR5Sku24mNaUuFCOVQLIJmGTThRcu1E7AMmY0vnIGj0Zi5irYCH1Cf6M/0G37B92FLrrpst/R8WOR2D0wcDjnXu6ZE+dKWgqC397O7oOHjx7XntSfPnv+Ys/fP+jbrDACeyJTmbmKuUUlNfZIksKr3CBPY4WX8fRi4V/eoLEy059pnuMw5RMtEyk4OWnkv+2wMxZJTSw6YhHhjMpxNWPRexalnK4FV2Wnas7ejfxG0AqWYNskXJNG+yUs0R35f6NxJooUNQnFrR2EQU7DkhuSQmFVjwqLORdTPsGBo5qnaIfl8kMVe+OUMUsy456LtlTvbpQ8tXaexm5ykdJuegvxf96goOR0WEqdF4RarA4lhWKUsUU7bCwNClJzR7gw0mVl4pobLsh1eO9KgnOd5lXdFRNu1rBN+h9aYdAKPx032uerhqAGr+A1NCGEE2jDR+hCDwR8gW/wHX54X72f3q33azW64613DuEevD//ALD4pDA=</latexit><latexit sha1_base64="8vxtnXiFvfGW0xbeZIJI9D9XEPA=">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</latexit><latexit sha1_base64="4miLyAx6p6t5RquYQ9AkM/HLRVc=">AAAB7XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwf5AO5RM5k4bmmSGJCMMY1/ArW/gTnwiX8DnMP1ZWOuBwOGchHvzxbngxgbBl1fb2d3bP6gf+kcN//jktNnomazQDLssE5kexNSg4Aq7lluBg1wjlbHAfjy7X/T9Z9SGZ+rJljlGkk4UTzmj1kWdcbMVtIOlyLYJ16YFa42b36MkY4VEZZmgxgzDILdRRbXlTODcHxUGc8pmdIJDZxWVaKJqueacXLokIWmm3VGWLNPfLyoqjSll7G5Kaqfmb7cI/+uGhU1vo4qrvLCo2GpQWghiM7L4M0m4RmZF6QxlmrtdCZtSTZl1ZDampFgqmc99xyX8S2Hb9K7bYdAOHwOowzlcwBWEcAN38AAd6AKDBF7hzXvx3r2PFb+atwZ5BhvyPn8A3biSog==</latexit><latexit sha1_base64="8Rw3IBTH0LhNxEUFDTC644OV9bg=">AAACGXicbZDNSgMxFIXv+FvrX3XrJiqigpQZN7oRBDcuXChYFTqlZNI7GsxkhuSOtAzzCL6GL+BW38CduPAFfA7T2oW1Xggczkm4J1+UKWnJ9z+9icmp6ZnZylx1fmFxabm2snBl09wIbIhUpeYm4haV1NggSQpvMoM8iRReR/cn/fz6AY2Vqb6kXoathN9qGUvByVnt2vYZO2Kh1MTCdRYSdqnolF0W7rEw4XQnuCrOyp3ubru26df9wbBxEQzFJgznvF37CjupyBPUJBS3thn4GbUKbkgKhWU1zC1mXNzzW2w6qXmCtlUMPlSyLed0WJwad1y1gfv7RcETa3tJ5G72W9q/Wd/8L2vmFB+2CqmznFCLn0VxrhilrE+HdaRBQarnBBdGuq5M3HHDBTmGI1ti7OkkK6sOTPAXw7i42q8Hfj248KECa7ABOxDAARzDKZxDAwQ8wjO8wKv35L157z8IJ7why1UYGe/jGy/vok0=</latexit><latexit sha1_base64="8Rw3IBTH0LhNxEUFDTC644OV9bg=">AAACGXicbZDNSgMxFIXv+FvrX3XrJiqigpQZN7oRBDcuXChYFTqlZNI7GsxkhuSOtAzzCL6GL+BW38CduPAFfA7T2oW1Xggczkm4J1+UKWnJ9z+9icmp6ZnZylx1fmFxabm2snBl09wIbIhUpeYm4haV1NggSQpvMoM8iRReR/cn/fz6AY2Vqb6kXoathN9qGUvByVnt2vYZO2Kh1MTCdRYSdqnolF0W7rEw4XQnuCrOyp3ubru26df9wbBxEQzFJgznvF37CjupyBPUJBS3thn4GbUKbkgKhWU1zC1mXNzzW2w6qXmCtlUMPlSyLed0WJwad1y1gfv7RcETa3tJ5G72W9q/Wd/8L2vmFB+2CqmznFCLn0VxrhilrE+HdaRBQarnBBdGuq5M3HHDBTmGI1ti7OkkK6sOTPAXw7i42q8Hfj248KECa7ABOxDAARzDKZxDAwQ8wjO8wKv35L157z8IJ7why1UYGe/jGy/vok0=</latexit><latexit sha1_base64="9xcQnj+K9N9lHlXUPtm5vZCRZn4=">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</latexit><latexit sha1_base64="5vPcAywGBzBbtTW3rmHDgI638IQ=">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</latexit><latexit sha1_base64="8vxtnXiFvfGW0xbeZIJI9D9XEPA=">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</latexit><latexit sha1_base64="8vxtnXiFvfGW0xbeZIJI9D9XEPA=">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</latexit><latexit sha1_base64="8vxtnXiFvfGW0xbeZIJI9D9XEPA=">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</latexit><latexit sha1_base64="8vxtnXiFvfGW0xbeZIJI9D9XEPA=">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</latexit><latexit sha1_base64="5vPcAywGBzBbtTW3rmHDgI638IQ=">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</latexit>

∂L

∂f= 0

<latexit sha1_base64="7xwMYpS6lejwTNM/LMAnO42aG40=">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</latexit><latexit sha1_base64="7xwMYpS6lejwTNM/LMAnO42aG40=">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</latexit><latexit sha1_base64="7xwMYpS6lejwTNM/LMAnO42aG40=">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</latexit><latexit sha1_base64="jTwafL38s4seBwJzgmInpAJbgnQ=">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</latexit>

f(x) =p(x)

q(x)<latexit sha1_base64="quzVQzouxbo/tOnoZlw/ZCJVKpE=">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</latexit><latexit sha1_base64="quzVQzouxbo/tOnoZlw/ZCJVKpE=">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</latexit><latexit sha1_base64="quzVQzouxbo/tOnoZlw/ZCJVKpE=">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</latexit><latexit sha1_base64="htVSx2ZZujuo2Hdt9ibMaWlHcBA=">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</latexit>

L(x) = −p(x) log f(x) + q(x)�

f(x)− 1�

<latexit sha1_base64="K298pScNYU/Ys1xSLKz/x92RmiM=">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</latexit><latexit sha1_base64="K298pScNYU/Ys1xSLKz/x92RmiM=">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</latexit><latexit sha1_base64="K298pScNYU/Ys1xSLKz/x92RmiM=">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</latexit><latexit sha1_base64="CIzs/gC4UpWwhuj1TBR+6hF4fYI=">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</latexit>

Requires shift in theoretical focus from solving problems to specifying problems




https://arxiv.org/abs/math/0510521



“What is the machine learning?”

For this loss function, an estimate of the likelihood ratio derived from sampled data and regularized by the

network architecture and training paradigm

“But where’s the physics?!”

In the choice of loss function, data samples,network architecture, and training paradigm

“…”


[h/t David Kaiser, MIT SERC; Suresh, Guttag, arXiv 2019]

(a) Data Generation

(b) Model Building and Implementation

1. Historical bias arises when there is a misalignment be-tween world as it is and the values or objectives to beencoded and propagated in a model. It is a normative con-cern with the state of the world, and exists even given per-fect sampling and feature selection.

2. Representation bias arises while defining and samplinga development population. It occurs when the develop-ment population under-represents, and subsequently failsto generalize well, for some part of the use population.

3. Measurement Bias arises when choosing and measur-ing features and labels to use; these are often proxies forthe desired quantities. The chosen set of features and la-bels may leave out important factors or introduce group-or input-dependent noise that leads to differential perfor-mance.

4. Aggregation bias arises during model construction, whendistinct populations are inappropriately combined. Inmany applications, the population of interest is heteroge-neous and a single model is unlikely to suit all subgroups.

5. Evaluation bias occurs during model iteration and evalu-ation. It can arise when the testing or external benchmarkpopulations do not equally represent the various parts ofthe use population. Evaluation bias can also arise from theuse of performance metrics that are not appropriate for theway in which the model will be used.

6. Deployment Bias occurs after model deployment, whena system is used or interpreted in inapppropriate ways.

Many Reasons to be Wary! “A Framework for Understanding Unintended Consequences of Machine Learning”

For collider physics, “bias” ≈ “systematic uncertainty”

https://computing.mit.edu/cross-cutting/social-and-ethical-responsibilities-of-computing/



Machine Learning for Jet Substructure

How can we leverage theory to advance machine learning for jets?

Jets from fragmentation

of quarks and gluons


My (Evolving) Perspective

Examples below are representative, not exhaustive; apologies![see HEPML-LivingReview for extensive bibliography]

When striving for “interpretable machine learning”we are essentially hoping that likelihood ratios can beapproximated via theoretically well-motivated forms

We can impose theoretical priors by judicious choice ofnetwork architecture that captures the underlying

structures and symmetries of our problem

Machine learning methods can only be as robust and reliableas the data samples used for training

We are making progress towards uncertainty quantification,using more elaborate loss functions and training paradigms

https://iml-wg.github.io/HEPML-LivingReview/


Compute Likelihood Ratios Directly?Yes, you can! And if you have enough calculational power, you should!

Shower Deconstruction Color Singlet Identification

Challenge is that in most cases, best estimate of likelihood ratiocomes from complex simulations with no closed form expression

[Soper, Spannowsky, PRD 2011]

χ({p, t}N) =P ({p, t}N |S)

P ({p, t}N |B)

[Buckley, Callea, Larkoski, Marzani, SciPost 2020]

<latexit sha1_base64="KiCPZsN2aBs/0cGFdsbFzzc54Nk=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYNRo9ELxwhkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj+7nffkKleSwfzCRBP6JDyUPOqLFSo9YvltyyuwBZJ15GSpCh3i9+9QYxSyOUhgmqdddzE+NPqTKcCZwVeqnGhLIxHWLXUkkj1P50ceiMXFhlQMJY2ZKGLNTfE1MaaT2JAtsZUTPSq95c/M/rpia89adcJqlByZaLwlQQE5P512TAFTIjJpZQpri9lbARVZQZm03BhuCtvrxOWldlr1K+blRK1bssjjycwTlcggc3UIUa1KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AKARjNQ=</latexit> <latexit sha1_base64="AoSV+QzrA75G+PPc1+8sGrVN1Oc=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexKRI/BgHiMaB6QLGF20psMmZ1dZmaFEPIJXjwo4tUv8ubfOEn2oIkFDUVVN91dQSK4Nq777eTW1jc2t/LbhZ3dvf2D4uFRU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj2sxvPaHSPJaPZpygH9GB5CFn1Fjpoda77RVLbtmdg6wSLyMlyFDvFb+6/ZilEUrDBNW647mJ8SdUGc4ETgvdVGNC2YgOsGOppBFqfzI/dUrOrNInYaxsSUPm6u+JCY20HkeB7YyoGeplbyb+53VSE177Ey6T1KBki0VhKoiJyexv0ucKmRFjSyhT3N5K2JAqyoxNp2BD8JZfXiXNi7JXKV/eV0rVmyyOPJzAKZyDB1dQhTuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4A3U+NiA==</latexit>

|MB|2

|MS |2'

1� cos θak + 1� cos θbk1� cos θab

,

O < 1

O > 1

Omin =1

2

××

<latexit sha1_base64="V0j1xwALP/d9qnTB1oMZW6NT5YM=">AAAB9XicbVDLSgNBEJyNrxhfUY9eBoPgKexKRI9BLx4jmAdkN2F20psMmX0w06uEJf/hxYMiXv0Xb/6Nk2QPmljQUFR1093lJ1JotO1vq7C2vrG5Vdwu7ezu7R+UD49aOk4VhyaPZaw6PtMgRQRNFCihkyhgoS+h7Y9vZ377EZQWcfSAkwS8kA0jEQjO0Eg9F0eArC96bgIq6ZcrdtWeg64SJycVkqPRL3+5g5inIUTIJdO669gJehlTKLiEaclNNSSMj9kQuoZGLATtZfOrp/TMKAMaxMpUhHSu/p7IWKj1JPRNZ8hwpJe9mfif100xuPYyESUpQsQXi4JUUozpLAI6EAo4yokhjCthbqV8xBTjaIIqmRCc5ZdXSeui6tSql/e1Sv0mj6NITsgpOScOuSJ1ckcapEk4UeSZvJI368l6sd6tj0VrwcpnjskfWJ8/z2aSuQ==</latexit>




Identifying Novel Jet ObservablesWhen likelihood ratio is not a function of standard high-level observables

Maximize

Decision

Ordering

BBN

Signal/Background Pairs

Same

Decision

Ordering?… …

…

… …

No

Yes

BBN

HLN

HL

HL

HL

HLN’

Black-Box

Guided

Search

[Faucett, JDT, Whiteson, PRD 2021; using Komiske, Metodiev, JDT, JHEP 2018]

Iteration (n) EFP κ β Chrom #

0 Mjet + pT – – –

1 2 1

22

2 0 2 2

3 0 – 1

A glimpse at an alternative history for field of jet substructure




Theory-Inspired Likelihood ParametrizationsFlexible frameworks for parton-shower-like modeling

JUNIPR DGLAP White Box Ginkgo

These methods are based on constrained generative models,hopefully generalizing better than generic methods

[Andreassen, Feige, Frye, Schwartz, EPJC 2019]

−2.9

−2.4

−3.4

−3.3

−3.0 −3.0

−2.4−3.6

−2.0 −2.9

−2.4−3.2

−2.7

−3.2

−2.9

−2.9 −2.6 −2.8

Pend · Pmother · Pbranch

Pt=18 = (10−0.7)(10−0.1)(10−2.0)

[Lai, Neill, Płoskoń, Ringer, arXiv 2020] [Cranmer, Drnevich, Macaluso, Pappadopulo, arXiv 2021]

Exhibit close relationship between generation and inference





Sidestepping Per-Event Likelihood Ratios?Could be helpful, though per-event information is usually complete

…therefore, (parametrized)per-event binary classifiers

can be used to constructasymptotically optimal

per-ensemble inference tools

[Nachman, JDT, arXiv 2021; see mixed sample discussion in Metodiev, Nachman, JDT, JHEP 2017]

Collider events are independent and identically distributed…

Ÿ

=1

p(xi|θA)

p(xi|θB)

p({x1, . . . , xN }|θA)

p({x1, . . . , xN }|θB)=

)

)==

NŸ

i=1




Jet Representations

[review in Kagan, arXiv 2020]

Pixelized Image

Calorimetry

… …

s1<latexit sha1_base64="7+SgB3EmiOq3NgrUGJ0ivQdQKjw=">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</latexit><latexit sha1_base64="7+SgB3EmiOq3NgrUGJ0ivQdQKjw=">AAAa4HicpVlbb9y4FZ7d3rbuLds+1X3h1kiaXYzHM+Ox42wRYJHdYFsgaVzHyaa1bIOSjiRiJFEmqcmMVfW5L0XRlxboH+pv6L/pIaWxdZ0Y7RjmUDzfd3h4eEgeauwkZFKNx//54MNvffs73/3eR9/f+sEPf/Tjn9z7+KdvJE+FA68dHnLx1qYSQhbDa8VUCG8TATSyQ/jGnn+p5d8sQEjG41O1SuA8on7MPOZQhU2v5OXk8t7OeDSdzCazIzIezcaHR9MDrBzMDg8ez8hkNDafnUH5Ob78+Of/tlzupBHEygmplGeTcaLOMyoUc0LIt6xUQkKdOfXhDKsxjUCeZ8bWnNzHFpd4XOB/rIhprTIyGkm5imxERlQFsinTjV2ys1R5R+cZi5NUQewUHXlpSBQneuDEZQIcFa6wQh3B0FbiBFRQR6F7trbuE+MsiRbFEv2OgyPvmApIEnKFPbngoYuNuZlLxXxXgJtnwrfzbDw6GqLHxrqYHI33H08nB4dHh9NHs4NJ3sG0wxRuqOMb6h2YvgCI69QqZ/wYdRh9XWQFNKxzDydF2YXmgsY+1Ec4K3V3zcnQlXpCh7quOA9lLQ4yydKYqWW90Rc0CZjTaI3SUDHB3w1tzueK2nKIRRpSsRx6IaeqHmC60ycxFxENJbuG80ymtsf8Ru8YpgG4Q1vQOdQVZB6s4ijZpanidc2SC/XA4RGuJ4nxG1NlMxshxxjyLxO9fOQpPy61BKskgFjmWSrCvKolcT0Fy2HAXL1G53KoS+PnJ5VoGBKHKag2F1M9JKiv2qyjzjgmwifJE4iNQsWdJzQMz7UdIAR49TFCnEaoP8JVYZYeisRuMeUuQd+GoOoLDT3ecEdGHQfXhFyrsIKAqhpgKbEGWm4qkrAYF1MU0dg16+triEHQ0KxGCUqx2N+yYnhXYjJLh42ezvxsco5PsFTSyXYmeV6HYT3BdWuCJ88srFsxT9ACGze+uWUzX85ZUmuLOYtdtL6hCUOKLbT7ih4D9FsWKJV8vrdnRCMu/D0MwD00ojBIKdzm3rLF59qshjZdgbX1OD0+7kVUZJb0PBqxcGVJ3HYSpaPU6GruJYXKiqP0NogLidS7YVGu1atARBlr2uAyz7sVu00x+HmWwcga+qP8zw0Zw8WeMZRBWwZXKVvQUI8Ox8MiuDKG/gEnOeYkoisb6oQVJIjEBZEK0MZo7zR3MUIsJwDHRHLLmXOcMO716yj2UFRBpWqRceqhn2qcjUyF45Cly1+ZiK1rubq6SilCLfNNiq+8hbkBrVE9sFvcDbBAYvfHwUoyR65nvE7GAaxjSgUODbMX+aUJoI61oQdew77MLztgoS/qsOcbVAq3qdIKwVMPdyaWYH6gPm0QbHEbgU9PmuqiBGOoPje6tL0syS9PL4pllkVM6qlpkaFJfvZeDp4w8io33xeWS30fcPPChwYMkwUMt1eXpTL9VITGazy3GjMCC7MMMnjTCtu1aN6WRWvZi7bMX8u+bsvUWnbalnl2IcLv5nK+FV1ku+3dKinFSZt5K1oz0Q1fYU4kmJ2ac48015s+fvOzaSVKfpdbn+BfGSnEipjrhvAnsjMlN2Gj1YLAnUWxBW4lqMscGKmiCjOxhr0uiKIHD3M2YvZ9xXCXxIitPk1b7msS9R5Zsopqm9LJuZhWWRnSLlpMr0WEUNE1r6hPC4c+5aHbveCxxVUBKFqPdWIEBQNtNIBm/xoRp3k/DYUdFEgkC3m8gbegYg3q4C8bzHJFL7uw193Y6y7sqhu76sIuurGLLqzqxqpOe0Hwbvi4mMgXoALuNubQoZgH3iQxX+onq0yQ6kDBQ3ELPNFP3UCJ+d3qFvnKPPZDWczrYN3QDXeodGjNWvPcY28DfNIAm8MUHH3FJByDVHTHuAlfnaeW1fW+q5taG1Ks9DWN4p22nlWQrDORSjZpEHfRIDZpkHfRIDdpUHfR0E5rKhqu76Lhjy0NIce5ijhuRHfy43pSDK04WnF6fxsrwOS7OaNond4G8ItYn5Bik1xav26PYZkIzLxa0F91YlkLd8k6gfM2cN4F9GjbzGIP0+gW/LqJve5R2gIWm0enrS3skljDqv4WBZMpgKbdF/vdDvZ4KlrYWT921cau+rBN0xHb7RFtg8SsIOwyZNg3xOosVo/efTLHE/Th1ErYpxf7eV+PffRZlT7rpJvuk97uk7t030efVend3Rt6j78SskfWZOO6++QNODpFMikTrmDBWrcX9Y4vwClyERvwLpolBrj8LD9zzolOySyTjAGib0TNREg7xajZf58aLPc36tIjNKpmd1Flytlm60qNoluluKPKltvMfavfyAeosND34L0WGv8ZfY836tPFfmEe1g50cWjMfaSrR7p4vLkjY/ZGT/wvdotuw8X/ZTiG73Mem2uYTUXjsrkAgY3lhXMOIiaT0UGUGoF+x1227ppWvIFWnvIqobhc8BTPSmJeVGEvIcS+vm/qN1cB6DtIY9haYLq+v0XwY+m3VLxA1lj16yu2F+9uShZ1XcU7+sp2x6P9A9zFS5zmWYskoLHiESZWaQjZRN+Oq6Q1GLcGWe2ryDz12Y1XflHel74CJ8SkSje/LFtxLjHjwf8e6SlKT3ulLqN+npmyB3FiXiqfQJ/8mDPMl0zZg3A4Zu666JHjzSPPdNEjhyWmm8r8nFG+d7Dt7FneB396kuu3Ej3SwMmzYGQNnVE34jP90tCPKB6v+G0NdW0TkMVrINZ6+hQQptqFz1/3IULu4+2boW03tb5OX7x6lme66AMoEQFFq1SUb13e21n/vkP6K2+mo8l4NPn9bOeLh+WPQB8NfjH45eDhYDJ4NPhi8JvB8eD1wBn4g78O/jH457a9/Zftv23/vYB++EHJ+dmg9tn+138BguEIhg==</latexit><latexit sha1_base64="7+SgB3EmiOq3NgrUGJ0ivQdQKjw=">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</latexit><latexit sha1_base64="7+SgB3EmiOq3NgrUGJ0ivQdQKjw=">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</latexit>

a1<latexit sha1_base64="2Ar+bIOaGZuklexVyFFE5MgBtWs=">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</latexit><latexit sha1_base64="2Ar+bIOaGZuklexVyFFE5MgBtWs=">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</latexit><latexit sha1_base64="2Ar+bIOaGZuklexVyFFE5MgBtWs=">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</latexit><latexit sha1_base64="2Ar+bIOaGZuklexVyFFE5MgBtWs=">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</latexit>



a2<latexit sha1_base64="D+KjXe8KEPinpUOYMX+jOmLeV9w=">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</latexit><latexit sha1_base64="D+KjXe8KEPinpUOYMX+jOmLeV9w=">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</latexit><latexit sha1_base64="D+KjXe8KEPinpUOYMX+jOmLeV9w=">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</latexit><latexit sha1_base64="D+KjXe8KEPinpUOYMX+jOmLeV9w=">AAAa4HicpVlbb9y4FZ7d3rbuLds+1X3h1kiaXYzHM+NLnC0CLLIbbAskjetLNq1lG5R0JBEjiTJJTWasqs99KYq+tED/UH9D/00PKY2t68RoxzCH4vm+w8PDQ/JQYychk2o8/s8HH37r29/57vc++v7GD374ox//5MHHP30jeSocOHN4yMVbm0oIWQxniqkQ3iYCaGSH8I09+1LLv5mDkIzHp2qZwEVE/Zh5zKEKm07o1fTqwdZ4dHhwuL+/S8ajyXQ8OdCV6WRvf3JIJqOx+WwNys/R1cc//7flcieNIFZOSKU8n4wTdZFRoZgTQr5hpRIS6syoD+dYjWkE8iIztubkIba4xOMC/2NFTGuVkdFIymVkIzKiKpBNmW7skp2nyju8yFicpApip+jIS0OiONEDJy4T4KhwiRXqCIa2EieggjoK3bOx8ZAYZ0m0KJbodxwcecdUQJKQK+zJBQ9dbMzNXCpm2wLcPBO+nWfoviF6bKyLyeF49+l0sn9weDB9gh7MO5h2mMItdXxLvQfTFwBxnVrljJ+iDqOvi6yAhnXuwaQou9Bc0NiH+gj3St1dczJ0pZ7Qoa4rzkNZi4NMsjRmalFv9AVNAuY0WqM0VEzwd0Ob85mithxikYZULIZeyKmqB5ju9FnMRURDyW7gIpOp7TG/0TuGaQDu0BZ0BnUFmQfLOEq2aap4XbPkQj1yeITrSWL8xlTZzEbIEYb860QvH3nKj0otwTIJIJZ5loowr2pJXE/BYhgwV6/RmRzq0vj5WSUahsRhCqrNxVQPCeqrNuuoM46J8EnyBGKjUHHnGQ3DC20HCAFefYwQpxHqj3BVmKWHIrFdTLlL0LchqPpCQ4833JFRx8E1IVcqrCCgqgZYSKyBlpuKJCzGxRRFNHbN+voaYhA0NKtRglIs9jesGN6VmMzSYaOnMz+fXOATLJR0sq1JntdhWE9w3ZrgyTML61bME7TAxo1vZtnMlzOW1NpizmIXrW9owpBic+2+oscA/ZYFSiWf7+wY0YgLfwcDcAeNKAxSCre5t2z+uTaroU1XYGU9To+PexEVmSU9j0YsXFoSt51E6Sg1upp7SaGy4ii9DeJCIvVuWJRr9SoQUcaaNrjM8+7EblMMfp5lMLKG/ij/c0PGcLFnDGXQlsF1yuY01KPD8bAIro2hf8BJjjmJ6NKGOmEJCSJxQaQCtDHaO81djBDLCcAxkdxy5gwnjHv9Ooo9FFVQqVpknHropxpnI1PhOGTp8hMTsXUt19fXKUWoZb5J8ZW3MLegFaoHdoe7BRZI7P4oWErmyNWM18k4gFVMqcChYfYqvzIB1LE29MBr2Nf5VQcs9EUd9nKNSuE2VVoheOrx1sQSzA/Upw2CLe4i8PlxU12UYAzV50aXtpcl+dXpZbHMsohJPTUtMjTJL97LwRNGXufm+9Jyqe8Dbl740IBhsoDhdnJVKtNPRWic4bnVmBGYm2WQwZtW2K5Es7YsWsletWX+SvZ1W6ZWstO2zLMLEX43l/Od6DLbbu9WSSlO2sw70YqJbvgKcyLB7NSce6S53vTxm59PK1Hyu9z6BP/KSCFWxFw3hD+RrSm5DRutFgTuLIrNcStBXebASBVVmIk17HVBFD14mLMRs+8rhrskRmz1adpyX5Oo98iSVVTblE7O5bTKypB22WJ6LSKEiq54RX1aOPQ5D93uBY8trgpA0XqsEyMoGGijATT714g4zftpKOygQCJZyOM1vDkVK1AHf9Fglit60YW96cbedGGX3dhlF3bejZ13YVU3VnXaC4J3w8fFRL4CFXC3MYcOxTzwNon5Uj9ZZYJUBwoeijvgsX7qBkrM75Z3yBPz2A9lMa+DdUM33KHSoTVrzXOPvQ3wcQNsDlNw9BWTcAxS0R3jJnx1nlpWV/uubmptSLHS1zSKd9p6VkGyzkQqWadB3EeDWKdB3keDXKdB3UdDO62paLi5j4Y/tjSEHOcq4rgR3cuPq0kxtOJoxen9bawAk+/mjKJ1ehvAL2J9QopNcmH9uj2GRSIw82pBf9WJZS3cFesEztrAWRfQo20ziz1Mo1vwmyb2pkdpC1hsHp22trALYg2r+lsUTKYAmnZf7nY72OOpaGH3+rHLNnbZh22ajthuj2gbJGYFYZchw74hVmexevTukhmeoI+nVsI+vdzN+3rso+9V6XuddNN90tt9cp/u++h7VXp394be46+E7JAV2bjuIXkDjk6RTMqEK1iw1u1FveNzcIpcxAa8i2aJAS4+y8+dC6JTMsskY4DoW1EzEdJOMWp236cGy921uvQIjaq9+6gy5d5660qNoluluKfKltvMfavfyEeosND36L0WGv8ZfU/X6tPFbmEe1vZ1cWDMfaKrh7p4ur4jY/ZaT/wvdotuw8X/ZTiG70sem2uYTUXjsjkHgY3lhXMGIiaT0X6UGoF+x122bptWvIFWnvIqobhc8BTPSmJeVGEvIcS+vm/qN1cB6DtIY9haYLp+uEHwY+m3VLxA1lj16yu2F+9uShZ1XcU7+sq2x6PdfdzFS5zmWfMkoLHiESZWaQjZRN+Oq6QVGLcGWe2ryDz12Y1XflHel74CJ8SkSje/LltxLjHjwf8e6SlKT3ulLqN+npmyB3FsXiofQ5/8iDPMl0zZg3A4Zu666JHjzSPPdNEjhwWmm8r8nFG+d7Dt7EXeB39+nOu3Ej3SwMmzYGQNnVE34jP90tCPKB6v+G0NdW0dkMUrINZ6+hQQptqFL8/6ECH38fbN0LbbWl+nr05e5Jku+gBKREDRKhXlG1cPtla/75D+ypvpaDIeTX6/t/XF4/JHoI8Gvxj8cvB4MBk8GXwx+M3gaHA2cAb+4K+Dfwz+uWlv/mXzb5t/L6AfflByfjaofTb/9V/Tzghs</latexit>



s2<latexit sha1_base64="aimGkCSaSL/CuW8DX/jXOPE7pFQ=">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</latexit><latexit sha1_base64="aimGkCSaSL/CuW8DX/jXOPE7pFQ=">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</latexit><latexit sha1_base64="aimGkCSaSL/CuW8DX/jXOPE7pFQ=">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</latexit><latexit sha1_base64="aimGkCSaSL/CuW8DX/jXOPE7pFQ=">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</latexit>

r1<latexit sha1_base64="2YWYnOtQjFGMNDA3DlPs39N6QeU=">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</latexit><latexit sha1_base64="2YWYnOtQjFGMNDA3DlPs39N6QeU=">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</latexit><latexit sha1_base64="2YWYnOtQjFGMNDA3DlPs39N6QeU=">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</latexit><latexit sha1_base64="2YWYnOtQjFGMNDA3DlPs39N6QeU=">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</latexit>





s3<latexit sha1_base64="R/2nL6KTDOvoTEtcUA8dgJ1Ukhg=">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</latexit><latexit sha1_base64="R/2nL6KTDOvoTEtcUA8dgJ1Ukhg=">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</latexit><latexit sha1_base64="R/2nL6KTDOvoTEtcUA8dgJ1Ukhg=">AAAa4HicpVlbb9y4FZ7d3rbuLds+1X3h1kiaXYzHM+Ox42wRYJHdYFsgaVzHyaa1bIOSjiRiJFEmqcmMVfW5L0XRlxboH+pv6L/pIaWxdZ0Y7RjmUDzfd3h4eEgeauwkZFKNx//54MNvffs73/3eR9/f+sEPf/Tjn9z7+KdvJE+FA68dHnLx1qYSQhbDa8VUCG8TATSyQ/jGnn+p5d8sQEjG41O1SuA8on7MPOZQhU2v5OX+5b2d8Wg6mU1mR2Q8mo0Pj6YHWDmYHR48npHJaGw+O4Pyc3z58c//bbncSSOIlRNSKc8m40SdZ1Qo5oSQb1mphIQ6c+rDGVZjGoE8z4ytObmPLS7xuMD/WBHTWmVkNJJyFdmIjKgKZFOmG7tkZ6nyjs4zFiepgtgpOvLSkChO9MCJywQ4KlxhhTqCoa3ECaigjkL3bG3dJ8ZZEi2KJfodB0feMRWQJOQKe3LBQxcbczOXivmuADfPhG/n2Xh0NESPjXUxORrvP55ODg6PDqePZgeTvINphyncUMc31DswfQEQ16lVzvgx6jD6usgKaFjnHk6KsgvNBY19qI9wVurumpOhK/WEDnVdcR7KWhxkkqUxU8t6oy9oEjCn0RqloWKCvxvanM8VteUQizSkYjn0Qk5VPcB0p09iLiIaSnYN55lMbY/5jd4xTANwh7agc6gryDxYxVGyS1PF65olF+qBwyNcTxLjN6bKZjZCjjHkXyZ6+chTflxqCVZJALHMs1SEeVVL4noKlsOAuXqNzuVQl8bPTyrRMCQOU1BtLqZ6SFBftVlHnXFMhE+SJxAbhYo7T2gYnms7QAjw6mOEOI1Qf4Srwiw9FIndYspdgr4NQdUXGnq84Y6MOg6uCblWYQUBVTXAUmINtNxUJGExLqYoorFr1tfXEIOgoVmNEpRisb9lxfCuxGSWDhs9nfnZ5ByfYKmkk+1M8rwOw3qC69YET55ZWLdinqAFNm58c8tmvpyzpNYWcxa7aH1DE4YUW2j3FT0G6LcsUCr5fG/PiEZc+HsYgHtoRGGQUrjNvWWLz7VZDW26AmvrcXp83IuoyCzpeTRi4cqSuO0kSkep0dXcSwqVFUfpbRAXEql3w6Jcq1eBiDLWtMFlnncrdpti8PMsg5E19Ef5nxsyhos9YyiDtgyuUragoR4djodFcGUM/QNOcsxJRFc21AkrSBCJCyIVoI3R3mnuYoRYTgCOieSWM+c4Ydzr11HsoaiCStUi49RDP9U4G5kKxyFLl78yEVvXcnV1lVKEWuabFF95C3MDWqN6YLe4G2CBxO6Pg5VkjlzPeJ2MA1jHlAocGmYv8ksTQB1rQw+8hn2ZX3bAQl/UYc83qBRuU6UVgqce7kwswfxAfdog2OI2Ap+eNNVFCcZQfW50aXtZkl+eXhTLLIuY1FPTIkOT/Oy9HDxh5FVuvi8sl/o+4OaFDw0YJgsYbq8uS2X6qQiN13huNWYEFmYZZPCmFbZr0bwti9ayF22Zv5Z93Zaptey0LfPsQoTfzeV8K7rIdtu7VVKKkzbzVrRmohu+wpxIMDs15x5prjd9/OZn00qU/C63PsG/MlKIFTHXDeFPZGdKbsJGqwWBO4tiC9xKUJc5MFJFFWZiDXtdEEUPHuZsxOz7iuEuiRFbfZq23Nck6j2yZBXVNqWTczGtsjKkXbSYXosIoaJrXlGfFg59ykO3e8Fji6sCULQe68QICgbaaADN/jUiTvN+Ggo7KJBIFvJ4A29BxRrUwV82mOWKXnZhr7ux113YVTd21YVddGMXXVjVjVWd9oLg3fBxMZEvQAXcbcyhQzEPvElivtRPVpkg1YGCh+IWeKKfuoES87vVLfKVeeyHspjXwbqhG+5Q6dCatea5x94G+KQBNocpOPqKSTgGqeiOcRO+Ok8tq+t9Vze1NqRY6WsaxTttPasgWWcilWzSIO6iQWzSIO+iQW7SoO6ioZ3WVDRc30XDH1saQo5zFXHciO7kx/WkGFpxtOL0/jZWgMl3c0bROr0N4BexPiHFJrm0ft0ewzIRmHm1oL/qxLIW7pJ1Audt4LwL6NG2mcUeptEt+HUTe92jtAUsNo9OW1vYJbGGVf0tCiZTAE27L/a7HezxVLSws37sqo1d9WGbpiO22yPaBolZQdhlyLBviNVZrB69+2SOJ+jDqZWwTy/2874e++izKn3WSTfdJ73dJ3fpvo8+q9K7uzf0Hn8lZI+sycZ198kbcHSKZFImXMGCtW4v6h1fgFPkIjbgXTRLDHD5WX7mnBOdklkmGQNE34iaiZB2ilGz/z41WO5v1KVHaFTN7qLKlLPN1pUaRbdKcUeVLbeZ+1a/kQ9QYaHvwXstNP4z+h5v1KeL/cI8rB3o4tCY+0hXj3TxeHNHxuyNnvhf7Bbdhov/y3AM3+c8Ntcwm4rGZXMBAhvLC+ccREwmo4MoNQL9jrts3TWteAOtPOVVQnG54CmelcS8qMJeQoh9fd/Ub64C0HeQxrC1wHR9f4vgx9JvqXiBrLHq11dsL97dlCzquop39JXtjkf7B7iLlzjNsxZJQGPFI0ys0hCyib4dV0lrMG4NstpXkXnqsxuv/KK8L30FTohJlW5+WbbiXGLGg/890lOUnvZKXUb9PDNlD+LEvFQ+gT75MWeYL5myB+FwzNx10SPHm0ee6aJHDktMN5X5OaN872Db2bO8D/70JNdvJXqkgZNnwcgaOqNuxGf6paEfUTxe8dsa6tomIIvXQKz19CkgTLULn7/uQ4Tcx9s3Q9tuan2dvnj1LM900QdQIgKKVqko37q8t7P+fYf0V95MR5PxaPL72c4XD8sfgT4a/GLwy8HDwWTwaPDF4DeD48HrgTPwB38d/GPwz217+y/bf9v+ewH98IOS87NB7bP9r/8Ct9UIiA==</latexit><latexit sha1_base64="R/2nL6KTDOvoTEtcUA8dgJ1Ukhg=">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</latexit>

r2<latexit sha1_base64="H/ax1x6wPdrxtnIu9kUbkBFPI/E=">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</latexit><latexit sha1_base64="H/ax1x6wPdrxtnIu9kUbkBFPI/E=">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</latexit><latexit sha1_base64="H/ax1x6wPdrxtnIu9kUbkBFPI/E=">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</latexit><latexit sha1_base64="H/ax1x6wPdrxtnIu9kUbkBFPI/E=">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</latexit>





Hierarchical Tree

Binary Splittings

Graphs

[e.g. Brehmer, Macaluso, Pappadopulo,Cranmer, NeurIPS 2020]

Pairwise Interactions

[e.g. Moreno, Cerri, Duarte, Newman, Nguyen,Periwal, Pierini, Serikova, Spiropulu, Vlimant, EPJC 2020]

O3

O1 O2E1

E2 E3

E4

E5 E6

Imposes implicit theoretical prior (typically a good thing!)Influences choice of network architecture




2 0 4 0 6 0 8 1 0


From Principles to Network Architectures

Permutation Equivariance Lorentz Equivariance

Infrared and Collinear SafetyLund Plane Emissions

[Dreyer, Qu, JHEP 2021]

Φ1

Φ`

100 100

y

�

~Φ(p1)

~Φ(pM)M

~E1

~EM

` `0

[Dolan, Ore, PRD 2021] [Bogatskiy, Anderson, Offermann, Roussi, Miller, Kondor, arXiv 2020]

:M

k,n

T (k,n) ! T (k1,n1) ⌦ T (k2,n2),

[Komiske, Metodiev, JDT, JHEP 2019]

…






Energy Flow NetworksArchitecture designed around symmetries and interpretability

[Komiske, Metodiev, JDT, JHEP 2019; see also Komiske, Metodiev, JDT, JHEP 2018; code at energyflow.network; special case of Zaheer, Kottur, Ravanbakhsh, Poczos, Salakhutdinov, Smola, NIPS 2017;

histogram pooling in Cranmer, Kreisch, Pisani, Villaescusa-Navarro, Spergel, Ho, ICLR SimDL 2021]

S�

J�

= F�

V1, V2, . . . , V`

�

<latexit sha1_base64="CLDgcpZzt8LjDFfYB3vgYAEsbYs=">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</latexit><latexit sha1_base64="CLDgcpZzt8LjDFfYB3vgYAEsbYs=">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</latexit><latexit sha1_base64="CLDgcpZzt8LjDFfYB3vgYAEsbYs=">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</latexit><latexit sha1_base64="m/9yOAvQkiM1IzQxy4IXGAhHNgk=">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</latexit>

Va

�

J�

=X

i∈J

Ei Φa(ni)

<latexit sha1_base64="+NXHWecmei+FQJ/kuD+SU735mn4=">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</latexit><latexit sha1_base64="+NXHWecmei+FQJ/kuD+SU735mn4=">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</latexit><latexit sha1_base64="+NXHWecmei+FQJ/kuD+SU735mn4=">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</latexit><latexit sha1_base64="MYHo9ojyANJmJN/z1V+vmWKEZZg=">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</latexit>

Linear weights(IRC safe)

PermutationinvariantLatent space of dim 𝓁

Easy to plot!E

2 0 4 0 6 0 8 1 0

Learned 𝓁 = 256 latent space for quark/gluon discrimination:



https://energyflow.network/


https://simdl.github.io/files/40.pdf


Learning from the Machine vs.

q g

−R −R/2 0 R/2 R

Translated Rapidity y

R

R/2

0

−R/2

−R

TranslatedAzimuthalAngle

φ

Filter 1

0.0

0.2

0.4

0.6

0.8

1.0

−R −R/2 0 R/2 R

Translated Rapidity y

R

R/2

0

−R/2

−R

TranslatedAzimuthalAngle

φ

Filter 2

0.0

0.2

0.4

0.6

0.8

1.0

0 R/8 R/4 3R/8 R/2

Radial Position θ

0.0

0.2

0.4

0.6

0.8

1.0

Rad

ialProfile

Φ(θ)

EFN2: Quark vs. Gluon

Pythia 8.230,√s = 14 TeV

R = 0.4, pT ∈ [500, 550] GeV

Learned Filter Φ1(θ)

Learned Filter Φ2(θ)

Closed-Form Φ(θ)

[Komiske, Metodiev, JDT, JHEP 2019;cf. Larkoski, JDT, Waalewijn, JHEP 2014; using Berger, Kucs, Sterman, PRD 2003; Ellis,Vermilion,Walsh, Hornig, Lee, JHEP 2010]

cf. Angularities: f(θ) = θβFor 𝓁 = 2, EFN learns radial moments: X

i∈jet

zif(θi)

<latexit sha1_base64="owte8LrCteMhJ5mCVmC09N3Jy7E=">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</latexit><latexit sha1_base64="owte8LrCteMhJ5mCVmC09N3Jy7E=">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</latexit><latexit sha1_base64="owte8LrCteMhJ5mCVmC09N3Jy7E=">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</latexit><latexit sha1_base64="ts5vAnChiyPhbqIko/i5BLewEQo=">AAACTXicbVDBSiNBFOyJurrR3Y169NIYBAUJMyqJ3sS9eMyCUSEzDD2dN0mv3T1D95vFOMxv+Ruehb3qH3iTZTsxwqpb0FBU1ev3qCSXwqLv33u1ufmFT4tLn+vLK1++fmusrp3brDAcejyTmblMmAUpNPRQoITL3ABTiYSL5Or7xL/4BcaKTJ/hOIdIsaEWqeAMnRQ3uqEtVFwKGgpNQ4RrLH8CVhUtw+nnfTNMotJv+VPs+q2D4HC/3XFkL/CP2vvVTSwqmm6HOAJksdiJG83XNP1Ighlpkhm6ceMxHGS8UKCRS2ZtP/BzjEpmUHAJVT0sLOSMX7Eh9B3VTIGNyulxFd1yyoCmmXFPI52q/06UTFk7VolLKoYj+96biP/z+gWmh1EpdF4gaP6yKC0kxYxOaqQDYYCjHDvCuBHuVspHzDCOruw3W1IYa5VXdVdM8L6Gj+R8rxX4reDHQfP4ZFbREtkgm2SbBKRDjskp6ZIe4eSW/CYP5NG78568Z+/PS7TmzWbWyRvUFv8CJ8+xXw==</latexit>

Traditional QCD observables emphasize homogeneous angular scalingBut EFN reveals that likelihood ratio exhibits collinear/wide-angle separation



https://arxiv.org/abs/hep-ph/0303051



Learning from the Machine vs.

q g

[Komiske, Metodiev, JDT, JHEP 2019;cf. Larkoski, JDT, Waalewijn, JHEP 2014; using Berger, Kucs, Sterman, PRD 2003; Ellis,Vermilion,Walsh, Hornig, Lee, JHEP 2010]

cf. Angularities: f(θ) = θβFor 𝓁 = 2, EFN learns radial moments: X

i∈jet

zif(θi)

<latexit sha1_base64="owte8LrCteMhJ5mCVmC09N3Jy7E=">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</latexit><latexit sha1_base64="owte8LrCteMhJ5mCVmC09N3Jy7E=">AAACTXicbVDLattAFL1yH0ndl/vYdTPUFFIoRkpCHjvTbrJ0oU4ClhCj8ZU9ycxIzFyVOkK/1d/outBt8wfZhdCxnEKT9MDA4Zxz515OVirpKAx/Bp179x88XFt/1H385Omz570XLw9dUVmBY1Gowh5n3KGSBsckSeFxaZHrTOFRdvpp6R99RetkYb7QosRE85mRuRScvJT2RrGrdFpLFkvDYsJvVJ8gNQ2r4/bziZ1lSR0OwhYfwsF2tLe1s+vJZhTu72w1Z6lsWL4R0xyJp/J92uv/TbO7JLom/eFraDFKe+fxtBCVRkNCcecmUVhSUnNLUihsunHlsOTilM9w4qnhGl1St8c17J1XpiwvrH+GWKv+O1Fz7dxCZz6pOc3dbW8p/s+bVJTvJbU0ZUVoxGpRXilGBVvWyKbSoiC18IQLK/2tTMy55YJ82Te25Lgwumy6vpjodg13yeHmIAoH0eft/vDjqiFYhzfwFjYggl0YwgGMYAwCvsMv+A3nwY/gIrgMrlbRTnA98wpuoLP2B6B7sbM=</latexit><latexit sha1_base64="owte8LrCteMhJ5mCVmC09N3Jy7E=">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</latexit><latexit sha1_base64="ts5vAnChiyPhbqIko/i5BLewEQo=">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</latexit>

EFN outperformed a domain expert (i.e. me)

But we reverse engineered the machine (and learned something about QCD)

0.0 0.2 0.4 0.6 0.8 1.0

Quark Jet Efficiency

0.0

0.2

0.4

0.6

0.8

1.0

Gluon

Jet

Rejection



R = 0.4, pT ∈ [500, 550] GeV

EFN2

C(A,B)

BDT(λ(1), λ(2), λ(1/2))

Angularity λ(1)

Angularity λ(2)

Angularity λ(1/2)

Bette

r






More Network Architectures for InterpretabilityRendering the black box more transparent

[Agarwal, Hay, Iashvili, Mannix, McLean, Morris, Rappoccio, Schubert, JHEP 2021] [Kasieczka, Marzani, Soyez, Stagnitto, JHEP 2020]

Input Feature Relevance Analytic Calculations

Imposing specific theoretical structures might reduce performancebut might also yield better robustness/generalizability




Classifier1 0

Signal Background

Quark/Gluon Classification“Hello, World!” of Jet Physics

[see e.g. Gras, Höche, Kar, Larkoski, Lönnblad, Plätzer, Siódmok, Skands, Soyez, JDT, JHEP 2017;Komiske, Metodiev, Schwartz, JHEP 2017; Komiske, Metodiev, JDT, JHEP 2018]

vs.Quark

Cq = 4/3Gluon

Cg = 3 = 9/3

Find such thatQuark

Gluon

h

!

<latexit sha1_base64="4jjUhxK/BwEv+hur9m3Cz8iSMp0=">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</latexit><latexit sha1_base64="4jjUhxK/BwEv+hur9m3Cz8iSMp0=">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</latexit><latexit sha1_base64="4jjUhxK/BwEv+hur9m3Cz8iSMp0=">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</latexit><latexit sha1_base64="8SO8vlNfFPHIXZ+2O3AixnXFXH4=">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</latexit>

h� �

= 1<latexit sha1_base64="xE+8GpNQNyhqEOMDmxBvKCxtyy0=">AAACQnicbVBNSxtBGH43ftSmVaPtzctgKChImNVqkoMQLAWPCsYI2SXMTmY3Q2Zn15lZISz7i/wb/oEe1XsPvZVePTibKPjRF2Z4eJ7n/eAJUsG1wfjWqczNLyx+WPpY/fR5eWW1trZ+rpNMUdaliUjURUA0E1yyruFGsItUMRIHgvWC8Y9S710xpXkiz8wkZX5MIslDTomx1KD2M/emQ/oqCvwcN/DBfnsP7+DGPnab7bYFGB+09naLUeEFPNpC3mVGhs+/ZbbRIXIHtXppLAu9B+4TqHe+wrROBrXf3jChWcykoYJo3XdxavycKMOpYEXVyzRLCR2TiPUtlCRm2s+nlxbom2WGKEyUfdKgKfuyIyex1pM4sM6YmJF+q5Xk/7R+ZsKWn3OZZoZJOlsUZgKZBJXZoSFXjBoxsYBQxe2tiI6IItTYhF9tCdlExmlRtcG4b2N4D853Gy5uuKff652jWUKwBBuwCVvgQhM6cAwn0AUK1/AL7uDeuXH+OH+dfzNrxXnq+QKvynl4BHb1rJk=</latexit><latexit sha1_base64="xE+8GpNQNyhqEOMDmxBvKCxtyy0=">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</latexit><latexit sha1_base64="xE+8GpNQNyhqEOMDmxBvKCxtyy0=">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</latexit><latexit sha1_base64="BO8+nCPq18cnFPBtjI3aDdgledQ=">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</latexit>

h� �

= 0<latexit sha1_base64="4hluKcepKlAfJrfNRk6i7kcWxe8=">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</latexit><latexit sha1_base64="4hluKcepKlAfJrfNRk6i7kcWxe8=">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</latexit><latexit sha1_base64="4hluKcepKlAfJrfNRk6i7kcWxe8=">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</latexit><latexit sha1_base64="q+Qb+5tOrRSE6qeJVX12u3ieLtA=">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</latexit>

<latexit sha1_base64="ND2Qv+3kLP3cDupwYPj92MhSDco=">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</latexit>

h�

J�

=

1 +p�

J |G�

p�

J |Q�

!

−1

(Neyman-Pearson lemma)

Best you can do:

Likelihood ratio yields optimal binary classifier (and vice versa)





Classifier1 0

Signal Background

Quark/Gluon Classification“Hello, World!” of Jet Physics

[see e.g. Gras, Höche, Kar, Larkoski, Lönnblad, Plätzer, Siódmok, Skands, Soyez, JDT, JHEP 2017;Komiske, Metodiev, Schwartz, JHEP 2017; Komiske, Metodiev, JDT, JHEP 2018]

vs.Quark

Cq = 4/3Gluon

Cg = 3 = 9/3

Likelihood ratio yields optimal binary classifier (and vice versa)

What do you mean by “quark” and “gluon”?

Jets are clusters of colorless hadrons!

Parton shower “truth” is but a (useful) fiction!





[Komiske, Metodiev, JDT, JHEP 2018; cf. ATLAS, PRD 2019; using Metodiev, Nachman, JDT, JHEP 2017; Metodiev, JDT, PRL 2018]see also Blanchard, Flaska, Handy, Pozzi, Scott, PLMR 2013; Katz-Samuels, Blanchard, Scott, JMLR 2016]

Topic Modeling to Disentangle Data Samples

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Soft Drop Multiplicity nSD

0.00

0.05

0.10

0.15

0.20

ProbabilityDensity

EFP-Extracted Fractions


R = 0.4, pT ∈ [500, 550] GeV

Z+jet

Dijets

Quark

Gluon

Topic 1

Topic 2

While you can’t unambiguously label individual jets, you canextract quark and gluon distributions from hadron-level measurements

Key concept from natural languageprocessing: “anchor words”








Parameterized Data SamplesIncorporating nuisance parameters into training

Regardless of the approach, inspires renewed theoretical focuson uncertainty modeling for Monte Carlo generation

Learn to Pivot… …or Learn to Profile?

[Louppe, Kagan, Cranmer, NeurIPS 2017] [Ghosh, Nachman, Whiteson, arXiv 2021]




Loss Function for Bayesian AnalysesTreating network parameters as having a prior distribution

Jet Energy Scale Event Generation

Use ELBO loss to capture both statistical and systematic uncertaintiesWorth developing a frequentist version of this approach?

[Kasieczka, Luchmann, Otterpohl, Plehn, SciPost 2020] [Bellagente, Haußmann, Luchmann, Plehn, arXiv 2021]

0.0

0.2

0.4

0.6

0.8

1.0

Normalized

×10−1

BINN

Training Data

+/- σpred

20 40 60

pT,e+ [GeV]

0.9

1.0

1.1

BINN

Data



Use ELBO loss to capture both statistical and systematic uncertaintiesWorth developing a frequentist version of this approach?

Loss Function for Bayesian AnalysesTreating network parameters as having a prior distribution


<latexit sha1_base64="KEZ3fh6QsVmMKwLSft3M13nD0VI=">AAACW3icbZBNb9NAEIYnLh8lFBq+TlwWIqRWQpGNQHBBiuDCsUikrRSHaLweN6vurq3dMWpk8u/4Exy4cuAKP4C1EwFtGWmlR++8o5l9s0orz3H8tRdtXbl67fr2jf7NnVu3dwd37h76snaSJrLUpTvO0JNWliasWNNx5QhNpukoO33b9o8+kfOqtB94WdHM4IlVhZLIQZoPPqbKskgfiTzlBTGK9Kn4S9Xehj8Hkc64yZFxtS9SW/JrkRo8m28MrfOPobO38v58MIxHcVfiMiQbGI7vQ1cH88H3NC9lbciy1Oj9NIkrnjXoWElNq35ae6pQnuIJTQNaNORnTZfDSjwJSi6K0oUXftWp/040aLxfmiw4DfLCX+y14v9605qLV7NG2apmsnK9qKi14FK0oYpcOZKslwFQOhVuFXKBDiWH6M9tKWhpTbXqh2CSizFchsNno+TFKH7/fDh+s04ItuEhPIY9SOAljOEdHMAEJHyBH/ATfvW+RVtRP9pZW6PeZuYenKvowW9IdLZu</latexit>Zdθ θ p(θ|data) 6= max

θ

p(data|θ)

Nothing intrinsically wrong with Bayesian analyses,but have to be aware of cases with strong prior dependence

If needed, can treat neural networks as static objectsand calibrate them as if they were ordinary observables

[see further discussion in Cranmer, Pavez, Louppe, arXiv 2015; Nachman, SciPost 2020]




Training Paradigm for Preserved Uncertainties When the goal is to maintain statistical properties

[Nachman, JDT, PRD 2020; building on Andersen, Gutschow, Maier, Prestel, EPJC 2020]

Classifier

1 0

Sample with wrong physics but all positive weights

Sample with correct physics but some negative weights

<latexit sha1_base64="dxaYhJyRgnmGu1Ql0Qq0LPx8Mz4=">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</latexit>

× w(x)⇒

<latexit sha1_base64="dxaYhJyRgnmGu1Ql0Qq0LPx8Mz4=">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</latexit>

pwrong(x)× w(x) = pcorrect(x)Plain reweighting yields all

positive weights with correctasymptotic probability density

Improved resampling throughauxiliary neural network yields correct statistical uncertainties

<latexit sha1_base64="rBiXLyd8ksjn3BORDdaFPN6e2PI=">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</latexit>

✓

δp

p

◆2

=

hw2i

hwi2




Training Paradigm for Preserved Uncertainties When the goal is to maintain statistical properties

[Nachman, JDT, PRD 2020; building on Andersen, Gutschow, Maier, Prestel, EPJC 2020]

Improved resampling throughauxiliary neural network yields correct statistical uncertainties

<latexit sha1_base64="rBiXLyd8ksjn3BORDdaFPN6e2PI=">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</latexit>

✓

δp

p

◆2

=

hw2i

hwi2

Case Study in Jet Physics at Large Hadron Collider

Original sample: large weight cancellations

Reweighting recovers desired distribution

Resampling recoversdesired uncertainties









Theory Perspective on ML for Jets

Looking forward to your thoughts and discussion!


Backup Slides


[from my talk at ML4Jets 2018, with apologies]

P1 P2 P5 P8 R1

Using All Five LHC Interaction Points

https://indico.cern.ch/event/745718/contributions/3146636/

http://www.theorycoffeeco.com/


E.g. Quark/Gluon ClassificationIn various limits, likelihood ratio is monotonically related to…

Away from these limits, the likelihood ratio isnot typically simple, elegant, or interpretable (but we can hope!)

log1

θ

log1

z

log1

z

log1

θ

Aemit

softdrop

, slope =

β

soft

collinear

angularcut

0 1 2 3 4 5

N

0.0

0.1

0.2

0.3

0.4

0.5

Quark

vs.

GluonAUC

Multiplicity

N-subjettiness: τ(β)N


R = 0.4, pT ∈ [1000, 1100] GeV

β = 0.5

β = 1.0

β = 2.0

Soft-dropped Multiplicity IRC Safe Multiplicity

[Frye, Larkoski, JDT, Zhou, JHEP 2017] [Larkoski, Metodiev, JHEP 2019]




E.g. Detector Unfolding

Simulation

Syn

theti

cN

atu

ral

Detector-level

Data

Particle-level

Generation

Truth

Pull Weights

Push Weights

Step 1: Reweight Sim. to Data

Step 2: Reweight Gen.

νn−1

ωn

−−→ νn<latexit sha1_base64="USI/aHUkKNmen4gwHqyTXZ7rTGs=">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</latexit><latexit sha1_base64="USI/aHUkKNmen4gwHqyTXZ7rTGs=">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</latexit><latexit sha1_base64="USI/aHUkKNmen4gwHqyTXZ7rTGs=">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</latexit><latexit sha1_base64="aANR4xqa5G8N4A3PGP5I4W3UWFE=">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</latexit>

νn−1

Data−−−→ ωn

<latexit sha1_base64="IV5Rkcz2AJeiGq38yKsp/wTcRwM=">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</latexit><latexit sha1_base64="IV5Rkcz2AJeiGq38yKsp/wTcRwM=">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</latexit><latexit sha1_base64="IV5Rkcz2AJeiGq38yKsp/wTcRwM=">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</latexit><latexit sha1_base64="i+iM1Y2AuIdG+9czSK1qRLZgOsY=">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</latexit>

[Andreassen, Komiske, Metodiev, Nachman, JDT, PRL 2020; + Suresh, ICLR SimDL 2021;Komiske, McCormack, Nachman, arXiv 2021; see unfolding comparison in Petr Baron, arXiv 2021]

Multi-dimensional unbinned detector corrections via iterated application of likelihood ratio trick

Use ML to computereweighting factors

OmniFold






E.g. Detector Unfolding

10−5

10−4

10−3

10−2

10−1

100

Normalized

Cross

Section

Preliminary

ALEPH E.P.J. C (2004)

ALEPH Raw 1994 Data

Pythia 6 + Geant 3 Sim.

Pythia 6 Gen.

IBU ln(1− T ) + stat.

UniFold ln(1− T ) + stat.

−8 −6 −4 −2

Thrust ln(1− T )

0.85

1.0

1.15

Ratioto

Gen

.

[talk by Badea, ICHEP 2020; cf. ALEPH, EPJC 2004][see also Badea, Baty, Chang, Innocenti, Maggi, McGinn, Peters, Sheng, JDT, Lee, PRL 2019; H1, DIS2021]

Back to the Future with ALEPH Archival Data

thrust axis

intrinsicallyunbinned!

binnedarchive

[Andreassen, Komiske, Metodiev, Nachman, JDT, PRL 2020; + Suresh, ICLR SimDL 2021;Komiske, McCormack, Nachman, arXiv 2021; see unfolding comparison in Petr Baron, arXiv 2021]

https://indico.cern.ch/event/868940/contributions/3814556/

https://doi.org/10.1140/epjc/s2004-01891-4


https://www-h1.desy.de/psfiles/confpap/DIS2021/H1prelim-21-031.pdf






Energy Flow RepresentationEmphasizes infrared and collinear safety

Detection

p

p

Composite Hadrons

Quarks & Gluons

[see e.g. Sveshnikov, Tkachov, PLB 1996; Hofman, Maldacena, JHEP 2008; Mateu, Stewart, JDT, PRD 2013;Belitsky, Hohenegger, Korchemsky, Sokatchev, Zhiboedov, PRL 2014; Chen, Moult, Zhang, Zhu, PRD 2020][complementary perspective on IRC unsafe information in Chakraborty, Lim, Nojiri, Takeuchi, JHEP 2020]

Theory

E

Energy Flow:E ' lim

t→∞

niT0i(t, vtn)Robust to hadronization and detector effects

Well-defined for massless gauge theories








Machine Learning Collinear QCD

2 0 4 0.6 0.8 1.0−4 −3 −2 −1 0



R = 0.4, pT ∈ [500, 550] GeV

Latent Dimension 256

Filter

Best fit, slope = 1.61

0 0 2 0.4 0.6 0

Radial Distance, lnθ

R

−1 0 −4

0.4

0.6

Pixel

Size,

lnA

πR

2

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Collinear

dPi→ig '

2αs

πCi

dθ

θ

dz

z

Soft

Cq = 4/3Cg = 3 z

θ

[Komiske, Metodiev, JDT, JHEP 2019]

vs.

q g



0.0 0.2 0.4 0.6 0.8 1.0

Azimuthal Angle ϕ

0.0

0.2

0 π/2 π 3π/2 2π

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5Latent Dimension 256

LogRadialDistance

lnR θ

Coordinate transformation to the emission plane

[Komiske, Metodiev, JDT, JHEP 2019; see also Dreyer, Salam, Soyez, JHEP 2018]

En Route to the Lund Plane vs.

q g



theory perspective on machine learning for jets · 2021. 6. 3. · jesse thaler (mit) — theory...

Documents