ミンコフスキー汎関数を用いた深層学習(深層学習から何を学ぶか）
KEK 野尻美保子(Deep Leaning 2020, 10/29, 2020)

自己紹介
•

専門

コライダー物理・ダークマター（理論）

LHC, ILC, 宇宙線によるダーク

マターの探索•コライダー物理 高いエネルギーの粒子を衝突させることによって、より高いエネルギーの物質像を探る。

•

LHC ヒッグス粒子を発見(2012) さらにヒッグス粒子の精密測定、暗黒物質の探索などを通じて素粒子の性質を究明•

e-Print 2010.13469, JHEP 07 (2020) 111, JHEP 07 (2019) 135 JEP
10(2018) 181 等に基づいてジェットの深層学習の解釈可能性を調べている。

•

最近やってる別のこと

Jeans 方程式をGAN で解いて、暗黒物質密度を決める。（また呼んでください。）•

なぜ深層学習をはじめたか（もごもご）

LHC実験とは
DONEC QUIS NUNC

コライダー物理と機械学習・深層学習
実験のト リガーの高ルミノシティ

理論の予言

システム

時では、 回のビーム・クロッシングに対して大量のデータイベントが発生する。これら大量のイベントのなかから、物理的に重要なイベントを選択
タを収集するためには、トリガー・システムの役割が重要になってくる。
ここでは、実験のト リガー・スキームを概観した後、実験
ガー・システムについて述べる。

トリガー、特に、

ト リガー・スキーム

新物理（ダークセクターなど）

実験でのトリガー、

システムは図のように、 段階のトリガー・レベ

イベント・フィルター から成り立つ。各レベルで条件を順に付加していき、最終的に目的•トだけをオフライン用のデータとして記録する。
「一つしかない自然の理解」系統誤差のタイミン
LHC実験はデータ取得も大変

グのよい理解が重要
•

目標は

0.1%

Interaction rate
~1 GHzBunch crossing
rate 40 MHz

ハドロンコライダー特有の難しさ•

測定側

CALO MUON TRACKING
Pipelinememories

LEVEL 1
TRIGGER< 75 (100) kHz

自動化・高速化(1GHz でイベントがDerandomizers

来る、衝突点が100 あるとか） HL-LHC 30倍のデータがくる

Readout drivers(RODs)

Regions of Interest
LEVEL 2
TRIGGER

Readout buffers(ROBs)~ 1 kHzEvent builder•

理論側: QCD の理解•

機械学習：BDT (決定木）2002 ~ 深層学習(CNN等）５年ほど特に粒子飛跡の解明で大きな成果

Full-event buffers
andprocessor sub-farms

EVENT FILTER~ 100 Hz

Data recording

図

1999年

におけるトリガースキーム

このトリガー・レベルでは、カロリメータとミューオン検出器からの情報だけが使LHC の物理（ミニマム）
DONEC QUIS NUNC
ハドロン化
（非摂動的）

パートンシャワー
(リサメーション）
ハードプロセス（摂動的）パートン(クオークやグルーオン）

ビームイベントに隠れる特殊な粒子!
H→ qq

t-> qlνt-> qlν

深層学習のジェット物理への応用の理想
ぐちゃぐちゃのイベントの中から素敵なものだけ見つけ出してきてほしいイベントに隠れるカラーを持たない粒子
H→ qq

深層学習のジェット物理への応用の理想
ぐちゃぐちゃのイベントの中から素敵なものだけ見つけ出してきてほしいLHCイベント

ジェットとジェットイメージ

Jets as Images

11

Jet ImageJet imageφ(η, φ, pT)
Rij=(ηi-ηj) 2+(φi-φj)2

Jet

η 2−4
• A jet induces a distribution of energy over

ol

パートン（q g）→ジェット
Flatten• パートンシャワー• クオークやグルーオンがハードなプロセスから生まれるとそ

こからたくさんのクオーク、グルーオンが生成される仕組み• 運動量の関係• q 12→ q 22•

Receiver Operating→ q32 .. (mass ordering)

p→p1+ p2, (p2)-1 = (2 p1p2)-1 散乱断面積は
16( collinear)
あるいは p2~0 で発散
arXiv:1511:05190

p1 p2 が同方向

(Soft Collinear Singlality)• θ1> θ２>θ３…. (angular ordering) Higgs 粒子

Understan

C0αLog2 (q1/q2)+C1αLog (q1/q2)

W 粒子 Since the selection of physics-driv(weak boson) -> qq parton shower は、qq の間にだけでる
sure that the representations we

ax-Poolingvariables. We introduce a new merage of most activating jets for a given neuralat

TaggingarningHiggs的

QCD 的

カラーシングレット粒子の崩壊はロングレンジでみるとQCDで粒子を放出しない。

meaning that physical variables
discriminant, and check for improv•

パートンからハドロン(モデルあれど理論なし）•

Quark gluon 同士をつなぐカラーか伸びると qqbar がうまれてハドロンになる

ストリングモデル•

ストリングモデルとクラスターモデル

Jet clustering （粒子を相互の関係によってまとめて、ひとかたまりのオブジェクト

CA Clustering

にまとめる
•

subjets

Parton が２つにわかれても、ハドロンになってもアウトプットが同じであるよう

particlesjet

なアルゴリズムを探す( IRC safe )粒子(ハドロン）

IRC safe
Figure 1. Schematic of CA clustering andthat fail to satisfy the soft drop criteria,
ジェット
親粒子

presented in section 7. Section 8 prese今後のLHC 高輝度化•

より高いエネルギーをもつ top quark や、Higgs boson などに興味が移ってくる•QCD ジェットと boost された top Higgs などの重たい粒子がよく似ている。

ここに機械学習や深層学習を使うことが注目されている。

s section, we specify the basis of IRC safe observables that we use to identify structure int. For simplicity, we will exclusively use the N -subjettiness observables [24–26], however
hoice is not special. One could equivalently use the originally-defined N -point energy
ation functions [27], or their generalization to di↵erent angular dependence [28]. Our
eepofLearning
using theN -subjettinessobservbles
analysis
is mostly
practical:
Our analysis shows that Deep Conv
— convolutional
networks
in particularin—thiscurrently
represent
the stateof the art inthe evaluation
new physics processes comparedst imagetasks. We
apply a deeparchitecture
to Jetandforthe recognition
N -subjettinessobservables
is convolutionalsignificantly
less thanforImages,
the energy
correlation
ジェットの特徴量をつかった
BDT
enhancing the discovery potential oform model
selection. Below, we visualize a simple architecture
to great success.
ons.
We alsoemphasize that the particular choice usedof observables
below is to just ensuresuggests that the deep convolutiona
Core 1
physics-motivated variables.
hey
span thespacefor emissions
inresponsea jet.
There
maylevelbeAaofmore optimal
foundactuallythat architectures
withphaselarge filters
capturedthewitha higherIntroduction
NeuralNetwork
Crash Course
spectral function of jet substructure Spectral Analysis of Jet Substructure: H
ジェットの運動量,
質量,… physics
learnedfilters from thebut
convolutional
layers exhibit
a two
prong
and location
basedecuracy.
of a Thebasis
of observables,
optimization
of
the
basis
is
beyondthispaper.
エネルギーでweight するのは、
ucture that sheds light on phenomenologicalstructures
withinjets.
( )Networkabout
Crash
What
he N -subjettiness observable ⌧N is a measure ofNeuralthe radiation
N Course:
axes in thejet, is an Artificial Neural N
N-subjettiness,
IRC safe にするため
fied by an angular exponent > 0:
The artificial neural network is a biology Core
inspired2framework of mono
1 Xfunction.
( )⌧N =pT i min R1i , R2i , . . . , RN i .
(2.1)
pT J
Basic architectural unit: neuron

機械学習 とPARTICLE
Physics Perfor
Deep Convolutional
Networks TAGGING

ReLU

Dropout

ReLU

Dropout

Local
Response
Normalization

Fully
Connected
ReLU Unit

i2Jet

···

Boosted Boson Type Tagging

として最小化···Jet 内の全ての粒子でsums expression, pT J is the transverse momentumof the jet of interest,
pwTi1i is the transverse
𝞃2 LARGE > Τ3 SMALLx1
N 個の中心があるPntum of particle i in the jet, and RKi , for
K = 1, 2, . . . , N , is the angle in pseudorapidity'(wij ofxjJet+Substructure:bi )ion Neural Network Crash Course A spectral function of jet substructure
Spectral Analysis of Jet Substructure: Higgs Spectral Analysis of Jet Substructure: '(·)
Sgluon Spectral Analysis
A Quickwin
xn
Jet ETmiss
tical Example
with
CNN:
Image
Recognition
Techniques
with
Jet
Image
Generic
overview
slide
bi
CNN
特徴量とか気にせず,がんがんジェットイメージを入れてやらせてみよう(2015~
Even more
non-linearity: Going Deep
Convolution

Jet Image

Max-Pool

Convolution

Flatten

Max-Pool-32L. Oliveira, M. Kagan, L. Mackey, B. Nachman, A. Schwartzman,(1511.05190)

Boosted Boson Type Tagging inputs weights bias
Convolved– Ariel
3 – Schartzman
Benjamin Nachman
andFeatureLayersConvolutions
Jet
ETmissBuild a network architecture
x1

SLAC, Stanford University

x2

March 26,
Benjamin Nachman
and2014
Ariel Schartzman
SLAC, Stanford University

March 26, 2014 Max-Pooling
W’→ WZ event

B. Nachman (SLAC)

Boosted Boson Type Tagging

activation

outputreduc

Receiver Operating Charŷ
UnderstandSince the selection of physics-driven
sure that the representations we lea
This kind of feed-forward network’s
outputŷ (x1 , · a· ·new
, xn )method
couldvariables.
We introduce

x3

approximate
an output of a function y (x1 , · · · , xn ) if proper w
1 / 21(pTij,
i、j)biases are assigned.

March 26, 2014

RepeatRepeatBasic building unit: 2D convolutional layer

meaning that physical variables hav

深層学習とJETの分類

SciPost Physics•

S

多くの実装: CNN, ResNeXT, Particle Net…
>> BDT>> cut base.

•

なんの効果でよくなるかが不明(というか気にしないスタンス）•

ジェット画像の特性上, 中心には多くのヒッFigure5: ROC Networks
curves for all algorithms
evaluatedon the sametest sample,
Graph
Neuraland Neural
Message
Passing

ensemble median of multiple trainings. More precise numbers as well
トがあるが、外側のヒットは少ない。分散もAUC
bands given by the ensemble analysis are given in Tab. 1.

大きい。CNN が非効率•

→graph network

Particle net “周辺”の object からoutput →そこからアウトプット→ ほんとにそこまでしないといけないのか。

Instead of extracting these performance measures from single models we can
For this purpose we train nine models for each tagger and define 84 ensemble tag
combining six of them. They allow us to evaluate the spread of the ensemble tagg
mean-of-ensemble and median-of-ensemble results. We find that ensembles lead
improvement in performance, depending on the algorithm. For the uncertainty ebackground rejection we remove the outliers. In Tab. 1 we see that the backgr
varies from around 1/600 to better than 1/1000. For the ensemble tagger th
ResNeXt, TreeNiN, and PFN approaches again lead to the best results. Phrof the improvement in the signal-to-background ratio they give factors ✏S /✏B
exceeding the current top tagging performance in ATLAS and CMS.
Altogether, in Fig. 5 and Tab. 1 we see that some of the physics-motivatedcompetitive with the technically much more advanced ResNeXt and Particle
This suggests that even for a straightforward task like top tagging in fat jets w
efficient physics-specific tools. While their performance does not quite matchthe-art standard networks, it is close enough to test both approaches on key reparticle physics, like treatment of uncertainties, stability with respect to detect

The obvious question in any deep-learning analysis is if the tagger captur
information. At this point we have checked that including full or partial i

Slide from I. Henrion

物理から考える：安心できる量と不安な量
摂動論
非摂動

•

IRC safe な量: soft or collinear emission に対して安定な量:subjet•

QCD補正に対して不安定な量,例えば運動量の低い粒子数はMC modeling に大きな差 (Pythia vs Hewig vs 実験データなど） 系統誤差に問題あり→Jet Imageの利用:

両者の間の区別が不明確

系統誤差の評価に問題はないか。実データで補正できるのか。
•

理論：ジェットの構造の中で理論の記述できる量を精査して、それだけ使えばよいのでは？

heN -subjettinessobservables
is significantly
lessshowthanfor showered
the energy
ythiav8.223 [36, 37] withdefault settings.
In App. B, we will
results
with correlation
wig[38, 39], however
the number
as the Pythia
samples.
Wev7.0.4
also emphasize
thatwiththeone-tenthparticular
choice ofofevents
observables
below
is to just ensurering the neutrinos
in phase
the showered
events,FastJetmay
v3.2.1be[40,a 41]
actuallyspan thespaceandforhadronizedemissionsin a wejet.useThere
more optimal
luster the jets. On the clustered anti-kT [42] jets with radius R = 0.8 and minimum pT
basis of observables, but optimizationof thebasis isusing
beyondthisprovided
paper.
00 GeV, we then measure the basisofN-subjettinessobservables
the
code
( )
-subjettinessobservable⌧a measuretheradiationabout
N axesastJet contrib v1.026. We emphasize
observablesofare
measuredon theparticles
as a in the jet,N isthatof
we exponent
do not apply any
detector
simulation.
y ofanconcept;
angular>MLP
0:
N-subjettinessclassification
The precise set of observables weX
measure on theare then jet that we use for discriminationo1( )wing. We measure
jet mass and the
N -subjettiness observables sufficient⌧Nthe=
pT icollection
min Rof(2.1)
1i , R2i , . . . , RN i .
pT J 6-body phase space. That is, we measure theFigure
ompletely determine up throughcollection4: ROC curves for top quark taggin

IRC safe な量だけではベストな結果はでない

i2Jet

N -subjettiness observables defined with kT axes:
right, for pT 2 [350, 400] GeV. Adding m
ression,n pT J is the transverse momentum of the jet of interest,o pT i is the transverse(0.5) (1) (2) (0.5) (1) (2) (0.5) (1) (2) (0.5) (1) (2) (1) (2)⌧1 , ⌧1i in, ⌧1the, ⌧2jet,, ⌧and⌧3, for, ⌧3K, ⌧=⌧42, ., .⌧.4 , N, ⌧4, is, ⌧5the, ⌧5angle. in (3.1)
m of particle
pseudorapidity
image
networks and the n-subjettiness netw
2 , ⌧2R,Ki
3 ,1,

will see that this collection of N -subjettinessobservables isDatta
more thansufficient to dearXiv 1704.08249
Larkoskibe all of the information useful for discrimination in the jet. Additionally, for comparison,
will measure a collection of standard observables that have been defined for discriminaof boosted, hadronic decays of Z bosons–fromjets initiated by QCD. We measure the
3
–
(1)
(2)
ubjettiness ratios ⌧2,1 and ⌧2,1 with one-pass winner-take-all (WTA) axes [32–34], and

CNN vs N-subjettniess MLPを比較する
(1)

(2)

(1)

(2)

eralized) energy correlation function ratios D2 and D2 [43] and N2 and N2 [28].
Liam Moore et al 1807.04769
discrimination power of these observables will provide a benchmark for the information
acted in the machine learning of the collection of N -subjettiness observables.
All deep learning analysis was carried out on the NVIDIA DIGITS DevBox, with four
とても高い次数の τ(n点相関)をいれるとorce GTX TitanX GPUs, built on the 28 nm Maxwell architecture. The specifications of
GPU are listed in Table 1. Only one GPU was used during training and testing.

なんとかCNNと同じになる。

なんでそんなものが必要なの？
Memory size
MemoryCUDA
cores

Base/Boost.
clock (MHz)

(GB)

3072

1000/1075

12

IRC

clock (Gbps)

Interface
width

MemoryBandwidth(GB/s)

7.0

384-bit336.5

unsafe な量はどういうふうに効いているの？お任せしちゃって大丈夫なの？

Table 1: Manufacturer specifications of the GTX TitanX.

Core2

ここまでのまとめ•

ジェットの中には、理論的に予言ができる量（IRC safe な量）と難しい量がある。特に難しいのは、ジェットの中にある、ランダムでエネルギーの低い粒子分布•

CNNはそういう量も使っているのではないか。

•

そのようなデータはジェットの外側に分布していて、変動も大きい。
ランダムな点分布を定量化する、よい量はないか。

When comparing patterns, one actually uses th
duced Minkowski functionals, namely the Minkoof the undミンコフスキー汎関数でmation fro
functionals for the pattern divided by whatis
When coMin
ジェットのソフトな粒子分布の定量化
pected for a set of non-overlapping discs. duced
Thes
functionals
pected forgiven
bygiven byof the underlying point interactions, including information from all possible groupings of points.
A(r)When comparing patterns, one actuallyuses the reAreaa(r)=2
duced Minkowski functionals, namely
therMinkowski
πN
functionals for the pattern divided by what is exP (r)These arepected for a set of non-overlapping discs.
Perimeterp(r) =
given by
2πN r

Euler a(r) = A(r)2 χ(r)
(3)
e(r)
=
πN
r
(i)
efinition of N (i) is also described as area. Define Pcharacteristic
as the surfacePof
the pixels
whose
N
(r)
p(r) =(4)t v 2 V i , and A(i) is the area of P (i) ,2πN r

The functi
figure 2.b.
sively on tdifferentiat

2.2

Sor

aim is
The functionals for aχ(r)Poisson process areOurshow
ing FPCA(5)e(r)=
figure2.b. The analysis
in this paperetionals, cluiNA(i) = ( R)2 ⇥ N(3.8) relies
scores on t
sivelyon thesefunctionals,
The functionals
for areduced
Poisson process
are shown inso wesamewillwith
we can dir
figure 2.b. The analysis in this paper relies exclui
differentiate
betweenthetwo.
[13]. For eour N (i = 0, 1, ...) essentially works as A(r). sively on these reduced functionals, so we will not

betweending to Hadwiger’s theorem, there are only d + differentiate
1 functionals!theR,two.
M (i) (i=1,...d+1), inon, that satisfies the feature associated with rigid
2.2body,
Sorting the patterns
2.2

Sorting the patterns

cipal comption. Forlate the pr
the area, pcatenate thuse these sc
clustering

Ouraimaim isistoautomatically
bypatterns
performautomaticallysorttion invariance: M (gB) = M (B) where g isOurelement
ofto
thegroupsortof patterns
rigid
motionG, by perf
ing FPCA onn their approximated Minkowski func• Ward’
ing
FPCA
on
their
approximated
Minkowskif
B convex ringRofall
finiteunionsofconvexbodiesinR.
which
N dim 空間→ N+1 個の独立汎関数がtionals, clustering
the 2:patterns
with theirindividual
Figure(a) The Minkowski
functionals
are calcuvarian
lated by imposing
discs on We
the point
pattern.
This indiv
tionals,thepatterns
with
theirscoresthe principal
components.
will
do theS
T onclusteringchose
new secondary structure can be characterized using
same
with
the
pairwisecorrellation
functionsothat
ditivity M (B1 B2 ) = M (B1 ) + M (B2 ) M (B
B
)
for
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and
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2
R
thedo
pa
topologicalwhichradii will1
2 the
1measures,
2 vary for differentWe
をみたし、並進、回転不変
scoresonprincipal
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to direwe can directly (b)comparemethodwithfunctionals
that offor a
The threeour
reducedMinkowski
lian
2-Dpairwise
Poissonprocess.
These functionals
are
same
with
correllation
functionsoet[13]. For
eachthepatternset,(random)
we willuse enoughprindue to the normalization by the same meaditional continuity M (Ki ) ! M (K) as K1 !cipal
K for
K, Kunitless
2 accountR,
i to• Fast W
components
for 95%of methodthevaria- with
we candirectlycompareourthasure one would expect fora set ofnon-overlapping
this rotion. For the Minkowski
functionals, we will calcudiscs

MF の他の物理応用
Additivity, Convexity, and Beyond

113

統計物理
左 多孔質体真ん中: 微乳濁液
左 コロイド体積の占有状況V, 表面の大きさ(S) 等に依存して物性が変わる

図はMecke andStoyan (2000)•

天文: 星の分布の定量化、 銀河分
布、シミュレーション結果の定量化、non-Gaussianity of CMB,
weak lensing..

Fig. 1. Porous media (left) can be described by overlapping grains (spheres, discs)
distributed in space. If the density of grains (white) decreases below a threshold, an
infinite cluster of connected pores (black area) is spanning through the whole system.
This cluster of pores enables the transport of fluids, for instance. The knowledge ofthe dependence of the so-called percolation threshold on the shape and distributionof the grains is essential for many applications. Inhomogeneous domains of thermodynamic stable phases of complex fluids may also be described by overlapping grains
6[9,35,38,39,43]. Such configurations resemble, for instance, the structureof microemulsions (figure in the middle) or an ensemble of hard colloidal particles (black points
in the figure on the right) surrounded by a fluid wetting layer (white). The interactions between these colloids, as well as the free energy of the homogeneous oil phasein a microemulsion are given by a bulk term (volume energy), a surface term (surface
tension), and curvature terms (bending energies) of the white region covered by theoverlapping shapes. Thus, the spatial structure of the phases, i.e., the morphology of
the white regions determines the configurational energy which determines itself thespatial structure due to the Boltzmann factor in the partition function of a canonicalensemble. A main feature of complex fluids is the occurrence of different length scales:
the clusters of the particles, i.e., the connected white regions are much larger than the‘microscopic’ radius of the discs and the typical nearest neighbor distance within acluster.

Kratochvil 1109.6334 Proving Cosmology
with Weak Lensing Minkowski Functionals

tions, the scientist faces the problem of reducing the information to a limited
number of relevant quantities. So far powerful methods have been developedin Fourier space, namely structure functions and more recently wavelet analysis. But techniques to analyze spatial information directly in real space may
be very useful for physicists in order to get more relevant spatial informationout of their data which may be complement to structure functions measured
1: Top left panel: example
a simulated 12-square-degree
convergence
map intechniques
the fiducial cosmology, withintrinsicmeasures haveby scatteringFIG.
techniques
inof Fourier
space.
Suchandellipticity noise from source galaxies and ✓ = 1 arcmin Gaussian smoothing. A source galaxy density of n
= 15/arcmin
at redshift z = 2 was assumed. Other three panels: the excursion sets above three di↵erent convergence thresholds , i.e. all
been developed
invalues
spatialstatistics
andThethereaderis referred to the
pixels withabove (below) the
threshold are black (white).
thresholdinterested
values are  = 0.0 (top right),
 = 0.02 (bottom
left), and  = 0.07 (bottom right). The Minkowski Functionals V , V , and V measure the area, boundary length, and Euler
characteristic(orgenus),
respectively,
of
the
black
regionsasafunctionofthreshold.
papers by D. Stoyan and W. Nagel in this volume. To this world also belong
the additive Minkowski
functionals
offer
robustmeafind excellent agreement out
to ` ⇠ 20, 000 for z = 1which
and
due tomay
the increased
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The deviation
fromthismay
relation
cananbeimportantunderstood thatquarks and gluons,and theMFstakesome change has been made at the given angular scale.the geometric features since typical neural networrole in the classification. We also test the model in top
Therefore, the analysis on these sequences of MFs can befrom a linear combination of the inputs.
a toy dark
sector and find that the classification performancejet tagging
considered as a persistent analysis on geometric features Although those MFs are global features of a bin
gnificantlydi↵eris similar or better
than a convolutional
neuralof jet substructures,
similar to [30].
Thesenetwork
sequences areage, they can be written as a sum of local contr
ginating from theusefulas inputs
to a neural network
for andanalyzing(CNN), with also
O(10)
times
less computational
power
from 2 ⇥ 2 subimages. Since there are only 16 co
ake an importantthegeometric
featuressince
typicalneuralnetworks
startmemorythe model
in topconsumption.
tions for the 2 ⇥ 2 subimages, we may use a lookfrom a linear
combination
ofconstituents
theはinputs.
• The
ミンコフスキー汎関数(MF)
CNNのフィルターの形で書くこと
morphologicalanalysisonjetispertion performance
Although those MFs are global features of a binary im-v for parameterizing the local contribution [31].
formedon theage,filtered
distributionofjet
constituents
proal neuralnetwork
theycanbewritten
as
a
sum
of
local
contributions
⇣
X X
ができる
ational jected
power
and
on (⌘, from) plane.
To
analyze
soft
and
hard
jet
con(k)(k)(k)(k)
2 ⇥ 2 subimages. Since there are only 16 configura- (A , L ,)=v P(i+n)(j+m)
stituents separately,the2 ⇥
jet2 subimages,
constituents
withusepTa look-up
below table
tions
the
we may2x2forのフィルター(ルックアップテーブル)を使った実装
i,j n,m2{0,1}onstituents is perv forfilteredparameterizing
the 27].
local contribution[31].
a threshold areout [20,
For the analysis
constituents pro⇣ pT thresh- ⌘of pixellated
jet images,
weuseX
the following
X
(k)
and
hard
jet
con(k)(k)(k)(k)
A&A 555, A38 (2013)
where
f
=
((1,
2),
(4,
8)),
and
P
1
(A
,
L
,
)
=
v
P
f
nm
ij is 1 or 0 if(i+n)(j+m)
default threshold of the detector simulation, 2,nmts witholds:
pT below(k)i,j n,m2{0,1}is active or not active, respectively.
4, analysis
and 8 GeV. The resulting binary
images on a two- (4)pixel of P
For the2
This algorithms are for square integer lattices,
lowing dimensionalpT thresh-Table
1.
Look-uptable
for
Minkowski
functionals.
integer
grid
are
used
for
the
morphological(k)where fnm = ((1, 2), (4, 8)), and Pij is 1 or 0 if (i, j)-th1
or simulation,
2,
may use hexagonal pixels or marching square alg
analysis. Sample
binary(k) images are in Fig. 1.
pixel of P
is active or not active, respectively.
mages on a twoTo understandthemorphologyofthe integer
binarylattices,images,but we[32] for approximating MFs without pixellations
This
algorithms
areforsquarethe morphological
Conf.
χsome
Conf. Equation
A 4 indicatesPχ that the MFs can be aof the A
imagesdilation
mayMFsuse hexagonal
pixels P
oraftermarchingsquarealgorithms
Fig. 1.we analyze themated by a deep convolutional neural network
approximatingMFs
pixellations.
by aimages,
square. [32]
Thefordilationis usefulforwithoutprobinggeometric
he binary100 the 0MFsof can
9 This is fully
1/4 discribed1
1/4
in the Appendix. Note th
approxiter somedilationfeaturesthat is Equation
visible at4 aindicatesangular that
resolution
the becharmateda deepconvolutional
neural(CNN).essentially
two components: 2 ⇥ 2 con
probingacteristic
geometric length
2 by of1 element.
1/4 network10
1/2consists
2 of−1/2
scale
the 1/4
structuring
For theThis is fully discribed in the Appendix. Note that Eq. 4between P (k) and f , and a scalar function v. The
lution of
the char- image
pixellated
we of
may
use
side
1 a square
1/4 2with
11
1/2 1
0
3analysis,
essentiallyconsists1/4
two components:⇥ 2 convolution
g element. For the
convolutional layers can model this , i.e., the welength 2k + 1 between
as a structuringof thedilationandP (k) and f ,element
and a scalarfunctionv. Thestackeda square with side
4
1/2
0
12
3/4
1
−1/4
(k)1
the
first
convolutional
layer is f and later layers
considertheresulting
image
as
P
,
convolutional layers can model this , i.e., the weights off the dilation andthe first
and later layers13
can beconsidered
5 convolutional
1/4 layer
1(k)is f 1/4
1/2as a multilayer
10 perceptron (MLP) that
(k)
(0)
P
= {a
+ b|a 2, b 2 Bperceptron},(1)modelsv. The filtering by pT is also a local operation t
consideredas P
a multilayer(MLP) that1 a local
0 operation that
14 canbe approximated
3/4 1 by −1/4
6 filtering by1/2
,
(1)
(k)
v.
The
p
is
also
T
a CNN. Dilation can be wr
B = {(i, j) | i, j 2 { k, k + 1, ...k 1, k}} (2)be approximatedby a CNN.
...k 1, k}} (2)71/2
2 Dilation−1/2 can be(0)written
15 ina step function
3/4 1of convolution−1/4
between the P (0)a step function of convolution between the P
and the(k)(0)
structuring
element
B,
i.e,
P
=
✓(P
(k)
(0)
8
3/4
1
−1/4
16
1
0
0
ij
(i+n)(j+structuring element B, i.e, P
= ✓(P
B ).

istic 1χミンコフスキー汎関数とCNN の関係ij(k)(i+n)(j+n)

nm

(k)

(k)

(k)

are covered by
Therefore, A(k) , L(k) , andare covered by a CNNTherefore, A , L , and
0.5forGeV
nd 1.0 GeV
the for the electronic calorimeters and 1.0 GeV for theimages.
But this expression con
Notes.
The
valuesof area A,
perimeterP, onandjetEuler
charactrainedon functional
jet images. But
this expression
contains
atrainedhadronic
erformed before
the calorimeters. This filtering is performed before theDark QCD jet is from a (SM)DARK
color neutral
parton,so that the energy deposits are con@ne
JETの場合
within a small area.
Avera
Lim, Nojiri in preparation••

Dark Jet pp →Z’ → qD qD→ dark Parton
shower → ρdiag→qq

ダーク

カラーシングレットなシャワー:粒子がたくセクター

さんあるが、いくつかのカラーシングレット
なクラスターになっている状態（ダーククオークとダークQCD)mρ=20GeVAverage plots

CNN は何を訓練しているか
pT,J 2 [150, 300] GeV mJ 2 [30, 70] GeV

Dark jet と QCD ジェットを判別させる

DJ(0)QCD(0)DJ(1)
DJ(3)
QCD(3)

PDF

Original distribution

QCD(1)DJ(5)QCD(5)

0

0

50

100

150

200

250

300

350

400

450

Minkowski Sequence

pT,J 2 [150, 300] GeV mJ 2 [30, 70] GeV
DJ(1)
QCD(1)DJ(3)Cut using CNN

Smaller MF endpoint suggests
Compact soft activities

DJ(5)QCD(5)

PDF

network in order to fully utilize relevant features of
als in the Dark QCD jet tagging.

QCD(3)

12 / 15

0

0

50

100

150

200

250

300

350

400

450

Minkowski Sequence(after CNN cut)

CNN allowing 10% rejection of signal

distribution of MS after rejecting 10% signal events by CNN
classifier (ReLU). Roughly 1.5% of QCD events remains after
the cut.

this regard, a relation network (RN) with IRC safe
raints [20] is a good complementary deep learning
ework to the MFs. The RN is based on two-point
y correlation S2,ab that accumulates energy correla0
sive
gauge
boson
Z
that mediates the interaction beat a given angular scale R.
tween the SM particles and the hidden sector. At the
ジェットの運動量などのX
hadronUcollider, qv may be produced through the process
S2,ab (R) =pT,i pT,j (R Rij ),(5)大域的な量
0 jet is from a (SM) color IRC
safe
でない情報は、
Dark
QCD
neutralparton,so that the ener
q
q̄
!
Z
!
q
q̄
.
The
secluded
gauge
group
containsvv
i2a,j2b
withina smallthat
area.confines qv and q̄v , and forms pions
an interaction
ここだけからくる⇡v and rho mesons ⇢v of the hidden sector.
The decay
e i and j are the jet constituents in a subset of jetmodes of the mesons dependon the realization of theituents, a and b,IRCrespectively.
Rij is the distance
SAFE な
ミンコフスキーLEADING SUBJET
hidden sector, and we consider a scenario that only ⇢v
een i and j on the (⌘, ) plane. The RN takes a disを除いた
汎関数
R
2点関数
0.1(k+1)
(k)
leave
a
visiblesignaturesvia
thedecay ⇢v ! q q̄ whileed S2,ab = 0.1k
dR S2,ab (R) as an input, where
2点関数the other mesons are not visible at the collider. The reagain the hadronic calorimeter resolution. The in(k)sulting semi-visible jet, which we call a dark jet, contains
roducts between S2,ab and the weights in the firstmany color-singlet quark pairs and missing particles thatconnected layer can be regarded as trainable twoMLP
appearafter the parton shower and hadronization in theenergy correlation. For example,
two-point energy
MLP
MLPthe dark jets are drasticallysecluded
sector.
Therefore,
lation functions [14, 16] can be written in terms ofdi↵erent from the ordinary QCD jets, both in hard subar combination of the S2 .
structure and in color structures.
e subjet indices a and b are hyperparmeters of thisFor the simulation of the dark jet, we use Pythia 8 [36]tecture, and we use the following subjets which are
Top jet
MG5+PY8+Delphes
and its hidden valley model implementation [35]. The
l in top tagging [20]: 1) the jet itself J, 2) theS2 (R)
R
mass spectrum is assigned as follows: mZ 0 = 1400 GeV,
med jet Jtrim , 3) the counterpartof the trimmed jet
S(R)
2,trimR
0.4
mqv = 10 GeV, and m⇡v = m⇢v = 20 GeV. The fraction
im , 4) the leading subjet J1 , 5) the counterpart ofRof ⇡v and ⇢v during the hadronization is set to be 1:3 aseading subjet J/J1 . To set the characteristic scales
the spin counting suggests. The pT of the qv is typicallye given jet, the RN also takes six kinematical pa700 GeV, but the selected jets have smaller pT because
ters of thosesubjets: the transverse momentum and
0.2⇡v take away some of the momentum fraction. The QCD
of J, Jtrim , and J1 .
jet samples are the leading pT jets of the process pp ! 2je combined architecture systematically aggregatesis generated using MadGraph5 2.6.6 [37] together withnitial local features, while CNN needs to find good
Pythia 8. Detector e↵ect is modeled by Delphes 3.4.1
features 0.0during
the training.
Note2.0that the aggre0.0
0.5
1.0
1.5
[38] with the default ATLAS detector card.
R less fluctuation. For exfeatures typically have
WeFigure
may useneuralnetwork(0)(k)in order to fully utilize relevant featur
e, the numberofactivepixelsA
hasfluctuation
2(left)
showstheA
distributions of dark jetsp
Minkowski
functionals(0) Dark(c) QCD jet/A(0) ⇠ 1/ A(0) but(a)each pixel-by-pixel fluctuation (b)and QCD jets.
While theinAthe
distributions
aretagging.
similar,

RELATION NETWORK+ MF

bin dR S2,ab (R)/dR S2 (R)

bq

bq̄

q q̄

MLP

DARK JET VS QCD
Jet PT 150~300GeV Jet mass 30~ 70GeV

MF only reject lots of QCD jet

100

MLP: A(n) , L(n) ,RN: S2 , xkin(n)

, xkin

m
rando

guess

RN: S2 , A(k) , xkin

CNN(ReLU)
ELU にするとすごく悪い
急な変化をする活性化関数

QCD jet mistag rate10

RN: S2 , A(k) , L(k) , (k) , xkin
CNN: jet image, xkin

1

CNN: jet image, A(k) , L(k) ,, xkin10 2

10 3

MF+IRC 安全な２点関数 CNNを必要としている。
Bottleneck 必要(k)活性化関数によらない。
10 4
0.0

0.2

0.4

0.6

0.8

1.0

Dark jet tagging efficiency

RN + MF also improve Top vs QCD (but improvement is smaller )FIG. 3.

ROC curves of various cla

Top jet の場合 やっぱり
100

RN+ MF がよい
tion.
by th
equiv
Th
MFs
overfimethphase
to fintrain
top jet tagging efficiency
repre

RN: S2 , xkin

m
rando

RN: S2 , A(0) , A(1) , xkin

guess

RN: S2 , A(k) , xkin

QCD jet mistag rate10

RN: S2 , A(k) , L(k) ,CNN: image, xkin

1

10 2

10 3

10 4
0.0

0.2

0.4

(k)

, xkin

0.6

0.8

1.0

The deviation
fromthismay
relationunderstood thatquarks and gluons,and theMFs
takecananbeimportantsome change has been made at the given angular scale.the geometric features since typical neural networrole in the classification. We also test the model in top
Therefore, the analysis on these sequences of MFs can befrom a linear combination of the inputs.
a toy dark
sector and find that the classification performancejet tagging
considered as a persistent analysis on geometric features Although those MFs are global features of a bin
gnificantlydi↵eris similar or better
than a convolutional
neuralof jet substructures,
similar to [30].
Thesenetwork
sequences areage, they can be written as a sum of local contr
ginating from theusefulas inputs
to a neural network
for andanalyzing(CNN), with also
O(10)
times
less computational
power
from 2 ⇥ 2 subimages. Since there are only 16 co
ake an importantthegeometric
featuressince
typicalneuralnetworks
startmemorythe model
in topconsumption.
tions for the 2 ⇥ 2 subimages, we may use a look-u
from a linear
combination
ofconstituents
theはinputs.
• The
ミンコフスキー汎関数(MF)
CNNのフィルターの形で書くこと
morphologicalanalysisonjetispertion performance
Although those MFs are global features of a binary im-v for parameterizing the local contribution [31].
formedon theage,filtered
distribution
ofasjetconstituents
proal neuralnetwork
theycanbewritten
a
sum
of
local
contributions
⇣
X X
ができる
ationaljected
power
and
on (⌘, from) plane.
To
analyze
soft
and
hard
jet
con(k)(k)(k)(k)
2 ⇥ 2 subimages. Since there are only 16 configura- (A , L ,)=v P(i+n)(j+m)
stituents separately,the2 jetconstituents
withusepTa look-up
below table
tions
the
⇥ 2 subimages,
we may2x2forのフィルター(ルックアップテーブル)を使った実装
i,j n,m2{0,1}onstituents is perv forfilteredparameterizing
the27].
local contribution[31].
a threshold areout [20,
For the analysis
constituents pro⇣ pT thresh- ⌘of pixellated
jet images,
weuseX
the following
X
(k)
and
hard
jet
con(k)(k)(k)(k)
A&A 555, A38 (2013)
where
f
=
((1,
2),
(4,
8)),
and
P
1
(A
,
L
,
)
=
v
P
f
nm
ij is 1 or 0 if(i+n)(j+m)
default threshold of the detector simulation, 2,nmts witholds:
pT belowi,j n,m2{0,1}pixel of P (k) is active or not active, respectively.
4, analysis
and 8 GeV.本来はジェットイメージはRN+MF
The resulting binary
images on a two-がもっている情報を使えるは
For the(4)
2This algorithms are for square integer lattices,
lowingdimensionalpT thresh-Table
1.
Look-uptable
for
Minkowski
functionals.
integer
grid
are
used
for
the
morphological(k)where fnm = ((1, 2), (4, 8)), and Pij is 1 or 0 if (i, j)-th1
or simulation
,
2,
ずだた。
may use hexagonal pixels or marching square alg
analysis. Sample
binary(k) images are in Fig. 1.
pixel of P
is active or not active, respectively.
mages on a twoTo understandthemorphology
ofsquarethe binary
images,but we[32] for approximating MFs without pixellations
This
algorithms
arefor
integerlattices,the morphological
Conf.
χsome
Conf. Equation
A 4 indicatesPχ that the MFs can be aof the A
imagesdilationmayusehexagonal
pixels P
oraftermarchingsquarealgorithms
1.MFs
ジェットイメージがエネルギーでウエイトされていること
Fig. 1.we analyze themated by a deep convolutional neural network[32] forapproximatingMFs
pixellations.
by aimages,
square. The
dilationis usefulforwithoutprobinggeometric
he binary10 resolution
0 the 0MFsof can
9 This is fully
1/4 discribed1
1/4
in the Appendix. Note tha
approxiter somedilationfeaturesthat is Equation
visible at4aindicatesangular thatthe becharmatedby ofa deepconvolutional
neural(CNN).essentially
two components: 2 ⇥ 2 conv
probingacteristic
geometriclength
2.scaleLoss
2 activation,
1 funcは連続関数であることが前提
1/4 network10
1/2consists
2 of−1/2
the 1/4
structuring
element.
For theThis is fully discribed in the Appendix. Note that Eq. 4between P (k) and f , and a scalar function v. The
lution pixellatedof the char- image
we mayuseside
1 a square
1/4 2with
11
1/2 1
0
3analysis,
essentiallyconsists1/4
of two components:⇥ 2 convolution
g element. For the
convolutional layers can model this , i.e., the we(k)length 2k + 1 between
as a structuring
elementof thedilation
and
P
and
f
,
and
a
scalarfunctionv.
Thestackeda square with side
3. 4でもMFはエネルギーに依存しない関数
1/2
0
12
3/4
1
−1/4
(k)1
the
first
convolutional
layer is f and later layers
considertheresulting
image
as
P
,
convolutional layers can model this , i.e., the weights off the dilation andthe first
and later layers13
can beconsidered
5 convolutional
1/4 layer
1(k)is f 1/4
1/2as a multilayer
10 perceptron (MLP) that
(k)
(0)
P
= {a
+ b|a 2, b 2 Bperceptron},(1)modelsv. The filtering by pT is also a local operation t
consideredas P
a multilayer(MLP) that1 a local
0 operation that
14 canbe approximated
3/4 1 by −1/4
6 filtering by1/2
,
(1)
(k)
v.
The
p
is
also
T
a CNN. Dilation can be wr
B = {(i, j) | i, j 2 { k, k + 1, ...k 1, k}} (2)be approximatedby a CNN.
...k 1, k}} (2)71/2
2 Dilation−1/2 can be(0)written
15 ina step function
3/4 of
1 convolution−1/4
between the P (0)a step function of convolution between the P
and the(k)(0)
structuring
element
B,
i.e,
P
=
✓(P
(k)
(0)
8
3/4
1
−1/4
16
1
0
0
ij
(i+n)(j+nstructuring element B, i.e, P
= ✓(P
B ).

istic 1χミンコフスキー汎関数とCNN の関係ij(k)(i+n)(j+n)

nm

(k)

(k)

(k)

are covered by
Therefore, A(k) , L(k) , andare covered by a CNNTherefore, A , L , and
0.5forGeV
nd 1.0 GeV
the for the electronic calorimeters and 1.0 GeV for theimages.
But this expression con
Notes.
The
valuesof area A,
perimeterP, onandjetEuler
charactrainedon functional
jet images. But
this expression
contains
atrained
erformed hadronic
before the calorimeters. This filtering is performed before the何が改善しているか•入力が少ない→ loss function の最小化が簡単

•

時間

•

等価なのか、相関なのか。CNN と RN はどうやら違う判断をしている。

1/10 メモリも1/10

0.0

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ΩXY = 0.970

Dark
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ΩXY = 0.791

Dark
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£10

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logit ± ŷ 0 (image, xkin )

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CNN vs RN

PDF [arb. scale]

1.0

logit ± ŷ 0 (S2 , A(n) , xkin ), diÆerent seed

ΩXY = 0.954

Dark
QCD jets
1£10

PDF [arb. scale]logit ± ŷ 0 (image, xkin ), diÆerent seed

RN vs RN

PDF [arb. scale]

CNN vs CNN
1.5

true minimum が見つけやすい。

0.5

0.0

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°1.0

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°1.0

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実データのキャリブレーション
(a)

(b)

Figure 14: (N (0) , N (0) (4 GeV)) distributions for (a) PY8 and (b) HW7

QCD jets•[MC を実データで補正]するときに、MFが実験

N (1) /N (0)トの粒子数と広がり）Pythia≠ Herwig≠実データ

0
10

0
10

8

8

6

4

2

データと同じになるようにweight をつけて補正

0

0

20

40

N

するとclassification の一致は良くなる。

60

80 0

N (1) /N (0)をきちんと表していない。（特にグルーオンジェッ

QCD jets

MG5+PY8+Delphes

0(0)MG5+HW7+Delphes

PDF [arb. scale]

既存のMCはQCD ジェットのIRCsafe でない分布

PDF [arb. scale]

•

6

4

2

0

0

20

40

N
(a)

60

80 0

0

(0)

(b)

Figure 15: (N (0) , N (1) /N (0) ) distributions for (a) PY8 and (b) HW7

RNS2
1.0

PY8
HW7
HW7!PY8

RNS2 ,N (0) ,N (0) (4 GeV)QCD jets

PDF

0.8

Pythia で訓練した分類器でThe separation of the top jets and QCD jets is worse for HW7 compared with PY8 discussed inprevious sections. The AUC of the top jet vs. QCD jet classification predicted by HW7 is smaller thanthat predicted by PY8. In figure 16, we show the ROC curves of each classifier trained on HW7 events.
The performance of the RNS2 is similar to that trained on PY8 events. Once N (0) is additionallyconsidered in the classification, the performance is improved. However, the improvement fromadding N (0) is significantly small in HW7, because the N (0) distributions of top jets and QCD jets

PY8
HW7
HW7!PY8

Herwig のQCD jet を分類

0.6

N0 分布(pT>0 と pT>4GeV) を補正した後

0.4
0.2

– 20 –

ratio

0.0
3

Generator を GAN で補正するときに、

2
1

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0.6

0.7

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(b)

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ミンコフスキー汎関数の分布だけ補正すれば良い?

教訓とやれそうなこと•自分の目でイベントをみよう•

CNN などのジェットイメージを使った訓練は IRC safeでない量も追加して分類を強化している。

•

CNNは理解した(1/10 まで input 圧縮. パフォーマンスは落ちない）解釈可能性、速度、安定性、いろいろなプロセスに使える。
•

ミンコフスキー汎関数のような「まとめ指標」の方が実データの検定にもよい。パートン→ハドロン化の空間分布をよりよく検定できるかも。