Jet tagging algorithm of graph network with HaarPooling message passing Fei Ma1Feiyi Liu1 2and Wei Li1 3 1Key Laboratory of Quark and Lepton Physics MOE and Institute of Particle Physics

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Jet tagging algorithm of graph network with HaarPooling message passing

Fei Ma,1Feiyi Liu,1, 2, ∗and Wei Li1, 3, †

1Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,

Central China Normal University, WuHan, 430079, China

2Institute for Physics, Eötvös Loránd University

1/A Pázmány P. Sétány, H-1117, Budapest, Hungary

3Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

(Dated: October 11, 2023)

Recently methods of graph neural networks (GNNs) have been applied to solving the problems in high en-

ergy physics (HEP) and have shown its great potential for quark-gluon tagging with graph representation of jet

events. In this paper, we introduce an approach of GNNs combined with a HaarPooling operation to analyze

the events, called HaarPooling Message Passing neural network (HMPNet). In HMPNet, HaarPooling not only

extracts the features of graph, but embeds additional information obtained by clustering of k-means of different

particle features. We construct Haarpooling from ﬁve different features: absolute energy log E, transverse mo-

mentum log pT, relative coordinates (∆η, ∆ϕ), the mixed ones (log E, log pT)and (log E, log pT,∆η, ∆ϕ).

The results show that an appropriate selection of information for HaarPooling enhances the accuracy of quark-

gluon tagging, as adding extra information of log PTto the HMPNet outperforms all the others, whereas adding

relative coordinates information (∆η, ∆ϕ)is not very effective. This implies that by adding effective particle

features from HaarPooling can achieve much better results than solely pure message passing neutral network

(MPNN) can do, which demonstrates signiﬁcant improvement of feature extraction via the pooling process.

Finally we compare the HMPNet study, ordering by pT, with other studies and prove that the HMPNet is also a

good choice of GNN algorithms for jet tagging.

DOI:10.1103/PhysRevD.108.072007

I. INTRODUCTION

As an event in high energy collisions, a jet refers to a col-

limated spray of hadrons observed by detectors as a signature

of quarks and gluons. In Large Hadron Collider (LHC), jets

with dynamic information combined from different detector

components are experimentally reconstructed by particle ﬂow

algorithms [1–3]. One of the prime study on jet is to specify

the origin of a jet from a type of elementary particle, called jet

tagging. Since the character of the source particles can be sur-

mised from the properties of jets, for example, jets initiated by

gluons generally have more extensive energy spread than by

quarks. The information of these initial elementary particles

could facilitate key tasks in high-energy physics (HEP) exper-

iments, such as searching for new particles and estimating the

Standard Model processes.

In past decades, researches on jet tagging via QCD the-

ory have never stopped and continuously improved for quark

and gluon jets [4–8], top jets [9–14] and jets from bottom

quarks [15–17]. Recently, methods of deep learning (DL)

have been applied to studying jet classiﬁcation, by construct-

ing a representation of event paired with a corresponding anal-

ysis method, such as particle calorimeter images with convo-

lutional neural networks (CNNs) [18–22], particle lists with

recurrent neural networks (RNNs) [23–26], and collections

of ordered inputs with dense neural networks (DNNs) [27–

29]. Moreover, energy ﬂow networks (EFNs) treat jet tag-

ging model under the framework of deep sets, which respect

∗fyliu@mails.ccnu.edu.cn

†liw@mail.ccnu.edu.cn

infrared and collinear safety by construction [30,31]. Inter-

action networks (INs) also have great potential in identifying

all-hadronic decays of high-momentum heavy particles [32].

Compared to previous traditional approaches, methods of DL

not only could better handle large amount of sophisticated

data generated by modern detectors, but also are powerful in

analyzing complex internal relations from limited input, lead-

ing to great advantages in dealing with jet tagging.

Previous researches have shown that graph neural net-

works (GNNs) can well handle collision events [33–35]. For

jet tagging, an event usually contains the information of a set

of particles with certain kinematic features. As a sensitive

probe for classiﬁcation, the geometrical relationship between

these particles can be represented by a geometrical pattern of

multiple entities, i.e., the structure of a graph. This graph rep-

resentation of jets is very ﬂexible as input to DL, which has

clear information of particles and does not require additional

sorting or information. In Refs. [36–38] the graph representa-

tion has been applied to jet classiﬁcation of high-momentum

heavy particles via message passing neutral network (MPNN),

an algorithm of GNNs. A similar representation called “par-

ticle cloud” treats a jet as an unordered set of particles, paired

with dynamic graph convolutional neural network (DGCNN)

as ParticleNet (PN) [39]. Methods of autoencoder based on

GNNs are also used to distinguish QCD jets and non-QCD

jets [40,41]. As a framework of GNNs, LundNet [42–44]

has been proposed for jet tagging in the Lund plane [45], by

transforming the Lund tree into a graph. And LorentzNet, a

symmetry-preserving model of GNNs, describes the particle

cloud representation of a jet by the neural network architec-

ture under Lorentz-equivariant [46,47]. These successful at-

tempts inspire us to deal with jet tagging problems via graph

arXiv:2210.13869v5 [hep-ex] 10 Oct 2023

representations and GNNs.

Graph pooling is a technique used to reduce the dimen-

sion and extract the features of graphs, which usually ap-

pear with the convolutional layers [48]. The most widely

used methods are graph clustering algorithms [49–52], as well

as some other ones which have been lately studied [53–56].

HaarPooling is a graph pooling operation to compress and

ﬁlter graph features [57], based on compressive Haar trans-

forms. One of its important characteristics is, the basis for

forming a Haar matrix is computed by a clustering step from

the input graph, which means additional input-related infor-

mation can be passed to the ML process via the Haar matrix.

For quark-gluon tagging using GNNs, HaarPooling makes it

possible to embed extra particle features to ﬁlter and enhance

the message passing.

In our work, we combine HaarPooling with MPNN to

build a new network structure, called HaarPooling Message

Passing neural network (HMPNet). On one hand, jet events

are transformed into a graph representation as input for GNN,

and the tagging can be achieved by training with the pro-

cess of message passing and self-updating [37,39]. On the

other hand, in the updating process of the algorithm, the ad-

ditional particle feature is embedded through the compres-

sive Haar basis matrix of pooling, which makes the extraction

and classiﬁcation of features more relevant to the input. This

means the pooling for compression also becomes an opera-

tion for adding ﬁne information of input. For test, we imple-

ment the HMPNet to the quark-gluon tagging of the process

pp →Z/γ∗+j+X→µ+µ−+j+X, and use different

particle features such as absolute energy log E, transverse mo-

mentum log pT, the relative coordinates (∆η, ∆ϕ), the mixed

ones (log E, log pT)and (log E, log pT,∆η, ∆ϕ)to generate

the Haar matrix by clustering the input, respectively. We anal-

yse the inﬂuences of different particle features, and compare

the results of log pTwith the counterparts of other algorithms,

which shows a remarkable improvement of performance.

The main structure of this paper is as follows. In Section

II.1, the graph representation of jets will be given. Section II.2

gives the method of MPNN. Section II.3 includes the concep-

tions of graph pooling and Haar matrix. In Section II.4, the

method of embedding particle features to Haar matrix is il-

lustrated. In Section II.5, we explain the detailed process of

HMPNet. In Section III.1, the input data and settings of HMP-

Net are listed. Section III.2 shows our major ﬁndings. Section

IV is the conclusion of this work.

II. METHODOLOGY

II.1. Graph representation of jets

In the language of GNNs, an undirected graph G=

{V,E,X,W} is deﬁned with nodes (vertices) V, edges E,

weights of nodes Xand of edges W. Each node vi∈ V has

its feature vector xi∈ X, and for the edge weight W, it is

always given in the form of an weight matrix dij in which the

element is given for the edge between i-th and j-th nodes in

the graph. And the number of nodes is deﬁned as N=|V|.

Usually, the information of a jet reconstructed from de-

tectors in high-energy collision includes: the three Cartesian

coordinates of the momentum (px, py, pz), the absolute en-

ergy E, the pseudorapidity η, the azimuthal angle ϕ, the trans-

verse momentum pTand so forth. For the feature vectors xi,

we use 10 variables of jet information as components of xi

similar to Ref. [39], as shown in Table. I. The dynamic infor-

mation of objects includes log pT,log E, the relative energy

log E

E(jet)and the relative transverse log pT

pT(jet). In addition,

qdenotes the electric charge of object and the rest four fea-

tures are particles identity (PID) information. The dimension

of xiis N×dx, where dx= 10 is the dimension of the feature

space.

For graph representation, we also need to identify a pa-

rameter as the edge weight dij . From the point view of jet

axis, the relative distance ∆R=q∆η2

ij + ∆ϕ2

ij from the

jet center is a suitable choice, where the relative coordinates

∆ηij =ηi−ηjand ∆ϕij =ϕi−ϕjdenote the angle dif-

ference between the i-th with j-th particle in jet axis. By the

deﬁnition of ∆R, the edge weight is given by,

dij =q∆η2

ij + ∆ϕ2

ij .(1)

As an illustration, we show the graph events of the process

pp →Z/γ∗+j+X→µ+µ−+j+Xby Monte Carlo sim-

ulations in Fig. 1. As a graph representation with Nnodes,

each component of xiis a vector of dxelements of jet infor-

mation, with N= 9 and dx= 10. So dij is an N×N-

dimensional symmetric, matrix with all the diagonal elements

being 0. Since ϕis not encoded in the node features, the graph

representation is invariant under rotation in ϕ.

II.2. MPNN algorithm

The ﬂexible and complete feature of graph makes it a nat-

ural and promising representation of jets; on the other hand,

to choose a paired algorithm of GNNs also requires careful

thought. Message Passing Neural Networks(MPNN) is in-

troduced as a powerful and efﬁcient supervised algorithm of

GNNs which can learn geometric representations as well, es-

pecially the edge features dij [38,58]. By ﬁnding the opti-

mized parameters in the nonlinear network model via training,

one can obtain the classiﬁcation as output of MPNN, from the

input graph representation of jets.

To start the process of MPNN, the feature vectors xi∈

RN×dxare embedded into a matrix consisting of higher di-

mensional state vectors s(0)

i∈RN×dswith ds> dx, by an

embedding function fe:

s(0)

i=fe(xi).(2)

Here s(0)

iis only related to xiwithout any information of the

graph structure. To encode the whole event graph into each

node state vector, message vector m(t)

iis introduced to pass

the message of s(t−1)

iand edge weight dij via the message

TABLE I. Input variables used in the quark-gluon tagging task with PID information.

Feature of graph representation Variable Deﬁnition

log pTlogarithm of the particle’s pT

log Elogarithm of the particle’s energy

log pT

pT(jet)logarithm of the particle’s pTrelative to the jet pT

log E

E(jet)logarithm of the particle’s energy relative to the jet energy

xiqelectric charge of the particle

isElectron 1if the particle is an electron else 0

isMuon 1if the particle is a muon else 0

isChargedHadron 1if the particle is a charged hadron else 0

isNeutralHadron 1if the particle is a neutral hadron else 0

isPhoton 1if the particle is a photon else 0

dij ∆ηij difference in pseudorapidity between the i-th and j-th particle in jet axis

∆ϕij difference in azimuthal angle between the i-th and j-th particle in jet axis

Electron

log pTElectron

log EElectron

log PT

PT(jet)

Electron

log E

E(jet)E

qElectron

Electron Muon

Muon Photon

Photon Charge

hadron

Neutral

hadron

x15.2309 6.1336 -1.0594 -1.0526 -1 0 0 0 1 0

x24.8371 5.7385 -1.4532 -1.4476 1 0 0 0 1 0

x33.5316 4.4088 -2.7587 -2.7773 1 0 0 0 1 0

x43.8692 4.3737 -2.3959 -2.4031 0 0 0 0 0 1

x53.5075 4.0072 -2.7576 -2.7697 0 0 0 1 0 0

x62.6582 3.1625 -3.6069 -3.6143 0 0 0 1 0 0

x73.5755 4.0572 -2.6896 -2.7196 0 0 0 0 0 1

x8-0.4229 0.2991 6.7064 -6.5225 -1 1 0 0 0 0

x92.2112 2.8822 -4.0837 4.0796 1 0 1 0 0 0

123456789

1 0 3.158 2.357 2.101 3.628 1.196 4.125 3.027 2.538

2 3.158 0 2.129 0.536 2.739 2.753 4.511 0.462 2.665

3 2.357 2.129 0 1.633 2.821 2.452 3.656 2.321 1.342

4 2.101 0.536 1.633 0 2.039 3.322 4.326 0.639 2.578

5 3.628 2.739 2.821 2.039 0 0.755 2.096 2.315 2.318

6 1.196 2.753 2.452 3.322 0.755 0 1.745 2.347 1.477

7 4.125 4.511 3.656 4.326 2.096 1.745 0 4.321 2.811

8 3.158 0.462 2.321 0.639 2.315 2.347 4.321 0 2.129

9 2.538 2.665 1.342 2.578 2.318 1.477 2.811 2.129 0

dij

Node weight

Edge weight

Graph representation

456

FIG. 1. An event graph with node and edge weights for a speciﬁc simulated event of the process pp →Z/γ∗+j+X→µ+µ−+j+X.

passing function fmin the t-th iteration as

m(t)

i=X

j̸=i

m(t)

i←j=X

j̸=i

f(t)

m(s(t−1)

j,dij ),(3)

and update its state vector

s(t)

i=f(t)

u(s(t−1)

i,m(t)

i),(4)

where f(t)

uis the update function. This is how a node icollects

the messages sent from other nodes in the t-th iteration in the

message passing layer.

By the repetition of the message passing procedure, the

information of feature from each node and edge continuously

passes to the other ones, until each node state contains the in-

formation of all other nodes and relations in the entire graph

after Titerations. At this time, they can be regarded as the

event features automatically extracted from the input event

graph. Next, each node votes a number as the likeness of the

event to be signal-like, based on its own state vector. Mean-

while, the signal-like probability yis calculated by the voting

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JettaggingalgorithmofgraphnetworkwithHaarPoolingmessagepassingFeiMa,1FeiyiLiu,1,2,∗andWeiLi1,3,†1KeyLaboratoryofQuarkandLeptonPhysics(MOE)andInstituteofParticlePhysics,CentralChinaNormalUniversity,WuHan,430079,China2InstituteforPhysics,EötvösLorándUniversity1/APázmányP.Sétány,H-1117,Budapest,Hungary...

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Jet tagging algorithm of graph network with HaarPooling message passing Fei Ma1Feiyi Liu1 2and Wei Li1 3 1Key Laboratory of Quark and Lepton Physics MOE and Institute of Particle Physics

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