# Measurement of the absolute branching fraction of $\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta$ decays

M. Ablikim1, M. N. Achasov10,c, P. Adlarson64, S. Ahmed15, M. Albrecht4, R.
Aliberti28, A. Amoroso63A,63C, Q. An60,48, Anita21, X. H. Bai54, Y. Bai47, O.
Bakina29, R. Baldini Ferroli23A, I. Balossino24A, Y. Ban38,k, K. Begzsuren26,
J. V. Bennett5, N. Berger28, M. Bertani23A, D. Bettoni24A, F. Bianchi63A,63C,
J Biernat64, J. Bloms57, A. Bortone63A,63C, I. Boyko29, R. A. Briere5, H.
Cai65, X. Cai1,48, A. Calcaterra23A, G. F. Cao1,52, N. Cao1,52, S. A.
Cetin51B, J. F. Chang1,48, W. L. Chang1,52, G. Chelkov29,b, D. Y. Chen6, G.
Chen1, H. S. Chen1,52, M. L. Chen1,48, S. J. Chen36, X. R. Chen25, Y. B.
Chen1,48, Z. J Chen20,l, W. S. Cheng63C, G. Cibinetto24A, F. Cossio63C, X. F.
Cui37, H. L. Dai1,48, J. P. Dai42,g, X. C. Dai1,52, A. Dbeyssi15, R. B. de
Boer4, D. Dedovich29, Z. Y. Deng1, A. Denig28, I. Denysenko29, M.
Destefanis63A,63C, F. De Mori63A,63C, Y. Ding34, C. Dong37, J. Dong1,48, L. Y.
Dong1,52, M. Y. Dong1,48,52, S. X. Du68, J. Fang1,48, S. S. Fang1,52, Y.
Fang1, R. Farinelli24A, L. Fava63B,63C, F. Feldbauer4, G. Felici23A, C. Q.
Feng60,48, M. Fritsch4, C. D. Fu1, Y. Fu1, X. L. Gao60,48, Y. Gao38,k, Y.
Gao61, Y. G. Gao6, I. Garzia24A,24B, E. M. Gersabeck55, A. Gilman56, K.
Goetzen11, L. Gong37, W. X. Gong1,48, W. Gradl28, M. Greco63A,63C, L. M. Gu36,
M. H. Gu1,48, S. Gu2, Y. T. Gu13, C. Y Guan1,52, A. Q. Guo22, L. B. Guo35, R.
P. Guo40, Y. P. Guo9,h, Y. P. Guo28, A. Guskov29, S. Han65, T. T. Han41, T. Z.
Han9,h, X. Q. Hao16, F. A. Harris53, K. L. He1,52, F. H. Heinsius4, C. H.
Heinz28, T. Held4, Y. K. Heng1,48,52, M. Himmelreich11,f, T. Holtmann4, Y. R.
Hou52, Z. L. Hou1, H. M. Hu1,52, J. F. Hu42,g, T. Hu1,48,52, Y. Hu1, G. S.
Huang60,48, L. Q. Huang61, X. T. Huang41, Y. P. Huang1, Z. Huang38,k, N.
Huesken57, T. Hussain62, W. Ikegami Andersson64, W. Imoehl22, M. Irshad60,48,
S. Jaeger4, S. Janchiv26,j, Q. Ji1, Q. P. Ji16, X. B. Ji1,52, X. L. Ji1,48, H.
B. Jiang41, X. S. Jiang1,48,52, X. Y. Jiang37, J. B. Jiao41, Z. Jiao18, S.
Jin36, Y. Jin54, T. Johansson64, N. Kalantar-Nayestanaki31, X. S. Kang34, R.
Kappert31, M. Kavatsyuk31, B. C. Ke43,1, I. K. Keshk4, A. Khoukaz57, P.
Kiese28, R. Kiuchi1, R. Kliemt11, L. Koch30, O. B. Kolcu51B,e, B. Kopf4, M.
Kuemmel4, M. Kuessner4, A. Kupsc64, M. G. Kurth1,52, W. Kühn30, J. J. Lane55,
J. S. Lange30, P. Larin15, L. Lavezzi63A,63C, H. Leithoff28, M. Lellmann28, T.
Lenz28, C. Li39, C. H. Li33, Cheng Li60,48, D. M. Li68, F. Li1,48, G. Li1, H.
Li43, H. B. Li1,52, H. J. Li9,h, J. L. Li41, J. Q. Li4, Ke Li1, L. K. Li1, Lei
Li3, P. L. Li60,48, P. R. Li32,m,n, S. Y. Li50, W. D. Li1,52, W. G. Li1, X. H.
Li60,48, X. L. Li41, Z. B. Li49, Z. Y. Li49, H. Liang60,48, H. Liang1,52, Y.
F. Liang45, Y. T. Liang25, L. Z. Liao1,52, J. Libby21, C. X. Lin49, B.
Liu42,g, B. J. Liu1, C. X. Liu1, D. Liu60,48, D. Y. Liu42,g, F. H. Liu44, Fang
Liu1, Feng Liu6, H. B. Liu13, H. M. Liu1,52, Huanhuan Liu1, Huihui Liu17, J.
B. Liu60,48, J. Y. Liu1,52, K. Liu1, K. Y. Liu34, Ke Liu6, L. Liu60,48, Q.
Liu52, S. B. Liu60,48, Shuai Liu46, T. Liu1,52, X. Liu32,m,n, Y. B. Liu37, Z.
A. Liu1,48,52, Z. Q. Liu41, Y. F. Long38,k, X. C. Lou1,48,52, F. X. Lu16, H.
J. Lu18, J. D. Lu1,52, J. G. Lu1,48, X. L. Lu1, Y. Lu1, Y. P. Lu1,48, C. L.
Luo35, M. X. Luo67, P. W. Luo49, T. Luo9,h, X. L. Luo1,48, S. Lusso63C, X. R.
Lyu52, F. C. Ma34, H. L. Ma1, L. L. Ma41, M. M. Ma1,52, Q. M. Ma1, R. Q.
Ma1,52, R. T. Ma52, X. N. Ma37, X. X. Ma1,52, X. Y. Ma1,48, Y. M. Ma41, F. E.
Maas15, M. Maggiora63A,63C, S. Maldaner28, S. Malde58, Q. A. Malik62, A.
Mangoni23B, Y. J. Mao38,k, Z. P. Mao1, S. Marcello63A,63C, Z. X. Meng54, J. G.
Messchendorp31, G. Mezzadri24A, T. J. Min36, R. E. Mitchell22, X. H.
Mo1,48,52, Y. J. Mo6, N. Yu. Muchnoi10,c, H. Muramatsu56, S. Nakhoul11,f, Y.
Nefedov29, F. Nerling11,f, I. B. Nikolaev10,c, Z. Ning1,48, S. Nisar8,i, S. L.
Olsen52, Q. Ouyang1,48,52, S. Pacetti23B,23C, X. Pan9,h, Y. Pan55, A. Pathak1,
P. Patteri23A, M. Pelizaeus4, H. P. Peng60,48, K. Peters11,f, J. Pettersson64,
J. L. Ping35, R. G. Ping1,52, A. Pitka4, R. Poling56, V. Prasad60,48, H.
Qi60,48, H. R. Qi50, M. Qi36, T. Y. Qi2, T. Y. Qi9, S. Qian1,48, W.-B. Qian52,
Z. Qian49, C. F. Qiao52, L. Q. Qin12, X. S. Qin4, Z. H. Qin1,48, J. F. Qiu1,
S. Q. Qu37, K. H. Rashid62, K. Ravindran21, C. F. Redmer28, A. Rivetti63C, V.
Rodin31, M. Rolo63C, G. Rong1,52, Ch. Rosner15, M. Rump57, A. Sarantsev29,d,
Y. Schelhaas28, C. Schnier4, K. Schoenning64, M. Scodeggio24A, D. C. Shan46,
W. Shan19, X. Y. Shan60,48, M. Shao60,48, C. P. Shen9, P. X. Shen37, X. Y.
Shen1,52, H. C. Shi60,48, R. S. Shi1,52, X. Shi1,48, X. D Shi60,48, J. J.
Song41, Q. Q. Song60,48, W. M. Song27,1, Y. X. Song38,k, S. Sosio63A,63C, S.
Spataro63A,63C, F. F. Sui41, G. X. Sun1, J. F. Sun16, L. Sun65, S. S. Sun1,52,
T. Sun1,52, W. Y. Sun35, X Sun20,l, Y. J. Sun60,48, Y. K. Sun60,48, Y. Z.
Sun1, Z. T. Sun1, Y. H. Tan65, Y. X. Tan60,48, C. J. Tang45, G. Y. Tang1, J.
Tang49, V. Thoren64, I. Uman51D, B. Wang1, B. L. Wang52, C. W. Wang36, D. Y.
Wang38,k, H. P. Wang1,52, K. Wang1,48, L. L. Wang1, M. Wang41, M. Z. Wang38,k,
Meng Wang1,52, W. H. Wang65, W. P. Wang60,48, X. Wang38,k, X. F. Wang32,m,n,
X. L. Wang9,h, Y. Wang60,48, Y. Wang49, Y. D. Wang15, Y. F. Wang1,48,52, Y. Q.
Wang1, Z. Wang1,48, Z. Y. Wang1, Ziyi Wang52, Zongyuan Wang1,52, D. H. Wei12,
P. Weidenkaff28, F. Weidner57, S. P. Wen1, D. J. White55, U. Wiedner4, G.
Wilkinson58, M. Wolke64, L. Wollenberg4, J. F. Wu1,52, L. H. Wu1, L. J.
Wu1,52, X. Wu9,h, Z. Wu1,48, L. Xia60,48, H. Xiao9,h, S. Y. Xiao1, Y. J.
Xiao1,52, Z. J. Xiao35, X. H. Xie38,k, Y. G. Xie1,48, Y. H. Xie6, T. Y.
Xing1,52, X. A. Xiong1,52, G. F. Xu1, J. J. Xu36, Q. J. Xu14, W. Xu1,52, X. P.
Xu46, F. Yan9,h, L. Yan9,h, L. Yan63A,63C, W. B. Yan60,48, W. C. Yan68, Xu
Yan46, H. J. Yang42,g, H. X. Yang1, L. Yang65, R. X. Yang60,48, S. L.
Yang1,52, Y. H. Yang36, Y. X. Yang12, Yifan Yang1,52, Zhi Yang25, M. Ye1,48,
M. H. Ye7, J. H. Yin1, Z. Y. You49, B. X. Yu1,48,52, C. X. Yu37, G. Yu1,52, J.
S. Yu20,l, T. Yu61, C. Z. Yuan1,52, W. Yuan63A,63C, X. Q. Yuan38,k, Y. Yuan1,
Z. Y. Yuan49, C. X. Yue33, A. Yuncu51B,a, A. A. Zafar62, Y. Zeng20,l, B. X.
Zhang1, Guangyi Zhang16, H. H. Zhang49, H. Y. Zhang1,48, J. L. Zhang66, J. Q.
Zhang4, J. W. Zhang1,48,52, J. Y. Zhang1, J. Z. Zhang1,52, Jianyu Zhang1,52,
Jiawei Zhang1,52, L. Zhang1, Lei Zhang36, S. Zhang49, S. F. Zhang36, T. J.
Zhang42,g, X. Y. Zhang41, Y. Zhang58, Y. H. Zhang1,48, Y. T. Zhang60,48, Yan
Zhang60,48, Yao Zhang1, Yi Zhang9,h, Z. H. Zhang6, Z. Y. Zhang65, G. Zhao1, J.
Zhao33, J. Y. Zhao1,52, J. Z. Zhao1,48, Lei Zhao60,48, Ling Zhao1, M. G.
Zhao37, Q. Zhao1, S. J. Zhao68, Y. B. Zhao1,48, Y. X. Zhao25, Z. G. Zhao60,48,
A. Zhemchugov29,b, B. Zheng61, J. P. Zheng1,48, Y. Zheng38,k, Y. H. Zheng52,
B. Zhong35, C. Zhong61, L. P. Zhou1,52, Q. Zhou1,52, X. Zhou65, X. K. Zhou52,
X. R. Zhou60,48, A. N. Zhu1,52, J. Zhu37, K. Zhu1, K. J. Zhu1,48,52, S. H.
Zhu59, W. J. Zhu37, X. L. Zhu50, Y. C. Zhu60,48, Z. A. Zhu1,52, B. S. Zou1, J.
H. Zou1 (BESIII Collaboration) 1 Institute of High Energy Physics, Beijing
100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
9 Fudan University, Shanghai 200443, People’s Republic of China
10 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
11 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
12 Guangxi Normal University, Guilin 541004, People’s Republic of China
13 Guangxi University, Nanning 530004, People’s Republic of China
14 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
16 Henan Normal University, Xinxiang 453007, People’s Republic of China
17 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
18 Huangshan College, Huangshan 245000, People’s Republic of China
19 Hunan Normal University, Changsha 410081, People’s Republic of China
20 Hunan University, Changsha 410082, People’s Republic of China
21 Indian Institute of Technology Madras, Chennai 600036, India
22 Indiana University, Bloomington, Indiana 47405, USA
23 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN
Sezione di Perugia, I-06100, Perugia, Italy; (C)University of Perugia,
I-06100, Perugia, Italy
24 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of
Ferrara, I-44122, Ferrara, Italy
25 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
26 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
27 Jilin University, Changchun 130012, People’s Republic of China
28 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
29 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
30 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
31 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
32 Lanzhou University, Lanzhou 730000, People’s Republic of China
33 Liaoning Normal University, Dalian 116029, People’s Republic of China
34 Liaoning University, Shenyang 110036, People’s Republic of China
35 Nanjing Normal University, Nanjing 210023, People’s Republic of China
36 Nanjing University, Nanjing 210093, People’s Republic of China
37 Nankai University, Tianjin 300071, People’s Republic of China
38 Peking University, Beijing 100871, People’s Republic of China
39 Qufu Normal University, Qufu 273165, People’s Republic of China
40 Shandong Normal University, Jinan 250014, People’s Republic of China
41 Shandong University, Jinan 250100, People’s Republic of China
42 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
43 Shanxi Normal University, Linfen 041004, People’s Republic of China
44 Shanxi University, Taiyuan 030006, People’s Republic of China
45 Sichuan University, Chengdu 610064, People’s Republic of China
46 Soochow University, Suzhou 215006, People’s Republic of China
47 Southeast University, Nanjing 211100, People’s Republic of China
48 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
49 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
50 Tsinghua University, Beijing 100084, People’s Republic of China
51 (A)Ankara University, 06100 Tandogan, Ankara, Turkey; (B)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (C)Uludag University, 16059 Bursa,
Turkey; (D)Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
52 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
53 University of Hawaii, Honolulu, Hawaii 96822, USA
54 University of Jinan, Jinan 250022, People’s Republic of China
55 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
56 University of Minnesota, Minneapolis, Minnesota 55455, USA
57 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
58 University of Oxford, Keble Rd, Oxford, UK OX13RH
59 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
60 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
61 University of South China, Hengyang 421001, People’s Republic of China
62 University of the Punjab, Lahore-54590, Pakistan
63 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
64 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
65 Wuhan University, Wuhan 430072, People’s Republic of China
66 Xinyang Normal University, Xinyang 464000, People’s Republic of China
67 Zhejiang University, Hangzhou 310027, People’s Republic of China
68 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at Bogazici University, 34342 Istanbul, Turkey
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
d Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
e Also at Istanbul Arel University, 34295 Istanbul, Turkey
f Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
g Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
h Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
i Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA
j Currently at: Institute of Physics and Technology, Peace Ave.54B,
Ulaanbaatar 13330, Mongolia
k Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
l Also at School of Physics and Electronics, Hunan University, Changsha
410082, China
m Also at Frontiers Science Center for Rare Isotopes, Lanzhou University,
Lanzhou 730000, People’s Republic of China
n Also at Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou
730000, People’s Republic of China

###### Abstract

Based on 586 $\rm{pb^{-1}}$ of $e^{+}e^{-}$ annihilation data collected at a
center-of-mass energy of $\sqrt{s}=4.6\leavevmode\nobreak\ \rm{GeV}$ with the
BESIII detector at the BEPCII collider, the absolute branching fraction of
$\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta$ decays is measured for the first
time to be $\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta)=(0.414\pm
0.084\pm 0.028)\%$, where the first uncertainty is statistical and the second
is systematic. The result is compatible with a previous CLEO result on the
relative branching fraction $\frac{\mathcal{B}(\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta)}{\mathcal{B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})}$, and
consistent with theoretical predictions of SU(3) flavor symmetry.

Keywords: charmed baryon, $\Lambda_{c}^{+}$ decays, absolute branching
fraction, BESIII

††journal: Physics Letters B

##  1 Introduction

The lightest charmed baryon, $\Lambda_{c}^{+}$, was first observed in the
1970s [1, 2], but only in the past few years has there been significant
progress in precision measurements of its absolute decay rates [3, 4]. Driven
by these improvements, the theoretical calculations of charmed baryon decays
have greatly improved their treatment of non-perturbative effects [5, 6, 7,
8]. However, more experimental and theoretical efforts are needed to further
improve the understanding of non-factorizable effects, which are non-trivial
in the charmed baryon sector [7, 9]. More experimental data are also desired
to test existing theoretical predictions [10].

The hadronic decay $\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta$ offers
important information about the dynamics of charmed baryon decays [11]. The
charge-conjugate mode is always implied throughout this Letter. The branching
fraction (BF) $\mathcal{B}(\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta)=\mathcal{B}(\Lambda_{c}^{+}\to
p\overline{K}\vphantom{K}^{0}\eta)/2=(0.79\pm 0.13\leavevmode\nobreak\
(\rm{stat.})\pm 0.13\leavevmode\nobreak\ (\rm{sys.}))\%$ has only been
measured relative to $\Lambda_{c}^{+}\to pK^{-}\pi^{+}$ [12, 13]. However,
recent comprehensive theoretical analyses of different BFs for
$\Lambda_{c}^{+}$ decays to three-body final states, which assume SU(3) flavor
symmetry, predict that $\mathcal{B}(\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta)$ is in the range 0.35% to 0.45% [11, 14]. There is a
discrepancy of approximately 2 standard deviations between the existing
experimental result and those new theoretical calculations. Further,
$\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta$ is regarded as an important
channel to understand the properties of the intermediate $N(1535)$ resonance
[15, 16]. The $N(1535)$ resonance is puzzling because it decays to final
states that contain strange hadrons, like $\eta N$ and $K\Lambda$ [13, 15,
16], but it does not contain any $\bar{s}s$ component according to the naive
constituent quark model. Thus, further measurements of the $\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta$ decays may help to verify SU(3) flavor symmetry and
may reveal the intermediate states contributing to these decays.

In this Letter, we present the first absolute measurement of the BF for
$\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta$ decays based on an $e^{+}e^{-}$
collision data sample corresponding to an integrated luminosity of 586
$\rm{pb^{-1}}$, collected at the center-of-mass energy of
$\sqrt{s}=4.6\leavevmode\nobreak\ \rm{GeV}$ with the BESIII detector [17] at
the Beijing Electron Positron Collider (BEPCII) [18]. At this energy, the
$\Lambda_{c}^{+}$ baryon is always produced along with a
$\bar{\Lambda}{}_{c}^{-}$, and the energy remaining does not allow the
production of any additional hadrons. Besides, the previous analysis [19],
based on the double tag method and the simultaneous fit along the 12
$\Lambda_{c}^{+}$ decay channels, has reported the total number of
$\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}$ pairs in the same data sample, which
gives a chance to meansure the absolute BF of the other $\Lambda_{c}^{+}$
decay channels. In this analysis, we use the single tag (ST) method in which
the $\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta$ is reconstructed while the
$\bar{\Lambda}{}_{c}^{-}$ is not.

##  2 BESIII Detector and Monte Carlo Simulation

The BESIII detector is a magnetic spectrometer [17] located at BEPCII [18].
The cylindrical core of the BESIII detector consists of a helium-based
multilayer drift chamber (MDC), a plastic scintillator time-of-flight system
(TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed
in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The
solenoid is supported by an octagonal flux-return yoke with resistive-plate-
counter muon-identifier modules interleaved with steel. The acceptance of
charged particles and photons is 93% of the $4\pi$ solid angle. The charged-
particle momentum resolution at $1\leavevmode\nobreak\ {\rm GeV}/c$ is
$0.5\%$, and the $dE/dx$ resolution is $6\%$ for electrons from Bhabha
scattering. The EMC measures photon energies with a resolution of $2.5\%$
($5\%$) at $1$ GeV in the barrel (end cap) region. The time resolution of the
TOF barrel part is 68 ps, while that of the end cap part is 110 ps.

Samples of simulated events produced with the geant4-based [20] Monte Carlo
(MC) package, which includes the geometric description of the BESIII detector
and the detector response, are used to determine the detection efficiency and
estimate the backgrounds. The simulation includes the beam energy spread and
initial state radiation (ISR) in the $e^{+}e^{-}$ annihilations modeled with
the generator kkmc [21]. Two types of MC samples are generated in this
analysis. The signal MC samples, in which the $\Lambda_{c}^{+}$ decays non-
resonantly to $\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta$ while the
$\bar{\Lambda}{}_{c}^{-}$ decays into all possible channels, are used to
determine the detection efficiency. The inclusive MC sample, which consists of
the production of $\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}$ pairs and open-
charm mesons, the ISR production of vector charmonium(-like) states, and the
continuum processes incorporated in kkmc [21], is used to estimate the
backgrounds. Known decay modes are modeled with evtgen [22] using BFs taken
from the PDG [13], and the remaining unknown decays from the charmonium states
are modeled with lundcharm [23]. The final-state radiation from charged final-
state particles is incorporated using the photos package [24].

##  3 Event selection

In this analysis, all the charged tracks are required to satisfy
$|\cos\theta|<0.93$, where $\theta$ is the polar angle of the charged track
with respect to the axis of the MDC. For the proton, the distance of closest
approach to the interaction point (IP) must be within $\pm
10\leavevmode\nobreak\ \rm{cm}$ along the MDC axis ($V_{z}$) and within $\pm
1\leavevmode\nobreak\ \rm{cm}$ in the plane perpendicular to the MDC axis
($V_{r}$). The particle identification (PID) algorithm combines the $dE/dx$
information from the MDC and flight time from the TOF to evaluate the
likelihood $\mathcal{L}(h)(h=\pi,K,p)$ for the different final state hadron
hypotheses; to be classified as a proton we require the charged track
satisfies both $\mathcal{L}(p)>\mathcal{L}(\pi)$ and
$\mathcal{L}(p)>\mathcal{L}(K)$.

We reconstruct $K^{0}_{\mathrm{S}}\to\pi^{+}\pi^{-}$ candidates by combining
two oppositely charged tracks. For these tracks, the distance of closest
approach to the IP must be within $\pm 20\leavevmode\nobreak\ \rm{cm}$ along
the MDC axis. No requirement on either $V_{r}$ or the PID likelihood is
applied to maximize the efficiency. The two tracks are constrained to a common
decay vertex by applying a vertex fit [3], and the corresponding $\chi^{2}$ is
required to be less than 100. The momenta of the $\pi^{+}\pi^{-}$ pairs
corrected by the vertex fit are used in the following analysis. Further, the
distance of the decay vertex of the $K^{0}_{\mathrm{S}}$ from the IP is
required to be larger than twice the estimated resolution. The invariant mass
of the $\pi^{+}\pi^{-}$ pairs, $M_{\pi^{+}\pi^{-}}$, is required to lie in the
mass region $(0.487,0.511)\leavevmode\nobreak\ \rm{GeV/\textit{c}^{2}}$.

The $\eta$ candidates are formed by photon pairs with their invariant mass
satisfying the requirement $0.500\leavevmode\nobreak\
\rm{GeV/\textit{c}^{2}}<M_{\gamma\gamma}<0.565\leavevmode\nobreak\
\rm{GeV/\textit{c}^{2}}$. Here, the $\gamma$ candidates are reconstructed from
clusters in the EMC crystals that are not matched with any charged tracks. The
energy deposited in the nearby TOF counters is included in the photon
reconstruction. The deposited energy of any cluster is required to be larger
than 25 MeV in the barrel region ($|\cos\theta|<0.8$) or 50 MeV in the end cap
region ($0.86<|\cos\theta|<0.92$). The measured EMC time of the cluster, which
is defined as the shower time recorded by the EMC relative to the event start
time, is required to be within $(0,700)\leavevmode\nobreak\ \rm{ns}$ to
suppress electronic noise and energy deposits unrelated to the event. A
kinematic fit is applied to constrain $M_{\gamma\gamma}$ to the nominal mass
of the $\eta$ meson [13]. The momenta of the $\gamma\gamma$ pairs corrected by
the kinematic fit are used for further analysis.

The $\Lambda_{c}^{+}$ candidates are formed by all combinations of proton,
$K^{0}_{\mathrm{S}}$ and $\eta$ candidates in an event. Two kinematic
variables, the energy difference $\Delta E\equiv
E_{\Lambda_{c}^{+}}-E_{\rm{beam}}$ and the beam-constrained mass
$M_{\rm{BC}}\equiv\sqrt{E^{2}_{\rm{beam}}/c^{4}-|\vec{p}_{\Lambda_{c}^{+}}|^{2}/c^{2}}$,
are defined to identify $\Lambda_{c}^{+}$ candidates. Here,
$E_{\Lambda_{c}^{+}}$ and $\vec{p}_{\Lambda_{c}^{+}}$ are the reconstructed
energy and momentum of the $\Lambda_{c}^{+}$ candidates calculated in the
$e^{+}e^{-}$ rest frame, and $E_{\rm{beam}}$ is the average energy of the
$e^{+}$ and $e^{-}$ beams. All the candidates are required to satisfy
$-0.07\leavevmode\nobreak\ \rm{GeV}<\Delta E<0.07\leavevmode\nobreak\
\rm{GeV}$ and $2.25\leavevmode\nobreak\
\rm{GeV/\textit{c}^{2}}<M_{\rm{BC}}<2.3\leavevmode\nobreak\
\rm{GeV/\textit{c}^{2}}$. If more than one candidate in an event satisfies all
the above requirements, the best candidate is selected with the following
sequence. First, we select candidates that contain the proton with the largest
$\mathcal{L}(p)$. Second, if there are still multiple candidates, we select
those that contain the $K^{0}_{\mathrm{S}}$ with the largest decay length in
units of its resolution. Finally, if there are still multiple candidates after
the proton and $K^{0}_{\mathrm{S}}$ requirements, we select the candidate that
contains the $\eta$ with the smallest $\chi^{2}$ returned by the kinematic
fit. About 16.9% of total events in data have multiple candidates, and the
probability to select the correct candidate is 63.1% based on a MC study.

##  4 Signal yield and BF

Figure 1: One-dimensional projection plots of (a) $M_{\rm{BC}}$ and (b)
$\Delta E$ distributions, as well as background reduced projection plots of
(c) $M_{\rm{BC}}$ ($|\Delta E|<0.02\leavevmode\nobreak\ \rm{GeV}$) and (d)
$\Delta E$ ($2.28\leavevmode\nobreak\
\rm{GeV/\textit{c}^{2}}<M_{\rm{BC}}<2.295\leavevmode\nobreak\
\rm{GeV/\textit{c}^{2}}$) with the fit results. Points with error bars are
data, the solid red line is the total fit result, the blue-dashed line is the
background contribution, and the green histogram is the background
distribution from the inclusive MC sample.

To determine the signal yield of $\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta$,
an extended unbinned maximum likelihood fit is performed on the two-
dimensional distribution of $M_{\rm{BC}}$ versus $\Delta E$. Based on the
$K^{0}_{\mathrm{S}}$ sideband ($0.02\leavevmode\nobreak\
\rm{GeV/\textit{c}^{2}}<|m(\pi^{+}\pi^{-})-m(K^{0}_{\mathrm{S}})|<0.03\leavevmode\nobreak\
\rm{GeV/\textit{c}^{2}}$) from data, and inclusive MC studies, no peaking
background is found in either the $M_{\rm{BC}}$ or $\Delta E$ signal region.
In the fit, the $M_{\rm{BC}}$ and $\Delta E$ background shapes are modeled by
an ARGUS function [25] and a second-order Chebyshev polynomial function,
respectively. The end point of the ARGUS function is fixed to the beam energy
$E_{\rm beam}=2.3\leavevmode\nobreak\ \rm{GeV}$, while all the other
parameters in the background function are free. The $M_{\rm{BC}}$ and $\Delta
E$ background shapes are treated as uncorrelated based on MC validation. The
signal shape is the MC-derived two-dimensional shape convolved with Gaussian
functions in each dimension, where the Gaussian functions compensate for the
resolution difference between data and MC samples. The parameters of the
Gaussian functions for $\Delta E$ and $M_{\rm{BC}}$ are fixed to the values
determined from the corresponding one-dimensional fits in data.

Figure 1 shows the one-dimensional projection plots of the fit results. The
signal yield is obtained to be $N_{\rm{sig}}=42.0\pm 8.5$, where the
uncertainty is statistical only. The statistical significance for the signal
is $7.3\sigma$, which is evaluated by the change of the likelihood between the
nominal fit and the fit without the signal component [3, 26]. And the
significance including the systematic uncertainties is evaluate to be
$5.3\sigma$ [27]. The detection efficiency, which is estimated by using the
signal MC sample, is $\varepsilon=(17.57\pm 0.01)\%$. Thus, the absolute BF of
$\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta$ can be obtained by

$\displaystyle\mathcal{B}(\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta)=\frac{N_{\rm{sig}}}{2\times
N_{\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}}\times\mathcal{B}_{\rm{inter}}\times\varepsilon}\quad,$
(1)

where
$\mathcal{B}_{\rm{inter}}=\mathcal{B}(K^{0}_{\mathrm{S}}\to\pi^{+}\pi^{-})\times\mathcal{B}(\eta\to\gamma\gamma)$
is the product of the BFs of the intermediate states taken from the PDG [13],
$N_{\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}}=(105.9\pm 5.3)\times 10^{3}$ is
the total number of $\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}$ pairs in data
[19], and the factor 2 is the factor related to the charge conjugation. Using
Eq. 1, we calculate $\mathcal{B}(\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta)=(0.414\pm 0.084)\%$, where the uncertainty is
statistical only.

##  5 Systematic uncertainties

In this work, the systematic uncertainties on the BF measurement include the
uncertainties related to the PID and tracking of the proton, the
reconstruction of the $K^{0}_{\mathrm{S}}$ and $\eta$, the two-dimensional
fit, the MC modeling, $N_{\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}}$ and the BFs
of the intermediate states.

The uncertainties related to both the PID and tracking efficiencies of the
proton are estimated to be 1%, respectively, based on a study of control
samples of $e^{+}e^{-}\to p\bar{p}\pi^{+}\pi^{-}$ and $e^{+}e^{-}\to
p\bar{p}\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ events in data at
$\sqrt{s}>4.23\leavevmode\nobreak\ \rm{GeV}$ [19]. The uncertainty associated
with the $K^{0}_{\mathrm{S}}$ reconstruction is assigned to be 2.5% based on
studies of control samples of $J/\psi\to K^{*}(892)^{\mp}K^{\pm}$ and
$J/\psi\to\phi K^{0}_{\mathrm{S}}K^{\mp}\pi^{\pm}$ decays [19]. We use a
control sample of $D^{0}\to K^{-}\pi^{+}\pi^{0}$ events to estimate the
uncertainty due to the $\eta$ reconstruction [28, 29]. The
$\pi^{0}\to\gamma\gamma$ reconstruction efficiencies are obtained in data and
MC simulations, and an $\eta$ reconstruction uncertainty of 1.8% is assigned
by weighting the efficiency difference to the $\eta$ momentum distribution of
the signal samples.

Figure 2: Distributions of (a) $M_{pK^{0}_{\mathrm{S}}}$, (b)
$M_{K^{0}_{\mathrm{S}}\eta}$ and (c) $M_{p\eta}$ in data and signal MC
samples, in which points with error bars are data after the background
subtraction, and the red-solid and blue-dashed histograms show the unweighted
and weighted signal MC samples, respectively.

The uncertainty associated with the two-dimensional fit consists of the
uncertainties of the signal and background shapes. To estimate the
uncertainty, 20000 pseudo-data-sets are sampled with replacement data
according to the bootstrap method [30]. For each pseudo-data-set, a two-
dimensional fit is performed, in which the mean values and standard deviations
of the two Gaussian functions in the signal shape are randomly varied by $\pm
1\sigma$ around their nominal values, the endpoint of the ARGUS function is
randomly varied by $\pm 0.2\,\mathrm{MeV}$ around its nominal value, and the
background shape of the $\Delta E$ distributions is replaced by a first-order
Chebyshev polynomial function. The pull of the fit result ${\rm
p}(N_{\rm{sig}})$ is defined as

$\displaystyle{\rm
p}(N_{\rm{sig}})=\frac{N_{\rm{sig}}^{\rm{pseudo}}-{N_{\rm{sig}}^{\rm{real}}}}{\sigma_{N}^{\rm{pseudo}}},$
(2)

where $N_{\rm{sig}}^{\rm{real}}$ is the fit yield of real data,
$N_{\rm{sig}}^{\rm{pseudo}}$ and $\sigma_{N}^{\rm{pseudo}}$ are the fit yield
and its statistical uncertainty of the pseudo-data, respectively. The
distribution of ${\rm p}(N_{\rm{sig}})$ values is fitted with a Gaussian
function, and the mean value is obtained to be $-0.1364\pm 0.0074$. The bias
in ${\rm p}(N_{\rm{sig}})$ is 13.64% times the statistical uncertainty, which
is calculated to be 2.8%, and as no correction for this bias is applied, we
use 2.8% as the systematic uncertainty related with the two-dimensional fit.

The uncertainty related to the MC model of the signal process is evaluated by
reweighting the signal MC sample, which is generated assuming no intermediate
resonances, according to the $M_{pK^{0}_{\mathrm{S}}}$,
$M_{K^{0}_{\mathrm{S}}\eta}$ and $M_{p\eta}$ invariant mass distributions
observed in data. Here, $M_{ij}$ is the invariant mass of the combination of
particles $i$ and $j$. Figure 2 shows the distributions of these invariant
masses in data after the background subtraction using the sPlot method [31]
based on the nominal fitting. The detection efficiency based on the modified
MC sample is calculated to be 17.19%, and the relative difference to the
nominal efficiency, 2.2%, is taken as the corresponding systematic
uncertainty.

The uncertainty of $N_{\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}}$ in data is
reported to be 4.6% in Ref. [19]. The BFs of intermediate states from the PDG
[13] are $\mathcal{B}(K^{0}_{\mathrm{S}}\to\pi^{+}\pi^{-})=(69.20\pm 0.05)\%$
and $\mathcal{B}(\eta\to\gamma\gamma)=(39.41\pm 0.20)\%$. The uncertainty from
these two BFs, which is dominated by the latter one, is 0.5%.

The total systematic uncertainty, which is determined by adding all sources of
systematic uncertainties in quadrature, is calculated to be 6.8%. The
systematic uncertainties are summarized in Table 1.

Table 1: Systematic uncertainties. Source | Uncertainty (%)
---|---
Proton PID | 1.0
Proton Tracking | 1.0
$K^{0}_{\mathrm{S}}$ reconstruction | 2.5
$\eta$ reconstruction | 1.8
Two-dimensional fit | 2.8
Signal model | 2.2
$N_{\Lambda_{c}^{+}\bar{\Lambda}{}_{c}^{-}}$ | 4.6
$\mathcal{B}_{\rm inter}$ | 0.5
Total | 6.8

##  6 Summary

Based on 586 $\rm{pb^{-1}}$ of ${\rm e}^{+}e^{-}$ collision data collected at
$\sqrt{s}=4.6\leavevmode\nobreak\ \rm{GeV}$ with the BESIII detector and the
BEPCII collider, the absolute BF for $\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta$ is determined for the first time. The BF is measured
to be $\mathcal{B}(\Lambda_{c}^{+}\to pK^{0}_{\mathrm{S}}\eta)=(0.414\pm
0.084\pm 0.028)\%$, where the first uncertainty is statistical and the second
one is systematic. The statistical significance of this measurement is larger
than 7$\sigma$, and the significance becomes 5.3$\sigma$ when both statistical
and systematic uncertainties are taken into account. This result is compatible
with the CLEO result $\mathcal{B}(\Lambda_{c}^{+}\to
pK^{0}_{\mathrm{S}}\eta)=(0.8\pm 0.2)\%$ [12] within $1.5\sigma$. Our result
is consistent with theoretical predictions based on SU(3) symmetry [11, 14].
With present statistics, it is not practical to measure any intermediate
$N^{*}$ states, as suggested in Refs. [15, 16].

##  7 Acknowledgments

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key Basic Research Program of China under Contract No. 2020YFA0406400,
2020YFA0406300; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 11425524, 11575193 11625523, 11635010, 11735014, 11822506,
11835012, 11935015, 11935016, 11935018, 11961141012,11805086, U1732266; the
Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the
Youth Innovation Promotion Association of CAS under Contract No. 2016153;
Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under
Contracts Nos. U1732263, U1832207; CAS Key Research Program of Frontier
Sciences under Contracts Nos. QYZDJ-SSW-SLH003, QYZDB-SSW-SLH039, QYZDJ-SSW-
SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key Laboratory for
Particle Physics and Cosmology; Fundamental Research Funds for the Central
Universities of China; ERC under Contract No. 758462; German Research
Foundation DFG under Contracts Nos. 443159800, Collaborative Research Center
CRC 1044, FOR 2359, GRK 214; Istituto Nazionale di Fisica Nucleare, Italy;
Ministry of Development of Turkey under Contract No. DPT2006K-120470; National
Science and Technology fund; Olle Engkvist Foundation under Contract No.
200-0605; STFC (United Kingdom); The Knut and Alice Wallenberg Foundation
(Sweden) under Contract No. 2016.0157; The Royal Society, UK under Contracts
Nos. DH140054, DH160214; The Swedish Research Council; U. S. Department of
Energy under Contracts Nos. DE-FG02-05ER41374, DE-SC-0012069.

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# Towards Learning Causal Representations from Multi-Instance Bags

Weijia Zhang1 Xuanhui Zhang2 Hanwen Deng1&Min-Ling Zhang1 1School of Computer
Science and Engineering, Southeast University, Nanjing 210096, China
2School of Information Management, Nanjing University, Nanjing 210023, China

###### Abstract

Although humans can easily identify the object of interest from groups of
examples using group-level labels, most of the existing machine learning
algorithms can only learn from individually labeled examples. Multi-instance
learning (MIL) is a type of weakly supervised learning that deals with objects
represented as groups of instances, and is theoretically capable of predicting
instance labels from group-level supervision. Unfortunately, most existing MIL
algorithms focus on improving the performances of group label predictions and
cannot be used to accurately predict instance labels. In this work, we propose
the TargetedMIL algorithm, which learns semantically meaningful
representations that can be interpreted as causal to the object of interest.
Utilizing the inferred representations, TargetedMIL excels at instance label
predictions from group-level labels. Qualitative and quantitative evaluations
on various datasets demonstrate the effectiveness of TargetedMIL.

##  1 Introduction

(a)

(b)

(c)

(d)

Figure 1: (a) MNIST digits. (b) Reconstructions of each target digit per row
by TargetedMIL under bag level supervision. (c) FashionMNIST objects. (b)
Reconstructions of each target fashion object per row. These figures are best
viewed when zoomed.

Supervised learning has achieved great success in many applications. However,
an unignorable limitation for most of the existing supervised learning
algorithms is that they model complex objects as a single feature vector and
cannot make fine-grained predictions without corresponding supervision, e.g.,
learning to localize the region of interest within an image from coarse-
grained image-level supervision. Unfortunately, acquiring fine-grained labels
is often not only a tedious task but also prohibitively expensive, especially
for applications that require a high level of domain-specific expertise, such
as medical image classifications Sirinukunwattana et al. (2016).

Multi-instance learning (MIL) Dietterich et al. (1997) is a weakly supervised
learning paradigm originally proposed for drug activity prediction, where the
task is to predict whether a molecule is suitable for binding to a target
receptor. Since a molecule can take many low-energy conformations and whether
the molecule is suitable for making drugs depends on some specific but unknown
conformations, objects in MIL are represented by groups of instances called
bags where each instance is described by its own feature vector, instead of
represented by a single feature vector as in standard supervised learning.
Because only molecule-level drug binding suitability is known to human
experts, MIL algorithms are only coarsely supervised at the bag level, and
fine-grained labels for the instances are unknown.

Based on the bag-level supervision, the prediction tasks of MIL are two folds:
predicting the bag labels, e.g., whether a molecule is suitable for making
drugs or a medical image contains cancerous cells; and predicting the instance
labels, e.g., specific conformations are suitable or which cells are
cancerous. The task of instance label prediction arguably provides more
utility than the bag label prediction task as the capability of learning fine-
grained labels from coarse-grained supervision significantly reduces the
labeling costs and, at the same time, offers better interpretability.

Unfortunately, even though theoretical results have shown the feasibility of
learning accurate instance concepts from coarse-grained bag-level supervision
Doran and Ray (2016), most of the previous MIL algorithms focus on the task of
predicting bag labels Zhou et al. (2009); Wei et al. (2017), and are not able
to accurately predict instance labels.

In this paper, we tackle the instance label prediction problem in MIL from a
new perspective by synergizing causal representation learning Schölkopf et al.
(2021) with multi-instance learning. We propose the Targeted Multi-Instance
Learning (TargetedMIL) algorithm that is targeted to explicitly infer the
latent representations for the instance of interest within a bag using only
bag-level supervision. For example, as illustrated in Figure 1 (b) and (d),
each row is constructed from the representations inferred by TargetedMIL
learning one digit/fashion item of interest. We can see that only the one
belonging to the positive concept is reconstructed to its semantic meaning,
while the others are ignored.

By exploiting the recent advancement of identifiable latent variable models
Khemakhem et al. (2020) and causality-based multi-instance learning Zhang et
al. (2020), the inferred latent representations can be interpreted causally.
Conceptually, the reconstruction illustrated in the last row of Figure 1 (b)
is learned from bags that contain ‘9’ versus those that do not, and we can see
that the reconstruction clearly resembles the causal meaning of ‘9’: a circle
at the top with a short line originating from the bottom right of the circle.
Furthermore, the blurriness of the line stroke corresponds to the fact that it
can be written in varying angles, and its angle does not determine whether a
digit is ’9’.

This paper contributes towards bridging the gap between two intrinsically
connected yet previously isolated problems, causal representation learning and
multi-instance learning, in one coherent and synergistic framework based-on
Variational Autoencoder (VAE) Kingma and Welling (2014). We summarize our
contribution as follows:

  * •

We propose the TargetedMIL algorithm, an end-to-end MIL algorithm that
explicitly infers semantically meaningful latent representations that is
identifiable and can be interpreted as causal to the bag label.

  * •

Utilizing the inferred representations, TargetedMIL accurately predicts
instance labels from bag labels, which greatly reduces the labeling cost and
at the same time improves interpretability.

  * •

Qualitatively and quantitatively, we demonstrate that TargetedMIL achieves
state-of-the-art performances for instance label prediction on a variety of
datasets.

##  2 Related Works

####  2.0.1 Multi-Instance Learning

Most of the MIL algorithms can be categorized into two groups according to
whether they work at the bag space or at the instance space. Generally
speaking, bag space algorithms Zhou et al. (2009); Wei et al. (2017) work by
embedding multi-instance bags into a single feature vector representation and
then solving the single-instance learning problem in the embedded space.
Therefore, bag space algorithms are only capable of learning a bag label
prediction function and are not designed to predict instance labels.

Instance space MIL algorithms train classifiers that directly separate the
positive instances from the negative ones Andrews et al. (2002); Kandemir and
Hamprecht (2014); Haussmann et al. (2017). Algorithms belonging to this
category can be used to predict instance labels, although not all of them are
explicitly designed for the task. However, these algorithms only work with
pre-computed features and do not learn meaningful representations, nor are
they capable of end-to-end learning.

Recently, by combining permutation-invariant pooling operations with deep
neural networks, several deep learning-based MIL algorithms have been proposed
and can be used for instance label prediction. For example, Wang et al. (2018)
used a max-pooling layer with a fully connected neural network; Ilse et al.
(2018) introduced the attention mechanism as a permutation-invariant MIL
pooling operation and used the attention weights for instance label
prediction; Rymarczyk et al. (2021) proposed to integrate self-attention with
the attention mechanism for capturing the non-i.i.d. information among
instances. Unfortunately, none of the above methods learn semantically
meaningful representations.

####  2.0.2 VAE-Based Representation Disentanglement and MIL

Unsupervised learning of meaningful representations with VAE-based models has
attracted much attention Higgins et al. (2017); Chen et al. (2018). Since it
has been shown that unsupervised disentanglement is theoretically impossible
without inductive biases on both the model and the data Locatello et al.
(2019), various methods have been proposed for learning disentangled
representations using different forms of inductive biases or weak
supervisions.

One form of weakly-supervised disentanglement learning that is closely related
to MIL is group-based VAEs Bouchacourt et al. (2018); Hosoya (2019). Similar
to MIL, where instances are organized into bags, group-based algorithms
require the objects to be divided into groups: those within the same group
share the same content but have different styles that are independent of the
contents, while those among different groups should have different contents.

Recently, Zhang (2021) proposed the MIVAE algorithm based on connections
between MIL and group-based disentangled representation learning which
utilizes VAE for capturing the dependencies among instances as the shared
content factors. However, MIVAE also does not learn semantically meaningful
latents, nor does it provide identifiability on what latents it actually
learns.

##  3 Methods

###  3.1 Preliminaries

Let $\mathcal{X}=\mathbb{R}^{m}$ denote the instance space and
$\mathcal{Y}=\\{0,1\\}$ denote the label space. MIL algorithms deal with the
weakly supervised learning scenario where the data contains samples consisted
of $n$ training bags
$\mathcal{B}=\\{\mathbf{B}_{1},\cdots,\mathbf{B}_{i},\cdots,\mathbf{B}_{n}\\}$,
where each bag is a set that contains different numbers of $n_{i}$ instances,
i.e.,
$\mathbf{B}_{i}=\\{\bm{x}_{i1},\cdots,\bm{x}_{ij},\cdots,\bm{x}_{in_{i}}\\}$
with $\bm{x}_{ij}\in\mathcal{X}$. The learner is provided with multi-instance
bags $\mathbf{B}_{i}$ and their associated bag labels $y_{i}\in\mathcal{Y}$
during training. For each instance $\bm{x}_{ij}$, there also exists an
instance label $y_{ij}\in\mathcal{Y}$; however, the instance labels are
_unknown_ to the learners.

We follow the standard multi-instance assumption Foulds and Frank (2010) which
assumes that a bag is positive if at least one of its instances is positive
and is negative if all of its instances are negative. It is worth noting that
under the standard MIL assumption, bags are independent, but instances within
the bags do not have to be independent. In the rest of this paper, we will
also refer to the positive instances as the target instances and denote them
as $\bm{x}_{t}$ as they are our target for learning the latent
representations.

Formally, we would like to learn semantically meaningful latent representation
$\bm{z}_{t}\in\mathbb{R}^{d}$ of the observed target instances
$\bm{x}_{t}\in\mathbb{R}^{m}$ with $d\ll m$, such that $\bm{z}_{t}$ can be
viewed as the causal representation of $\bm{x}_{t}$ in the sense that
$\bm{z}_{t}$ captures the high-level semantics Schölkopf et al. (2021) that is
causal to the labels of the multi-instance bags Zhang et al. (2020). After
learning the latents, the focus of TargetedMIL is to predict the instance
labels of unseen test bags, which is an important weakly supervised learning
task that reduces labeling costs and improves interpretability simultaneously.
The bag labels can be derived from the instance labels based on the standard
multi-instance assumption.

###  3.2 Variational Autoencoder

Before presenting our proposed approach, we first briefly introduce
Variational Autoencoder (VAE) since it serves as the basis of TargetedMIL. VAE
can be considered as the combination of a generative latent variable model and
an associated inference model where both models are parameterized by deep
neural networks Kingma and Welling (2014). Specifically, VAE learns the joint
distribution
$p_{\bm{\theta}}(\bm{x},\bm{z})=p_{\bm{\theta}}(\bm{x}|\bm{z})p_{\bm{\theta}}(\bm{z})$
where $p_{\bm{\theta}}(\bm{x}|\bm{z})$ is the conditional distribution of
observing $\bm{x}$ given $\bm{z}$, $\bm{\theta}$ is the set of generative
parameters, and $p_{\bm{\theta}}(\bm{z})=\Pi_{i=1}^{d}p_{\bm{\theta}}(z_{i})$
is the factorized prior distribution of the latents. By introducing an
inference model $q_{\bm{\phi}}(\bm{z}|\bm{x})$, the set of parameters
$\bm{\phi}$ and $\bm{\theta}$ can be jointly optimized by maximizing the
evidence lower bound (ELBO) on the marginal likelihood which can be written
as:

$\displaystyle\mathcal{L}$
$\displaystyle=\mathbb{E}_{q_{\bm{\phi}}(\bm{z}|\bm{x})}[\log
p_{\bm{\theta}}(\bm{x}|\bm{z})]-D_{\text{KL}}(q_{\bm{\phi}}(\bm{z}|\bm{x})||p(\bm{z}))$
$\displaystyle=\log
p_{\bm{\theta}}(\bm{x})-D_{\text{KL}}(q_{\bm{\phi}}(\bm{z}|\bm{x})||p_{\bm{\theta}}(\bm{z}|\bm{x}))\leq\log
p_{\bm{\theta}}(\bm{x}),$

where $D_{\text{KL}}$ denotes the Kullback-Leighbler divergence between the
approximation and the true posterior, and $\mathcal{L}$ is a lower bound of
the marginal likelihood $p_{\bm{\theta}}(\bm{x})$ because of the non-
negativity of the KL-divergence. Here we slightly abuse the notation by
dropping the per data point summation in order to avoid cluttering. However,
it is clear that the likelihood is calculated as the sum of ELBOs for each
observed data sample Kingma and Welling (2014).

###  3.3 TargetedMIL

At the heart of the proposed TargetedMIL algorithm lies the claims that it is
relatively easier to learn causal representations from MIL than from single-
instance learning where the instances are i.i.d and individually supervised.
Intuitively, this is because that the paradigm of MIL contains two types of
inductive biases that naturally arise from the formulation of the multi-
instance problem, and these inductive biases enable the learning of
semantically meaningful representations that can be interpreted as causal to
the bag label. Firstly, as instances within the same bag capture inherent
structural information of complex objects and are not independent to each
other Zhou et al. (2009); Zhang (2021), it is possible to learn identifiable
latent representations from bags utilizing the recent advancement of
Identifiable VAE Khemakhem et al. (2020). Secondly, because the bag labels can
be considered as the effects and the target instance concepts can be
considered as the causes under the standard multi-instance assumption Zhang et
al. (2020), the latent target concepts can be viewed as the cause of the bag
label. Synergistically, the causal representations facilitate the downstream
task of instance label prediction from coarse-grained bag labels.

Formally, let us consider the following generative models:

$\displaystyle p_{\mathbf{\theta}}(\bm{x}_{t},\bm{z}_{t}|\bm{B})$
$\displaystyle=p_{\mathbf{f}}(\bm{x}_{t}|\bm{z}_{t})p_{\mathbf{T},\bm{\eta}}(\bm{z}_{t}|\bm{B})$
(1) $\displaystyle p_{\mathbf{f}}(\bm{x}_{t}|\bm{z}_{t})$
$\displaystyle=p_{\bm{\varepsilon}}(\bm{x}_{t}-f(\bm{z}_{t}))$ (2)
$\displaystyle
p_{\mathbf{T},\bm{\eta}}(\bm{z}_{t}|\bm{B})=\prod\limits_{i}h_{i}(z_{i})$
$\displaystyle g_{i}(\bm{B})\exp[\bm{T}_{i}(z_{i})^{T}\bm{\eta}_{i}(\bm{B})]$
(3)

where Equation 1 describes the generative process of $\bm{x}_{t}$ given the
underlying latent factors $\bm{z}_{t}$ and $\bm{B}$. Equation 2 implies that
the observed representation $\bm{x}_{t}$ is an additive noise function of
$\bm{z}_{t}$, i.e., $\bm{x}_{t}=f(\bm{z}_{t})+\bm{\varepsilon}$ where
$\bm{\varepsilon}\sim N(\bm{0},\bm{\Sigma})$. Equation 3 specifies that the
conditional prior distribution of
$p_{\mathbf{T},\bm{\eta}}(\bm{z}_{t}|\bm{B})$ belongs to the $d$-dimensional
exponential family distribution where $h_{i}$ is the base measure,
$g_{i}(\bm{B})$ is the normalizing constant,
$\bm{T}_{i}=[T_{i,1},\dots,T_{i,k}]^{T}$ are k-dimensional sufficient
statistics, and
$\bm{\eta}_{i}(\bm{B})=[\eta_{i,1}(\bm{B}),\dots,\eta_{i,k}(\bm{B})]^{T}$ are
parameters that depend on the bag.

######  Theorem 1.

Khemakhem et al. (2020) Assume we have data sampled from
$p_{\bm{\theta}}(\bm{x}_{t},\bm{z}_{t}|\bm{B})=p_{\bm{f}}(\bm{x}_{t}|\bm{z}_{t})p_{\bm{T},\bm{\eta}}(\bm{z}_{t}|\bm{B})$,
where $p_{\bm{f}}(\bm{x}_{t}|\bm{z}_{t})$ has additive noise as in Equation 2
and $p_{\bm{T},\bm{\eta}}(\bm{z}_{t}|\bm{B})$ is an exponential family
distribution with parameters $\bm{\theta}=(\bm{f},\bm{T},\bm{\eta})$ as in
Equation 3, assuming that the following assumption holds:

  1. i.

The set $\\{\bm{x}_{t}\in\mathcal{X}:\phi_{\bm{\varepsilon}}(\bm{x}_{t}=0\\}$
has measure zero, where $\phi_{\bm{\varepsilon}}$ is the characteristic
function of the density $p_{\bm{\varepsilon}}$ defined as
$p_{\bm{f}}(\bm{x}_{t}|\bm{z}_{t})=p_{\bm{\varepsilon}}(\bm{x}_{t}-\bm{f}(\bm{z}_{t}))$.

  2. ii.

The function $\bm{f}$ is injective.

  3. iii.

The sufficient statistics $T_{i,j}$ are differentiable almost everywhere, and
linearly independent on any subset of $\mathcal{X}$ of measure greater than
zero.

  4. iv.

There exist $kd+1$ distinct bags $\bm{B}^{0},\dots,\bm{B}^{kd}$ such that the
$kd\times kd$ matrix $\bm{E}$ is invertible, which is defined as

$\bm{E}=[\bm{\eta}(\bm{B}^{1})-\bm{\eta}(\bm{B}^{0});\dots;\bm{\eta}(\bm{B}^{2d})-\bm{\eta}(\bm{B}^{0})]$
(4)

Then, the parameters $\bm{\theta}=(\bm{f},\bm{T},\bm{\eta})$ are identifiable
up to an equivalence class. Specifically, if
$p_{\bm{\theta}}(\bm{x}_{t},\bm{z}_{t}|\bm{B})=p_{\bm{\theta}^{\prime}}(\bm{x}_{t},\bm{z}_{t}|\bm{B})$
where $\bm{\theta}^{\prime}$ is another set of parameters in the class, we
have the following equivalence between $\bm{\theta}$ and
$\bm{\theta}^{\prime}$: for any $\bm{x}_{t}\in\mathcal{X}$,
$\exists\bm{A},\bm{c}$ such that

$T(\bm{f}^{-1}(\bm{x}_{t}))=\bm{A}\bm{T}^{\prime}(\bm{f}^{\prime-1}(\bm{x}_{t}))+\bm{c},$
(5)

where $\bm{A}$ is an invertible matrix of $kd\times kd$ dimension and $\bm{c}$
is a vector of dimension $kd$.

The essence of Theorem 1 is that it guarantees that $\bm{f}$ can be identified
up to an affine transformation specified by $\bm{A}$ and $\bm{c}$. This
identifiablity of the latent representation $\bm{z}_{t}$ crucially depends on
the multi-instance bag $\bm{x}_{t}$ belongs to. Following the same reasoning
of Khemakhem et al. (2020), it is also possible to reduce $\bm{A}$ to a
permutation matrix. We leave the proof sketch to the supplementary due to the
page limit.

Note that in practice we choose $p(\bm{z}_{t}|\bm{B})$ to be a factorized
Gaussian distributions whose location and scale is conditioned on $\bm{B}$,
which is a special case of the exponential family distributions with two
dimensional sufficient statistics. The evidence lower bound of TargetedMIL can
then be written as:

$\displaystyle\mathcal{L}_{\text{ELBO}}=$
$\displaystyle\mathbb{E}_{q_{\bm{\phi}}(\bm{z}_{t}|\bm{x}_{t})}[\log
p_{\bm{\theta}}(\bm{x}_{t}|\bm{z}_{t})]$ $\displaystyle-
D_{KL}(q_{\phi}(\bm{z}_{t}|\bm{x}_{t})||p(\bm{z}_{t}|\bm{B}))$ (6)

In other words, for each bag the reconstruction probability is only calculated
for the target instance $\bm{x}_{t}$ and the target concept $\bm{z}_{t}$,
while the KL-divergence depends on the bag. However, it is obvious that we do
not know which instance of the bag is the target because only bag-level
supervision is available in MIL. This problem can be solved by incorporating a
max-pooling operation in combination with a binary classifier supervised by
the bag label. Specifically, we employ a linear classifier $f_{cls}$ as:

$f_{cls}(\bm{z})=\mathbf{w}\bm{z}+\bm{b},$ (7)

where $f_{cls}$ is parameterized by a neural network, along with a max-pooling
operation for learning the latent $\bm{z}_{t}$ and the corresponding
observation $\bm{x}_{t}$:

$\displaystyle\bm{z}_{t}=\arg\max\limits_{\bm{z}}f_{cls}(\bm{z}).$ (8)

The linear classifier $f_{cls}$ is supervised with the bag label $y$ using the
binary cross entropy loss:

$\mathcal{L}_{cls}=-y\cdot\log
f_{cls}(\bm{z}_{t})-(1-y)\log(1-f_{cls}(\bm{z}_{t}))$ (9)

Finally, the optimization objective of TargetedMIL can be expressed as the sum
of the ELBO and the classifier loss $\mathcal{L}_{cls}$,

$\displaystyle\mathcal{L}_{\text{TargetedMIL}}=\mathcal{L}_{\textrm{ELBO}}+\alpha\mathcal{L}_{cls}$
(10)

where $\alpha$ is a hyper parameter. The objective of TargetedMIL can be
optimized using stochastic gradient descent with the re-parameterization trick
Kingma and Welling (2014). After training, instance labels can be obtained by
first infer the latent representation using the encoder
$q_{\bm{\phi}}(\mathbf{z}|\boldsymbol{x})$ and then perform prediction with
the linear classifier $f_{cls}(\bm{z})$ using $\bm{z}$.

It is worth noting that all VAE-based approaches conduct approximate inference
as there exist gaps between the ELBOs and the marginal log-likelihoods. On the
one hand, the approximation gap comes from the capability of how well the
chosen variational distribution family can match the true posterior; on the
other hand, the amortization gap is caused by amortizing the variational
parameters over the entire dataset, instead of optimizing for each training
sample individually.

However, the approximation behavior of VAE does not prevent its success in
generative modeling. In our experiment, by qualitatively investigating the
latent representation learned by TargetedMIL and quantitatively comparing its
instance label prediction performances against state-of-the-arts, we
empirically show that this approximate inference behavior inherited from VAE-
based method does not prevent TargetedMIL from learning semantically
meaningful causal representations from multi-instance bags. Moreover, previous
results have shown that by incorporating more expressive approximate
posteriors instead of factorized conditional Gaussians, the model learns a
better distribution over the data. We leave the endeavors towards these
directions under the setting of multi-instance learning for future works.

##  4 Experiments

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: MNIST digits and TargetedMIL reconstructions for digits ‘4’, ‘7’,
‘9’. Figures are best viewed when zoomed.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: FashionMNIST objects and TargetedMIL reconstructions for ‘dress’,
‘sandal’, and ‘handbag’. Figures best viewed when zoomed.

In this section, we first perform qualitative evaluations of TargetedMIL, then
we quantitatively compare TargetedMIL against the following deep learning-
based MIL baselines. We compared against deep learning-based MIL algorithms
including mi-Net Wang et al. (2018), Attention-based MIL (Attn-MIL) Ilse et
al. (2018), Kernel Self-Attention-based MIL (KSA-MIL) Rymarczyk et al. (2021),
and a recently proposed VAE-based MIL algorithm MIVAE Zhang (2021).

We implement TargetedMIL with PyTorch. Experiments are conducted with a single
Nvidia RTX3090 GPU. Detailed experiment settings, the code, and the data for
reproducing the results are supplied within the supplementary materials.

###  4.1 Qualitative Evaluation

For inspecting the latent representation inferred by TargetedMIL, we utilize
two multi-instance datasets based on MNIST and FashionMNIST constructed
similar to the setting of the multi-instance 20 Newsgroup datasets Zhou et al.
(2009).

####  4.1.1 MNIST-bags

The MNIST dataset is a widely-used collection of handwritten digits that
contains 60,000 training images and 10,000 test images where each image is
represented by 28$\times$28 grayscale pixels. In order to investigate the
setting of predicting instance labels from bag level supervision, we create
the multi-instance MNIST-bags such that each bag contains a number of images
where the bag size is randomly drawn from a Gaussian distribution with fixed
mean and variance. The bag is positive if it contains a target digit, e.g.,
‘4’, and negative if otherwise. Using 10 different digits as targets, we
obtain 10 multi-instance MNIST-bag datasets.

####  4.1.2 FashionMNIST-bags

The FashionMNIST dataset is similar to MNIST in terms of sample sizes and
image resolutions but contains much more complex fashion objects. Our multi-
instance FashionMNIST-bags are created similarly as the MNIST-bags, i.e.., a
bag of images is positive if it contains a target fashion object,e.g.,
‘handbags’, and negative if otherwise. Using 10 different objects as targets,
we obtain 10 FashionMNIST-bags.

For both datasets, unless otherwise specified the reported results are
obtained by setting the mean bag size to $50$ with a standard deviation of
$10$. The numbers of positive instances in each positive bag are set to one
tenth of the bag size.

####  4.1.3 Qualitative Results

Our qualitative results show that TargetedMIL effectively learns semantically
meaningful representations that are causal to the bag label. Looking at the
originals and reconstructions at each row of images in Figure 2 and 3, we can
see that TargetedMIL reconstructs the target digit or fashion objects into
their high-level semantic meanings, and such semantics can be viewed as causal
to the bag label instead of spurious correlations.

Let us take a closer look at Figure 2 (a) and (b). In the reconstruction of
digit ‘4’ we can see that TargetedMIL learns the semantic meaning of the
handwritten ‘4’s: a ‘U’ shape at the top and a short line at the bottom. The
‘U’ shape is clear because it is the necessary condition for a digit to be
considered ‘4’; however, the short line at the bottom is blurry since it can
be written in different angles and its angle are irrelevant for determining
whether a digit is ‘4’. Furthermore, a ‘9’ at the 4th row, 10th column of (b)
has been reconstructed as ‘4’ with lower confidence than the real ‘4’s. This
shows that the learned representation indeed captures the causal information
of the digits rather than a failure of the model, as we can see that the
original ‘9’ is written in a way that resembles the ‘4’. Furthermore, in the
originals and reconstruction of ‘7’ (Figure 2 (c) and (d)) we can see that
there is a ‘7’ that has not been reconstructed at the 1st row, 8th column;
however, we can also see that original digit does not resemble the semantics
of ‘7’, even to us humans. Lastly, in Figure 2 (e) and (f), a ‘4’ at the 6th
row, 5th column of (f) has been reconstructed to ‘9’ with low confidence, and
again we can see that the original ‘4’ closely resembles the semantics of ‘9’.

On the visually more difficult FashionMNIST-bags in Figure 3, the
representations learned by TargetedMIL are also meaningful. Figure 3 (a) and
(b) show the originals and reconstructions for ‘boot’, we can see that the
reconstructions resemble the high-level semantic concept of boots.
Furthermore, a ‘scandal’ at the 8th row, 6th column has been reconstructed as
‘boot’, but it is also clear that the corresponding original image looks more
like ‘boot’ instead of ‘scandal’. From the originals and reconstructions of
‘scandals’ in Figure 3 (c) and (d), it is also clear that all true targets are
correctly identified as shoe-shaped objects that have some kind of holes.
Lastly, for the ‘bag’ objects in Figure 3 (e) and (f), we can see that the bag
handles are much blurrier than the body of the bags, which is reasonable since
many of the original bag objects have no handles, but almost all of them share
square-shaped main bodies. Reconstruction results for the other digits and
fashion objects are provided in the supplementary material due to the page
limit.

###  4.2 Quantitative Evaluation

Table 1: Quantitative results of instance label prediction performance on
MNIST and FashionMNIST datasets. The means $\pm$ standard deviations are
obtained from averaging the results of 10 MNIST/FashionMNIST-bag datasets
where each one is repeated for 5 times.

| MNIST Bags | FashionMNIST Bags
---|---|---
| F-score | Precision | Recall | AUC-PR | F-score | Precision | Recall | AUC-PR
mi-NET | 0.595$\pm$.078 | 0.824$\pm$.056 | 0.467$\pm$.069 | 0.702$\pm$.165 | 0.251$\pm$.290 | 0.427$\pm$.490 | 0.179$\pm$.208 | 0.329$\pm$.450
Attn-MIL | 0.712$\pm$.147 | 0.837$\pm$.235 | 0.620$\pm$.236 | 0.776$\pm$.222 | 0.398$\pm$.243 | 0.284$\pm$.191 | 0.712$\pm$.289 | 0.534$\pm$.194
KSA-MIL | 0.775$\pm$.092 | 0.920$\pm$.081 | 0.670$\pm$.183 | 0.845$\pm$.122 | 0.545$\pm$.288 | 0.423$\pm$.226 | 0.771$\pm$.405 | 0.617$\pm$.360
MIVAE | 0.901$\pm$.035 | 0.925$\pm$.027 | 0.885$\pm$.033 | 0.921$\pm$.056 | 0.701$\pm$.257 | 0.748$\pm$.228 | 0.652$\pm$.272 | 0.733$\pm$.271
TargetedMIL | 0.966$\pm$.018 | 0.984$\pm$.019 | 0.950$\pm$.030 | 0.981$\pm$.018 | 0.748$\pm$.206 | 0.888$\pm$.109 | 0.669$\pm$.248 | 0.822$\pm$.168

We now focus on quantitatively evaluating whether the latent representation
inferred by TargetedMIL is useful for instance label prediction. Specifically,
we use a real-world Colon Cancer histopathology image classification dataset
(please refer to supplementary for details of this dataset due to page limit)
in addition to MNIST-bags and FashionMNIST-bags to evaluate how TargetedMIL
would compare with state-of-the-art MIL algorithms. We emphasize that all of
the reported quantitative results are for instance label predictions from bag
level supervision. In other words, all algorithms are trained using only bag
labels, and the ground truth instance labels are used only for calculating the
instance label prediction performance metrics on the test set.

For evaluation metrics, we use the area under the Precision-Recall curve (AUC-
PR) instead of AUC due to class imbalance at the instance level. For example,
although the original MNIST and FashionMNIST datasets are balanced, only 5% of
the instances are positive in the MNIST-bags and FashionMNIST-bags.
Furthermore, we also report the f-score, precision, and recall metrics of the
compared methods. For the methods that directly output instance prediction
such as mi-NET, MIVAE, and TargetedMIL, the results are obtained from
thresholding their predicted scores. For methods based on the attention
mechanism, such as Attn-MIL and KSA-MIL, the results are obtained by first
normalizing the attention weights to $[0,1]$ and then applying the threshold
to determine the instance label. As there is usually no instance label
available in real application scenarios, we use $0.5$ uniformly as the
classification threshold.

####  4.2.1 Quantitative Results

Table 2: Instance label prediction performances of the Colon Cancer dataset. Experiments are repeated for 5 times and the average metric$\pm$standard deviation of 5-fold cross validations are reported. Method | F-score |  | AUC-PR
---|---|---|---
mi-Net | 0.392$\pm$.017 |  | 0.491$\pm$.028
Attn-MIL | 0.466$\pm$.037 |  | 0.536$\pm$.014
KSA-MIL | 0.510$\pm$.029 |  | 0.578$\pm$.025
MIVAE | 0.675$\pm$.033 |  | 0.747$\pm$.032
TargetedMIL | 0.802$\pm$.019 |  | 0.865$\pm$.028

Table 1 shows the quantitative results on MNIST and FashionMNIST-bags.
TargetedMIL performs significantly better than the compared methods on most
metrics. KSA-MIL achieves higher recall on FashionMNIST-bags; however, its
precision is significantly lower. Note that the large standard deviations on
FashionMNIST-bags are because some objects are much more difficult to classify
(e.g., distinguishing shirts from pullovers and coats), instead of unstable
performances. Among the compared MIL algorithms, TargetedMIL achieves the
lowest standard deviations. These results show that the representations
learned by TargetedMIL are semantically meaningful and useful for instance
label prediction.

Table 2 shows the instance prediction performances of the compared algorithms
on the Colon Cancer dataset. We can see that the trends of performances are
similar to the results of MNIST/ FashionMNIST-bags. For discriminative deep
learning- based algorithms, attention-pooling performs better than max-pooling
and incorporating the self-attention mechanism further improves the
performances. MIVAE, which is another VAE-based generative MIL algorithm,
performs better than discriminative ones but still performs significantly
worse than TargetedMIL because it does not learn meaningful representations
(for comparisons of the learned latents, please refer to ablation results in
the supplementary).

####  4.2.2 Sensitivity to Bag Sizes and Witness Rates

Figure 4:  The average AUC-PR results of different bag sizes (left) and
witness rates (right) on ten FashionMNIST-bags.

Figure 4 shows how different bag sizes and witness rates (number of positive
instances per bag) affect the instance prediction performances of different
MIL algorithms. From the bag size results, we can see that the performances of
all compared MIL algorithms decrease as the bag size increases, which is not
surprising as it affects the levels of inexactness in the supervision;
however, the performances drop of TargetedMIL is significantly less severe
than the compared ones. From the witness rate results, it is interesting to
observe that larger witness sometimes decreases performances for algorithms
that consider instances as independent (e.g., mi-Net and Attn-MIL). Again,
TargetedMIL performs consistently better than the compared algorithms.

There are two reasons behind the effectiveness of TargetedMIL. On the one
hand, the representations learned by previous MIL algorithms do not attempt to
capture the underlying semantic information that is causal to the bag label.
From this perspective, TargetedMIL can be viewed as possibly the first attempt
towards learning causal representations by utilizing the problem
characteristics of MIL. On the other hand, most of the existing algorithms
treat the inexact bag label solely as the source of ambiguity that prohibits
accurate prediction; however, finding some way to utilize this ambiguity is
crucial for MIL performances. Although there are some previous efforts Zhou et
al. (2009); Zhang et al. (2020); Rymarczyk et al. (2021), TargetedMIL is the
first to utilize the bag information for learning meaningful representations.
Therefore, TargetedMIL is a synergistic framework that utilizes the problem
characteristics of MIL for learning better latent representations, which in
turn benefits instance label prediction performances.

##  5 Conclusion

This work proposes a novel TargetedMIL algorithm which excels at learning
semantically meaningful representations and predicting instance labels from
bag-level supervision. Evaluation results show that TargetedMIL performs
significantly better than existing state-of-the-art deep MIL algorithms on
semi-synthetic and real-world datasets.

Many future directions are worth investigating, e.g., incorporating more
expressive hierarchical latent structures to capture the target concepts of
more complex tasks.

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