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@@ -0,0 +1,1225 @@
+MNRAS 000, 1–8 (0000)
+Preprint 31 January 2023
+Compiled using MNRAS LATEX style file v3.0
+𝑅𝑝 Attractors Static Neutron Star Phenomenology
+Vasilis K. Oikonomou1,2
+1 Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
+2 Institut für Theoretische Physik, Goethe Universität Frankfurt, Max-von-Laue-Str.1, 60438 Frankfurt am Main, Germany
+31 January 2023
+ABSTRACT
+In this work we study the neutron star phenomenology of 𝑅𝑝 attractor theories in the Einstein frame. The Einstein frame 𝑅𝑝
+attractor theories have the attractor property that they originate from a large class of Jordan frame scalar theories with arbitrary
+non-minimal coupling. These theories in the Einstein frame provide a viable class of inflationary models, and in this work we
+investigate their implications on static neutron stars. We numerically solve the Tolman-Oppenheimer-Volkoff equations in the
+Einstein frame, for three distinct equations of state, and we provide the mass-radius diagrams for several cases of interest of the
+𝑅𝑝 attractor theories. We confront the results with several timely constraints on the radii of specific mass neutron stars, and as
+we show, only a few cases corresponding to specific equations of state pass the stringent tests on neutron stars phenomenology.
+Key words: stars: neutron; Physical Data and Processes, cosmology: theory
+INTRODUCTION
+The direct gravitational wave observation GW170817 LIGO &
+Virgo Collaboration, et al. (2017, 2020) initiated what is nowadays
+known as gravitational wave astronomy. Neutron stars (NS) Haensel,
+Potekhin & Yakovlev (2007); Friedman & Stergioulas (2013); Baym,
+et al. (2018); Lattimer & Prakash (2004); Olmo, Rubiera-Garcia &
+Wojnar (2020) are at the core of astrophysical gravitational wave
+observations, and numerous scientific areas are jointly studying NS
+from their perspective, for example nuclear theory Lattimer (2012);
+Steiner & Gandolfi (2012); Horowitz, et al. (2005); Watanabe, Iida
+& Sato (2000); Shen, et al. (1998); Xu, et al. (2009); Hebeler, et
+al. (2013); Mendoza-Temis, et al. (2014); Ho, et al. (2015); Kanakis-
+Pegios, Koliogiannis & Moustakidis (2020); Tsaloukidis et al. (2022),
+high energy physics Buschmann, et al. (2021); Safdi, Sun & Chen
+(2019); Hook, et al. (2018); Edwards, et al. (2020); Nurmi, Schi-
+appacasse & Yanagida (2021), modified gravity Astashenok, et al.
+(2020, 2021); Capozziello, et al. (2016); Astashenok, Capozziello &
+Odintsov (2015, 2014, 2013); Arapoˇglu, Deliduman & Eksi (2011);
+Panotopoulos et al.
+(2021); Lobato et al.
+(2020); Numajiri et
+al. (2022) and astrophysics Altiparmak, Ecker & Rezzolla (2022);
+Bauswein, et al. (2020b); Vretinaris, Stergioulas & Bauswein (2020);
+Bauswein, et al. (2020a, 2017); Most, et al. (2018); Rezzolla, Most &
+Weih (2018); Nathanail, Most & Rezzolla (2021); Köppel, Bovard &
+Rezzolla (2019); Raaijmakers et al. (2021); Most, et al. (2021); Ecker
+& Rezzolla (2022); Jiang, et al. (2022). The perspective of modified
+gravity implications on NS has been for a long time in the mainstream
+of NS works, see for example Astashenok, Capozziello & Odintsov
+(2015, 2014) and also Refs. Pani & Berti (2003); Staykov, et al.
+(2014); Horbatsch, et al. (2015); Silva, et al. (2015); Doneva, et al.
+(2013); Xu, Gao & Shao (2020); Salgado, Sudarsky & Nucamendi
+(1998); Shibata, et al. (2014); Arapoğlu, Ekşi & Yükselci (2019);
+Ramazanoğlu & Pretorius (2016); Motahar, et al. (2019); Chew, et
+al. (2019); Blázquez-Salcedo, Scen Khoo & Kunz (2020); Motahar,
+et al. (2017); Odintsov & Oikonomou (2021, 2022a); Oikonomou
+(2021); Pretel et al. (2022); Pretel & Duarte (2022); Cuzinatto et
+al. (2016) for scalar-tensor descriptions of NS phenomenology. The
+main effect of modified gravity descriptions of NS is the significant
+elevation of the maximum NS masses, with modified gravity bring-
+ing this maximum mass near or inside the mass-gap region with
+𝑀 ≥ 2.5 𝑀⊙. Regarding non-minimally coupled scalar field theo-
+ries, there exists a vast class of viable inflationary potentials which
+have the remarkable property of being attractors Kallosh, Linde
+& Roest (2014a); Kallosh & Linde (2013); Ferrara, et al. (2013);
+Kallosh, Linde & Roest (2013); Linde (2015); Cecotti & Kallosh
+(2014); Carrasco, Kallosh & Linde (2015); Carrasco, et al. (2015);
+Kallosh, Linde & Roest (2015); Roest & Scalisi (2015); Kallosh,
+Linde & Roest (2014b); Ellis, Nanopoulos & Olive (2013); Cai,
+Gong & Pi (2014); Yi & Gong (2016); Akrami, et al. (2018); Qum-
+mer, Jawad & Younas (2020); Fei, Yi & Yang (2020); Kanfon, Mavoa
+& Houndjo (2020); Antoniadis, et al. (2020); García-García, et al.
+(2019); Cedeño, et al. (2019); Karamitsos (2019); Canko, Gialamas
+& Kodaxis (2020); Miranda, et al. (2019); Karam, Pappas & Tam-
+vakis (2019); Nozari & Rashidi (2018); García-García, et al. (2018);
+Rashidi & Nozari (2018); Gao, Gong & Fei (2018); Dimopoulos,
+Wood & Owen (2018); Miranda, Fabris & Piattella (2017); Karam,
+Pappas & Tamvakis (2017); Nozari & Rashidi (2017); Gao & Gong
+(2018); Geng, Lee & Wu (2017); Odintsov & Oikonomou (2020,
+2016, 2017); Järv, et al. (2020). The attractor terminology is justi-
+fied due to the fact that distinct non-minimally coupled scalar-tensor
+inflationary theories, lead to the same Einstein frame inflationary
+phenomenology, which is compatible with the latest Planck data
+Planck Collaboration (2020). The question always when studying
+these attractor models is whether these models can be distinguished
+in some way, phenomenologically. From an inflationary point of
+view, and regarding the large wavelength Cosmic Microwave Back-
+ground modes, a discrimination between these models is impossible.
+However, this discrimination is possible if NS are studied. Indeed,
+the phenomenologically indistinguishable attractor models can be
+discriminated in NS and vice versa, with the latter feature being phe-
+© 0000 The Authors
+arXiv:2301.12136v1  [gr-qc]  28 Jan 2023
+
+2
+Oikonomou
+nomenal. That is, if some models are indistinguishable with respect
+to their NS phenomenology, they can be distinguished if their infla-
+tionary properties are studied. To address these issues in a concrete
+way, in this work we shall study 𝑅𝑝 attractor theories. The inflation-
+ary phenomenology of these theories is studied in the recent literature
+Odintsov & Oikonomou (2022b) see also Motohashi (2015); Renzi,
+Shokri & Melchiorri
+(2009) for subcases of the original 𝑅𝑝 at-
+tractors theories. For a spherically symmetric metric we derive and
+solve numerically the Einstein frame Tolman-Oppenheimer-Volkoff
+(TOV) equations, using an LSODA based double shooting python 3
+numerical integration Stergioulas (2019). We derive the Jordan frame
+𝑀 −𝑅 graphs for the 𝑅𝑝 attractors, for three different piecewise poly-
+tropic Read, et al. (2009a,b) equations of state (EoS), WFF1 Wiringa,
+Fiks & Fabrocini (1988), the SLy Douchin & Haensel (2001), and
+the APR EoS Akmal, Pandharipande & Ravenhall (1998), using the
+Arnowitt-Deser-Misner (ADM) definition of Jordan frame masses
+of NS Arnowitt, Deser & Misner (1960). The NSs temperature is
+significantly lower than the Fermi energy of the constituent particles
+of NSs, thus NS matter can be in principle described by a single-
+parameter EoS that may describe perfectly cold matter at densities
+higher than the nuclear density. However, a serious problem emerges,
+having to do with the uncertainty in the EoS, which is larger, and
+the pressure as a function of the baryonic mass density cannot be
+accurately defined and is uncertain to one order of magnitude at least
+above the nuclear density. Moreover, the exact nature of the phase of
+matter at the NSs core is highly uncertain. Hence, a parameterized-
+type EoS at high densities is an optimal choice for an EoS, thus
+rendering the piecewise polytropic EoS a suitable choice. In order
+to construct the piecewise polytropic EoS, astrophysical constraints
+are taken into account, both observational and theoretical, like the
+causality constraints, see Read, et al. (2009a,b), to also confirm the
+causality fulfilment for all the piecewise polytropic EoS we shall use
+in this paper. For the construction of the piecewise polytropic EoS
+one uses a low-density part with 𝜌 < 𝜌0, which is basically chosen to
+be a tabulated and well-known EoS for the crust, and furthermore, the
+piecewise polytropic EoS also has a large density part with 𝜌 ≫ 𝜌0.
+We finally confront the resulting NS phenomenologies with several
+recent constraints on the radii of specific mass NS Altiparmak, Ecker
+& Rezzolla (2022); Raaijmakers et al. (2021); Bauswein, et al. (2017)
+and as we show, only a few scenarios and EoS are compatible with
+the constraints on NS radii. Obviously, the gravitational wave astron-
+omy era has changed the way of thinking on theoretical astrophysics,
+since several models of scalar-tensor gravity which in the recent past
+could be considered as viable, nowadays may no longer be valid.
+1 INFLATIONARY PHENOMENOLOGY OF 𝑅𝑃
+ATTRACTORS
+The full analysis of the generalized 𝑅𝑝 attractors is given in Ref.
+Odintsov & Oikonomou (2022b), so we refer the reader for details.
+Here we shall briefly discuss the inflationary phenomenological prop-
+erties of 𝑅𝑝 attractors in order to stress their importance among other
+cosmological attractors Kallosh, Linde & Roest (2014a); Kallosh &
+Linde (2013); Ferrara, et al. (2013); Kallosh, Linde & Roest (2013);
+Linde (2015); Cecotti & Kallosh (2014); Carrasco, Kallosh & Linde
+(2015); Carrasco, et al. (2015); Kallosh, Linde & Roest (2015); Roest
+& Scalisi (2015); Kallosh, Linde & Roest (2014b); Ellis, Nanopou-
+los & Olive (2013); Cai, Gong & Pi (2014); Yi & Gong (2016);
+Akrami, et al. (2018); Qummer, Jawad & Younas (2020); Fei, Yi
+& Yang (2020); Kanfon, Mavoa & Houndjo (2020); Antoniadis,
+et al. (2020); García-García, et al. (2019); Cedeño, et al. (2019);
+Karamitsos (2019); Canko, Gialamas & Kodaxis (2020); Miranda,
+et al. (2019); Karam, Pappas & Tamvakis (2019); Nozari & Rashidi
+(2018); García-García, et al. (2018); Rashidi & Nozari (2018); Gao,
+Gong & Fei (2018); Dimopoulos, Wood & Owen (2018); Miranda,
+Fabris & Piattella (2017); Karam, Pappas & Tamvakis (2017); Nozari
+& Rashidi (2017); Gao & Gong (2018); Geng, Lee & Wu (2017);
+Odintsov & Oikonomou (2020, 2016, 2017); Järv, et al. (2020). The
+𝑅𝑝 attractors constitute a class of their own among other attrac-
+tors, and all the 𝑅𝑝 attractors in the Einstein frame correspond to
+generalizations of the following Einstein frame potential,
+𝑉(𝜑) = 𝑉0 𝑀4
+𝑝𝑒−2
+√︃
+2
+3 𝜅 𝜑
+�
+𝑒
+√︃
+2
+3 𝜅 𝜑 − 1
+�
+𝑝
+𝑝−1
+,
+(1)
+where 𝑀𝑝 =
+1
+√
+8𝜋𝐺 is the reduced Planck mass and 𝐺 is Newton’s
+gravitational constant. The inflationary properties of the above theory
+have been addressed in the recent literature, see for example Moto-
+hashi (2015); Renzi, Shokri & Melchiorri (2009). The scalar-tensor
+theory with the potential (1) corresponds to the Jordan frame 𝐹(𝑅)
+gravity,
+𝐹(𝑅) = 𝑅 + 𝛽𝑅𝑝 ,
+(2)
+with 𝛽 is a free parameter with its physical dimensions in natural
+units being [𝛽] = [𝑚]2−2𝑝. The 𝑅𝑝 attractors have the following
+scalar potential in the Einstein frame,
+𝑉(𝜑) = 𝑉0 𝑀4
+𝑝𝑒−2
+√︃
+2
+3𝛼 𝜅 𝜑
+�
+𝑒
+√︃
+2
+3𝛼 𝜅 𝜑 − 1
+�
+𝑝
+𝑝−1
+,
+(3)
+where 𝑀𝑝 is the reduced Planck mass, and for 𝛼 = 1 we obtain
+the scalar theory with scalar potential (3). Now the question is why
+these models are classified as attractor models, what justifies the
+terminology attractors? It is the class of scalar-tensor Jordan frame
+theories which correspond to the Einstein frame potential (3) that
+justify the use of the terminology attractors. Basically, the potential
+(3) can be the Einstein frame potential for a large class of Jordan
+frame scalar-tensor theories, as we now evince. The 𝜙-Jordan frame
+action is,
+S𝐽 =
+∫
+𝑑4𝑥
+� Ω(𝜙)
+2𝜅2 𝑅 − 𝜔(𝜙)
+2
+𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 − 𝑉𝐽 (𝜙)
+�
+,
+(4)
+with the scalar field describing a non-canonical scalar field in
+the Jordan frame, and the coupling function has the general form
+Ω(𝜙) = 1+𝜉 𝑓 (𝜙) with 𝜉 and 𝑓 (𝜙) being the arbitrary dimensionless
+coupling and an arbitrary dimensionless function respectively. The
+𝑅𝑝 attractors have the following 𝜙-Jordan frame scalar potential,
+𝑉𝐽 (𝜙) = 𝑉0 (Ω(𝜙) − 1)
+𝑝
+𝑝−1 ,
+(5)
+and more importantly, the kinetic term function 𝜔(𝜙) has the follow-
+ing form,
+𝜔(𝜙) = 1
+4𝜉
+� 𝑑Ω(𝜙)
+𝑑𝜙
+�2
+Ω(𝜙)
+.
+(6)
+Hence the large class of the 𝑅𝑝-attractors correspond to the Jordan
+frame theories which are described by Eqs. (5) and (6). Notice that
+the Jordan frame functions 𝑓 (𝜙) are arbitrary and we shall not need
+to specify these. By performing the conformal transformation of the
+Jordan frame metric 𝑔𝜇𝜈,
+˜𝑔𝜇𝜈 = Ω(𝜙)𝑔𝜇𝜈 ,
+(7)
+MNRAS 000, 1–8 (0000)
+
+𝑅𝑝 Attractors Static Neutron Star Phenomenology
+3
+Figure 1. The constraints CSI, CSII and CSIII. This figure is inspired and
+based after editing on Credit: ESO/L.Calçada: https://www.eso.org/
+public/images/eso0831a/.
+we get the Einstein frame action,
+S𝐸 =
+√︁
+− ˜𝑔
+�
+˜𝑅
+2𝜅2 − ˜𝑔𝜇𝜈𝜕𝜇𝜑𝜕𝜈𝜑 − 𝑉(𝜑)
+�
+,
+(8)
+with ˜𝑔𝜇𝜈 denoting the Einstein frame metric tensor, and the “tilde”
+indicates Einstein frame quantities. Also the Einstein frame potential
+𝑉(𝜙) and the Jordan frame potential 𝑉𝐽 (𝜙) are related as follows,
+𝑉(𝜑) = Ω−2(𝜙)𝑉𝐽 (𝜙) .
+(9)
+Notice that the general relation which connects the Jordan frame
+scalar field 𝜙 with the canonical Einstein frame scalar field 𝜑 is,
+� 𝑑𝜑
+𝑑𝜙
+�2
+= 3
+2
+� 𝑑Ω(𝜙)
+𝑑𝜙
+�2
+Ω(𝜙)
++ 𝜔(𝜙)
+Ω(𝜙) ,
+(10)
+hence for the 𝑅𝑝 attractors, in which case the kinetic term function
+𝜔(𝜙) is chosen to be that of Eq. (6), we finally have the important
+relation of the non-minimal scalar coupling function to gravity,
+Ω(𝜙) = 𝑒
+√︃
+2
+3𝛼 𝜑 ,
+(11)
+with the parameter 𝛼 being defined to be,
+𝛼 = 1 + 1
+6𝜉 .
+(12)
+Notice that by substituting Eq. (11) in Eq. (9) we obtain the gen-
+eralized 𝑅𝑝-attractor potential of Eq. (3). Furthermore, the impor-
+tant case with 𝛼 = 1 is realized when 𝜉 → ∞, or similarly when
+Ω(𝜙) ≪ 3
+2
+�
+𝑑Ω(𝜙)
+𝑑𝜙
+�2
+𝜔(𝜙)
+. The 𝑅𝑝 attractors yield a viable inflationary
+phenomenology, see Ref. Odintsov & Oikonomou (2022b), with the
+spectral index of the primordial scalar perturbations as a function of
+the canonical scalar field being,
+𝑛𝑠 =
+� �
+3𝛼 + (3𝛼 − 2)𝑝2 + (8 − 6𝛼)𝑝 − 8
+�
+𝑒2
+√︃
+2
+3
+√︃
+1
+𝛼 𝜅 𝜑
+(13)
+− 2(𝑝 − 1)(−3𝛼 + (3𝛼 − 2)𝑝 + 8)𝑒
+√︃
+2
+3
+√︃
+1
+𝛼 𝜅 𝜑 + (3𝛼 − 8)(𝑝 − 1)2�
+× 3𝛼(𝑝 − 1)2
+�
+𝑒
+√︃
+2
+3
+√︃
+1
+𝛼 𝜅 𝜑 − 1
+�2
+,
+and the tensor-to-scalar ratio is,
+𝑟 =
+16
+�
+(𝑝 − 2)𝑒
+√︃
+2
+3
+√︃
+1
+𝛼 𝜅 𝜑 − 2𝑝 + 2
+�2
+3𝛼(𝑝 − 1)2
+�
+𝑒
+√︃
+2
+3
+√︃
+1
+𝛼 𝜅 𝜑 − 1
+�2
+.
+(14)
+Also the free parameter 𝑉0 of the potential is constrained to have
+values
+𝑉𝑠 ∼ 9.6 × 10−11 ,
+(15)
+a results which originates from the constraints of the Planck data on
+the Einstein frame amplitude Δ2𝑠 of the scalar perturbations,
+Δ2
+𝑠 =
+1
+24𝜋2
+𝑉(𝜑 𝑓 )
+𝑀4𝑝
+1
+𝜖(𝜑 𝑓 ) .
+(16)
+For the purposes of this paper, we shall consider several limiting
+cases for the values of the parameter 𝛼, mainly the cases 𝛼 ≠ 1,
+and the case 𝛼 = 1, which corresponds to the strong 𝜉 coupling
+theory. Also in order to have a viable inflationary phenomenology,
+the parameter 𝑝 which is the exponent in the 𝑅𝑝 attractors potential,
+has to take values in the range 1.91 ≤ 𝑝 ≤ 1.99. It proves that this is
+irrelevant for NS studies, so we shall assume that 𝑝 = 1.91 without
+loss of generality. In the next section we shall specify the values of
+the various functions involved in the TOV equations of NS.
+2 NEUTRON STARS WITH 𝑅𝑃 ATTRACTORS
+For the purpose of studying NS in Einstein frame, we shall use the
+Geometrized physical units system 𝐺 = 𝑐 = 1, and we shall adopt
+the notation of Ref. Pani & Berti (2003).
+The Jordan frame scalar-tensor theory has the following form,
+S =
+∫
+𝑑4𝑥
+√−𝑔
+16𝜋
+�
+Ω(𝜙)𝑅 − 1
+2𝑔𝜇𝜈𝜕𝜇𝜙𝜕𝜈𝜙 −𝑈(𝜙)
+�
++ 𝑆𝑚(𝜓𝑚, 𝑔𝜇𝜈) ,
+(17)
+and by performing the following conformal transformation,
+˜𝑔𝜇𝜈 = 𝐴−2𝑔𝜇𝜈 , 𝐴(𝜙) = Ω−1/2(𝜙) ,
+(18)
+we obtain the Einstein frame action,
+S =
+∫
+𝑑4𝑥
+√︁
+− ˜𝑔
+�
+˜𝑅
+16𝜋 −1
+2 ˜𝑔𝜇𝜈𝜕𝜇𝜑𝜕𝜈𝜑−𝑉(𝜑)
+16𝜋
+�
++𝑆𝑚(𝜓𝑚, 𝐴2(𝜑)𝑔𝜇𝜈) ,
+(19)
+with 𝜑 denoting the Einstein frame canonical scalar field as in the
+previous section, and
+𝑉(𝜑) = 𝑈(𝜙)
+Ω2
+.
+(20)
+For the 𝑅𝑝 attractors with general 𝛼, the important function 𝐴(𝜑)
+has the following form,
+𝐴(𝜑) = 𝑒− 1
+2
+√︃
+2
+3𝛼 𝜑 ,
+(21)
+therefore, the function 𝛼(𝜙) which is defined as follows,
+𝛼(𝜑) = 𝑑 ln 𝐴(𝜑)
+𝑑𝜑
+,
+(22)
+takes the form,
+𝑎(𝜑) = −1
+2
+√︂
+2
+3𝛼 .
+(23)
+MNRAS 000, 1–8 (0000)
+
+CS I
+-0.99
+CS II
+R1.4Mo
+-0.81
+CS III
+-0.04
+-0.034
+Oikonomou
+Table 1. CSI vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for
+NS Masses 𝑀 ∼ 2𝑀⊙
+𝑅𝑝 Attractor Model
+APR
+SLy
+WFF1
+𝛼 = 1
+𝑀 = 2.00 𝑀⊙
+𝑀 = 2.01 𝑀⊙
+𝑀 = 0.31 𝑀⊙
+𝛼 = 1
+𝑅 = 11.10km
+𝑅 = 11.17km
+𝑅 = 11.06km
+𝛼 = 0.1
+𝑀 = 2.02 𝑀⊙
+𝑀 = 2.00 𝑀⊙
+𝑀 = 2.00 𝑀⊙
+𝛼 = 0.1
+𝑅 = 11.52km
+𝑅 = 11.818km
+𝑅 = 11.012km
+𝛼 = 8
+𝑀 = 2.00 𝑀⊙
+𝑀 = 2.09 𝑀⊙
+𝑀 = 0.32 𝑀⊙
+𝛼 = 8
+𝑅 = 11.08km
+𝑅 = 10.983km
+𝑅 = 11.114km
+Table 2. CSI vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for
+NS Masses 𝑀 ∼ 1.4𝑀⊙
+𝑅𝑝 Attractors Model
+APR
+SLy
+WFF1
+𝛼 = 1
+𝑀 = 0.58 𝑀⊙
+𝑀 = 1.41 𝑀⊙
+𝑀 = 0.25 𝑀⊙
+𝛼 = 1
+𝑅 = 11.48km
+𝑅 = 11.74km
+𝑅 = 11.89km
+𝛼 = 0.1
+𝑀 = 1.39 𝑀⊙
+𝑀 = 1.39 𝑀⊙
+𝑀 = 0.07 𝑀⊙
+𝛼 = 0.1
+𝑅 = 11.55km
+𝑅 = 12.04km
+𝑅 = 11.79km
+𝛼 = 8
+𝑀 = 0.64 𝑀⊙
+𝑀 = 1.42 𝑀⊙
+𝑀 = 0.28 𝑀⊙
+𝛼 = 8
+𝑅 = 11.45km
+𝑅 = 11.73km
+𝑅 = 11.46km
+Finally, the Einstein frame scalar potential is given in Eq. (3), which
+we also quote it here for reading convenience,
+𝑉(𝜑) = 𝑉0 𝑒−2
+√︃
+2
+3𝛼 𝜑
+�
+𝑒
+√︃
+2
+3𝛼 𝜑 − 1
+�
+𝑝
+𝑝−1
+,
+(24)
+and in Geometrized units, the constraint on 𝑉0 given in Eq. (15)
+becomes,
+𝑉0 ≃ 7.62 × 10−12 .
+(25)
+For the study of NS physics, we shall consider the following spheri-
+cally symmetric metric,
+𝑑𝑠2 = −𝑒𝜈(𝑟)𝑑𝑡2 +
+𝑑𝑟2
+1 − 2𝑚(𝑟)
+𝑟
++ 𝑟2(𝑑𝜃2 + sin2 𝜃𝑑𝜙2) ,
+(26)
+which describes a static NS, where the function 𝑚(𝑟) describes the
+total gravitational mass of the NS and 𝑟 stands for the circumferential
+radius. In the following, we shall calculate numerically the functions
+𝜈(𝑟) and
+1
+1− 2𝑚(𝑟)
+𝑟
+following a simple procedure, in which the central
+value of 𝜈(𝑟) and of the scalar field will be arbitrary and will be
+optimally calculated numerically by using a double shooting method.
+The double shooting aims to find the optimal values of the central
+values of 𝜈(𝑟) and of the scalar field, which guarantee that the metric
+at numerical infinity becomes identical to the Schwarzschild metric.
+This procedure is different compared to standard General Relativity
+(GR) NS, because in GR, the metric at the surface of the star abruptly
+becomes the Schwarzschild metric. This is not true in the scalar-
+tensor theories, because the scalar potential and the non-minimally
+coupling function 𝐴(𝜑) have non-trivial effects on the NS beyond the
+Figure 2. The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR
+and SLy EoSs, for 𝛼 = 1
+surface of the star (scalarization). The Einstein frame TOV equations
+take the following form,
+𝑑𝑚
+𝑑𝑟 = 4𝜋𝑟2𝐴4(𝜑)𝜀 + 𝑟
+2 (𝑟 − 2𝑚(𝑟))𝜔2 + 4𝜋𝑟2𝑉(𝜑) ,
+(27)
+𝑑𝜈
+𝑑𝑟 = 𝑟𝜔2+
+2
+𝑟(𝑟 − 2𝑚(𝑟))
+�
+4𝜋𝐴4(𝜑)𝑟3𝑃−4𝜋𝑉(𝜑)𝑟3�
++
+2𝑚(𝑟)
+𝑟(𝑟 − 2𝑚(𝑟)) ,
+(28)
+𝑑𝜔
+𝑑𝑟 = 4𝜋𝑟 𝐴4(𝜑)
+𝑟 − 2𝑚(𝑟)
+�
+𝛼(𝜑)(𝜖 − 3𝑃) + 𝑟𝜔(𝜖 − 𝑃)
+�
+− 2𝜔(𝑟 − 𝑚(𝑟))
+𝑟(𝑟 − 2𝑚(𝑟))
+(29)
++
+8𝜋𝜔𝑟2𝑉(𝜑) + 𝑟 𝑑𝑉 (𝜑)
+𝑑𝜑
+𝑟 − 2𝑚(𝑟)
+,
+𝑑𝑃
+𝑑𝑟 = −(𝜖 + 𝑃)
+� 1
+2
+𝑑𝜈
+𝑑𝑟 + 𝛼(𝜑)𝜔
+�
+,
+(30)
+𝜔 = 𝑑𝜑
+𝑑𝑟 ,
+(31)
+with 𝛼(𝜑) being defined in Eq. (22). Also note that the energy density
+𝜖 and the pressure 𝑃 of the matter fluid are Jordan frame quantities.
+We shall solve the TOV equations for both the interior and the exterior
+of the NS, with the following set of initial conditions being used,
+𝑃(0) = 𝑃𝑐 , 𝑚(0) = 0 , 𝜈(0) , = −𝜈𝑐 , 𝜑(0) = 𝜑𝑐 , 𝜔(0) = 0 .
+(32)
+Both 𝜈𝑐 and 𝜑𝑐 will be determined using a double shooting method,
+and the numerical analysis shall be performed for three distinct piece-
+wise polytropic EoS, with the central part being described by the
+SLy, WFF1 or the APR EoS. For the calculation of the ADM mass
+in the Jordan frame we shall use the following definition Odintsov &
+Oikonomou (2021, 2022a); Oikonomou (2021),
+𝑀 = 𝐴(𝜑(𝑟𝐸))
+�
+𝑀𝐸 −
+𝑟2
+𝐸
+2 𝛼(𝜑(𝑟𝐸)) 𝑑𝜑
+𝑑𝑟
+�
+2 + 𝛼(𝜑(𝑟𝐸))𝑟𝐸
+𝑑𝜑
+𝑑𝑟
+� �
+1 − 2𝑀𝐸
+𝑟𝐸
+��
+.
+(33)
+where 𝑟𝐸 denotes the Einstein frame circumferential radius of the
+NS, and also we define 𝑑𝜑
+𝑑𝑟 = 𝑑𝜑
+𝑑𝑟
+���𝑟=𝑟𝐸
+. Finally, the circumferential
+MNRAS 000, 1–8 (0000)
+
+MM -R Diagramm
+25
+WFF1 EoS a=1
+APR EoS a=1
+2D
+SLy EoSa=1
+15
+LD
+0.5
+0.D
+9
+1f
+11
+12
+13
+R (krm)𝑅𝑝 Attractors Static Neutron Star Phenomenology
+5
+Figure 3. The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR
+and SLy EoSs, for 𝛼 = 8.
+radii of the NS in the Jordan and Einstein frames are related as
+𝑅 = 𝐴(𝜑(𝑅𝑠)) 𝑅𝑠. We shall measure the Jordan frame mass in solar
+masses 𝑀⊙ and the Jordan frame radius in kilometers.
+2.1 Results of the Numerical Analysis
+Let us now present the results of our numerical analysis on the NS
+phenomenology of the 𝑅𝑝 attractors. We considered three character-
+istic cases of attractors, corresponding to three values of 𝛼, namely
+𝛼 = 1, 𝛼 = 0.1 and 𝛼 = 8. All these values of 𝛼 produce a vi-
+able inflationary phenomenology as was shown in Ref. Odintsov &
+Oikonomou (2022b). Here we shall present the 𝑀 − 𝑅 graphs for the
+𝑅𝑝 attractors for the three values of 𝛼. Accordingly the results will
+be confronted with three distinct constraints on NS radii for specific
+mass NS. Specifically we shall use the following constraints, devel-
+oped in Refs. Altiparmak, Ecker & Rezzolla (2022), Raaijmakers et
+al. (2021) and Bauswein, et al. (2017) to which we shall refer to as
+CSI, CSII and CSIII respectively. The CSI indicates that the radius of
+an 1.4𝑀⊙ mass NS should be 𝑅1.4𝑀⊙ = 12.42+0.52
+−0.99 and furthermore,
+the radius of an 2𝑀⊙ mass NS should be 𝑅2𝑀⊙ = 12.11+1.11
+−1.23 km. Ac-
+cordingly, CSII indicates that the radius of an 1.4𝑀⊙ mass NS should
+be 𝑅1.4𝑀⊙ = 12.33+0.76
+−0.81 km. Lastly, CSIII indicates that the radius of
+an 1.6𝑀⊙ mass NS should be larger than 𝑅1.6𝑀⊙ = 12.42+0.52
+−0.99 km
+and the radius of a NS with maximum mass should be larger than
+𝑅𝑀𝑚𝑎𝑥 > 10.68+0.15
+−0.04 km. The constraints CSI, CSII and CSIII are
+pictorially represented in Fig. 11. Using a double shooting LSODA
+python 3 numerical integration method Stergioulas (2019), and also
+by setting the numerical infinity at 𝑟 ∼ 67.943 km, at this point we
+shall present our results, which can be seen in the 𝑀 − 𝑅 plots and
+the tables appearing in this work. Note that the numerical infinity
+plays an important role for the double shooting method, in order for
+the scalar field effects to be switched off at the numerical infinity.
+To start with, in Figs. 2, 4 and 3 we present the 𝑀 − 𝑅 graphs of
+the 𝑅𝑝 attractors for 𝛼 = 1, 𝛼 = 0.1 and 𝛼 = 8 NS respectively, for
+1 This media was originally created by the European Southern Observatory
+(ESO). I edited the figure for demonstrative purposes. Their website states:
+”Unless specifically noted, the images, videos, and music distributed on the
+public ESO website, along with the texts of press releases, announcements,
+pictures of the week, blog posts and captions, are licensed under a Creative
+Commons Attribution 4.0 International License, and may on a non-exclusive
+basis be reproduced without fee provided the credit is clear and visible.”
+Figure 4. The 𝑀 − 𝑅 graphs for the 𝑅𝑝 attractor model for the WFF1, APR
+and SLy EoSs, for 𝛼 = 0.1.
+Figure 5. The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve),
+𝛼 = 0.1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for
+the WFF1 EoS.
+the WFF1 EoS (red curve), the APR EoS (green curve) and the SLy
+EoS (blue curve). In all the cases, the maximum masses of the NS
+are larger compared to the GR case. Also it is notable that the 𝛼 = 1
+case is quite similar to the 𝛼 = 8 case, however strong differences are
+observed for the 𝛼 = 0.1 case. Also in Figs. 5, 6 and 7 we present
+for each EoS the 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red
+curves), 𝛼 = 0.1 (green curves), 𝛼 = 8 (blue curves) and the GR
+(magenta curves) for the WFF1 EoS (upper left plot) the SLy EoS
+(upper right) and the APR EoS (bottom plot). Now let us present the
+confrontation of the 𝑅𝑝 attractor NS with the constraints CSI, CSII
+and CSIII.
+The results of our analysis regarding the confrontation of the 𝑅𝑝
+inflationary attractors models with the observational constraints on
+NS, namely CSI, CSII, AND CSIII are presented in Tables 1-5.
+For the case with 𝛼 = 1, the SLy EoS is compatible with all the
+constraints, with regard to the APR, it is not compatible with CSII, the
+first constraint of CSI, but it is compatible with the second constraint
+of CSII and the CSIII constraints. Also the WFF1 case is incompatible
+with all the constraints. For the case with 𝛼 = 0.1, the SLy EoS is
+compatible with all the constraints, and interestingly enough, for this
+case the APR is also compatible with all the constraints. However,
+in this case the WFF1 EoS satisfies the second constraint of CSI and
+also satisfies all the constraints of CSIII. Finally, for the case with
+𝛼 = 1, the SLy EoS is compatible with all the constraints, with regard
+MNRAS 000, 1–8 (0000)
+
+MM -R Diagramm
+25
+WFF1 EoS a=8
+APR EoS a=8
+2D
+SLy EoS a=8
+15
+LD
+0.5
+0.D
+9
+1f
+11
+12
+13
+R (krm)MM -R Diagramm
+25
+*- WFF1 EoS a=0.1
+*- APR EoS a=0.1
+2D
+SLy EoS a=0.1
+15
+LD
+0.5 
+0.0
+9
+1f
+11
+12
+13
+R (krm)MM -R Diagramm
+25
+*-WFF1EoSa=1
+WFF1 EoS a=0.1
+2D
+WFF1 EoSa=8
+*- WFF1EoS GR
+15
+LD
+0.5 
+0.D
+9
+1f
+11
+12
+13
+R (krm)6
+Oikonomou
+Figure 6. The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve),
+𝛼 = 0.1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for
+the SLy EoS .
+Table 3. CSIII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for
+NS Masses 𝑀 ∼ 1.6𝑀⊙
+𝑅𝑝 Attractors Model
+APR
+SLy
+WFF1
+𝛼 = 1
+𝑀 = 1.60 𝑀⊙
+𝑀 = 1.60 𝑀⊙
+𝑀 = 1.61 𝑀⊙
+𝛼 = 1
+𝑅 = 11.30km
+𝑅 = 11.63km
+𝑅 = 10.41km
+𝛼 = 0.1
+𝑀 = 1.61 𝑀⊙
+𝑀 = 1.60 𝑀⊙
+𝑀 = 1.59 𝑀⊙
+𝛼 = 0.1
+𝑅 = 11.61km
+𝑅 = 12.05km
+𝑅 = 11.05km
+𝛼 = 8
+𝑀 = 1.61 𝑀⊙
+𝑀 = 1.60 𝑀⊙
+𝑀 = 1.58 𝑀⊙
+𝛼 = 8
+𝑅 = 11.28km
+𝑅 = 12.05km
+𝑅 = 10.40km
+to the APR, it is not compatible with CSII, and the first constraint of
+CSI, but it is compatible with the second constraint of CSII and the
+CSIII constraints.
+Also the WFF1 case is incompatible with all the constraints, save
+the first constraint of CSIII. Hence, the viable NS phenomenologies
+that pass all the tests imposed by the constraints CSI, CSII and CSIII,
+are provided by all the SLy cases for all the values of the parameter
+𝛼, and also by the APR EoS, only when 𝛼 = 0.1. Thus apparently,
+obtaining a viable NS phenomenology nowadays is not as easy it was
+before the GW170817 event. Also regarding the 𝑅𝑝 attractors, these
+can be discriminated in NS, for different values of 𝛼, especially for
+0.1 < 𝛼 < 1. However, as 𝛼 grows larger than unity, it seems that
+𝑅𝑝 attractors provide an almost identical NS phenomenology. This
+is a notable feature for the class of 𝑅𝑝 attractors. Before closing,
+we need to discuss an important issue, having to do with the NS
+phenomenology of inflationary potentials, with regard to the tidal
+deformability of NSs, the radial perturbations of static NSs and finally
+the overall stability of NSs, by also taking into account the constraints
+imposed by the GW170817 event. This issue however extends further
+from the aims and scopes of this article, since a whole article could
+be devoted to these issues, see for example Refs. Brown (2022) and
+Yang et al. (2022), in which these issues are addressed in the context
+of scalar-tensor gravity Brown (2022) and in unimodular gravity
+Yang et al. (2022).
+Figure 7. The 𝑀 − 𝑅 graphs of the 𝑅𝑝 attractors for 𝛼 = 1 (red curve),
+𝛼 = 0.1 (green curve), 𝛼 = 8 (blue curve) and the GR (magenta curve) for
+the APR EoS.
+CONCLUDING REMARKS
+In this article we studied the NS phenomenology of the 𝑅𝑝 infla-
+tionary attractor scalar-tensor models in the Einstein frame. The 𝑅𝑝
+attractors constitute a class of models in the Einstein frame, which
+originate from a large number of different models in the Jordan frame
+These distinct Jordan frame models result to the same phenomenol-
+ogy in the Einstein frame and this feature justifies the terminology
+inflationary attractors. Our aim was to investigate whether these at-
+tractor models can be distinguished when NSs are considered. As
+we showed the NS phenomenology corresponding to different values
+of the parameter 𝛼 which characterizes the attractors, is in general
+different for 𝛼 < 1, however the models for 𝛼 > 1 show many
+similarities and generate almost identical 𝑀 − 𝑅 diagrams. We also
+confronted the NS phenomenology of the 𝑅𝑝 attractors to several
+NS constraints, which we named CSI, CSII and CSIII. The con-
+straint CSI was developed in Ref. Altiparmak, Ecker & Rezzolla
+(2022) and indicates that the radius of an 1.4𝑀⊙ mass NS has to
+be 𝑅1.4𝑀⊙ = 12.42+0.52
+−0.99 while the radius of an 2𝑀⊙ mass NS has
+to be 𝑅2𝑀⊙ = 12.11+1.11
+−1.23 km. The constraint CSII was developed
+in Ref. Raaijmakers et al. (2021) and indicates that the radius of
+an 1.4𝑀⊙ mass NS has to be 𝑅1.4𝑀⊙ = 12.33+0.76
+−0.81 km and the
+constraint CSIII was developed in Ref. Bauswein, et al. (2017) and
+indicates that the radius of an 1.6𝑀⊙ mass NS has to be larger than
+𝑅1.6𝑀⊙ = 12.42+0.52
+−0.99 km while the radius of the maximum mass NS
+has to be larger than 𝑅𝑀𝑚𝑎𝑥 > 10.68+0.15
+−0.04 km. Our analysis indicated
+that for 𝑅𝑝 attractors, for the case with 𝛼 = 1, only the SLy EoS is
+compatible with all the constraints, while the APR is not compatible
+with CSII, the first constraint of CSI, but it is compatible with the
+second constraint of CSII and the CSIII constraints. Also the WFF1
+case is incompatible with all the constraints.
+For the case with 𝛼 = 0.1, which is the most interesting case phe-
+nomenologically, the SLy EoS is compatible with all the constraints,
+and for this case the APR is also compatible with all the constraints.
+However, in this case the WFF1 EoS satisfies the second constraint
+of CSI and also satisfies all the constraints of CSIII. Finally, for
+the case with 𝛼 = 1, only the SLy EoS is compatible with all the
+constraints while the APR is not compatible with CSII, and the first
+constraint of CSI, but it is compatible with the second constraint of
+CSII and the CSIII constraints. Finally, the WFF1 case is incompat-
+ible with all the constraints, save the first constraint of CSIII. Our
+results indicate two main research lines, firstly that NS phenomenol-
+MNRAS 000, 1–8 (0000)
+
+MM -R Diagramm
+25
+SLy EoS a=1
+SLy EoS a=0.1
+2D
+SLy EoS a=8
+SLy EoS GR
+15
+LD
+0.5
+0.D
+9
+11
+12
+13
+R (kri)MM -R Diagramm
+25
+APR EoS a=1
+APR EoS a=0.1
+2D
+APR EoS a=8
+APR EoS GR
+15
+LD
+0.5
+0.D
+9
+11
+12
+13
+R (kri)𝑅𝑝 Attractors Static Neutron Star Phenomenology
+7
+Table 4. CSII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for
+NS Masses 𝑀 ∼ 1.4𝑀⊙
+𝑅𝑝 Attractors Model
+APR
+SLy
+WFF1
+𝛼 = 1
+𝑀 = 0.52 𝑀⊙
+𝑀 = 1.41 𝑀⊙
+𝑀 = 0.25 𝑀⊙
+𝛼 = 1
+𝑅 = 11.56km
+𝑅 = 11.74km
+𝑅 = 11.89km
+𝛼 = 0.1
+𝑀 = 1.39 𝑀⊙
+𝑀 = 1.39 𝑀⊙
+𝑀 = 0.07 𝑀⊙
+𝛼 = 0.1
+𝑅 = 11.55km
+𝑅 = 12.04km
+𝑅 = 11.79km
+𝛼 = 8
+𝑀 = 0.53 𝑀⊙
+𝑀 = 1.42 𝑀⊙
+𝑀 = 0.25 𝑀⊙
+𝛼 = 8
+𝑅 = 11.60km
+𝑅 = 11.738km
+𝑅 = 11.944km
+Table 5. CSIII vs the 𝑅𝑝 Attractors for the SLy, APR and WFF1 EoSs for
+Maximum NS Masses
+𝑅𝑝 Attractors Model
+APR
+SLy
+WFF1
+𝛼 = 1
+𝑀 = 2.41 𝑀⊙
+𝑀 = 2.24 𝑀⊙
+𝑀 = 2.33 𝑀⊙
+𝛼 = 1
+𝑅 = 9.91km
+𝑅 = 9.99km
+𝑅 = 9.30km
+𝛼 = 0.1
+𝑀 = 2.41 𝑀⊙
+𝑀 = 2.27 𝑀⊙
+𝑀 = 2.32 𝑀⊙
+𝛼 = 0.1
+𝑅 = 10.40km
+𝑅 = 10.09km
+𝑅 = 11.06km
+𝛼 = 8
+𝑀 = 2.41 𝑀⊙
+𝑀 = 2.27 𝑀⊙
+𝑀 = 2.34 𝑀⊙
+𝛼 = 8
+𝑅 = 9.91km
+𝑅 = 10.72km
+𝑅 = 9.28km
+ogy for scalar-tensor theories is not easily rendered viable, since a
+large number of astrophysical and cosmological constraints have to
+be satisfied in order for the viability of the model to be guaranteed.
+Thus a simple parameter assigning is not the correct way to study
+NS nowadays, both cosmology and astrophysics constrain in a rigid
+way NSs. Secondly, several inflationary attractors which are indistin-
+guishable at the cosmological level, may be discriminated to some
+extent when their NS phenomenology is considered. This research
+line is not the general rule though, so work is in progress toward
+comparing a large sample of cosmological attractors with respect to
+their NS phenomenology. Finally, let us note that the scalar-tensor
+inflationary framework we used in this work cannot be considered
+more advantageous compared to other modified gravity theories, it is
+one of the many possible modified gravity descriptions of the nature
+of NSs.
+ACKNOWLEDGMENTS
+This work was supported by MINECO (Spain), project PID2019-
+104397GB-I00 (S.D.O). This work by S.D.O was also partially
+supported by the program Unidad de Excelencia Maria de Maeztu
+CEX2020-001058-M, Spain.
+Data availability. No new data were generated or analysed in
+support of this research.
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2NAyT4oBgHgl3EQf1fm1/content/tmp_files/2301.00737v1.pdf.txt

@@ -0,0 +1,968 @@
+1
+Rotational Abstractions for Verification of Quantum
+Fourier Transform Circuits
+1st Arun Govindankutty Department of Electrical and Computer Engineering
+North Dakota State University
+Fargo, USA
+[email protected]
+2nd Sudarshan K. Srinivasan Department of Electrical and Computer Engineering
+North Dakota State University
+Fargo, USA
+[email protected]
+3rd Nimish Mathure Department of Electrical and Computer Engineering
+North Dakota State University
+Fargo, USA
+[email protected]
+Abstract—With the race to build large-scale quantum com-
+puters and efforts to exploit quantum algorithms for efficient
+problem solving in science and engineering disciplines, the
+requirement to have efficient and scalable verification methods
+are of vital importance. We propose a novel formal verification
+method that is targeted at Quantum Fourier Transform (QFT)
+circuits. QFT is a fundamental quantum algorithm that forms the
+basis of many quantum computing applications. The verification
+method employs abstractions of quantum gates used in QFT that
+leads to a reduction of the verification problem from Hilbert
+space to the quantifier free logic of bit-vectors. Very efficient
+decision procedures are available to reason about bit-vectors.
+Therefore, our method is able to scale up to the verification of
+QFT circuits with 10,000 qubits and 50 million quantum gates,
+providing a meteoric advance in the size of QFT circuits thus
+far verified using formal verification methods.
+Index
+Terms—Formal
+verification,
+Quantum
+algorithms,
+Quantum computing, Quantum Fourier transform, Quantum
+circuit verification.
+1
+I. INTRODUCTION
+The race to build large scale Quantum computers with
+1,000 qubits and beyond is in full steam [1] [2]. The IBM
+Condor quantum computer with 1,000 qubits is expected to
+be released in 2023 [3]. After Condor, IBM plans to use
+chip-to-chip couplers to build even larger quantum computing
+systems [4], with a goal of building a system with 1 million
+qubits [5]. Google’s road map is to built a quantum computer
+with 1 million qubits as well in the near future [6]. There
+are numerous other quantum computers being developed by
+corporations such as Xanadu, Rigetti, IonQ, and D-Wave, to
+name a few. The development of cryogenic control circuits
+needed for quantum computing is also accelerated as demon-
+strated by Intel (Horse Ridge chip) [7], which realizes quantum
+computing and communication applications [8].
+1This paper is a preprint of a paper submitted to IET Quantum Computing.
+If accepted, the copy of record will be available at the IET Digital Library.
+The Quantum Algorithm Zoo website tracks algorithms
+in this domain and currently lists 430 citations of various
+Quantum algorithms [9].
+The 80/20 design rule is well know in computing, i.e.,
+20% of the design cycle time is spend in the actual design,
+while 80% is spent in validation and verification. Without
+verification technologies that can scale, the useful deployment
+of these large-scale quantum systems will be significantly
+hampered. It is imperative therefore to develop verification
+methods for quantum circuits, which is the focus of this
+work. Formal verification has become a standard in the
+semiconductor industry with its ability to provide correctness
+guarantees and flag hard-to-find corner case bugs. There are
+various formal verification methods proposed for quantum
+circuits [10].
+However, for example, the largest Quantum Fourier Trans-
+form (QFT) circuit verified as reported in literature has only
+31 qubits [11]. Scalable verification methods are thus the need
+of the hour.
+Contributions: One of the approaches to achieve scalability
+in formal verification is to develop domain-specific methods.
+In this work, we target one of the fundamental quantum
+algorithms, the Quantum Fourier Transform (QFT). In com-
+puting and engineering, transformations play a vital role in
+problem solving and analysis. Quantum computing uses QFT
+to tackle various problems. QFT is an integral part of numer-
+ous quantum algorithms including Shor’s factoring algorithm,
+quantum phase estimation algorithm, and computing discrete
+logarithm to name a few [12] [13]. The real-world applications
+where QFT is employed include portfolio optimization in
+computational finance [14], Monte Carlo pricing of financial
+derivatives [15], quantum meteorology for building interferom-
+eters [16], materials examination and analysis [17], analysis of
+image data [18] in medical applications, and risk analysis [19]
+among others.
+We have developed a formal verification method that can be
+arXiv:2301.00737v1  [quant-ph]  2 Jan 2023
+
+2
+used to efficiently verify Quantum Fourier Transform (QFT)
+circuits for up to 10,000 qubits and 50 million gates. Our
+specific contributions are as follows:
+1) Abstractions of the Hadamard (H) gate and the control
+rotation gate (Rn) that exploits the rotational impact of
+these gates on the incoming qubit.
+2) A correctness framework that exploits these abstractions
+and allows the verification problem to be reduced from
+Hilbert space (complex vector space) to the quantifier
+free logic of bit-vectors (QF BV).
+3) Theorems with proofs to show that the abstractions are
+sound, i.e., if the abstract QFT circuit is verified to be
+correct, then the correctness of the QFT circuit under
+verification is guaranteed
+While we have developed our approach with QFT as the
+target, the key ideas used in the abstractions can be applied
+to a much larger class of quantum circuits, which is what we
+plan to do for future work.
+The rest of the paper is organized as follows. Section II
+covers background on quantum circuits and QFT circuits.
+Section III overviews the related work on formal methods
+for verification of quantum circuits. Section IV describes
+the key contributions of the proposed work, including the
+gate abstractions and the correctness framework. Section V
+addresses the correctness of the abstractions and the overall
+approach. Experimental results are provided and discussed
+in Section VI. Conclusions and future work are outlined in
+section VII.
+II. BACKGROUND
+In this section, we review background on qubits, quantum
+gates, and QFT circuits. A detailed description of these topics
+can be found in [12]. Information in the quantum computing
+domain is represented by qubits. A qubit is the basic unit
+of information analogous to a bit in classical computing. In
+general, qubits are represented by a linear combination of
+ortho-normal (orthogonal and normalized) vectors |0⟩ and |1⟩.
+The vectors are linearly independent i.e., we cannot express
+one as the linear combination of the other. The independent
+vectors are shown below.
+|0⟩ =
+�1
+0
+�
+, and |1⟩ =
+�0
+1
+�
+The above ortho-normal vectors can be used to represent any
+vectors in the vector space by using vector addition and scaling
+(linear combination), and thus they are called the basis vectors.
+A standard representation of a qubit |q⟩ is shown below where,
+α and β are complex numbers such that α2 + β2 = 1.
+|q⟩ = α|0⟩ + β|1⟩
+Quantum gates are unitary operators that act on qubits and
+produce a required output. A quantum algorithm is a step by
+step process that utilizes quantum mechanical properties to
+solve a particular problem. Quantum algorithms are run on
+computation models for quantum computing and this work is
+based on the quantum circuit model, which is the most widely
+used method [20].
+QFT is analogous to the Discrete Fourier Transform (DFT)
+in the classical domain and efficiently performs the quantum
+mechanical model’s Fourier transform. The QFT operates on
+the input qubit states (ortho-normal basis vectors |0⟩, ....., |N−
+1⟩) and transforms them to the corresponding output states.
+The transformation is shown below [12].
+|j⟩ →
+1
+√
+N
+N−1
+�
+k=0
+e2πijk/N|k⟩
+In the above, |j⟩, N, i, and k represents the input qubit,
+the number of QFT points, imaginary number (√−1), and the
+iteration variable, respectively. Here N = 2n, where n is the
+number of qubits in the QFT.
+In the transformed domain, this resultant state (transformed
+|j⟩) can be represented as a sum of individual components
+whose frequencies are integer multiples of
+2π
+N . The same
+equation can be re-organized to obtain the equivalent trans-
+formation happening in each qubit independently, which we
+exploit in this work.
+Implementation of QFT as a circuit can be achieved by a
+series of cascaded Hadamard (H) gates and controlled rotation
+(Rn) gates. The H gates and Rn gates are defined below.
+H =
+1
+√
+2
+�1
+1
+1
+eπi
+�
+=
+1
+√
+2
+�1
+1
+1
+−1
+�
+Rn =
+�1
+0
+0
+e2πi/2n
+�
+The H gate introduces equal superposition of the input basis
+vectors for the qubit. The Rn gates are responsible for the
+frequency harmonics. QFT circuits are constructed by first
+applying the H gate to all qubits. Qubit 1 of a QFT circuit
+with m qubits should have gates R2, ..., Rm acting on it, with
+control inputs qubit 2, ..., qubit m taken before the H gate is
+applied to the control qubits, respectively. Qubit 2 should have
+gates R2, ..., Rm−1 acting on it with control inputs qubit 3, ...,
+qubit m taken before the H gate is applied, respectively, and
+so on. Figure 1(a) shows the transformations happening while
+QFT is performed on a 3 qubit system.
+III. RELATED WORK
+Formal verification of quantum algorithms and circuits has
+been an active area of research. In this section, we overview
+these related works and how they contrast with our approach.
+The main takeaway is that the approaches have not demon-
+strated the efficiency and scalability that we have been able to
+achieve. In this sense, our approach is a meteoric advance in
+the size of quantum circuits thus far verified.
+Yamashita and Markov [22] have proposed an equivalence
+checking approach for quantum circuits. In equivalence check-
+ing, the circuit to be verified is compared with a reference
+
+3
+Fig. 1. (a) 3-qubit QFT circuit [21]. (b) Abstract Hadamard gate. (c) Abstract rotation gate. (d) 3-qubit QFT abstract circuit representation.
+circuit. There are two prominent contrasts with our approach.
+The first contrast is related to equivalence checking in general,
+where a golden (already verified, trusted) circuit is required as
+the reference circuit. For example, to verify a QFT circuit with
+10,000 qubits and 50 million gates, a trusted QFT circuit of
+the same size is required. Therefore, to enable equivalence
+checking, methods that can verify functional correctness of
+a given circuit is mandatory. This is the gap that we address.
+Equivalence checking is useful in synthesis optimizations. Our
+approach is property based and does not require a reference
+circuit of the same size for verification. If a QFT circuit
+with 10,000 qubits and 50 million gates satisfies our proposed
+correctness property, it is guaranteed to be correct. The second
+contrast is that if they are not able to reduce the problem to
+a boolean space, then a hybrid approach is used [23], where
+the verification problem is solved in the Hilbert space. We use
+rotational abstractions to reduce the problem fully to a Boolean
+space, solvers for which are orders of magnitude more efficient
+and scalable. We also exploit the fact that our approach is
+domain-specific to QFT circuit verification to enable this. The
+largest circuits they verified have 5,000 gates and requires
+about 59 seconds. In contrast, we are able to verify circuits
+with 8,000 gates in 0.04 seconds, 5 million gates in about 60
+seconds, and 50 million gates in 2,380 seconds.
+Amy [11] use complex path-sums to model quantum gates
+for verification. They perform reductions on the resulting
+circuit, which are implemented using rewrite rules. The re-
+ductions are performed using the Haskell theorem prover.
+The rewrite rules are guaranteed to reduce the circuit to
+a normal form, which is then used to check correctness.
+They verify a 16-qubit and a 31-qubit QFT, which required
+1.250 seconds and 16.020 seconds for circuits without errors,
+respectively. In contrast, our approach required 0.02 seconds
+and 0.03 seconds for 16-qubit and 32-qubit QFT circuits,
+respectively. They employ a dyadic arithmetic technique, the
+current implementation of which causes an integer overflow
+for QFT circuits larger than 31 qubits. Therefore, with this
+current implementation, they are unable to handle QFT circuits
+larger than 31 qubits. We are able to handle upto 10,000 qubits.
+Liu et al. [24] formalize quantum hoare logic in the Is-
+abelle/HOL theorem prover and use it to prove the correctness
+of Grover’s search algorithm for infinite size input. They report
+that the proof required 5 person months of effort. They do not
+describe how this proof can be used to verify a given quantum
+circuit that implements Grover’s algorithm. In contrast, our
+approach is fully automated for verification of any QFT circuit.
+They have not addressed QFT verification.
+Feng et al. [25] have developed a model checking algorithm
+that can check the Quantum CTL (QCTL) properties on
+quantum Markov chains. The method is used to check the
+correctness of the BB84 protocol when n=1, the corresponding
+circuit for which has 8 qubits and 24 quantum gates. They have
+not addressed QFT verification either.
+IV. ROTATIONAL ABSTRACTIONS
+There are three key ideas in developing the abstractions for
+the Hadamard (H) gate and the controlled rotation (Rn) gate.
+The first key idea is with regard to the basis vectors. If
+a QFT circuit works correctly when the input qubits are the
+basis vectors |0⟩ or |1⟩, then the circuit is guaranteed to work
+correctly for any qubit inputs [26]. Therefore, for verification
+purposes, we only consider the cases where the input qubits
+are |0⟩ or |1⟩.
+The second key idea is with regard to quantum gates and
+is as follows. If the input qubits are limited to basis vectors,
+then both the H gate and the Rn gate can be modelled as gates
+causing rotation on the basis vectors. The H gate has only one
+input. We call this the control input qc as shown in Figure 1(b),
+
+H
+R2
+R3
+H
+R2
+H
+HA
+HA
+R2A
+ReA
+HA
+R2A
+R.A
+HA4
+because if the input is |1⟩, then the H gate function can be
+represented as a rotation on |1⟩. If this control input is |0⟩, then
+no rotation is performed. The Rn gate has two inputs (control
+and data) and one output, we call the control input qc, the data
+input qd, and the output qo (as shown in Figure 1(c)). If qc is
+|1⟩, then Rn performs a rotation on qd. Otherwise, if qc is |0⟩,
+then no rotation is performed.
+The third key idea is with regard to the amount of rotation
+performed by the quantum gates on data input qubits and the
+resulting output qubit states, and is as follows. The H gate
+induces a π (2π/2) rotation on |1⟩ and does not rotate |0⟩.
+The Rn gate induces a 2π/2n (π/2n−1 ) rotation on |1⟩ and
+does not rotate |0⟩. For examle, R4 induces a rotation of π/8.
+Thus, the rotation performed by the gates on |1⟩ are negative
+powers of 2 with reference to 2π .
+The QFT circuit structure is such that the control inputs to
+the quantum gates are always initial qubit states and are used
+only to make the decision, whether to rotate or not.
+Thus, we can abstract the basis vector input values |0⟩ and
+|1⟩ using Boolean values 0 and 1.
+The qubits once transformed by these rotations are input
+to the next quantum gate and finally the output state of the
+circuit.
+If the 2πi term is factored out of the exponent, the final
+output state of each qubit (after transformation) can be ab-
+stractly represented using fractional bit-vectors that essentially
+capture the amount of rotation on |1⟩. The fractional bit-vector
+⟨.b1b2b3⟩ corresponds to rotation value 2π ∗ (b1 ∗ 2−1 + b2 ∗
+2−2+b3∗2−3). For example, the bit-vector ⟨.101⟩ corresponds
+to rotation value of 2π(1/2+0+1/8). Abstractions of the H gate
+and the Rn gate can be obtained by defining their rotational
+impact on the fractional bit-vectors, and an abstracted QFT
+circuit can be obtained by using these abstracted gates. In a
+QFT circuit with m qubits, the smallest amount of rotation
+will be 2π/2m. Therefore, the fractional bit-vectors used to
+represent qubits in the abstracted QFT circuit will have to
+have m bits.
+The abstract H gate is defined below and has one input qubit
+qc, which is Boolean type. Output qubit qo is a bit-vector of
+size equal to m, the number of qubits.
+Definition 1. (Abstract Hadamard Gate) If
+qc=1, then
+qo
+← ⟨.100...0⟩m,
+else
+qo ← ⟨.000...0⟩m.
+The abstract Rn gate is defined below and has two qubit
+inputs qc and qd. The control input qc is type Boolean, the
+data input qd and the output qubit qo are both fractional bit-
+vectors of size m, the number of qubits.
+Definition 2.
+(Abstract Rn Gate) If
+qc=1,
+then qo ←
+qd +m ⟨.00..01m−n0...0⟩m, else
+qo ← qd.
+In the above, +m represents fixed-point modulo addition
+w.r.t m. The abstracted QFT circuit is obtained by replacing
+the H gates and Rn gates of the original circuit with the
+abstracted gates. Input qubits are declared as type Boolean
+and all other qubits are declared as type bit-vector of size m.
+The abstracted QFT circuit with 3 qubits is shown in Figure
+1(d). When the abstract H gate is applied, the qubits at the
+output of the H gates of the QFT circuit in Figure 1(d) will
+have the following values:
+q1
+1 ← ⟨.b100⟩
+q1
+2 ← ⟨.b200⟩
+q1
+3 ← ⟨.b300⟩
+The QFT correctness property is given next. Let QFT-
+Absi(b1, b2, ..., bm) denote the output state of the ith qubit
+of an abstracted version of a QFT circuit, where m is the
+number of qubits and b1, b2, ..., bm are Boolean variables.
+Property 1.
+(QFT Correctness Property) A QFT circuit is
+functionally correct if, for all 1 ≤ i ≤ m, i is an integer,
+QFT-Absi(b1, b2, ..., bm) = ⟨.bibi+1...bm0...0⟩m.
+Based on the correctness property above, for the QFT
+circuit from Figure 1(a) to be correct, the state of qubits at
+the output should be as follows:
+q3
+1 = ⟨.b1b2b3⟩
+q3
+2 = ⟨.b2b30⟩
+q3
+3 = ⟨.b300⟩
+The abstracted gates, abstracted QFT circuit, and Property
+1 are expressible in the Quantifier Free logic of Bit Vectors
+(QF BV). A number of SMT solvers exist that can very
+efficiently check properties in this logic. Therefore, verification
+of a given QFT circuit can be performed by encoding the
+abstracted circuit and correctness property in this logic (using
+the SMT LIB language). An SMT solver will check the
+property automatically and indicate if the property is satisfied
+or not. If the property is satisfied, then the QFT circuit is
+guaranteed to be correct (as will be established in the next
+section). If the property is not satisfied, the tool will generate
+a counter example, which can be used to trace the error(s) in
+the circuit.
+V. ABSTRACTION CORRECTNESS
+Fig. 2. QFT circuit showing error scenarios.
+In this section, we provide a proof of correctness of our ver-
+ification approach. The overall approach is that we enumerate
+through all possible classes of errors in QFT circuits and show
+how the verification approach will flag each error class. The
+error classes are depicted in Figure 2. We call bit-vector values
+as data values as well.
+
+H
+R3
+R3
+R2
+R2
+H5
+TABLE I
+VERIFICATION RESULTS
+QFT Benchmark
+Correct Circuit
+Incorrect Gate Error
+Incorrect Control Error
+No Error
+Error Depth
+Error Depth
+Gate-2
+Gate-n
+Gate-2
+Gate-n
+Verification Stats.
+Verification Stats.
+Verification Stats.
+Verification Stats.
+Verification Stats.
+Qubits(n)
+Gates
+M(MB)
+T(s)
+M(MB)
+T(s)
+M(MB)
+T(s)
+M(MB)
+T(s)
+M(MB)
+T(s)
+16
+136
+19.0
+0.02
+27.2
+0.04
+27.3
+0.02
+19.0
+0.01
+27.2
+0.02
+32
+528
+19.0
+0.03
+27.2
+0.02
+27.5
+0.02
+19.0
+0.02
+19.0
+0.01
+64
+2,080
+19.0
+0.03
+27.3
+0.03
+27.6
+0.02
+19.1
+0.02
+19.1
+0.03
+128
+8,256
+19.3
+0.04
+27.4
+0.07
+27.9
+0.04
+19.3
+0.03
+19.3
+0.02
+256
+32,896
+20.1
+0.19
+27.7
+0.06
+28.6
+0.06
+20.0
+0.08
+20.0
+0.08
+512
+131,328
+22.1
+0.26
+28.3
+0.29
+29.8
+0.2
+22.2
+0.29
+22.2
+0.2
+1,024
+524,800
+29.1
+1.37
+29.5
+0.77
+32.2
+0.92
+29.5
+1.46
+29.5
+1.29
+2,048
+2,098,176
+56.1
+9.85
+56.9
+5.52
+73.7
+5.87
+56.9
+9.66
+56.9
+9.47
+4,096
+8,390,656
+169.3
+95.75
+169.6
+51.78
+203.3
+53.57
+169.6
+73.65
+169.6
+79.68
+8,192
+33,558,528
+592.1
+1,109.0
+593.6
+640.53
+596.2
+643.9
+593.6
+641.03
+593.6
+639.57
+10,000
+50,005,000
+888.7
+2,379.88
+890.7
+1,523.99
+894.5
+1,568.79
+890.6
+1,571.29
+890.6
+1,524.65
+Lemma 1. If a QFT circuit has an error, where an incorrect
+input is fed to an H gate, verification of the abstracted version
+of the QFT circuit will either generate a type error or will not
+satisfy Property 1.
+If the input to the abstract H gate is a bit-vector input, this
+will be flagged as a type error as the H gate expects a Boolean
+input. If Boolean input qubit bj is expected whereas bk is fed
+for qubit qj, then the LHS of Property 1 for qj will be ⟨.bk...⟩
+and RHS will be ⟨.bj...⟩. Therefore, Property 1 will not be
+satisfied.
+Lemma 2. If a QFT circuit has an error, where an incorrect
+input is fed to an Rn gate, verification of the abstracted version
+of the QFT circuit will either generate a type error or will not
+satisfy Property 1.
+If a control value is fed to the data input of an Rn gate
+or if a data value is fed to the control input of an Rn gate,
+a type error will be generated. If bj is expected whereas bk
+is fed for the control input of an Rn gate acting on qubit qj,
+then the LHS of Property 1 for qj will be ⟨....bk...⟩ and RHS
+will be ⟨....bj...⟩. Therefore, Property 1 will not be satisfied.
+If an incorrect data value is fed to an Rn gate, this will result
+in a missing Rn gate on a qubit and this case is dealt with
+subsequently.
+The error above is shown in Figure 2. R3 gate with input q2
+1
+should have b3 as its control input. Instead b2 is erroneously
+fed as the control input.
+Lemma 3. If a QFT circuit has an error, where an H gate
+is missing on a qubit or there is more than one H gate acting
+on a qubit, verification of the abstracted version of the QFT
+circuit will generate a type error.
+In the abstracted version of a QFT circuit, the input of an
+H gate is a control value and the output is a data value. Thus,
+if there is more than one H gate acting on a qubit, the H gates
+after the first one will receive data inputs and this will result
+in a type error. If there are no H gates acting on a qubit, the
+subsequent Rn gates will not get a data value at its data input
+and this will again result in a type error.
+An example of a missing H gate error is shown in Figure
+2. The H gate on q2 is missing.
+Lemma 4. If a QFT circuit has an error where an incorrect
+set of Rn gates are acting on a qubit, i.e., required Rn gates are
+missing or additional Rn gates are present or both, verification
+of the abstract version of the QFT circuit will not satisfy
+Property 1.
+Qubit 1 of a QFT circuit with m qubits should have gates
+R2, ..., Rm acting on it. Qubit 2 should have gates R2, ...,
+Rm−1 acting on it and so on. Thus, there is only one Rn gate
+of a certain n value required to act on each qubit. If a required
+Rn gate is missing, then its rotational impact on the fractional
+bit-vector value abstracting the qubit will not be observed in
+Property 1. If a qubit has additional erroneous Rn gates acting
+on it, then the required rotation of the qubit will be incorrect
+and this will be reflected on the final fractional bit-vector value
+of the qubit. In both the above cases, Property 1 will not be
+satisfied. Note that an Rn gate can be replaced with two Rn−1
+gates, with the same control inputs. For example, R2 can be
+substituted with two R3 gates. If the total rotational impact of
+a sequence of Rn gates is what is expected, even though it
+does not conform with the Rn gate sequence described above,
+Property 1 will be satisfied because the fractional bit-vector
+abstraction accurately captures the rotations.
+An example of an incorrect Rn gate is shown in Figure 2,
+where the gate on q1
+1 should be R2 instead of R3.
+Lemma 5. If a QFT circuit has a combination of errors from
+the error classes described in Lemmas 1-4, verification of the
+abstracted version of the QFT circuit will generate a type error
+or will not satisfy Property 1.
+As can be seen from Lemmas 1-4, the effect that flags each
+error class is disjoint, i.e., there is no overlap in these effects
+for type errors or Property 1. Thus a combination of errors
+will also be flagged as a type error or will not satisfy Property
+1.
+Theorem 1.
+(QFT-Rotational Abstraction Correctness) If a
+QFT circuit has an error, verification of the abstracted version
+of the QFT circuit will generate a type error or will not satisfy
+Property 1.
+A QFT circuit has only two types of gates, the H gate and
+the Rn gate. Based on this, there are only four classes of
+
+6
+errors possible: Incorrect input to a H gate, incorrect input to
+an Rn gate, missing or additional H gates in the circuit, and
+incorrect set of Rn gates acting on a qubit. The fifth case of an
+erroneous QFT circuit is any combination of the above. From
+Lemmas 1-5, we see that in all the above cases, verification of
+the abstracted version of the QFT circuit will generate a type
+error or will not satisfy Property 1.
+VI. RESULTS AND DISCUSSIONS
+Table I gives the verification results. The verification bench-
+marks were generated by varying the number of qubits in
+the QFT circuit from 16 qubits to 10,000 qubits. The table
+gives the number of quantum gates in each of the QFT
+benchmark circuits as well (column 2: Gates). The verification
+experiments were conducted on an Intel(R) Core(TM) i9 -
+12900K CPU @ 3.2 GHz with 32 GB RAM and Ubuntu 64-
+bit operating system. The z3 version 4.8.12 SMT solver [27]
+was used to check Property 1 for all benchmarks.
+In the table, ”T(s)” indicates verification time in seconds,
+which is the z3 run time. ”M(MB)” gives the z3 run time
+memory consumption in megabytes. ”Correct Circuit” gives
+the verification statistics for the QFT circuits without errors.
+For these circuits Property 1 is proved to be satisfied. Property
+1 allows for each qubit output to be verified independently.
+Therefore, the verification of all the qubit output in the circuit
+were done in parallel and the memory and time reported
+correspond to the worst case.
+”Incorrect Gate Error” are circuits with gates errors and is
+described as follows. The Gate-2 error here indicates that the
+R3 gate is incorrectly acting on qubit 1 instead of R2. The
+Gate-n error here indicates that the Rn−1 gate is incorrectly
+acting on qubit 1 instead of Rn. ”Incorrect Control Error”
+are circuits with incorrect control input to an Rn gate. The
+Gate-2 error here indicates that the R2 gate in qubit 1 is
+incorrectly controlled by qubit 3 instead of qubit 2. The Gate-
+n error here indicates that the Rn gate in qubit 1 is incorrectly
+controlled by qubit n-1 instead of qubit n. For the circuits with
+errors, verification of Property 1 generates a counterexample.
+The time and memory reported corresponds to the verification
+of the first qubit output that caused a counterexample to be
+generated.
+Figures 3 and 4 plot the verification time and memory from
+Table I versus the number of quantum gates, respectively. In
+these graphs, both the x-axis and y-axis use a log scale. As
+can be seen from these graphs, with increase in the number
+of gates, both memory and verification time increase linearly
+for both correct circuits and circuits with errors. The most
+complex circuit with 10,000 qubits and 50 million gates is
+verified in only about 37 minutes. This demonstrates the high
+efficiency and scalability of our approach. The time taken to
+verify circuits with errors is less than that of correct circuits.
+However, there is not an order-of-magnitude reduction that is
+often observed in formal verification.
+Figure 5 shows both verification time and memory as the
+position of the gate error is moved from qubit 1 to qubit
+10,000 on the QFT circuit with 10,000 qubits. The x-scale
+increases linearly, whereas the y-scale is logarithmic. The
+graph indicates the variation of time and memory with the
+vertical location of errors. We see that as the error moves
+from qubit 1 to qubit 10,000, both time and memory reduce
+exponentially.
+Fig. 3.
+Execution time requirement capture for QFT verification versus
+quantum gate count. Correct circuit, control input error and value error at
+qubit positions 2 and 10000 captured for elucidation.
+Fig. 4. Execution memory requirement capture for QFT verification versus
+quantum gate count. Correct circuit, control input error and value error at
+qubit positions 2 and 10000 captured for elucidation.
+VII. CONCLUSIONS AND FUTURE WORK
+Our proposed approach for verification of Quantum Fourier
+Transform (QFT) circuits achieves a meteoric advance in the
+efficiency and scalability of quantum circuits thus far verified.
+We have been able to verify a QFT circuit with 10,000
+qubits and over 50 million gates in only about 37 minutes.
+We exploit the fact that our approach is domain specific
+to QFT verification. This is a common theme to achieve
+scalability in formal verification. For example, there are a
+large number of formal verification techniques dedicated to
+the verification of multipliers. We also exploit the idea that
+the rotations performed by the gates are negative powers of 2
+
+Execution time capture
+103
+Correct Circuit
+Incorrect Control Error at gate 2
+101.
+Incorrect Control Erro at gate n
+10-1
+102
+103
+104
+105
+106
+107
+Correct Circuit
+102
+Incorrect Gate Error at gate 2
+Incorrect Gate Error at gate n
+100
+102
+103
+104
+105
+106
+107
+Number of quantum gates in QFT circuitExecution memory capture
+103
+Correct Circuit
+Incorrect Control Error at gate 2
+Incorrect Control Erro at gate n
+102
+102
+103
+104
+105
+106
+107
+103
+Correct Circuit
+Incorrect Gate Error at gate 2
+Incorrect Gate Error at gate n
+102
+102
+103
+104
+105
+106
+107
+Number of quantum gates in QFT circuit7
+Fig. 5.
+Resource utilization (time and memory) capture versus qubit count
+for erroneous QFT circuit.
+and can therefore be encoded as fractional bit-vectors, thus
+reducing the verification obligations from Hilbert space to
+Boolean space. For future work, our goal is to extend these
+ideas to other quantum algorithms to advance efficiency and
+scalability of formal verification so as to cope with the size
+and complexity of quantum hardware roadmaps of the near
+future.
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+ 
+Accuracy and Fidelity Comparison of Luna and 
+DALL-E 2 Diffusion-Based Image Generation 
+Systems 
+Michael Cahyadi  
+School of Computer Science  
+Bina Nusantara University 
+Jakarta, Indonesia 
+[email protected] 
+Muhammad Rafi 
+School of Computer Science  
+Bina Nusantara University 
+Jakarta, Indonesia 
+[email protected] 
+ 
+ 
+ 
+ 
+ 
+William Shan 
+School of Computer Science  
+Bina Nusantara University 
+Jakarta, Indonesia 
+[email protected] 
+ 
+ 
+Abstract — We qualitatively examine the accuracy and 
+fideltiy between two diffusion-based image generation systems, 
+namely DALL-E 2 and Luna, which have massive differences in 
+training datasets, algorithmic approaches, prompt resolvement, 
+and output upscaling. In our research we conclude that DALL-
+E 2 significantly edges Luna in both alignment and fidelity 
+comparisons 
+I. 
+INTRODUCTION 
+Image generation systems is one of the many avenues 
+artificial intelligence research projects have been pursuing 
+ways to improve generative methods. Image generation 
+systems have immense potential to compliment and extend 
+human creativity[1], but on the other hand there are issues 
+with the field such as potential abuse to spread misinformation 
+and harrassment[2], bias against certain cultural groups[3], 
+and harmful associations against marginalized societies [4]. 
+The field of image generation using artificial intelligence 
+technologies has made great advancements in the last few 
+years, with recent models capable of generating images with 
+near human-like characteristics. Variations of Generative 
+Adversarial Networks (GAN)[5] were among the first models 
+to generate high-quality images, but recently there has been 
+more focus by researchers and the public[6] on diffusion-
+based models trained using massive datasets. 
+The open nature of research information regarding 
+diffusion-based image generation models have also led to an 
+increase of image generation systems made by individuals, 
+which may not have sufficient guardrails against abuse[7]. 
+Image generation systems also use CLIP latents to 
+associate certain human concepts and understand them in 
+creative contexts[8]. With artificial intelligence systems 
+continuing to get better in non-analytical areas such as artistic 
+creativity that mimic closely human cognitive architectures, 
+researchers might soon get closer into the realm of General 
+Artificial Intelligence (GAI)[9]. 
+As artificial intelligence becomes a more pervaise tool in 
+day-to-day workflows, there needs to be an evaluation 
+regarding the quality of outputs generated by image 
+generation systems. Accurately judging the alignment and 
+perceived fidelity of generated outputs from these image 
+generation systems can help researchers and developers build 
+better systems that are aware of the pitfalls of current systems 
+in the market. 
+While there exists many diffusion-based image generation 
+systems, both open and closed sourced, we decided to search 
+for two systems who adopt resolvement approaches that lead 
+in their industry in terms of accuracy and widescale 
+implementation in the image generation technology 
+community. Latent diffusion models and CLIP-guided[10] 
+diffusion models represent the forefront methodologies for 
+image-generation technologies with alignment and fidelity 
+results surpassing previous GAN-based systems. 
+This paper also ultimately aims to examine the difference 
+in accuracy between images generated by diffusion-based 
+systems that are made by a large company using a large 
+training data set and a system created by an individual with 
+more limited training resources and less guardrails towards 
+abuse. To that effect, we consider the following two image 
+generation models for comparing our results: 
+1. 
+DALL-E 2[8]  is an image generation system created 
+by OpenAI which can generate high-resolution images that 
+combine various concepts and art styles. The project was built 
+in PyTorch using ViT-H/16 text encounters with the training 
+data of 650M images scraped from the internet and aligned by 
+CLIP[10].  
+2. 
+Luna is an image generation system built by Arfy 
+Slowly, a Senior Software Engineer at Google Research. The 
+project was built in Tensorflow and published on GitHub as 
+an open-source project. The system uses a latent diffusion 
+model[11] to condition the model on text prompts, however 
+the training dataset used is unknown.  
+ 
+II. 
+RELATED WORKS 
+There have been many papers that try to compare the 
+performance of two image generation systems, whether 
+quantitatively or qualitatively. The metrics that are used to 
+verify the accuracy of image generation systems mostly rely 
+on image fidelity and its benchmark against real-world 
+equivalents. 
+While quantitative methods have been laid out to gauge 
+the accuracy of image generation systems such as the Fréchet 
+Inception Distance by Heusel et al.[12], the metric is used to 
+compare GAN performance at image generation using real-
+life samples, which differs from our attempts to gauge the 
+Henry Lucky 
+School of Computer Science  
+Bina Nusantara University 
+Jakarta, Indonesia 
+[email protected] 
+Jurike Moniaga 
+School of Computer Science  
+Bina Nusantara University 
+Jakarta, Indonesia 
+[email protected] 
+
+accuracy of image generation systems at prompt resolving 
+novel concepts that have little to no real-life examples. 
+Existing evaluations of diffusion-based image generation 
+models are mostly based on the accuracy of the image 
+generated in specific fields such as in artificially generated 
+faces[13]. But no paper has qualitatively evaluated the 
+inherent accuracy between the prompt given to the model to 
+the generated image. 
+Qualititative methods of performance analysis are usually 
+done by human surveyors such as in research by Saharia et 
+al.[14]. This is due to the subjective nature of art[15], unlike 
+measurable things such as image fidelity, that cannot be 
+measured with common metric calculations.  
+In Saharia et al., the method used in evaluating image 
+accuracy consists of 2 questions given to human raters 
+inquriing about the fidelity and alignment (the accuracy 
+between human interpretation of a given concept and the 
+output given by an artificial intelligence[16]) of the system’s 
+output. 
+III. 
+METHODOLOGY 
+A. Prompt Creation 
+The prompts list below are modified prompts from 
+Google’s Drawbench Benchmark[14], whic covers a variety 
+of concepts, art styles, and common pitfalls of image 
+generation systems to generate images that can be a point of 
+evaluation for the alignment and fidelity of the image 
+generation systems. We have also listed the reasoning towards 
+why we choose each prompt. 
+The prompts detailed above have been screened by 
+running them through a Google image search and seeing how 
+easily images for these concepts could be retrieved; from this 
+process, we eliminated two prompts and modified another. 
+Number 
+Contents 
+Prompt 
+Explanation 
+1 
+A 
+photorealistic 
+image of a machine 
+resembling 
+a 
+human being and 
+able to replicate 
+certain 
+human 
+movements 
+and 
+functions 
+automatically. 
+Prompt is used to evaluate 
+the ability of the image 
+generation system to build 
+photorealistic images that 
+don’t cross the uncanny 
+valley[17]. 
+2 
+ 
+A half-robot and 
+man 
+entity 
+with 
+chainsaws for their 
+head and hands in 
+the 
+style 
+of 
+Japanese anime. 
+ 
+Prompt is used to evaluate 
+the 
+bias 
+in 
+machine 
+learning algorithms that are 
+trained with data from 
+westernized-culture[18].  
+3 
+Rbefraigerator. 
+The prompt is used as a 
+way to determine the image 
+generation system’s ability 
+to 
+navigate 
+word 
+misspellings[19]. 
+4 
+A car on top of a 
+spoon. 
+The prompt is used to 
+examine the ability of 
+image generation systems 
+to generate images novel in 
+concept[19]. 
+Number 
+Contents 
+Prompt 
+Explanation 
+5 
+Two bicycles and 
+one 
+car 
+on 
+an 
+empty grass field. 
+The prompt is used to 
+examine the ability of 
+image generation systems 
+to generate images with 
+accurate 
+positional 
+information[19]. 
+6 
+In late afternoon in 
+January in Jakarta, 
+a man stands in the 
+shadow of a tree. 
+The prompt is used to 
+examine the ability of 
+image generation systems 
+to 
+accurately 
+create 
+shadows that correspond 
+with 
+differing 
+lighting 
+conditions[19] 
+7 
+A Sumatran tiger 
+under the sea.  
+The prompt is used to 
+examine the ability of the 
+image generation system to 
+generate images that have 
+conflicting concepts[19]. 
+8 
+Art 
+nouveau 
+stained 
+glass 
+window 
+art 
+depicting 
+Woody 
+from Toy Story. 
+The prompt is used to 
+examine the ability of the 
+image generation system to 
+generate images with pop 
+culture 
+products 
+in 
+a 
+medium 
+not 
+usually 
+associated 
+with 
+the 
+product[20]. 
+ 
+B. Image Generation 
+Below each set of 4 images per-prompt on each model, we 
+will outline general observations from the researchers as with 
+the cited reasons of why each model behave in such a way. 
+The image results are compiled for further analysis in the 
+paper in methodologies outlined in later subsections. 
+The researchers will generate every 32 images from each 
+system, noting that DALL-E 2 generates four images per run. 
+For DALL-E 2, access was provided to the system during 
+September 2022 after a request for beta-testing research was 
+approved by the company. DALL-E 2 was accessed from the 
+OpenAI Beta website, with 8 credits dispensed every month 
+for non-commercial research use only. DALL-E 2 outputs 4 
+photos in one-prompt execution. DALL-E 2 outputs 
+1024x1024 pixel images and Luna outputs 512x512 pixel 
+images.  
+For Luna, we use the provided Colab notebook by Google 
+to run the system. Considerations we’re made to run Luna 
+using on-premises hardware, however due to Tensorflow’s 
+requirement of NVIDIA CUDA cores we decided to opt for 
+cloud solutions instead due to faster compute times and as a 
+better benchmark against DALL-E 2 which is a cloud-based 
+system hosted in Microsoft Azure. Luna outputs 4 photos in 
+one-prompt execution. 
+C. Analysis 
+As laid out in the related works section, due to art being 
+inherently subjective in nature[15], normal metrics cannot be 
+
+applied when analyzing the inherent accuracy of prompt 
+creations from image generation systems.  
+The methodologies to evaluate these images are based on 
+Drawbench by Saharia et al. [14] who used human raters to 
+judge prompt accuracy of Imagen, Google’s in-house 
+proprietary image generation system, against existing 
+competitors such as OpenAI DALL-E 2[8] and GLIDE[21]. 
+For the benchmark analysis, we conduct an independent 
+human evaluation run for each category. For each prompt, the 
+rater is shown two sets of images with one from DALL-E 2, 
+and second from Luna. Each set contains eight non-cherry-
+picked generations from the corresponding model. The human 
+rater will be asked two questions. 
+1. Which set of images better represents the text 
+caption: [Text Caption]? Question subjectively evaluates 
+image-text alignment. 
+2. Which set of images is of higher quality? Question 
+subjectively evaluates image fidelity. 
+For each question, the rater is asked to select from two 
+choices: 
+1. I prefer set A. 
+2. I prefer set B. 
+The paper aggregates the scores from different raters and then 
+score it in a percentage value which will be presented in the 
+form of a candle graph. The authors did not perform any post 
+filtering of the data to identify unreliable raters, both for 
+expedience of the analysis process and because the task was 
+straightforward to explain and execute. 
+ 
+IV. 
+RESULTS 
+After carefully compiling the results of the survey from 
+human raters over a span of two weeks. We analyze the 
+results according to generally acceptable benchmarks for 
+alignment and fidelity scores. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 1.        Alignment and Fidelity comparison between DALL-E 2 and Luna 
+using methodologies outlined in Saharia et al. plotted into a candle graph: User 
+preference rates for prompt alignment and image fidelity. 
+ 
+Results show that when the output images are given to 
+human raters and evaluated using methods outlined in Saharia 
+et al, the results show that DALL-E 2 on average received a 
+higher image-text alignment (62.1%) and image fidelity 
+(83.4%) rating than Luna. 
+FID scores can be a more objective measurement of 
+fidelity of machine-generated images, but previous research 
+has shown that FID scores are not reflective of perceptual 
+quality[22].  
+ 
+ 
+ 
+ 
+ 
+Fig. 2.        MS-COCO 256 × 256 FID-30K for DALL-E 2[8] and Luna (which 
+is based on stable diffusion, marked as LDM-KL[11]). Lower score is better. 
+While quantitative measurements are outside the scope of 
+this paper, FID scores cited from research papers of the 
+respective models show that DALL-E 2 outperforms other 
+methods on MS-COCO 256 x 256 with zero-shot FID-30K 
+with a score of 10.39, significantly outperforming systems 
+based on Latent Diffusion Models (LDM-KL) such as Luna. 
+The results line up with human raters’ indication of individual 
+samples fidelity ratings. 
+ 
+Fig. 3.        Selected image samples from the resolvement process of Prompt 
+4 by Luna (left) and DALL-E 2 (right). Images we’re picked from a set of 4 
+each generated per system.  
+It’s observed that both prompt systems have difficulties in 
+prompt resolvement of novel concepts, such as a car on a 
+spoon. While Luna seems to struggle with the concept, 
+DALL-E 2 interpret it as a request for a toy car, and not a real 
+car. 
+ 
+Fig. 4.        Selected image samples from the resolvement process of Prompt 
+5 by Luna (left) and DALL-E 2 (right). Images we’re picked from a set of 4 
+each generated per system.  
+It's also observed that Luna has difficulty assigning the 
+correct number of items in an image given a prompt that 
+contains numerical amounting values. While the issue is also 
+present in DALL-E 2, prior research has proven that the 
+system can atleast count to four objects[19]. 
+100% 
+50% 
+0% 
+DALLE-2 
+Luna 
+Alignment 
+Fidelity 
+NParams 
+Model                   FID-30K 
+DALL-E 2 
+          10.39                       650M 
+    Luna (LDM-KL)            12.63                         n/a 
+
+Researchers behind DALL-E 2 has also disclosed issues 
+regarding compositionality[8], which is the ability to 
+comprehend the merging of multiple object properties such as 
+shape and positioning within the image. Which is why the 
+placement of the objects inside of the picture generated might 
+look too symetrical. 
+ 
+Fig. 5.        Selected image samples from the resolvement process of Prompt 
+3 by Luna (left) and DALL-E 2 (right). Images we’re picked from a set of 4 
+each generated per system.  
+Resolvement of misspelled prompts[19] has also proved a 
+challenge for LDM-based systems such as Luna with DALL-
+E 2 accurately representing the misspelled prompt as a 
+“refrigerator” and Luna failing to generate a comprehensible 
+image. This is theorized to be the result of significantly better 
+prompt alignment within DALL-E 2’s generation system that 
+enables it to edge out Luna in this prompt. 
+Resolvement of misspelled prompts[19] has also proved a 
+challenge for LDM-based systems such as Luna with DALL-
+E 2 accurately representing the misspelled prompt as a 
+“refrigerator” and Luna failing to generate a comprehensible 
+image. This is theorized to be the result of significantly better 
+prompt alignment within DALL-E 2’s generation system that 
+enables it to edge out Luna in this prompt. 
+ 
+Fig. 6.        Selected image samples from the resolvement process of Prompt 
+2 by Luna (left) and DALL-E 2 (right). Images we’re picked from a set of 4 
+each generated per system.  
+Resolvement of prompts with non-westernized artstyles 
+both failed to generate anything resembling the inputted 
+prompt. Machine learning systems have consistently hit 
+difficulties in detecting and generating styles that are 
+uncommon outside of western culture such as the artstyle of 
+Japanese anime[23] This is possibly the result of bias within 
+large compiled datasets that’s mainly trained on webscrapes 
+of mostly western-aligned content[24]. This challenge will 
+also present itself more in bigger datasets, which will 
+complicate efforts to effectively scale computer vision and 
+generative datasets without significant alignment. 
+The possibility of training differences affecting the 
+performance of image generation systems are also observed to 
+be correlated. Comparing FID-30K scores and observing the 
+interception distance of FID-2K scores between the two 
+systems have yielded interesting technical observations. The 
+figures show that Luna experiences a distinct lowered amount 
+of TPU (Tensor Processing Units) training days compared to 
+DALL-E 2, which can negatively impact the alignment quality 
+and perceived fidelity of the image as less itterations are 
+performed within a specific timeframe. Luna as an 
+individually built system also possibly suffered from time 
+limitations during training. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 6. 
+Comparison of TPU training time needed to achieve a 20 basis 
+point FID-2K rating between regular U-Nets (Luna or LDM-KL) vs efficient 
+U-Nets (DALLE-2). 
+ 
+The differences mainly come down to complexity, which 
+might have caused worse FID-2K scores due the amount of 
+time used to train LDM-KL based systems compared to 
+OpenAI’s approach with DALL-E 2[25]. Time differences 
+may be attributed to the difference in libraries used, as Luna 
+uses Tensorflow and DALL-E 2 uses PyTorch, the latter of 
+which has been shown to be more performant than the former 
+resulting in faster compute times[26]. 
+ 
+V. 
+CONCLUSIONS 
+The round of experimentation showcases the effectiveness 
+of frozen large pretrained language models as text encoders 
+for the text-to-image generation, but differences exist between 
+the capabilities of large models such as DALL-E 2 and smaller 
+scale models such as Luna.  
+Dramatically increasing the size of these language models 
+have significantly more impact than scaling the U-Net size on 
+overall performance on alignment and fidelity. This 
+encourages future research directions on exploring even 
+bigger language models as text encoders, both by companies 
+and individuals.  
+But increasing datasets has also several kinks other than 
+purely technical complications as there are ethical challenges 
+relating to large datasets used for the image generation 
+systems, particularly regarding subject data awareness and 
+consent[27], [28] and some datasents even reflect stereotypes, 
+offensive viewpoints, and derogatory associations of various 
+marginalized identity groups[24]. 
+While Luna was edged out in both alignment and fidelity 
+measurements both in qualitative benchmarks through human 
+raters and quantitative benchmarks through zero-shot FID-2K 
+and FID-30K scores, it has reached a remarkable level of 
+accuracy for a system that is built by an individual and trained 
+using a limited dataset. 
+We also find considerable performance penalties incurred 
+by Luna’s use of Tensorflow compared to DALL-E 2’s use of 
+FID-2K 
+Training Days 
+DALLE-2 equiv. 
+Luna equiv. 
+
+DcoeceBrrees
+RERD
+Fceeecfor
+TRBBEER
+DBB40
+30
+20
+0
+1
+2
+3
+4
+5
+6
+7PyTorch which resulted in a slower comparative TPU training 
+days compared to the latter, which affects training accuracy. 
+We ultimately conclude that while differences exist 
+between large systems made by corporations and smaller 
+individual made systems, the advent of diffusion-based image 
+generation systems have lowered the barrier to enter the image 
+generation field significantly. The advancement in research of 
+generative AI technologies need to be paired with safeguards 
+and acknowledgement of ethical concerns, working towards a 
+safer implementation of systems. 
+ 
+ACKNOWLEDGMENT 
+We give thanks to Arfy Slowy from the Google Brain 
+Research Team in Singapore and Imre Bard from the OpenAI 
+Alignment Research team for helping early discussions, and 
+providing 
+many 
+helpful comments and suggestions 
+throughout the project. We thank you the team at Kaggle and 
+OpenAI for the free tiers given for testing and exploratory 
+research purposes. Special thanks to Agneta Viola for 
+reviewing grammatical and linguistical errors. We thank 
+Herendra Kurniawan for their consistent and critical help with 
+TPU resource allocation and Kaggle notebook initialization.  
+ 
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+  
+ 
+

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+page_content='Accuracy and Fidelity Comparison of Luna and  DALL-E 2 Diffusion-Based Image Generation  Systems  Michael Cahyadi  School of Computer Science  Bina Nusantara University  Jakarta, Indonesia  michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
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+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='id  Muhammad Rafi  School of Computer Science  Bina Nusantara University  Jakarta, Indonesia  muhammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='rafi007@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='id  William Shan  School of Computer Science  Bina Nusantara University  Jakarta, Indonesia  william.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='sitanggang@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='id  Abstract — We qualitatively examine the accuracy and  fideltiy between two diffusion-based image generation systems,  namely DALL-E 2 and Luna, which have massive differences in  training datasets, algorithmic approaches, prompt resolvement,  and output upscaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' In our research we conclude that DALL-  E 2 significantly edges Luna in both alignment and fidelity  comparisons  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  INTRODUCTION  Image generation systems is one of the many avenues  artificial intelligence research projects have been pursuing  ways to improve generative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Image generation  systems have immense potential to compliment and extend  human creativity[1], but on the other hand there are issues  with the field such as potential abuse to spread misinformation  and harrassment[2], bias against certain cultural groups[3],  and harmful associations against marginalized societies [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The field of image generation using artificial intelligence  technologies has made great advancements in the last few  years, with recent models capable of generating images with  near human-like characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Variations of Generative  Adversarial Networks (GAN)[5] were among the first models  to generate high-quality images, but recently there has been  more focus by researchers and the public[6] on diffusion-  based models trained using massive datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The open nature of research information regarding  diffusion-based image generation models have also led to an  increase of image generation systems made by individuals,  which may not have sufficient guardrails against abuse[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Image generation systems also use CLIP latents to  associate certain human concepts and understand them in  creative contexts[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' With artificial intelligence systems  continuing to get better in non-analytical areas such as artistic  creativity that mimic closely human cognitive architectures,  researchers might soon get closer into the realm of General  Artificial Intelligence (GAI)[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  As artificial intelligence becomes a more pervaise tool in  day-to-day workflows, there needs to be an evaluation  regarding the quality of outputs generated by image  generation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Accurately judging the alignment and  perceived fidelity of generated outputs from these image  generation systems can help researchers and developers build  better systems that are aware of the pitfalls of current systems  in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  While there exists many diffusion-based image generation  systems, both open and closed sourced, we decided to search  for two systems who adopt resolvement approaches that lead  in their industry in terms of accuracy and widescale  implementation in the image generation technology  community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Latent diffusion models and CLIP-guided[10]  diffusion models represent the forefront methodologies for  image-generation technologies with alignment and fidelity  results surpassing previous GAN-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  This paper also ultimately aims to examine the difference  in accuracy between images generated by diffusion-based  systems that are made by a large company using a large  training data set and a system created by an individual with  more limited training resources and less guardrails towards  abuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' To that effect, we consider the following two image  generation models for comparing our results:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  DALL-E 2[8]  is an image generation system created  by OpenAI which can generate high-resolution images that  combine various concepts and art styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' The project was built  in PyTorch using ViT-H/16 text encounters with the training  data of 650M images scraped from the internet and aligned by  CLIP[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Luna is an image generation system built by Arfy  Slowly, a Senior Software Engineer at Google Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' The  project was built in Tensorflow and published on GitHub as  an open-source project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' The system uses a latent diffusion  model[11] to condition the model on text prompts, however  the training dataset used is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  RELATED WORKS  There have been many papers that try to compare the  performance of two image generation systems, whether  quantitatively or qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' The metrics that are used to  verify the accuracy of image generation systems mostly rely  on image fidelity and its benchmark against real-world  equivalents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  While quantitative methods have been laid out to gauge  the accuracy of image generation systems such as the Fréchet  Inception Distance by Heusel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' [12], the metric is used to  compare GAN performance at image generation using real-  life samples, which differs from our attempts to gauge the  Henry Lucky  School of Computer Science  Bina Nusantara University  Jakarta, Indonesia  henry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='lucky@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='id  Jurike Moniaga  School of Computer Science  Bina Nusantara University  Jakarta, Indonesia  jurike@binus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='edu  accuracy of image generation systems at prompt resolving  novel concepts that have little to no real-life examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Existing evaluations of diffusion-based image generation  models are mostly based on the accuracy of the image  generated in specific fields such as in artificially generated  faces[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' But no paper has qualitatively evaluated the  inherent accuracy between the prompt given to the model to  the generated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Qualititative methods of performance analysis are usually  done by human surveyors such as in research by Saharia et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' This is due to the subjective nature of art[15], unlike  measurable things such as image fidelity, that cannot be  measured with common metric calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  In Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=', the method used in evaluating image  accuracy consists of 2 questions given to human raters  inquriing about the fidelity and alignment (the accuracy  between human interpretation of a given concept and the  output given by an artificial intelligence[16]) of the system’s  output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Prompt Creation  The prompts list below are modified prompts from  Google’s Drawbench Benchmark[14], whic covers a variety  of concepts, art styles, and common pitfalls of image  generation systems to generate images that can be a point of  evaluation for the alignment and fidelity of the image  generation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' We have also listed the reasoning towards  why we choose each prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The prompts detailed above have been screened by  running them through a Google image search and seeing how  easily images for these concepts could be retrieved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' from this  process, we eliminated two prompts and modified another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Number  Contents  Prompt  Explanation  1  A  photorealistic  image of a machine  resembling  a  human being and  able to replicate  certain  human  movements  and  functions  automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Prompt is used to evaluate  the ability of the image  generation system to build  photorealistic images that  don’t cross the uncanny  valley[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  2  A half-robot and  man  entity  with  chainsaws for their  head and hands in  the  style  of  Japanese anime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Prompt is used to evaluate  the  bias  in  machine  learning algorithms that are  trained with data from  westernized-culture[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  3  Rbefraigerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The prompt is used as a  way to determine the image  generation system’s ability  to  navigate  word  misspellings[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  4  A car on top of a  spoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The prompt is used to  examine the ability of  image generation systems  to generate images novel in  concept[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Number  Contents  Prompt  Explanation  5  Two bicycles and  one  car  on  an  empty grass field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The prompt is used to  examine the ability of  image generation systems  to generate images with  accurate  positional  information[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  6  In late afternoon in  January in Jakarta,  a man stands in the  shadow of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The prompt is used to  examine the ability of  image generation systems  to  accurately  create  shadows that correspond  with  differing  lighting  conditions[19]  7  A Sumatran tiger  under the sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The prompt is used to  examine the ability of the  image generation system to  generate images that have  conflicting concepts[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  8  Art  nouveau  stained  glass  window  art  depicting  Woody  from Toy Story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The prompt is used to  examine the ability of the  image generation system to  generate images with pop  culture  products  in  a  medium  not  usually  associated  with  the  product[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Image Generation  Below each set of 4 images per-prompt on each model, we  will outline general observations from the researchers as with  the cited reasons of why each model behave in such a way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The image results are compiled for further analysis in the  paper in methodologies outlined in later subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The researchers will generate every 32 images from each  system, noting that DALL-E 2 generates four images per run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  For DALL-E 2, access was provided to the system during  September 2022 after a request for beta-testing research was  approved by the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' DALL-E 2 was accessed from the  OpenAI Beta website, with 8 credits dispensed every month  for non-commercial research use only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' DALL-E 2 outputs 4  photos in one-prompt execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' DALL-E 2 outputs  1024x1024 pixel images and Luna outputs 512x512 pixel  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  For Luna, we use the provided Colab notebook by Google  to run the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Considerations we’re made to run Luna  using on-premises hardware, however due to Tensorflow’s  requirement of NVIDIA CUDA cores we decided to opt for  cloud solutions instead due to faster compute times and as a  better benchmark against DALL-E 2 which is a cloud-based  system hosted in Microsoft Azure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Luna outputs 4 photos in  one-prompt execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Analysis  As laid out in the related works section, due to art being  inherently subjective in nature[15], normal metrics cannot be  applied when analyzing the inherent accuracy of prompt  creations from image generation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The methodologies to evaluate these images are based on  Drawbench by Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' [14] who used human raters to  judge prompt accuracy of Imagen, Google’s in-house  proprietary image generation system, against existing  competitors such as OpenAI DALL-E 2[8] and GLIDE[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  For the benchmark analysis, we conduct an independent  human evaluation run for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' For each prompt, the  rater is shown two sets of images with one from DALL-E 2,  and second from Luna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Each set contains eight non-cherry-  picked generations from the corresponding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' The human  rater will be asked two questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Which set of images better represents the text  caption: [Text Caption]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Question subjectively evaluates  image-text alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Which set of images is of higher quality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Question  subjectively evaluates image fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  For each question, the rater is asked to select from two  choices:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' I prefer set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' I prefer set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The paper aggregates the scores from different raters and then  score it in a percentage value which will be presented in the  form of a candle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' The authors did not perform any post  filtering of the data to identify unreliable raters, both for  expedience of the analysis process and because the task was  straightforward to explain and execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  RESULTS  After carefully compiling the results of the survey from  human raters over a span of two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' We analyze the  results according to generally acceptable benchmarks for  alignment and fidelity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Alignment and Fidelity comparison between DALL-E 2 and Luna  using methodologies outlined in Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' plotted into a candle graph: User  preference rates for prompt alignment and image fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Results show that when the output images are given to  human raters and evaluated using methods outlined in Saharia  et al, the results show that DALL-E 2 on average received a  higher image-text alignment (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='1%) and image fidelity  (83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='4%) rating than Luna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  FID scores can be a more objective measurement of  fidelity of machine-generated images, but previous research  has shown that FID scores are not reflective of perceptual  quality[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' MS-COCO 256 × 256 FID-30K for DALL-E 2[8] and Luna (which  is based on stable diffusion, marked as LDM-KL[11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Lower score is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  While quantitative measurements are outside the scope of  this paper, FID scores cited from research papers of the  respective models show that DALL-E 2 outperforms other  methods on MS-COCO 256 x 256 with zero-shot FID-30K  with a score of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='39, significantly outperforming systems  based on Latent Diffusion Models (LDM-KL) such as Luna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The results line up with human raters’ indication of individual  samples fidelity ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Selected image samples from the resolvement process of Prompt  4 by Luna (left) and DALL-E 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Images we’re picked from a set of 4  each generated per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  It’s observed that both prompt systems have difficulties in  prompt resolvement of novel concepts, such as a car on a  spoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' While Luna seems to struggle with the concept,  DALL-E 2 interpret it as a request for a toy car, and not a real  car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Selected image samples from the resolvement process of Prompt  5 by Luna (left) and DALL-E 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Images we’re picked from a set of 4  each generated per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content="  It's also observed that Luna has difficulty assigning the  correct number of items in an image given a prompt that  contains numerical amounting values." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' While the issue is also  present in DALL-E 2, prior research has proven that the  system can atleast count to four objects[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  100%  50%  0%  DALLE-2  Luna  Alignment  Fidelity  NParams  Model                   FID-30K  DALL-E 2  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='39                       650M  Luna (LDM-KL)            12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='63                         n/a  Researchers behind DALL-E 2 has also disclosed issues  regarding compositionality[8], which is the ability to  comprehend the merging of multiple object properties such as  shape and positioning within the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Which is why the  placement of the objects inside of the picture generated might  look too symetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Selected image samples from the resolvement process of Prompt  3 by Luna (left) and DALL-E 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Images we’re picked from a set of 4  each generated per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Resolvement of misspelled prompts[19] has also proved a  challenge for LDM-based systems such as Luna with DALL-  E 2 accurately representing the misspelled prompt as a  “refrigerator” and Luna failing to generate a comprehensible  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' This is theorized to be the result of significantly better  prompt alignment within DALL-E 2’s generation system that  enables it to edge out Luna in this prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Resolvement of misspelled prompts[19] has also proved a  challenge for LDM-based systems such as Luna with DALL-  E 2 accurately representing the misspelled prompt as a  “refrigerator” and Luna failing to generate a comprehensible  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' This is theorized to be the result of significantly better  prompt alignment within DALL-E 2’s generation system that  enables it to edge out Luna in this prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Selected image samples from the resolvement process of Prompt  2 by Luna (left) and DALL-E 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Images we’re picked from a set of 4  each generated per system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Resolvement of prompts with non-westernized artstyles  both failed to generate anything resembling the inputted  prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Machine learning systems have consistently hit  difficulties in detecting and generating styles that are  uncommon outside of western culture such as the artstyle of  Japanese anime[23] This is possibly the result of bias within  large compiled datasets that’s mainly trained on webscrapes  of mostly western-aligned content[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' This challenge will  also present itself more in bigger datasets, which will  complicate efforts to effectively scale computer vision and  generative datasets without significant alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The possibility of training differences affecting the  performance of image generation systems are also observed to  be correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Comparing FID-30K scores and observing the  interception distance of FID-2K scores between the two  systems have yielded interesting technical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' The  figures show that Luna experiences a distinct lowered amount  of TPU (Tensor Processing Units) training days compared to  DALL-E 2, which can negatively impact the alignment quality  and perceived fidelity of the image as less itterations are  performed within a specific timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Luna as an  individually built system also possibly suffered from time  limitations during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Comparison of TPU training time needed to achieve a 20 basis  point FID-2K rating between regular U-Nets (Luna or LDM-KL) vs efficient  U-Nets (DALLE-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  The differences mainly come down to complexity, which  might have caused worse FID-2K scores due the amount of  time used to train LDM-KL based systems compared to  OpenAI’s approach with DALL-E 2[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Time differences  may be attributed to the difference in libraries used, as Luna  uses Tensorflow and DALL-E 2 uses PyTorch, the latter of  which has been shown to be more performant than the former  resulting in faster compute times[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  CONCLUSIONS  The round of experimentation showcases the effectiveness  of frozen large pretrained language models as text encoders  for the text-to-image generation, but differences exist between  the capabilities of large models such as DALL-E 2 and smaller  scale models such as Luna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Dramatically increasing the size of these language models  have significantly more impact than scaling the U-Net size on  overall performance on alignment and fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' This  encourages future research directions on exploring even  bigger language models as text encoders, both by companies  and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  But increasing datasets has also several kinks other than  purely technical complications as there are ethical challenges  relating to large datasets used for the image generation  systems, particularly regarding subject data awareness and  consent[27], [28] and some datasents even reflect stereotypes,  offensive viewpoints, and derogatory associations of various  marginalized identity groups[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  While Luna was edged out in both alignment and fidelity  measurements both in qualitative benchmarks through human  raters and quantitative benchmarks through zero-shot FID-2K  and FID-30K scores, it has reached a remarkable level of  accuracy for a system that is built by an individual and trained  using a limited dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  We also find considerable performance penalties incurred  by Luna’s use of Tensorflow compared to DALL-E 2’s use of  FID-2K  Training Days  DALLE-2 equiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  Luna equiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  DcoeceBrrees  RERD  Fceeecfor  TRBBEER  DBB40  30  20  0  1  2  3  4  5  6  7PyTorch which resulted in a slower comparative TPU training  days compared to the latter, which affects training accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  We ultimately conclude that while differences exist  between large systems made by corporations and smaller  individual made systems, the advent of diffusion-based image  generation systems have lowered the barrier to enter the image  generation field significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' The advancement in research of  generative AI technologies need to be paired with safeguards  and acknowledgement of ethical concerns, working towards a  safer implementation of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  ACKNOWLEDGMENT  We give thanks to Arfy Slowy from the Google Brain  Research Team in Singapore and Imre Bard from the OpenAI  Alignment Research team for helping early discussions, and  providing  many  helpful comments and suggestions  throughout the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' We thank you the team at Kaggle and  OpenAI for the free tiers given for testing and exploratory  research purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' Special thanks to Agneta Viola for  reviewing grammatical and linguistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' We thank  Herendra Kurniawan for their consistent and critical help with  TPU resource allocation and Kaggle notebook initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content='  REFERENCES  [1]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAzT4oBgHgl3EQf8_7Q/content/2301.01914v1.pdf'}
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5dE1T4oBgHgl3EQf6gWk/content/tmp_files/2301.03524v1.pdf.txt

@@ -0,0 +1,1596 @@
+Physics-separating artificial neural networks for predicting
+sputtering and thin film deposition of AlN in Ar/N2 discharges
+on experimental timescales
+Tobias Gergs,1, ∗ Thomas Mussenbrock,1, † and Jan Trieschmann2, 3, ‡
+1Chair of Applied Electrodynamics and Plasma Technology,
+Department of Electrical Engineering and Information Science,
+Ruhr University Bochum, 44780 Bochum, Germany
+2Theoretical Electrical Engineering, Department of Electrical and Information Engineering,
+Kiel University, Kaiserstraße 2, 24143 Kiel, Germany
+3Kiel Nano, Surface and Interface Science KiNSIS,
+Kiel University, Christian-Albrechts-Platz 4, 24118 Kiel, Germany
+(Dated: January 10, 2023)
+Abstract
+Understanding and modeling plasma-surface interactions frame a multi-scale as well as multi-
+physics problem. Scale-bridging machine learning surface surrogate models have been demonstrated
+to perceive the fundamental atomic fidelity for the physical vapor deposition of pure metals. How-
+ever, the immense computational cost of the data-generating simulations render a practical appli-
+cation with predictions on relevant timescales impracticable. This issue is resolved in this work
+for the sputter deposition of AlN in Ar/N2 discharges by developing a scheme that populates
+the parameter spaces effectively. Hybrid reactive molecular dynamics / time-stamped force-bias
+Monte Carlo simulations of randomized plasma-surface interactions / diffusion processes are used
+to setup a physics-separating artificial neural network. The application of this generic machine
+learning model to a specific experimental reference case study enables the systematic analysis of
+the particle flux emission as well as underlying system state (e.g., composition, mass density, stress,
+point defect structure) evolution within process times of up to 45 minutes.
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+1
+arXiv:2301.03524v1  [cond-mat.mtrl-sci]  9 Jan 2023
+
+I.
+INTRODUCTION
+In most technological applications of plasmas (e.g., thin film sputter deposition, catalysis)
+surfaces and, hence, plasma-surface interactions (e.g., growth, sputtering, surface chemical
+reactions) are involved [1–4]. Analyzing, understanding, and modeling the last is considered
+to be essential for a knowledge-driven process design. However, the physics of these two
+states of matter (i.e., plasma, solid-state) demand for descriptions on length as well as time
+scales that differ in orders of magnitudes (see Figure 1) [5–8].
+Common scale bridging solutions include event dependent coefficients, lookup-tables, and
+analytic formulas (e.g., Berg-model [9, 10], Sigmund–Thompson theory [11–13]). However,
+they altogether lack a fundamental atomic fidelity.
+An issue that has been addressed by applying machine learning (ML) models.
+They
+have been shown to be capable of describing physical processes relevant to plasma science
+with high accuracy while mitigating statistical noise, generalizing successfully [5, 14–21].
+In particular, a series of ML plasma-surface interaction (PSI) surrogate models have been
+proposed for the sputter deposition of Ti1−xAlx thin films. First, a multi-layer-perceptron
+(MLP) was trained to predict the Ar+ ion bombardment induced sputtering of a Ti0.5Al0.5
+composite target [5]. Second, a more advanced artificial neural network (ANN) combining a
+dedicated mapper network with the decoder of a β-variational autoencoder (β-VAE [22–26])
+was established for Ti1−xAlx composite targets [20]. Therein, the stoichiometry has been
+introduced as a basic surface state descriptor. Both studies are based on transport of ions
+in matter (TRIM) simulation data. Further, a physics-separating artificial neural network
+Figure 1. Schematic of the physical time and length scales for thin film sputter depositions.
+2
+
+Heavy particle
+dynamics
+mm
+Electron
+dynamics
+Nanostructured thin
+film deposition
+Surface processes(PSNN) was proposed to describe the PSIs at the substrate as well as target in a generalized
+manner for Al and Ar as material system and working gas, respectively. The PSNN consists
+of two conditional variational autoencoders (CVAEs [21, 26, 27]). One describes the PSIs
+(e.g., sputtering, ion bombardment induced damage formation). The other one describes
+the conversion of the defect structure (i.e., ring statistical connectivity profile [28–30]) to the
+surface state (i.e., stoichiometry, mass density, biaxial stress, tensile stress). It was demon-
+strated that both (i.e., defect structure, surface state) are sufficient for a complete system
+description that may evolve in time. However, being based on molecular dynamics (MD)
+simulations for data generation, the latter was limited to the impingement of two consecutive
+particle doses (in total: 2.42 × 1015 particles/cm2) due to the immense computational cost.
+Hence, the input parameter space (i.e., particle flux composition, ion energy, surface state)
+was found to be sampled insufficiently to setup a long-term evolution ML PSI surrogate
+model for the sputter deposition of metal thin films.
+In this work, the concept of a ML surface surrogate model is advanced by – among other
+aspects – proposing a randomized data generating scheme which enables PSNNs to predict
+the reactive sputter deposition of AlN thin films in Ar/N2 discharges for up to hours. The
+considered process is relevant for the preparation of hard coatings, protective wear (e.g.,
+transition metal aluminium nitride, transition metal aluminium oxynitride), and energy
+harvesting (scavenging) [31–35]. This manuscript is structured as follows: The considered
+scenario is presented in Section II. In Section III, applied methods and parameters are
+described. The results are presented and discussed in Section IV. Finally, conclusions are
+drawn in Section V.
+II.
+SETUP
+The general scenario of an Ar/N2 plasma discharge interacting with AlN surfaces is con-
+sidered. While the gas discharge and sputtered particle transport dynamics are considered
+predetermined, the focus is on the substrate side AlN thin film deposition. The target side
+sputtering of AlN is not of main concern, but is included up to the maximum considered ion
+energy (i.e., 300 eV). The key aspect for robust and reliable data-driven ML model develop-
+ment is to efficiently populate the parameter space relevant for representing the dynamics
+of PSI and diffusion. This is achieved by random sampling of a given number of initial
+3
+
+Figure 2. Illustration of the PSI setup. The atom configuration is rendered with the Open Visu-
+alization Tool (OVITO) [36]. Al and N atoms are colored gray and light blue, respectively.
+AlN bulk systems, which are subsequently subject to a series of diffusion process and PSI
+simulations (e.g., ion bombardment). The corresponding evolution is recorded and used for
+ML. A brief description of the procedure is as follows:
+System state
+A bulk wurtzite AlN supercell is considered with a point defect structure
+that includes up to 5 % Ar, 10 % Al, and 10 % N interstitials as well as 20 % Al and 20
+% N vacancies. The defect structure is assumed to define the system sufficiently [21, 37].
+Complementing properties are determined after the atom configuration is relaxed.
+The
+system is characterized by the mass density ρ, lattice constant a, heat of formation ∆Hf,
+bulk modulus B0, its derivative B′
+0, and 12 point defect populations ρvAl, ρAlN, ρAli, ρvN,
+ρNAl, ρ(N-N)Al, ρNi, ρ(N-N)N, ρ(N-N)i, ρArAl, ρArN, ρAri. The Kr¨oger-Vink notation is used for
+the defect types (subscripts) [38]. The defect populations define the total number of atoms
+in the system:
+ntot = (1 + ρvAl + ρvN − ρAli − ρNi − ρAri − 2ρ(N-N)i − ρ(N-N)N − ρ(N-N)Al)−1nideal
+tot
+(1)
+nideal
+tot
+refers to the total number of atoms in the ideal AlN supercell (8 atoms per unit cell).
+The point defect structure defines the Al, N, and Ar concentrations cAl, cN, and cAr, which
+4
+
+Fout
+Tinare denoted as the composition:
+cAl = 0.5nideal
+tot
+ntot
+− ρvAl + ρAli + ρAlN − ρNAl − ρ(N-N)Al − ρArAl
+(2a)
+cN = 0.5nideal
+tot
+ntot
+− ρvN + ρNi + 2ρ(N-N)i + ρ(N-N)N + 2ρ(N-N)Al − ρAlN − ρArN
+(2b)
+cAr = ρAri + ρArAl + ρArN
+(2c)
+The first terms on the right hand side of Eqs. (2a) and (2b) refer to the ideal configuration,
+as 0.5nideal
+tot
+is the number of Al or N atoms when point defects are absent. The mass density
+is determined by the lattice constants, the total number of atoms ntot, and the composition:
+ρ = mAlcAl + mNcN + mArcAr
+√
+3nuca2c
+ntot
+(3)
+mAl, mN, and mAr are the masses of Al, N, and Ar atoms, respectively. nuc is the number
+of unit cells (detailed later) and the lattice constant c = 1.6a is kept constant (anisotropic
+deformations are suppresesed).
+Plasma-Surface Interaction and Diffusion
+For each initialized system, seven diffusion
+and PSI simulations are performed alternately (detailed later).
+First, the effect of bulk
+diffusion processes on the system state is studied. For this a temperature T is imposed.
+Second, an AlN surface is obtained by cleaving the bulk system either in [100] or [002]
+direction. Third, the effect of individual particles s (i.e., Al, N, N2, Ar) bombarding the
+AlN surface with specified kinetic energies Ekin is investigated. The contribution from the
+plasma onto the surface is characterized by the particle fluxes Γin
+s , the kinetic energy of the
+particles Ekin, and the species s. The emitted fluxes are denoted by Γout
+s .
+The first and the last are used to setup two individual machine learning regression models
+(i.e., PSI-CVAE, Diffusion-CVAE) that eventually are used to form a PSNN.
+III.
+METHODS
+First, the data generating hybrid reactive molecular dynamics (RMD) / time-stamped
+force-bias Monte Carlo (tfMC) simulations are described.
+Second, the data processing,
+training workflow and included metric are introduced. Third, the structure and information
+flow of the PSNN is outlined. Fourth, physics-constraints and their implementation are
+introduced. Fifth, the hyperparameter (HP) optimization is descried. Sixth, the production
+run is presented.
+5
+
+Figure 3.
+Schematic of the workflow and information flow for the data generating hybrid
+RMD/tfMC simulations.
+A.
+Hybrid reactive molecular dynamics / time-stamped force-bias Monte Carlo
+RMD, tfMC, and hybrid RMD/tfMC simulations are performed with the open-source
+Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [39–43]. The in-
+teractions of AlN complexes are described by the third-generation charge-optimized many-
+body (COMB3) potential that is tapered with the Ziegler-Biersack-Littmark (ZBL) potential
+(COMB3/ZBL potential) to account for high-energy collisions by including screened nuclear
+repulsions [44–46]. The COMB3 formalism is outlined in [44]. The COMB3 AlN parame-
+terization and combination with the ZBL potential is described in [46]. Its predecessor was
+setup for nanostructures as well as heterogeneous interfaces and revisited to describe plasma-
+surface interactions more accurately (e.g., ion bombardment induced damage production)
+[46, 47]. The atomic charges are equilibrated by applying the charge transfer equilibration
+(QTE+) method to account for meaningful charge exchange during PSIs (e.g., ion bombard-
+ment, sputtering) [48]. In the following, charge equilibration refers to the application of the
+QTE+ method with a timestep of 10−2 fs. The exponents of the 1s Slater type orbitals used
+for the overlap integral computations are 0.668 ˚A−1 and 1.239 ˚A−1 for Al and N, respectively
+[48].
+a.
+System state initialization
+It has been argued and demonstrated that the defect
+structure is sufficient to describe a system [21, 37]. Hence, the initial atom configuration
+is constructed by specifying the point defect structure.
+The Ar and Al (N) interstitial
+population ρAri and ρAli are sampled from a normal distribution N(0, σ) with the standard
+6
+
+System state Ss,
+System state Ss
+Bulk cleavage
+PSI
+PSI:
+Diffusion:
+Diffusion
+Molecular dynamics
+Monte Carlo
+data
+data
+Bulk reinforcement
+System state Ss.
+System state S.
+flux Fout
+temperature Tdeviations 3σ = 5 % and 3σ = 10 %, respectively. The N interstitial populations account
+for single as well as split interstitials (N-N) [49]. They are distinguished from each other
+at the end of the surface state initialization. The Al and N vacancy population ρvAl and
+ρvN are sampled from a normal distribution N(0, σ) with a standard deviation 3σ = 20 %.
+Initially, no anti-sites (i.e., NAl, (N-N)Al, AlN, ArAl, ArN) are defined.
+The surface orientation (i.e., AlN(002), AlN(100)) is determined by a coin flip. In either
+case, a bulk supercell consisting of 8 × 5 × 7 orthorhombic unit cells is constructed with
+the lattice constants a=3.136 ˚A and c = 1.6a. The total number of atoms in the ideal AlN
+supercell (8 atoms per unit cell) is nideal
+tot
+= 2240. The targeted total number of atoms ntot
+is calculated as a function of the point defect population following Eq. (1). The absolute
+number of point defects is obtained by multiplying the total number of atoms with the
+individual point defect population.
+First, Al and N vacancies are created by removing the required number of Al and N
+atoms from the system. Second, interstitials are taken care of by randomly inserting new
+atoms (i.e., Al, N, Ar) into the simulation domain. The Ar atoms’ coordinate in surface
+normal direction is constrained to fall in between 5 ˚A above and below the lower and upper
+boundary of the simulation domain, respectively. If the new atoms overlap with each other
+or old atoms, they are deleted. This second step is repeated until the correct number of Al,
+N and Ar atoms are generated.
+The atom configuration is then relaxed. Minor discontinuities of the COMB3 interaction
+potential hinder the successful application of a single conjugate gradient descent algorithm.
+This issue is addressed by performing multiple energy minimizations (relaxations) as de-
+picted in Figure 4.
+The alternation between charge equilibration (i.e., applying QTE+)
+and relaxation is meant to increase the computational efficiency. It is easier to relax an
+expanding than shrinking atom configuration. Hence, the system is compressed when the
+instantaneous pressure falls below -1 MPa.
+The resultant point defect structure is determined by comparing the position of each
+atom mapped into the unit cell with the Al as well as the N atom sites of the ideal AlN(002)
+or AlN(100) structures (periodic images are taken into account). The distance tolerance is
+defined by the halved Al-N bond length 1.9/2 ˚A. Nitrogen split interstitials (N-N)i, (N-N)N
+or anti-sites (N-N)Al are identified by searching for interatomic distances between N atoms
+that fall below 1.5 ˚A (the N-N bond length equals 1.3 ˚A [49]). The number of Al and N
+7
+
+Figure 4. Workflow of the bulk relaxation. Relaxation: Application of the conjugate gradient
+descent algorithm implemented in LAMMPS. The tolerance for the residual force on any atom
+is 1 eV/˚A. Charge equilibration: Performing one time step while the QTE+ method is applied.
+∆U, ∆V and p refer to the change of the potential energy, volume, and instantaneous pressure
+value, respectively. Compression: Application of the strain −10−6 along each direction. Volume
+relaxation: In addition to the atom site relaxation, the simulation box dimensions are adjusted
+isotropically to remove the residual stress from the system.
+vacancies are computed at last to fulfill the particle balances:
+nvAl = 0.5nideal
+tot
+− ntot,Al + nAli + nAlN − nNAl − n(N−N)Al − nArAl
+(4a)
+nvN = 0.5nideal
+tot
+− ntot,N + nNi + n(N−N)N + 2n(N−N)i + nNAl + 2n(N−N)Al − nAlN − nArN (4b)
+The symbol n describes the absolute number of point defects, while the indexes denote
+the particular point defect type. When the provided distance tolerance results in negative
+8
+
+(start)
+interstitial relaxation
+ charge equilibration
+relaxation
+AU < 0.1 eV
+no
+yes
+charge equilibration
+OU
+end
+△U > 0.1 eV
+yes
+compression
+-1 MPa
+no
+yes
+charge equilibration
+volume relaxation
+charge equilibration
+relaxationnumbers for vacancies, Frenkel pairs (i.e., vacancies plus interstitials) are added to even
+out this diagnostic artifact. However, this procedure is applied rarely and is only meant to
+guarantee physically meaningful results (i.e., non-negative numbers of vacancies).
+At last, the minimum of the potential energy and corresponding lattice constant is ob-
+tained by fitting the third-order Birch-Murnaghan equation of state (EOS) to the p-V/ntot
+and U/ntot-V/ntot-curve of the just relaxed structure [50, 51]. The system dimensions are
+scaled isotropically to evaluate ten strains distributed equidistantly in between −10−2 and
+10−2. The system is compressed before it is expanded. The atom sites are relaxed for each
+probed atom configuration.
+b.
+Diffusion
+The tfMC method is applied for the simulation of the diffusion processes
+[39–41]. The maximal displacement length of the lightest atom (i.e., N) is ∆ = 0.19˚A, that
+is approximately 10 % of the typical nearest neighbor distance for AlN. The temperature is
+sampled from a uniform distribution in between 300 and 1000 K. Simultaneously, the QTE+
+method is applied with a time step of 10−2 fs, tolerance of 1 V, and damping constant of
+0.45 [48]. The simulation is run for 104 steps. The resultant atom configuration is relaxed
+and evaluated as detailed previously (the interstitial relaxation is skipped).
+c.
+Plasma-surface interaction
+The surface is established by cleaving the bulk system
+either in [100] or [002] direction and elongating the simulation domain in surface normal
+direction by additional 35 ˚A. This value is defined to be the sum of 11 ˚A and 24 ˚A, that
+are attributed to the COMB3 cutoff radii and serve as a buffer for recognizing reflected or
+sputtered particles, respectively. The atom sites are adjusted by alternating between charge
+equilibration (i.e., QTE+) for fixed atom configuraiton and relaxation (i.e., performing a
+conjugate gradient descent minimization until the residual force perceived by each atom
+falls below 1 eV/˚A) for a fixed charge distribution until the potential energy is changed by
+less than 0.1 eV per iteration. The surface slab is displaced randomly in both surface parallel
+directions to include different impingement sites (the impinging particles are centered above
+the surface slab).
+Following this procedure, the system can be subdivided into four regions as depicted
+in Figure 5: i) Excluded atoms whose surface normal coordinate falls below a threshold
+zth = hz − 15 ˚A − (hz − 15 ˚A)Ekin/Ekin,max. zth is decreased linearly for increasing kinetic
+energies of impinging particles Ekin, ranging from 0 eV to 300 eV (Ekin,max = 300 eV). hz is
+the surface slab height. Excluded atoms are not allowed to interact with any other atom,
+9
+
+Figure 5. Illustration of the PSI setup. The atom configuration is rendered with OVITO [36]. Al
+and N atoms are colored gray and light blue, respectively. The regions containing i) excluded, ii)
+immobile, iii) temperature-controlled, and iv) all remaining atoms are colored transparent gray,
+light blue, red, and not at all respectively.
+effectively reducing the surface slab thickness to reduce the computational cost. These atoms
+are also excluded from the charge equilibration. The interactions of reflected or sputtered
+particles with other atoms are excluded too, when their surface normal coordinate exceeds
+hz + 12 ˚A. However, they are kept in the simulation domain to evaluate them later. The
+members of this group are updated dynamically (i.e., every 2000 steps). ii) Immobile atoms
+whose surface normal coordinate falls below a threshold zth + 5 ˚A. These are not evolved in
+time to anchor the surface slab in the simulation domain. iii) Mobile atoms whose distance
+to the left or right periodic boundary in the surface parallel directions falls below 5 ˚A are
+coupled to a Langevin thermostat with a damping constant of 100 fs to gradually remove
+their kinetic energy, targeting 0 K. Ar atoms are always excluded. iv) All remaining atoms.
+10
+
+4
+i)
+24 A
+12 A
+5A
+5A
+iv)
+15 A
+iii)
+iii)
+5 A
+i)
+ZthAn impinging particle is created 11 ˚A above the surface and centered laterally. Its species
+(i.e., Al, N, N2, Ar) is determined randomly, whereas projectiles are assumed to be charge
+neutral prior interaction. The likelihood for each candidate is distributed equally among
+them.
+Its kinetic energy ranging from 0 eV to 300 eV is found by squaring a sample
+from a uniform distribution U(0
+√
+eV,
+√
+300 eV). The atoms are assumed to hit the surface
+perpendicularly (the surface parallel components of its velocity vector equals zero). The time
+step equals 0.25 fs and is eventually lowered to secure that the maximum displacement and
+change in kinetic energy of any atom does not exceed 0.1 ˚A and 0.01 eV, respectively. The
+simulation is run for 1 ps repeatedly until the temperature of the mobile atoms falls below 100
+K. Impinging atoms that bypass the lower simulation domain boundary due to channeling
+are reinserted at a random position in surface normal direction (lateral coordinates are
+maintained) within the surface slab (overlapping with another atom by less than 0.5 ˚A leads
+to a repetition).
+To again transfer from a surface to a bulk configuration, the atom sites are relaxed
+as outlined previously.
+The random shifts in surface parallel directions outlined in the
+beginning of this section are reversed. The change of the surface normal coordinate of the
+uppermost temperature controlled atom ∆zup is used as a reference to invert the particle
+impingement induced thermal expansion of the mobile surface slab. The surface normal
+coordinates z of all mobile atoms are updated by z → z − ∆zup(z − zth − 5 ˚A)/(zup − zth −
+5 ˚A) (assuming a linear expansion). All atoms that exceed the original surface slab height
+prior to the particle impingement are removed from the system. This includes reflected
+particles, sputtered particles, and in general atoms atop the surface (e.g., adatoms). The
+last are assumed to contribute to the film growth of following layers, but do not effect the
+subsurface region. Hence, they are neglected when making a prediction for the bulk system
+by reestablishing a periodic boundary in surface normal direction. This procedure has been
+validated by comparing lattice constants and stresses obtained with density functional theory
+based molecular dynamics thermal spike simulations to experimentally measured reference
+values for metal aluminium nitrides [37, 52, 53]. The resultant atom configuration is relaxed
+and further evaluated as detailed previously (the interstitial relaxation is omitted).
+11
+
+B.
+Data preparation, training and metrics
+a.
+Data set splitting
+The data sets consisting of 6496 diffusion processes and 4470 PSIs
+are shuffled and split for the HP optimization to train, validate and test the ML model with
+80 %, 10 %, and 10 % of the available data, respectively. The size of the training, validation
+and test set are referred to by ntrain
+data , nval
+data, and ntest
+data, respectively.
+b.
+Data normalization
+Min-max normalization is utilized, whereas the minimum and
+maximum values are taken from the training set to avoid data leakage.
+c.
+Data augmentation
+The normalized data is augmented to virtually extend the train-
+ing database and, therewith, setup a more robust ML model [54–56]. In this work, a gener-
+alized version of the constrained mixup augmentation is utilized. Input and output samples
+are determined by ˆxij = λxi + (1 − λ)xj and ˆyij = λyi + (1 − λ)yj, respectively [57]. Hence,
+a hypothesis of linear superposition is provided to the network. Its validity is reflected by
+the probability distribution function (i.e., Beta(α)) of the λ value. α approaching zero, one,
+or infinity resembles a coinflip, uniform distribution, or 0.5, respectively. In this work, sam-
+ples i and j are only mixed up when the length of the vector pointing from one to another
+falls below rc, that is
+��nx
+k=1(xi − xj)2 < rc. nx is the number of input parameters. rc
+is considered a HP. The augmented data set size equals the original training data set size
+(ntrain
+data,aug = ntrain
+data ). The training data is augmented anew once per epoch.
+d.
+Training procedure and metrics
+Backpropagation of the mean absolute errors
+(MAEs) is used to update the internal degrees of freedom of the ANNs (e.g., weights)
+once per batch.
+The stochastic gradient descent algorithm adaptive moment estimation
+(Adam) is applied [58]. The applied batch size nbatch is defined to match the set up and
+the ideal batch size nbatch,ideal included as HP as close as possible, but required to fulfill
+|nbatch − ntrain
+data,aug%nbatch| ≤ nbatch,ideal. Hence, all data samples contribute almost equally to
+the learning progress.
+The learning rate rl is initialized with rl-0 and kept constant for a simulated annealing
+phase, that is outlined later. Afterwards, it is divided by ten whenever the validation MAE
+falls below its previous minimum value over the course of nl-patience epochs. Early stopping
+stops the training when there is no further reduction of the validation MAE after 2.5nl-patience
+epochs.
+12
+
+Figure 6. Schematic of the CVAE structure. Input variables enter from the left and predictions
+are extracted on the right of the graph. The coordinates zls of the latent space are indicated by
+the center white box. The figure is taken from [21].
+e.
+Hyperparameter study
+10-fold Monte Carlo cross validation (MCCV) is utilized to
+determine a more accurate final validation MAE. Training, validation, and test data sets are
+randomly selected according to the given split. In 10-fold MCCV, an ensemble of 10 ANNs
+is trained with 10 different random splits. It used in the selection of the best HPs using an
+evolution strategy (described later). The coefficient of determination R2 is introduced as an
+additional, secondary metric. The test set is meant to provide an unbiased measure for the
+ML model’s performance. The last is determined even more thoroughly by applying a 100-
+fold MCCV for the eventually selected set of HPs to compute the final training, validation
+and test errors.
+f.
+Production run
+The test set is not required for the production run.
+Thus, it is
+combined with the training set, that is 90 % of the data. An ensemble of ten ML models is
+set up and trained to reduce the bias introduced to the model by splitting the data into the
+two subsets [59].
+C.
+Physics-separating artificial neural network methodology
+a.
+Conditional variational autoencoders
+The proposed PSNN combines two regression
+ML models (i.e., PSI-CVAE, Diffusion-CVAE), that are implemented as conditional vari-
+ational autoencoders (CVAEs) [21, 26, 27].
+Their network architecture and information
+13
+
+Input 
+Input 
+Mis (ylac)
+Decoder
+Output y
+Encoder
+S
+is (ylc)
+2
+Latent
+Train
+space
+Input y
+phase?
+True
+False
+N(0, I)
+N(0, I)flow are shown schematically in Figure 6. CVAEs resemble β-variational autoencoders (β-
+VAEs [22–26]), whose encoder and decoder are conditioned on the regression input variables.
+These are set up symmetrically. The number of hidden layer nhl and nodes per layer nnpl
+are considered as HPs. The activation functions for any hidden and output layer are set as
+rectified linear unit (ReLU) and linear, respectively. The encoder projects information of the
+regression output variables yi to an nls dimensional latent space representation. Similarity
+to a standard normal distribution in latent space is enforced by introducing an additional
+Kullback-Leibler (KL) divergence. [22, 23]. The HP β is used to scale the KL loss. It is
+additionally scaled with a simulated annealing factor that is increased logarithmically from
+10−3 to 1 per batch over the course of nSA (also a HP) epochs. The decoder is conditioned
+on the regression input and the latent space (a standard normal distribution after successful
+training) tries to reconstruct the regression output. The decoder resembles the regression
+model to be utilized for prediction (after training is completed). CVAEs are described in
+detail in [26, 27]. The outputs of the CVAEs are passed through physics-constraint enforcing
+custom layers.
+b.
+Physics-constraints
+The physics-constraints enforced by the last output layer sim-
+plify the regression problem to be solved by the individual CVAEs and are described in the
+following. First, the suppression of extrapolation is outlined. Second, the particle conversa-
+tion for prediction on bulk diffusion processes are introduced. Note that predicted quantities
+are denoted by primes (e.g., y′).
+1. Extrapolation Suppression Constrained predictions were utilized in previous works
+to secure physically plausible predictions (e.g., an Ar concentration in the range of 0 % to
+100 %) [21]. This procedure is developed further and generalized in this work. In general ML
+models are well suited for interpolation but often fail to extrapolate beyond known input
+data.
+Hence, predictions below (beyond) the minimal (maximal) training reference ymin
+(ymax) are suppressed by folding them back three times to facilitate a more stable system
+state evolution and guarantee positive quantities when required (e.g., mass density, sputter
+yield):
+y′ → 2ymin − y′
+if y′ ≤ ymin
+(5a)
+y′ → 2ymax − y′
+if y′≥ ymax
+(5b)
+2. Particle conservation (diffusion) The absolute number of Ar, Al, and N atoms must be
+14
+
+conserved during bulk diffusion processes, which are modeled by the Diffusion-CVAE. This
+also demands a balance of the individual point defect populations. Using three corresponding
+constraints (e.g., based on Eqs. (2c)-(2a)) to determine them reduces the number of the ML
+model’s output descriptors, but may eventually contradict the extrapolation suppression
+constraint introduced in the preceding paragraph. For example, the conservation of Ar atoms
+prior and post diffusion (prediction) could be realized by determining the Ar population
+occupying Al lattice sites ρ′
+ArAl = ntot/n′
+tot(ρAri +ρArAl +ρArN)−ρ′
+Ari −ρ′
+ArN and using Eq. (1).
+However, some predictions may require n′
+ArAl to be negative (i.e., nAri + nArAl + nArN <
+n′
+Ari + n′
+ArN), even though the number of Ar atoms occupying Al lattice sites cannot be
+negative.
+Enforcing the constraint outlined in the preceding paragraph may resolve the
+issue, but being evaluated sequentially again may lead to a violation of particle conversation
+during diffusion processes.
+Thus, a more careful point defect balancing is required and
+introduced in the following.
+All Ar point defect population predictions (i.e., ρ′
+Ari + ρ′
+ArAl + ρ′
+ArN) are multiplied with a
+correction factor fAr:
+fAr = ntot
+n′
+tot
+ρAri + ρArAl + ρArN
+ρ′
+Ari + ρ′
+ArAl + ρ′
+ArN + 10−7
+(6)
+The deviation of predicted Al and N atoms prior/post diffusion is defined by ∆nAl and ∆nN,
+respectively:
+∆nAl =ntot(ρAli + ρAlN − ρvAl − ρNAl − ρ(N-N)Al − ρArAl)
+− n′
+tot(ρ′
+Ali + ρ′
+AlN − ρ′
+vAl − ρ′
+NAl − ρ′
+(N-N)Al − fArρ′
+ArAl)
+(7a)
+∆nN =ntot(ρNi + 2ρ(N-N)i + ρ(N-N)N − ρvN + ρNAl + 2ρ(N-N)Al − ρAlN − ρArN)
+− n′
+tot(ρ′
+Ni + 2ρ′
+(N-N)i + ρ′
+(N-N)N − ρ′
+vN + ρ′
+NAl + 2ρ′
+(N-N)Al − ρ′
+AlN − fArρ′
+ArN)
+(7b)
+with ntot as a function of the point defect population following Eq. (1). All defect populations
+15
+
+but anti-sites are compensated for the particle balancing using these deviations:
+ρ′
+vAl → n′
+totρ′
+vAl − ∆nAl
+ntot
+if ∆nAl< 0
+(8a)
+ρ′
+Ali → n′
+totρ′
+Ali + ∆nAl
+ntot
+if ∆nAl> 0
+(8b)
+ρ′
+vN → n′
+totρ′
+vN − ∆nN
+ntot
+if ∆nN < 0
+(8c)
+ρ′
+(N-N)N →
+n′
+totρ′
+(N-N)N +
+ρ′
+(N-N)N
+ρ′
+Ni+ρ′
+(N-N)N ∆nN
+ntot
+if ∆nN > 0
+(8d)
+ρ′
+Ni →
+n′
+totρ′
+Ni +
+ρ′
+Ni
+ρ′
+Ni+ρ′
+(N-N)N ∆nN
+ntot
+if ∆nN > 0
+(8e)
+All point defect populations, which have not been altered up this point, are scaled with the
+quotient n′
+tot/ntot to account for the changed total number of atoms, ensuring consistent
+predictions.
+c.
+Physics-separating artificial neural network
+Each CVAE (i.e., PSI-CVAE, Diffusion-
+CVAE) describes one physical process, separating one from another. The (trained) decoders
+are combined to form a PSNN that allows for an evolution in time by passing the surface
+state Ss from one surrogate model to another. The information flow of the PSNN is depicted
+in Figure 7. It resembles closely the information flow inherent to the physical simulations
+(Fig. 3. The input to the PSI-Decoder is a single particle sampled from the particle flux of
+the plasma, characterized by the particles’ kinetic energy Ekin, species s, and surface state Ss.
+It predicts the updated surface state S′
+s and emitted flux for each species Γout ′
+s
+. The former is
+fed together with the temperature T to the Diffusion-Decoder, which predicts a new surface
+state S′′
+s . It is passed on to the PSI-Decoder, establishing an recurrent link within the PSNN.
+Note that the direct correspondence of the physical simulations and the separated PSI-
+Decoder and Diffusion-Decoder structure allows for an efficient parameter space exploration,
+as outlined in Section III A. However, relying on single PSIs, the predictions after training
+may be subject to vastly different plasma conditions and are not limited to the specific flux
+ratios or ion energy distributions used for setting up the data set.
+16
+
+Figure 7. Schematic of the PSNN structure and information flow. Plasma dynamics can be imposed
+by hand, simulation or experiment.
+D.
+Hyperparameter study
+The HP of each CVAE (i.e., PSI-CVAE, Diffusion-CVAE) are optimized by applying
+an individual anisotropic self-adaptive evolution strategy with intermediate recombination
+(µ/µI, λ)-σSA-ESs. µ, λ, and σ refer to the number of parents, populations size, and step
+sizes (mutation strengths), respectively.
+Generalized and topic-wise related descriptions
+of this method can be found in [21, 60–62]. The HPs considered in this work and their
+initialization ranges are listed in Table I.
+The evolution strategies are inialized as (7/7I, 70)-σSA-ESs.
+The population sizes λ
+are reduced by one per generation over the course of 63 generations and, afterwards, kept
+constant. The numbers of parents are determined by µ = λ/7 (integer values are enforced)
+[63]. The ESs are conducted for 200 generations and, hence, end as (1, 7)-σSA-ESs.
+E.
+Production run: Reference experiment
+First, the experimental scenario considered for production as well as validation is outlined.
+Second, the fluxes onto the AlN surfaces required for the ML simulation are calculated and
+used to introduce an estimated process time.
+a.
+Reference experiment
+Ries et al.
+used a large-area multi-frequency capacitively
+coupled plasma (MFCCP) to sputter deposit AlN with Ar and N2 as working gases (Ar/N2
+gas inlet ratio equal 8/1) [64]. The electrical asymmetry effect was taken advantage of to
+17
+
+System state Ss:
+System state S"
+Plasma dynamics
+PSI-Decoder
+Diffusion-Decoder
+System state $
+flux Fouti
+temperature TTable I. The HPs to be optimized, their initialization range, and final values for the PSI-CVAE as
+well as Diffusion-CVAE.
+HP
+Init. range PSI-CVAE Diffusion-CVAE
+rc
+[0.0,1.0]
+0.50
+0.44
+α
+[10−5,1.0]
+0.47 ·10−2
+0.17
+rl-0
+[10−3,10−2] 9.14 ·10−3
+1.08 ·10−3
+nl-patience
+[4,7]
+9
+7
+λL2
+[0,10−4]
+2.23 ·10−7
+1.22 ·10−7
+nhl
+[1,5]
+1
+3
+nnpl
+[8,128]
+107
+155
+nls
+[1,6]
+1
+5
+β
+[10−1,10]
+56.69
+0.14
+nSA
+[1,102]
+102
+102
+nbatch,ideal
+[16,64]
+37
+53
+decouple the ion flux from the ion energy. The former was kept approximately constant and
+the latter was controlled by applying voltage waveform tailoring (i.e., adjusting the relative
+phase shift between the two excitation frequencies).
+Four cases with mean ion energies
+Eion of 47 eV, 53 eV, 57 eV, and 81 eV were considered. The predominant AlN surface
+orientation was found to vary as function of the mean ion energy: AlN(002) for Eion =47
+eV and Eion =53 eV as well as AlN(100) for Eion =57 eV and Eion =81 eV [64].
+In this work, the species most relevant for PSI (i.e., Al, N+, N+
+2 , Ar+) are sampled from
+the experimentally determined fluxes impinging onto the substrate (cf. next subsection). The
+kinetic energy Ekin of ions and Al neutrals are sampled from measured ion energy distribution
+functions (IEDFs) (depicted in Figure 11 of [64]) and from Monte Carlo transport simulations
+(Al in pure Ar, but assumed invariant), respectively [64–66]. A threshold for the IEDFs
+is imposed to avoid sampling from noise. Monte Carlo accept-reject sampling (rejection
+sampling) is used to determine the individual particle energies.
+The evolution and response of both monocrystalline systems (i.e., AlN(002), AlN(100))
+predicted by the PSNN is to be studied as a function of the mentioned four ion energy
+18
+
+distribution functions (IEDFs). Each case starts with ideal, defect free AlN and is run until
+a steady-state is reached, that is approximately 45 minutes (experimental process time).
+All cases are re-run 100 times to evaluate their statistics accurately. The final results are
+averaged over the last minute. Intrinsic stresses are determined as a function of the predicted
+lattice constants by utilizing the third-order Birch-Murnaghan EOS of the ideal, defect free
+AlN reference system as proposed and validated in [37]. The final stresses and compositions
+predicted by the PSNN are compared to experimentally measured reference values [64].
+Spurious Fe and O concentrations observed in the experiment are substituted with Al and
+N concentrations to define a comparable reference for the simulation.
+b.
+Flux and process time estimations
+The process time tp for npi particle impingements
+may be estimated by the sum over all reciprocal impingement rates, tp = �npi
+i=1 1/(ΓiA′
+RMD).
+Γin
+i
+and A′
+RMD denote the experimental particle flux onto the AlN surface and predicted
+RMD AlN surface area, respectively.
+A′
+RMD is computed as a function of the predicted
+lattice constant a′ and imposed surface orientation.
+The ion flux onto the target and substrate is assumed to be approximately equal for
+the given geometry and approximated by Γin
+ion = hnevB, with assumed edge-to-center ration
+h = 0.61, electron density ne = 5·1015 m−3, and Bohm velocity vB = 3.21·103 m/s (for Ar+
+ions and a given electron temperature of kBTe = 3 eV) [7, 67].
+The flux of Al neutrals onto the substrate is calculated from Γin
+Al = ctYAr+(352 eV) Γin
+ion =
+3.47·1018 m−2s−1 with the collisional transport coefficient ct = 0.6 obtained from Monte Carlo
+transport simulations (Al in pure Ar, but assumed invariant) as well as an Ar sputtering
+yield YAr+(352 eV) = 0.579 (clean Al target) [64, 65]. The Al flux from the target is obtained
+by multiplying the ion flux Γin
+ion with the Ar+ sputtering yield YAr+.
+The considered ion fluxes (i.e., Γin
+N+, Γin
+N+
+2 , Γin
+Ar+) are determined by assuming that the
+composition of the ion fluxes onto the substrate Γin
+ion resemble the volumetric composition.
+The total gas density is given by ng,tot = p/(kBTg) with the Boltzmann constant kb, and gas
+temperature Tg = 650 K [67]. The species specific gas densities are calculated as relative
+fractions assuming ng,N/ng,N2 = 1/9 and (ng,N + ng,N2)/ng,Ar = 1/8 [64]. Hence, the working
+gas approximately consists of 1.16 % N, 10.47 % N2, and 88.37 % Ar. The ion (Bohm) flux
+onto the substrate is split up accordingly: ΓAl+ = 3.11·1014 m−2s−1, ΓN+ = 1.14·1017 m−2s−1,
+ΓN+
+2 = 1.03 · 1018 m−2s−1, and ΓAr+ = 8.65 · 1018 m−2s−1. The contribution due to Al+ is
+neglected due to their rare occurrence.
+19
+
+IV.
+RESULTS
+A.
+Hyperparameter study
+Following the outlined evolution strategy with MCCV, an optimum set of HPs is de-
+termined and listed in Table I. As apparent, data augmentation by means of constrained
+mixup augmentation is beneficial for the Diffusion-CVAE (α = 0.17). This means that the
+hypothesis of linear superposition is accepted to some extend for the diffusion processes but
+declined for the PSIs (α = 0.47·10−2). Kernel regularization is found to be disadvantageous
+for either ML model (λL2 ≈ 10−7). The network structure of the Diffusion-CVAE (i.e., 3
+hidden layer with 155 nodes per layer) allows for higher order of complexity than the PSI-
+CVAE’s one (i.e., 1 hidden layer with 107 nodes per layer). It is also interesting to note that
+the optimum number of simulated annealing epochs is 100 for both ML models, which is
+the imposed upper boundary for this HP for the evolution strategies. Hence, the simulated
+annealing step is assumed to be of great use for the training procedure.
+In addition to the MAE, the performance of the PSI-CVAE and Diffusion-CVAE with
+their final set of HPs listed in Table I can be assessed by the coefficient of determination
+R2. It is calculated on the training, validation, and test set to equal 0.87, 0.86, and 0.87 for
+the PSI-CVAE as well as 0.94, 0.93, and 0.93 for the Diffusion-CVAE, respectively. These
+values ≳ 0.9 signify an accurate model approach (R2 = 1 signifies fully explained variance
+in the data). The negligible difference between the three subsets indicates that the ML
+models learned successfully to generalize on the training data. This finding is analyzed more
+thoroughly in the following by comparing the unnormalized mean absolute errors (MAEs) of
+each system property (e.g., mass density). It is important to note though that the reference
+data does not resemble any kind of ground truth but contains statistical fluctuations (e.g.,
+a single ion hitting the surface on a different surface sites is likely to inflict different kinds
+of defect structures) which intrinsically provide limits for the MAEs.
+The MAE of all considered defect populations are shown in Figure 8. The PSI-CVAE and
+Diffusion-CVAE is found to predict the defect structure accurately with errors that are of
+the order/below 0.1 %. The error of the PSI-CVAE’s predictions on the training, validation,
+and test set are barely distinguishable from each other, resembling excellent generalization.
+The Diffusion-CVAE is found to perform best on the training set, showing minor signs of
+20
+
+Figure 8. MAE of the unnormalized predictions on the point defect populations and data sets.
+Point defect types are listed on the x-axis.
+Table II. MAE of the unnormalized predictions on all data sets for the PSI-PSNN.
+Property
+Train. set Val. set Test set
+a (˚A)
+0.002
+0.002
+0.002
+∆Ef (eV)
+0.005
+0.005
+0.005
+B (GPa)
+3.029
+3.067
+3.055
+B′ (GPa)
+0.913
+0.922
+0.931
+Γout
+Al /Γin
+s (.)
+0.015
+0.016
+0.015
+Γout
+N /Γin
+s (.)
+0.089
+0.089
+0.087
+Γout
+Ar /Γin
+s (.)
+0.218
+0.219
+0.219
+Γout
+N2 /Γin
+s (.)
+0.139
+0.139
+0.140
+overfitting. However, the difference between the validation and test set is negligible.
+The high accuracy prediction of the PSI-CVAE and the Diffusion-CVAE on the lattice
+constant a, the formation energy ∆Ef, the bulk modulus B, and its derivative B′ are pre-
+sented in Table II and Table III, respectively. The almost interchangeable performance on
+the training, validation, and test set shows again that the models successfully learned to
+21
+
+Training set
+Validation set
+Test set
+X
+0.125
+PSI-CVAE
+Diffusion-CVAE
+0.100
+0.075
+0.050
+MA
+X
+0.025
+X
+0.000Table III. MAE of the unnormalized predictions on all data sets for the Diffusion-PSNN.
+Property Train. set Val. set Test set
+a (˚A)
+0.001
+0.001
+0.001
+∆Ef (eV)
+0.007
+0.008
+0.008
+B (GPa)
+2.905
+3.094
+3.124
+B′ (GPa)
+0.918
+0.963
+0.973
+generalize on the provided data. However, the MAEs of the emitted Al, N, Ar and N2 flux
+per incident flux, as listed in Table II, are relatively large when compared to typical sputter
+yields as well as reflection ratios in the considered regime of kinetic energies (i.e., Ekin in
+[0 eV, 300 eV]). It is argued that these larger errors do not signify bad performance, but
+are rather a consequence of the data assembly for the PSIs. One PSI data sample contains
+the information on a single PSI, which leads to the emission of, for example, none, one, or
+maybe two particles. This will be perceived as noise to the ML model, which consequently
+learns to predict the mean number of emitted particles per PSI for a given surface state.
+This inherently leads to relatively large MAEs but ultimately is exactly what the PSI-CVAE
+is meant to learn.
+B.
+Production run
+The production run resembles the reference experiment of AlN thin-film deposition for
+four discharge conditions as previously discussed. In the following, they are investigated for
+two surface orientations (100) and (002). Initially the emitted particle fluxes are discussed:
+Particles are emitted from the surface due to reflection of the incident particle or sputtering
+of surface atoms. It is observed that most fluxes reach a steady-state after a few seconds.
+Minor changes on the minute time-scale are observed only for three cases (i.e., Eion =47 eV:
+(002), Eion =53 eV: (002), Eion =57 eV: (100)) due to a change of the Al sticking probability
+of approximately 0.5 %. This transient variation is a side-effect of slowly evolving system
+states, described in detail later. All Al sticking coefficients are in between 98-99 %.
+The emitted per incident particle fluxes averaged over the last, 45th minute are shown
+22
+
+Figure 9. The emission of all film forming species per incident fluxes are presented for all considered
+IEDFs as well as surface orientations. Circle and error bars represent mean values and root-mean-
+squared deviations, respectively.
+in Figure 9 for all film forming flux combinations (i.e., the emission of Ar is omitted). No
+significant difference between the two surface orientations is recognizable, which is attributed
+to the considered ion energy regime of 30 to 100 eV. Higher ion energies are expected to
+present surface orientation dependent sputtering yields.
+The impingement of N+ and Ar+ ions leads to an almost similar removal of Al atoms,
+whereas Ar+ ions achieve a slightly increased Al sputtering yield (i.e., Γout
+Al /Γin
+N+ < Γout
+Al /Γin
+Ar+).
+This is attributed to elastic collisions of bombarding N+ ions with N surface atoms, distribut-
+ing the momentum more rapidly and evenly among them than Al atom. The displacement
+of N atoms in the subsurface regions leads to the temporary formation of (N-N)N close to
+the surface, where they eventually leave as N2. Higher ion energies lead to deeper collision
+cascades spawned with higher momenta. The proportionality with the mean ion energy
+indicates that for neither IEDF a relevant proportion of N+ ions directly form temporary
+(N-N)N at the surface (and desorb as N2).
+Bombarding N+
+2 ions are split apart when they hit the surface and, thus, inhibit a reduced
+individual momentum compared to the initially shared one. This favors an even stronger
+distribution of the momenta in the surface slab and, thus, lessens the likelihood of sputtering
+23
+
+0.3
+Fout: (002)
+(002)
+Tout.
+out.
+(002)
+Al
+N
+N2
+47 eV
+0.2
+53 eV
+0.1
+57 eV
+81 eV
+0.0
+0.3
+Tout.
+out.
+:(100)
+(100)
+(100)
+Al
+N2
+47 eV
+0.2
+53 eV
+0.1
+57 eV
+81 ev
+0.0Figure 10. Transient evolution of the most relevant point defect populations for all considered
+IEDFs as well as surface orientations. Error bars and the height of transparent region resemble
+the mean plus / minus the RMSD.
+Al atoms in the considered ion energy regime. Moreover, for smaller ion energies a shallower
+subsurface region is affected, which enables incident N+
+2 ions to directly form temporary
+(N-N)N at the surface before leaving as N2. This is reflected by the decreased flux ratio
+Γout
+N2 /Γin
+N+
+2 for increased ion energies.
+The transient evolutions of the most relevant point defect populations are shown in Fig-
+24
+
+2.4
+Eion
+pvn: (002)
+P(N-N)n: (002)
+47eV
+.8
+ 53 eV
+1.2
+57eV
+81eV
+0.0
+2.4
+Eion
+pvA1: (002)
+PAl: (002)
+(%)
+1.8
+47 eV
+53 eV
+1.2
+5 57 eV
+0.6
+81 eV
+0.0
+2.4
+Eion
+Pvn: (100)
+P(N-N)N: (100)
+47 eV
+1.8
+ 53 eV
+1.2
+57eV
+0.6
+81 eV
+0.0
+2.4
+Eion
+pvAl: (100)
+pAl:: (100)
+(%)
+1.8
+47eV
+53 eV
+fect
+1.2
+57 eV
+0.6
+_81eV
+0.0
+0
+10
+20
+30
+40
+0
+10
+20
+30
+40
+t (min)
+t (min)ure 10 for all considered IEDFs and surface orientations. The deposition onto AlN(002) with
+Eion =47 eV takes up to 30 minutes to reach a steady-state. The ongoing ion bombardment
+spawns collision cascades in the subsurface region, which once they have worn off may leave
+vacancies and interstitials behind. Sputtering events or the desorption of N2 remove atoms
+from the surface and, thus, facilitate the accumulation of vacancies. The Al and N vacancy
+populations are approximately equal. The Al interstitial population is greater than the N
+split interstitial population, and both exceed the corresponding vacancy populations. This
+is due forward sputtering (peening) of surface atoms as well as incorporation of energetic
+particles (i.e., N+, N2, small proportion of Al), which eventually either reside as interstitials
+or recombine with vacancies. IEDFs with slightly higher mean ion energies (i.e., 53 eV, 57
+eV) converge to a similar point defect structure with marginally increased N split and Al
+interstitial populations, but require significantly less time for equilibration. These require
+a few minutes and seconds for Eion =53 eV and Eion =57 eV, respectively. Therefore it is
+assumed that for Eion =47 eV scarcely sampled ions with relatively high kinetic energies
+push the systems to their final state. The likelihood for encountering such ions is naturally
+increased when increasing the mean ion energy.
+This effect is enhanced by a change of
+the IEDF shapes (i.e., narrow unimodal → narrow bimodal → broad unimodal). A more
+significantly increased mean ion energy of 81 eV leads to the evolution to a different sys-
+tem state with less Al and N vacancies (ρvAl ≈ ρvN) and more interstitials (ρAli > ρ(N-N)N).
+The evolution of the vacancy populations inhibits intermediate maxima after a few seconds.
+Subsequently, vacancies are removed due to recombination as described before and reach a
+steady-state after 10 seconds. The evolution of the point defect structures are depicted in
+Figure 10 for up to 45 minutes (and are available in the appendix for up to 100 seconds).
+The deposition onto AlN(100) leads to similar system dynamics for Eion =47 eV and
+Eion =53 eV. The Al and N vacancy populations are approximately equal too. But a greater
+number of N split and smaller number of Al interstitials are observed. Scarce Al atoms
+hitting the surface with relative high kinetic energies of up to 30 eV provide an insufficient
+momentum when penetrating the AlN(100) surfaces to be persistently incorporated, i.e.
+they end up atop the surface. Increasing the mean ion energy to 57 eV leads to a system
+evolution that requires up to 10 minutes to reach a steady-state that differs significantly
+from the previous one. The equilibration on the minute-time scale is again attributed to the
+contribution of only a small proportion of the incident ions with sufficient kinetic energies,
+25
+
+Figure 11. Transient evolution of the mass density for all considered IEDFs as well as surface
+orientations. Error bars and the height of transparent region resemble the mean plus / minus the
+root-mean-squared deviations.
+which are pushing the systems to their final state (cf. Eion =53 eV: (002)). The (N-N)N
+populations remain unchanged. The Al and N vacancy populations are doubled. Hence,
+the probability for the recombination of surface near Al vacancies and incident Al atoms is
+increased too. The final Al interstitial population are therefore even more than doubled. The
+point defect structure is characterized predominantly by Al and N Frenkel pairs (vacancies
+plus interstitials). The evolution to this new system state is caused by a change of the
+IEDF shapes.
+The IEDF with Eion =53 eV (narrow bimodal IEDF) and Eion =57 eV
+(broad unimodal IEDF) reaches up to 60 eV and 75 eV, respectively. The cases with the
+highest mean ion energies of 83 eV converge to a similar system state with slightly increased
+interstitial populations, but it takes only a few seconds.
+The evolution of the mass densities are presented in Figure 11. The equilibration time of
+the individual cases shows a consistent behavior. However, it is interesting to note that all
+cases converge to a similar mass density for the surface orientation (002). The accumulation
+of interstitials is balanced out by a corresponding volumetric expansion. In case of AlN(100),
+two final point defect structures were discussed in the preceding paragraph.
+These two
+system states are reflected by two distinctly separate mass densities. Higher ion energies
+26
+
+3.175
+(002)
+Eion
+47
+eV
+3.150
+53 eV
+3.125
+57 eV
+3.100
+81 eV
+3.075
+3.175
+Eion
+(100)
+47
+eV
+3.150
+53 eV
+3.125
+57
+eV
+3.100
+81
+eV
+3.075
+0
+10
+20
+30
+40
+t (min)Figure 12. The final composition (i.e., Al and Ar concentration cAl and cAr, respectively) for all
+considered IEDFs as well as surface orientations averaged over the last minute are compared to
+experimental reference values [64]. Circles and error bars represent mean values and root-mean-
+squared deviations.
+lead to a great number of Al as well as N Frenkel pairs, which do not alter the mass of the
+atomic system but cause stress and correspondingly a volumetric relaxation. Hence, smaller
+mass densities are observed.
+The composition of the deposited AlN(002) and AlN(100) thin films averaged over the
+last minute are shown in Figure 12 in comparison to experimental reference values [64]. A
+good agreement with the experiment is achieved when predicting stoichiometric AlN thin
+films even though the Ar concentration of 2.5 ± 0.1 % for Eion = 47 eV and Eion = 53 eV is
+not reproduced.
+The stresses predicted by the PSNN and measured in the experiment are presented in
+Figure 13 (a). An increasingly compressive stress is observed for greater mean ion energies in
+either case due to the enhanced ion bombardment induced point defect formation. Vacancies
+and interstitials cause tensile and compressive stresses, respectively. The interplay of all
+point defects define the film stresses in the ML simulation. However, the contributions due
+to Al interstitials dominate the stress formation due to their larger size and high formation
+energies [68]. This finding is illustrated by a similar dependence of the stresses and the
+negated Al interstitial populations (multiplied by -1) on the mean ion energy Eion, as shown
+in Figures 13. The preferential surface orientation was found to change from (002) to (100)
+in the experiment when increasing the mean ion energies from 47-53 eV to 57-81 eV [64].
+27
+
+45
+CAl
+30
+c
+Experiment
+CAr
+15
+Simulation: (002)
+Simulation: (100)
+48
+56
+64
+72
+80
+Eion (eV)Figure 13. The (a) final stress and (b) negated Al interstitial population for all considered IEDFs
+as well as surface orientations averaged over the last minute are compared to experimental reference
+values [64]. Circles and error bars represent mean values and root-mean-squared deviations.
+By comparison with the ML prediction for the (002) surface orientation, it can be inferred
+that the predicted stresses for the two IEDFs with smaller mean ion energies (i.e., 47 eV, 53
+eV) are overestimated. However, from comparison with the prediction for the (100) surface
+orientation, the two IEDFs with greater mean ion energies (i.e., 57 eV, 81 eV) are in excellent
+agreement with the experiment. The change of the predominant surface orientation (i.e.,
+(002)→(100)) observed in the experiment may be attributed to the reduced compressive
+stresses predicted to reach up to -12 GPa for (002), compared to -8 GPa for (100).
+28
+
+0
+a)
+Experiment
+Simulation: (002)
+-3
+(GPa)
+Simulation: (100)
+stress α 
+-12
+48
+56
+64
+72
+80
+Eion (eV)
+0.0
+(b)
+Simulation: (002)
+Simulation: (100)
+-0.6
+.1.2
+-pAli
+-1.8
+2.4
+48
+56
+64
+72
+80
+Eion (eV)V.
+CONCLUSION
+This work is meant to further advance the development of data-driven plasma-surface
+interaction models with atomic fidelity [21]. Reactive processes (i.e., sputtering and depo-
+sition of AlN in an Ar/N2 discharges) are taken into account. A data-generating scheme
+is proposed that overcomes the burden of computationally too demanding simulations (i.e.,
+hybrid RMD/tfMC) and, hence, undersampled parameter spaces. The latter are effectively
+populated by evolving randomly sampled system states Ss by means of random PSIs (i.e.,
+species s, kinetic energy Ekin) and diffusion processes (i.e., temperature T). The effect of a
+single PSI on the deposited film is estimated by cleaving and reinforcing the corresponding
+bulk structure in surface normal direction [37]. A PSNN is used to separate the PSIs from
+the diffusion processes, which allows for a more efficient data-generation and enforcement of
+physics-constraints (e.g., particle conservation during bulk diffusion).
+The trained PSNN model is applied to an experimental reference sputter deposition of
+AlN by taking the corresponding particle fluxes and IEDFs with mean ion energies in the
+range of 47-81 eV into account [64]. Ar+ ions are found to remove more Al than N atoms
+from the surface.
+The inverse is observed for N+ ions, which spawn collision cascades
+that distribute their momenta more rapidly with the N surface atoms. This facilitates the
+temporary formation of (N-N)N at the very surface that eventually leave as N2. N+
+2 ions
+are split up when they hit the surface and, thus, spawn two collision cascades with reduced
+individual momenta compared to the initially shared one.
+A diminishing amount of Al
+atoms is sputtered and a shallower subsurface region is effected. The latter allows for the
+direct formation of (N-N)N at the surface and subsequent emission as N2. Atomic nitrogen is
+rarely sputtered by either ion species. Higher mean ion energies decrease the outgoing flux
+of N2 due N+
+2 ion bombardment but increase the formation of persistent, deeper (N-N)N.
+The predicted film depositions take either a few seconds or up to 30 minutes to reach their
+respective steady-state. Long equilibration times are observed when rare ions whose kinetic
+energy originates from the high energy tail of the IEDF push the systems to their final states.
+The latter is found to be dependent on the imposed surface orientation. In particular, a
+greater Al interstitials population is predicted for AlN(002) than for AlN(100). This point
+defect type predominantly determines the compressive stress evolution in the deposited AlN
+thin films.
+The stresses predicted by the PSNN are quantitatively and qualitatively in
+29
+
+good agreement with the experimental reference values in spite of neglecting for instance
+thermal stresses or point defect annihilation at grain boundaries. The ML model predicts
+stoichiometric AlN that is observed in the experiment too.
+In summary, 200 million plasma-surface interactions and diffusion processes were pre-
+dicted with high physical fidelity (hybrid RMD/tfMC). This enabled the evolution of 800
+AlN systems (100 × four IEDFs × two surface orientations) in time for up to 45 minutes.
+It took about 34 hours to perform all machine learning predictions with a single GPU.
+Hence, predictions can be readily extended to cover up the total experimental deposition
+time of up to hours when required. In contrast, conducting the same case study with hybrid
+RMD/tfMC simulations is unattainable as it would take more than approximately 8 million
+CPU years.
+ACKNOWLEDGEMENT
+Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
+– Project-ID 138690629 – TRR 87 and – Project-ID 434434223 – SFB 1461. The authors
+thank Dr.-Ing. S. Ries from Ruhr University Bochum, S. Karimi Aghda, M. Sc. from RWTH
+Aachen University, and L. Vialetto, Ph.D. from Kiel University for fruitful discussions.
+DATA AVAILABILITY
+The data that support the findings of this study are available from the corresponding
+author upon reasonable request.
+ORCID
+T. Gergs: https://orcid.org/0000-0001-5041-2941
+T. Mussenbrock: https://orcid.org/0000-0001-6445-4990
+J. Trieschmann: https://orcid.org/0000-0001-9136-8019
+30
+
+APPENDIX
+Figure 14.
+Schematic of the CVAE network structure.
+The shape of the data is provided in
+parenthesis. Machine learning operations are indicated by colored arrows. The inputs and outputs
+for the PSI-CVAE are given by x = {Ekin, s, Ss} and y = {Γout
+s
+, Ss}, respectively. The inputs
+and outputs for the Diffusion-CVAE are given by x = {T, Ss} and y = {Ss}, respectively.
+31
+
+Concat
+Dense
+Calc
+EnforceConstraints
+2
+μls,r
+Train
+phase?
+True
+False
+Ols,r
+N(0, 1)
+N(0,1)
+m
+m
+nhl = 2
+nhl = 2
+(nis)
+(SIu)Figure 15. Transient evolution of the most relevant point defect populations for all considered
+IEDFs as well as surface orientations. Error bars and the height of transparent region resemble
+the mean plus / minus the root-mean-squared deviations.
+32
+
+2.4
+Eion
+pvn: (002)
+P(N-N)n: (002)
+47eV
+1.8
+ 53 eV
+1.2
+57 eV
+81eV
+0.0
+2.4
+Eion
+pvA1: (002)
+PAl: (002)
+(%)
+1.8
+47 eV
+53 eV
+1.2
+57eV
+0.6
+81 eV
+0.0
+2.4
+Eion
+Pvn: (100)
+P(N-N)N: (100)
+47 eV
+ 53 eV
+1.2
+57eV
+0.6
+81 eV
+0.0
+2.4
+Eion
+pvAr: (100)
+PAl;: (100)
+1.8
+47 eV
+53 eV
+fect
+1.2
+557 eV
+0.6
+81eV
+0.0
+0
+25
+50
+75
+100
+0
+25
+50
+75
+100
+t (s)
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+

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+arXiv:2301.11957v1  [math.OC]  27 Jan 2023
+A continuity result for the adjusted normal cone operator
+Marco Castellania, Massimiliano Giulia
+aDepartment of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via
+Vetoio, L’Aquila, Italy
+Abstract
+The concept of adjusted sublevel set for a quasiconvex function was introduced in [5] and the local
+existence of a norm-to-weak∗ upper semicontinuous base-valued submap of the normal operator
+associated to the adjusted sublevel set was proved. When the space is finite dimensional, a globally
+defined upper semicontinuous base-valued submap is obtained taking the intersection of the unit
+sphere, which is compact, with the normal operator, which is closed. Unfortunately, this technique
+does not work in the infinite dimensional case.
+We propose a partition of unity technique to
+overcome this problem in Banach spaces. Application is given to a quasiconvex quasioptimization
+problem through the use of a new existence result for generalized quasivariational inequalities which
+is based on the Schauder fixed point theorem.
+Keywords:
+Cone upper semicontinuity, Normal operator, Quasivariational inequality,
+Quasioptimization
+1. Introduction
+The notion of upper semicontinuity seems to be unappropriate for cone-valued maps and hence
+modified definitions were been introduced and studied [5, 8, 12]. The normal cone operator to the
+adjusted sublevel sets of a quasiconvex function f defined on a Banach space was introduced in [5]
+and it was proved to be both quasimonotone and cone upper semicontinuous. In particular, the
+authors showed that the normal cone operator admits a locally defined base-valued submap being
+norm-to-weak∗ upper semicontinuous. In [4], when the space is Euclidian, the authors obtained a
+globally defined upper semicontinuous base-valued submap taking the convex hull of the normalized
+Email addresses: [email protected] (Marco Castellani), [email protected]
+(Massimiliano Giuli)
+Preprint submitted to Journal of LATEX Templates
+January 31, 2023
+
+normal operator which is the intersection of the unit sphere, which is compact, with the normal
+operator, which is closed. Since the unit sphere is not compact in the infinite dimensional case, this
+approach is unsuccessful in a Banach space X.
+The first aim of this paper is to overcome this problem by using a partition of unity tech-
+nique.
+Theorem 3 states the existence of a norm-to-weak∗ upper semicontinuous submap A :
+X \ arg min f ⇒ X∗ such that each A(x) is a nonempty weak∗ compact convex set not containing
+the origin which generates the normal cone to the adjusted sublevel set at x. Subsequently, we
+establish an existence result (Theorem 5) for a generalized quasivariational inequality which im-
+proves the famous Tan’s result [14]. Finally, combining both results, we present an application to
+quasioptimization problems.
+In the last part of this introduction, we present some preliminary notions and results. Let X be
+a real Banach space with norm ∥ · ∥, X∗ its topological dual with norm ∥ · ∥∗, and ⟨·, ·⟩ the duality
+pairing between X∗ and X. From now on, unless otherwise indicated, the spaces X and X∗ will be
+equipped by the strong (norm) topology s and the weak∗ topology w∗, respectively. The closed unit
+balls in X and X∗ are denoted by B and B∗, respectively. Given a nonempty set A ⊆ X and x ∈ X,
+dist(x, A) = inf{∥y−x∥ : y ∈ A} is the distance of x from A and B(A, r) = {x ∈ X : dist(x, A) ≤ r}
+is the neighbourhood of A with radius r ≥ 0.
+A subset K of X∗ is a cone if for each x∗ ∈ K and scalar t > 0, the product tx∗ ∈ K (note
+that some authors define cone with the scalar t ranging over all non-negative scalars). Clearly the
+empty set is a cone.
+Let K ⊆ X∗ be a cone. A convex subset A of K is called a base if K = {tx∗ : t ≥ 0, x∗ ∈ A}
+and 0 ̸∈ w∗- cl A, where w∗- cl denotes the closure with respect to the weak∗ topology. Clearly the
+empty set is a base of the empty cone. Vice versa, if K admits a nonempty base then K is a convex
+cone such that {0} ⊊ K. In particular, if the base is compact then K is closed.
+The domain and the graph of a set-valued map Φ : X ⇒ X∗ are denoted by dom Φ and gph Φ,
+respectively. The map Φ is norm-to-weak∗ upper semicontinuous at x ∈ X if for every open set Ω
+such that Φ(x) ⊆ Ω, there exists a neighbourhood Ux of x such that Φ(x′) ⊆ Ω, for all x′ ∈ Ux.
+The map Φ is norm-to-weak∗ closed at x ∈ dom f if for each x∗ ∈ Φ(x) and for each net {(xα, x∗
+α)}
+with x∗
+α ∈ Φ(xα) which converges to (x, x∗) in the s × w∗ topology, we have that x∗ ∈ Φ(x). The
+map is norm-to-weak∗ closed if its graph is closed with respect to the topology s × w∗. The Closed
+Graph Theorem states that a closed-valued map Φ with values in a compact set is norm-to-weak∗
+2
+
+upper semicontinuous if and only if it is norm-to-weak∗ closed.
+When we are dealing with a cone-valued map Φ, the concept of norm-to-weak∗ upper semicon-
+tinuity is not appropriate to picture the behaviour of Φ and it is convenient to slightly alter the
+definition. The cone-valued map Φ is called
+• norm-to-weak∗ cone upper semicontinuous at x ∈ X if for every open cone Ω such that
+Φ(x) ⊆ Ω ∪ {0}, there exists a neighbourhood Ux of x such that Φ(x′) ⊆ Ω ∪ {0}, for all
+x′ ∈ Ux;
+• norm-to-weak∗ base upper semicontinuous at x ∈ X if there exist a neighbourhood Ux of x
+and a set-valued map A : Ux ⇒ X∗ such that A(x′) is a base of Φ(x′) for each x′ ∈ Ux and A
+is norm-to-weak∗ upper semicontinuous at x.
+Some remarks are needed. If Φ is norm-to-weak∗ base upper semicontinuous at x ∈ X then there
+exists a neighbourhood Ux of x such that Φ(x′) ̸= {0} for each x′ ∈ Ux. Instead, if Φ is norm-to-
+weak∗ cone upper semicontinuous at x ̸∈ dom Φ then there exists a neighbourhood Ux of x such
+that Φ(x′) ⊆ {0} for each x′ ∈ Ux. Therefore, if Φ is norm-to-weak∗ cone upper semicontinuous
+and Φ(x) admits a base for each x ∈ X then dom Φ is closed. Moreover the norm-to-weak∗ base
+upper semicontinuity of Φ at x implies the norm-to-weak∗ cone upper semicontinuity at the same
+point. The reverse implication holds if Φ(x) admits a base and Φ(x′) ̸= {0} for all x′ in a suitable
+neighbourhood of x [5].
+The norm-to-weak∗ cone upper semicontinuity of a map implies its norm-to-weak∗ closedness if
+the map admits a compact base at every point [7, Proposition 2.3]. The same proof works for local
+closedness.
+Theorem 1. Let Φ : X ⇒ X∗ be a cone-valued map which is norm-to-weak∗ cone upper semicon-
+tinuous at x ∈ dom Φ. If Φ(x) has a compact base then Φ is norm-to-weak∗ closed at x.
+2. The result
+Let f : X → R∪{+∞} be an extended-valued function. Define for any λ ∈ R∪{+∞} the sublevel
+and the strict sublevel set of f at level λ by Sλ = {x ∈ X : f(x) ≤ λ} and S<
+λ = {x ∈ X : f(x) < λ},
+respectively. Clearly S∞ = X and S<
+∞ = dom f. The function f is quasiconvex if Sλ is convex for
+all λ ∈ R. Now, we recall the notion of adjusted level set introduced in [5].
+3
+
+Definition 2.1. Let f : X → R ∪ {+∞} and x ∈ X. The adjusted sublevel set of f at x is
+Sa
+f (x) =
+
+
+
+Sf(x)
+if x ∈ arg min f
+Sf(x) ∩ B(S<
+f(x), ρx)
+if x /∈ arg min f
+where ρx = dist(x, S<
+f(x)).
+Note that S<
+f(x) ⊆ Sa
+f (x) ⊆ Sf(x) for all x ∈ X; moreover the convexity of the adjusted sublevel
+sets characterizes the quasiconvexity of the function.
+Theorem 2 (Proposition 2.4 in [5]). The extended-valued function f is quasiconvex if and only if
+Sa
+f (x) is convex, for every x ∈ X.
+To any function f we associate the set-valued map N a : X ⇒ X∗ defined by
+N a(x) = {x∗ ∈ X∗ : ⟨x∗, y − x⟩ ≤ 0, ∀y ∈ Sa
+f (x)}
+In [5, Proposition 3.5] the authors showed that N a is norm-to-weak∗ base upper semicontinuous
+under regularity assumptions on f. Combining Theorem 1 and Proposition 3.5 in [5], the following
+result can be easily deduced.
+Corollary 2.1. Let f be quasiconvex and lower semicontinuous at x ∈ dom f \ arg min f. If there
+exists λ < f(x) such that int Sλ ̸= ∅ then N a is closed at x.
+Such a result has been proved in [4] and, with weaker assumptions but in a finite dimensional
+case, in [1]. Taking advantage of Corollary 2.1, the authors deduce [4, Proposition 4.4] the upper
+semicontinuity of the normalized map N a ∩ S : Rn \ arg min f ⇒ B, being the unit sphere S in
+Rn compact. Moreover, the assumptions in [4, Proposition 4.4] guarantee that the convex hull of
+N a ∩ S is an upper semicontinuous base-valued submap of N a. Our aim is to extend their result to
+the infinite dimensional case. Since the sphere is not weak∗ compact in the dual of a Banach space,
+the previous technique does not work.
+Theorem 3. Let f : X → R ∪ {+∞} be proper, quasiconvex and lower semicontinuous. Assume
+that for each x ∈ X \ arg min f there exists λ < f(x) such that int Sλ ̸= ∅. Then there exists a
+norm-to-weak∗ upper semicontinuous set-valued map A : X \ arg min f ⇒ B∗ such that A(x) is a
+compact base of N a(x), for all x.
+4
+
+Proof. For the first step of the proof, we argue as in [5, Lemma 3.6]. Let z ∈ X \ arg min f
+be fixed.
+Choose z0 ∈ X and λ ∈ R such that λ < f(z) and z0 ∈ int S<
+λ .
+Since f is lower
+semicontinuous, there exists ε > 0 such that
+z0 + 2εB ⊆ S<
+λ ⊆ S<
+f(x),
+∀x ∈ z + εB
+Thus, for every x ∈ z + εB and for every
+x∗ ∈ N <(x) = {x∗ ∈ X∗ : ⟨x∗, y − x⟩ ≤ 0, ∀y ∈ S<
+f(x)}
+we obtain the following:
+⟨x∗, z0 + 2εu − x⟩ ≤ 0,
+∀u ∈ B
+It follows that
+2ε∥x∗∥∗ = 2ε sup
+u∈B
+⟨x∗, u⟩
+≤
+⟨x∗, x − z0⟩
+=
+⟨x∗, z − z0⟩ + ⟨x∗, x − z⟩
+≤
+⟨x∗, z − z0⟩ + ε∥x∗∥∗
+Thus,
+⟨x∗, z − z0⟩ ≥ ε∥x∗∥∗,
+∀x ∈ z + εB, x∗ ∈ N <(x)
+Set Hz = {x∗ ∈ X∗ : ⟨x∗, z − z0⟩ = ε}. Obviously, for every x ∈ z + εB we have N <(x) ∩ Hz ⊆ B∗
+and, since N a(x) ⊆ N <(x), the set N a(x) ∩ Hz ⊆ B∗ is a compact base for the cone N a(x). Now,
+following the proof of [5, Proposition 3.5], we get the norm-to-weak∗ upper semicontinuity of the
+set-valued map Az : z + εB ⇒ X∗ defined by Fz(x) = N a(x) ∩ Hz, for all x ∈ z + εB.
+The last step of the proof consists in finding the selection A as convex combination of the local
+maps Az through a partition of unity technique.
+Since X \ arg min f is paracompact, there exists a locally finite open covering U = {Ui : i ∈ I}
+where every Ui ∈ U is a subset of some ball z + εB: let us denote by Ai the map Az corresponding
+to the ball z + εB. Moreover, there is a partition of unity {λi : i ∈ I} subordinate to U such that
+each λi : X \ arg min f → [0, 1] is continuous, the finite sum �
+i∈I λi(y) = 1 for any y and λi(y) = 0
+for each y ̸∈ Ui. For every x ∈ X \ arg min f, let I(x) = {i ∈ I : λi(x) > 0}, which is nonempty and
+finite, and define the map A : X \ arg min f ⇒ X∗ as follows
+A(x) =
+�
+i∈I(x)
+λi(x)Ai(x)
+5
+
+Clearly A(x) is a compact base of N a(x), for all x. Moreover, since the values of A are all contained
+in the compact ball B∗, the norm-to-weak∗ upper semicontinuity of A is equivalent to prove that
+the graph of A is closed with respect to the s × w∗ topology. Assume that the net {xα} converges
+to x. Since all the λi are continuous, it is not restrictive to assume that I(x) ⊆ I(xα) for all α and
+we get:
+A(xα) =
+�
+i∈I(x)
+λi(xα)Ai(xα) +
+�
+i∈I(xα)\I(x)
+λi(xα)Ai(xα)
+Moreover, from the continuity of the functions λi, we deduce
+lim
+α
+�
+i∈I(xα)\I(x)
+λi(xα) = 1 − lim
+α
+�
+i∈I(x)
+λi(xα) = 0
+(1)
+Now, let {x∗
+α} be a net which weakly∗ converges to x∗ and such that x∗
+α ∈ A(xα) for any α. Then,
+there exist x∗
+i,α ∈ Ai(xα) for every i ∈ I(xα) such that
+x∗
+α =
+�
+i∈I(x)
+λi(xα)x∗
+i,α +
+�
+i∈I(xα)\I(x)
+λi(xα)x∗
+i,α
+(2)
+The second addend of (2) weakly∗ converges to zero since, thanks to (1), it converges to zero in
+norm
+������
+�
+i∈I(xα)\I(x)
+λi(xα)x∗
+i,α
+������
+∗
+≤
+�
+i∈I(xα)\I(x)
+λi(xα)∥x∗
+i,α∥∗ ≤
+�
+i∈I(xα)\I(x)
+λi(xα)
+On the other hand, without loss of generality, we may assume that {x∗
+i,α} weakly∗ converges to
+some x∗
+i , for every i ∈ I(x). Since Ai has closed graph, we obtain x∗
+i ∈ Ai(x) and x∗ ∈ A(x) follows
+from (2) taking the weak∗ limit.
+✷
+3. An application
+In this section, our aim is to consider a special optimization problem, called quasioptimization
+problem, and to provide an existence result for this problem through the study of an associated
+generalized quasivariational inequality where Theorem 3 plays a key role. We start establishing a
+new existence result for a generalized quasivariational inequality without requiring any assumption
+of monotonicity.
+Let C be a nonempty subset of X and T : C ⇒ X∗ and K : C ⇒ C be two set-valued maps;
+the generalized quasivariational inequality GQV I(T, K) consists in finding
+x ∈ K(x) such that ∃x∗ ∈ T (x) with ⟨x∗, y − x⟩ ≥ 0,
+∀y ∈ K(x)
+6
+
+One of the most classic existence results for GQV I(T, K) in the infinite dimensional setting is due
+to Tan and it was originally stated for locally convex topological vector spaces. We recall that the
+set-valued map K : C ⇒ C is said to be lower semicontinuous if for every open set Ω the lower
+inverse image {x ∈ C : K(x) ∩ Ω ̸= ∅} is open in C. Moreover K is called compact if K(C) is
+contained in a compact subset of C.
+Theorem 4 (Theorem 1 in [14]). Let C be compact and convex and K be closed and lower semi-
+continuous with nonempty convex values. Assume that T is norm-to-norm upper semicontinuous
+with nonempty norm compact convex values, then GQV I(T, K) has a solution.
+The existence of solutions for GQV I(T, K) can be obtained with a weaker continuity assumption
+on T than in Theorem 4 if the space X is normed. To this purpose, we need to recall the notion
+of inside point of a convex set that appeared in 1956 in a paper by Michael [13]. The convex set
+S ⊆ C is a face of C if x1, x2 ∈ C, t ∈ (0, 1) and tx1 + (1 − t)x2 ∈ S imply x1, x2 ∈ S. Let FC be
+the (possibly empty) collection of all proper closed faces of cl C, which is the closure of C
+Definition 3.1. A point x ∈ C is an inside point if it is not in any proper closed face of cl C.
+Denote by
+I(C) = C \
+�
+S∈FC
+S
+the set of the inside points of C.
+A comparison with other notions of relative interior is given in [9, 10]. Thanks to this concept
+of interior point, we can define the following family of convex sets
+D(X) = {C ⊆ X : C is convex and I(cl C) ⊆ C}
+It was proved [13] that D(X) contains all the convex sets which are either closed, or with nonempty
+interior, or finite dimensional. In particular, when X is finite dimensional the class D(X) coincides
+with the family of all convex sets. Now we are in position to state and prove our existence result.
+Let us denote by fix K the set of the fixed points of K.
+Theorem 5. Let C be convex and K be a compact and lower semicontinuous set-valued map with
+nonempty values in D(X), and fix K closed. Assume that T is norm-to-weak∗ upper semicontinuous
+with nonempty weak∗ compact convex values, then GQV I(T, K) has a solution.
+7
+
+Proof. Notice that K admits a continuous selection thanks to [10, Theorem 3.2]. Hence the
+Schauder fixed point theorem as formulated in [11, Proposition 6.3.2] guarantees fix K ̸= ∅.
+Let us consider the set-valued map F : fix K ⇒ X defined as
+F(x) =
+�
+x∗∈T (x)
+{y ∈ X : ⟨x∗, y − x⟩ < 0} =
+�
+y ∈ X :
+max
+x∗∈T (x)⟨x∗, y − x⟩ < 0
+�
+Clearly, F has convex values. To prove that F has open graph in fix K × X, it is sufficient to show
+that the function m : fix K × X → R defined as
+m(x, y) =
+max
+x∗∈T (x)⟨x∗, y − x⟩
+is upper semicontinuous.
+First, fix K is compact since closed subset of the compact set which
+contains K(C). From [2, Lemma 17.8], the subset T (fix K) is weak∗ compact; hence, it is norm
+bounded. Thanks to [2, Corollary 6.40] the duality pairing ⟨·, ·⟩ restricted to T (fix K)× X is jointly
+continuous, where X has its norm topology and X∗ has its weak∗ topology; hence, [2, Lemma 17.30]
+guarantees the upper semicontinuity of m.
+By contradiction, assume that F(x) ∩ K(x) ̸= ∅ for all x ∈ fix K. Fix (x0, y0) ∈ gph K and
+define the map K0 : C ⇒ C as
+K0(x) =
+
+
+
+K(x)
+if x ̸= x0
+{y0}
+if x = x0
+K0 is compact and lower semicontinuous, and K0(x) ∈ D(X) for every x ∈ C. From [10, Theorem
+3.2] the map K0 admits a continuous selection, hence K is locally selectionable (see Definition
+1.10.1 in [3]). From [3, Proposition 1.10.4] we deduce that also F ∩ K is locally selectionable and
+[3, Proposition 1.10.2] guarantees that F ∩ K has a continuous selection f : fix K → C. Therefore,
+the set-valued map Υ : C ⇒ C defined as
+Υ(x) =
+
+
+
+K(x)
+if x /∈ fix K
+{f(x)}
+if x ∈ fix K
+is lower semicontinuous [10, Lemma 2.3] with values in the class D(X). Hence [10, Theorem 3.2]
+guarantees that f can be extended to a continuous selection ϕ for Υ. The Schauder fixed point
+theorem guarantees that ϕ has a fixed point, that is, there exists x ∈ C such that x = ϕ(x) ∈ Υ(x).
+Clearly x ∈ fix K and this implies x = f(x) ∈ F(x) which is absurd.
+Therefore, there exists
+8
+
+x ∈ fix K such that F(x) ∩ K(x) = ∅, that is,
+min
+y∈K(x) max
+x∗∈T (x)⟨x∗, y − x⟩ ≥ 0
+Invoking the Sion’s minimax theorem we deduce that
+max
+x∗∈T (x) min
+y∈K(x)⟨x∗, y − x⟩ ≥ 0
+which means that x solves the generalized quasivariational inequality.
+✷
+Remark 3.1. Let us compare our result with Theorem 4 due to Tan. The first difference is about
+the setting: Tan’s result works in a locally convex topological vector space instead Theorem 5 is
+stated in a Banach space. Nevertheless, the other assumptions of Theorem 5 are rather weaker
+than the ones in Theorem 4. Maybe, the most significant improvement consists in requiring the
+norm-to-weak∗ upper semicontinuity of T instead of the stronger norm-to-norm upper semiconti-
+nuity. Moreover, the values of T are assumed weakly∗ compact instead of norm compact. Also the
+assumptions on K are weaker. In Theorem 4 the map K is closed, which implies the closedness of
+K(x), for all x. Conversely, in Theorem 5 we require only the closedness of fix K, that is necessary
+for the closedness of K, and K(x) may not be closed but belonging to the class D(X) only. Lastly,
+we do not assume the compactness of C, not even its closedness, but only the fact that K(C) is
+contained in a compact set.
+Taking advantage of Theorem 5 and the good properties of the normal operator Na, our last
+aim is to obtain an existence result for a quasioptimization problem through the study of a suitable
+associated generalized quasivariational inequality.
+A quasioptimization problem is an optimization problem in which the constraint set is subject
+to modifications depending on the considered point. Given C ⊆ X nonempty, K : C ⇒ C and
+f : C → R, a quasioptimization problem consists in finding
+x ∈ K(x) such that f(x) ≤ f(y),
+∀y ∈ K(x)
+Clearly, if K(x) = C for all x ∈ C, quasioptimization problem reduces to a classical optimization
+problem.
+Theorem 6. Let C be convex and K be a compact and lower semicontinuous set-valued map with
+nonempty values in D(X), and fix K closed. Assume that f is continuous and quasiconvex, then
+the quasioptimization problem has a solution.
+9
+
+Proof. Let T : C ⇒ X∗ be defined as
+T (x) =
+
+
+
+B∗
+if x ∈ arg min f
+A(x)
+if x /∈ arg min f
+where A is the norm-to-weak∗ upper semicontinuous set-valued map obtained in Theorem 3. Since
+arg min f is closed and A(x) ⊆ B∗, then T is norm-to-weak∗ upper semicontinuous. In this way,
+thanks to Theorem 5, it follows that GQV I(T, K) has a solution x ∈ C. Clearly, if x ∈ arg min f,
+then f(x) ≤ f(y) for all y ∈ K(x). Instead, if x /∈ arg min f, then it results that
+x∗ ∈ T (x) = A(x) ⊆ N a(x) \ {0}
+Hence, x is a solution to the generalized variational inequality associated to the operator N a \ {0}
+and the feasible set K(x). Thanks to [6, Proposition 3.2], the thesis follows.
+✷
+Theorem 6 extends Proposition 4.5 in [4] which is stated in a finite dimensional space and
+requires also the compactness of C and the closedness of K.
+References
+[1] S. Al-Homidan, N. Hadjisavvas, L. Shaalan, Transformation of quasiconvex functions to elim-
+inate local minima, J. Optim. Theory Appl. 177 (2018) 93–105.
+[2] C.D. Aliprantis, K.C. Border, Infinite dimensional analysis. A hitchhikers guide, Springer-
+Verlag, third ed., Berlin, 2006.
+[3] J.P. Aubin, A. Cellina, Differential inclusions. Set-valued maps and viability theory, Springer-
+Verlag, Berlin, 1984.
+[4] D. Aussel, J. Cotrina, Quasimonotone quasivariational inequalities: existence results and ap-
+plications, J. Optim. Theory Appl. 158 (2013) 637–652.
+[5] D. Aussel, N. Hadjisavvas, Adjusted sublevel sets, normal operator, and quasi-convex program-
+ming, SIAM J. Optim. 16 (2005) 358–367.
+[6] D. Aussel, J.J. Ye, Quasiconvex programming with locally starshaped constraint region and
+applications to quasiconvex MPEC, Optimization 55 (2006) 433–457.
+10
+
+[7] M. Bianchi, N. Hadjisavvas, R. Pini, Continuity and maximal quasimonotonicity of normal
+cone operators, Stud. Univ. Babeş-Bolyai Math. 67 (2022) 31–45.
+[8] J. Borde, J.-P Crouzeix, Continuity properties of the normal cone to the level sets of a quasi-
+convex function, J. Optim. Theory Appl. 66 (1990) 415–429.
+[9] M. Castellani, M. Giuli, An existence result for quasiequilibrium problems in separable Banach
+spaces, J. Math. Anal. Appl. 425 (2015) 85–95.
+[10] M. Castellani, M. Giuli, Existence of quasiequilibria in metric vector spaces, J. Math. Anal.
+Appl. 484 (2020) 123751.
+[11] A. Granas, J. Dugundji, Fixed point theory, Springer-Verlag, New York, 2003.
+[12] D.T. Luc, J.-P. Penot, Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353
+(2001) 4095–4121.
+[13] E. Michael, Continuous selections. I, Ann. of Math. 63 (1956) 361–382.
+[14] N.X. Tan, Quasi-variational inequalities in topological linear locally convex Hausdorff spaces,
+Math. Nachr. 122 (1985) 231–245.
+11
+

5dFKT4oBgHgl3EQf-S4-/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf,len=276
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='11957v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='OC]  27 Jan 2023  A continuity result for the adjusted normal cone operator  Marco Castellania, Massimiliano Giulia  aDepartment of Information Engineering, Computer Science and Mathematics, University of L’Aquila, Via  Vetoio, L’Aquila, Italy  Abstract  The concept of adjusted sublevel set for a quasiconvex function was introduced in [5] and the local  existence of a norm-to-weak∗ upper semicontinuous base-valued submap of the normal operator  associated to the adjusted sublevel set was proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' When the space is finite dimensional, a globally  defined upper semicontinuous base-valued submap is obtained taking the intersection of the unit  sphere, which is compact, with the normal operator, which is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Unfortunately, this technique  does not work in the infinite dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  We propose a partition of unity technique to  overcome this problem in Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Application is given to a quasiconvex quasioptimization  problem through the use of a new existence result for generalized quasivariational inequalities which  is based on the Schauder fixed point theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Keywords:  Cone upper semicontinuity, Normal operator, Quasivariational inequality,  Quasioptimization  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Introduction  The notion of upper semicontinuity seems to be unappropriate for cone-valued maps and hence  modified definitions were been introduced and studied [5, 8, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The normal cone operator to the  adjusted sublevel sets of a quasiconvex function f defined on a Banach space was introduced in [5]  and it was proved to be both quasimonotone and cone upper semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' In particular, the  authors showed that the normal cone operator admits a locally defined base-valued submap being  norm-to-weak∗ upper semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' In [4], when the space is Euclidian, the authors obtained a  globally defined upper semicontinuous base-valued submap taking the convex hull of the normalized  Email addresses: marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='castellani@univaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='it (Marco Castellani), massimiliano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='giuli@univaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='it  (Massimiliano Giuli)  Preprint submitted to Journal of LATEX Templates  January 31, 2023  normal operator which is the intersection of the unit sphere, which is compact, with the normal  operator, which is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Since the unit sphere is not compact in the infinite dimensional case, this  approach is unsuccessful in a Banach space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  The first aim of this paper is to overcome this problem by using a partition of unity tech-  nique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Theorem 3 states the existence of a norm-to-weak∗ upper semicontinuous submap A :  X \\ arg min f ⇒ X∗ such that each A(x) is a nonempty weak∗ compact convex set not containing  the origin which generates the normal cone to the adjusted sublevel set at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Subsequently, we  establish an existence result (Theorem 5) for a generalized quasivariational inequality which im-  proves the famous Tan’s result [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Finally, combining both results, we present an application to  quasioptimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  In the last part of this introduction, we present some preliminary notions and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let X be  a real Banach space with norm ∥ · ∥, X∗ its topological dual with norm ∥ · ∥∗, and ⟨·, ·⟩ the duality  pairing between X∗ and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' From now on, unless otherwise indicated, the spaces X and X∗ will be  equipped by the strong (norm) topology s and the weak∗ topology w∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The closed unit  balls in X and X∗ are denoted by B and B∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Given a nonempty set A ⊆ X and x ∈ X,  dist(x, A) = inf{∥y−x∥ : y ∈ A} is the distance of x from A and B(A, r) = {x ∈ X : dist(x, A) ≤ r}  is the neighbourhood of A with radius r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  A subset K of X∗ is a cone if for each x∗ ∈ K and scalar t > 0, the product tx∗ ∈ K (note  that some authors define cone with the scalar t ranging over all non-negative scalars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Clearly the  empty set is a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Let K ⊆ X∗ be a cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' A convex subset A of K is called a base if K = {tx∗ : t ≥ 0, x∗ ∈ A}  and 0 ̸∈ w∗- cl A, where w∗- cl denotes the closure with respect to the weak∗ topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Clearly the  empty set is a base of the empty cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Vice versa, if K admits a nonempty base then K is a convex  cone such that {0} ⊊ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' In particular, if the base is compact then K is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  The domain and the graph of a set-valued map Φ : X ⇒ X∗ are denoted by dom Φ and gph Φ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The map Φ is norm-to-weak∗ upper semicontinuous at x ∈ X if for every open set Ω  such that Φ(x) ⊆ Ω, there exists a neighbourhood Ux of x such that Φ(x′) ⊆ Ω, for all x′ ∈ Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  The map Φ is norm-to-weak∗ closed at x ∈ dom f if for each x∗ ∈ Φ(x) and for each net {(xα, x∗  α)}  with x∗  α ∈ Φ(xα) which converges to (x, x∗) in the s × w∗ topology, we have that x∗ ∈ Φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The  map is norm-to-weak∗ closed if its graph is closed with respect to the topology s × w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The Closed  Graph Theorem states that a closed-valued map Φ with values in a compact set is norm-to-weak∗  2  upper semicontinuous if and only if it is norm-to-weak∗ closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  When we are dealing with a cone-valued map Φ, the concept of norm-to-weak∗ upper semicon-  tinuity is not appropriate to picture the behaviour of Φ and it is convenient to slightly alter the  definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The cone-valued map Φ is called  norm-to-weak∗ cone upper semicontinuous at x ∈ X if for every open cone Ω such that  Φ(x) ⊆ Ω ∪ {0}, there exists a neighbourhood Ux of x such that Φ(x′) ⊆ Ω ∪ {0}, for all  x′ ∈ Ux;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  norm-to-weak∗ base upper semicontinuous at x ∈ X if there exist a neighbourhood Ux of x  and a set-valued map A : Ux ⇒ X∗ such that A(x′) is a base of Φ(x′) for each x′ ∈ Ux and A  is norm-to-weak∗ upper semicontinuous at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Some remarks are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' If Φ is norm-to-weak∗ base upper semicontinuous at x ∈ X then there  exists a neighbourhood Ux of x such that Φ(x′) ̸= {0} for each x′ ∈ Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Instead, if Φ is norm-to-  weak∗ cone upper semicontinuous at x ̸∈ dom Φ then there exists a neighbourhood Ux of x such  that Φ(x′) ⊆ {0} for each x′ ∈ Ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Therefore, if Φ is norm-to-weak∗ cone upper semicontinuous  and Φ(x) admits a base for each x ∈ X then dom Φ is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Moreover the norm-to-weak∗ base  upper semicontinuity of Φ at x implies the norm-to-weak∗ cone upper semicontinuity at the same  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The reverse implication holds if Φ(x) admits a base and Φ(x′) ̸= {0} for all x′ in a suitable  neighbourhood of x [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  The norm-to-weak∗ cone upper semicontinuity of a map implies its norm-to-weak∗ closedness if  the map admits a compact base at every point [7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The same proof works for local  closedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let Φ : X ⇒ X∗ be a cone-valued map which is norm-to-weak∗ cone upper semicon-  tinuous at x ∈ dom Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' If Φ(x) has a compact base then Φ is norm-to-weak∗ closed at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The result  Let f : X → R∪{+∞} be an extended-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Define for any λ ∈ R∪{+∞} the sublevel  and the strict sublevel set of f at level λ by Sλ = {x ∈ X : f(x) ≤ λ} and S<  λ = {x ∈ X : f(x) < λ},  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Clearly S∞ = X and S<  ∞ = dom f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The function f is quasiconvex if Sλ is convex for  all λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Now, we recall the notion of adjusted level set introduced in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  3  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let f : X → R ∪ {+∞} and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The adjusted sublevel set of f at x is  Sa  f (x) =  \uf8f1  \uf8f2  \uf8f3  Sf(x)  if x ∈ arg min f  Sf(x) ∩ B(S<  f(x), ρx)  if x /∈ arg min f  where ρx = dist(x, S<  f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Note that S<  f(x) ⊆ Sa  f (x) ⊆ Sf(x) for all x ∈ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' moreover the convexity of the adjusted sublevel  sets characterizes the quasiconvexity of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Theorem 2 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='4 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The extended-valued function f is quasiconvex if and only if  Sa  f (x) is convex, for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  To any function f we associate the set-valued map N a : X ⇒ X∗ defined by  N a(x) = {x∗ ∈ X∗ : ⟨x∗, y − x⟩ ≤ 0, ∀y ∈ Sa  f (x)}  In [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='5] the authors showed that N a is norm-to-weak∗ base upper semicontinuous  under regularity assumptions on f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Combining Theorem 1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='5 in [5], the following  result can be easily deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let f be quasiconvex and lower semicontinuous at x ∈ dom f \\ arg min f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' If there  exists λ < f(x) such that int Sλ ̸= ∅ then N a is closed at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Such a result has been proved in [4] and, with weaker assumptions but in a finite dimensional  case, in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Taking advantage of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='1, the authors deduce [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='4] the upper  semicontinuity of the normalized map N a ∩ S : Rn \\ arg min f ⇒ B, being the unit sphere S in  Rn compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Moreover, the assumptions in [4, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='4] guarantee that the convex hull of  N a ∩ S is an upper semicontinuous base-valued submap of N a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Our aim is to extend their result to  the infinite dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Since the sphere is not weak∗ compact in the dual of a Banach space,  the previous technique does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let f : X → R ∪ {+∞} be proper, quasiconvex and lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Assume  that for each x ∈ X \\ arg min f there exists λ < f(x) such that int Sλ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Then there exists a  norm-to-weak∗ upper semicontinuous set-valued map A : X \\ arg min f ⇒ B∗ such that A(x) is a  compact base of N a(x), for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' For the first step of the proof, we argue as in [5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let z ∈ X \\ arg min f  be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Choose z0 ∈ X and λ ∈ R such that λ < f(z) and z0 ∈ int S<  λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Since f is lower  semicontinuous, there exists ε > 0 such that  z0 + 2εB ⊆ S<  λ ⊆ S<  f(x),  ∀x ∈ z + εB  Thus, for every x ∈ z + εB and for every  x∗ ∈ N <(x) = {x∗ ∈ X∗ : ⟨x∗, y − x⟩ ≤ 0, ∀y ∈ S<  f(x)}  we obtain the following:  ⟨x∗, z0 + 2εu − x⟩ ≤ 0,  ∀u ∈ B  It follows that  2ε∥x∗∥∗ = 2ε sup  u∈B  ⟨x∗, u⟩  ≤  ⟨x∗, x − z0⟩  =  ⟨x∗, z − z0⟩ + ⟨x∗, x − z⟩  ≤  ⟨x∗, z − z0⟩ + ε∥x∗∥∗  Thus,  ⟨x∗, z − z0⟩ ≥ ε∥x∗∥∗,  ∀x ∈ z + εB, x∗ ∈ N <(x)  Set Hz = {x∗ ∈ X∗ : ⟨x∗, z − z0⟩ = ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Obviously, for every x ∈ z + εB we have N <(x) ∩ Hz ⊆ B∗  and, since N a(x) ⊆ N <(x), the set N a(x) ∩ Hz ⊆ B∗ is a compact base for the cone N a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Now,  following the proof of [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='5], we get the norm-to-weak∗ upper semicontinuity of the  set-valued map Az : z + εB ⇒ X∗ defined by Fz(x) = N a(x) ∩ Hz, for all x ∈ z + εB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  The last step of the proof consists in finding the selection A as convex combination of the local  maps Az through a partition of unity technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Since X \\ arg min f is paracompact, there exists a locally finite open covering U = {Ui : i ∈ I}  where every Ui ∈ U is a subset of some ball z + εB: let us denote by Ai the map Az corresponding  to the ball z + εB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Moreover, there is a partition of unity {λi : i ∈ I} subordinate to U such that  each λi : X \\ arg min f → [0, 1] is continuous, the finite sum �  i∈I λi(y) = 1 for any y and λi(y) = 0  for each y ̸∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' For every x ∈ X \\ arg min f, let I(x) = {i ∈ I : λi(x) > 0}, which is nonempty and  finite, and define the map A : X \\ arg min f ⇒ X∗ as follows  A(x) =  �  i∈I(x)  λi(x)Ai(x)  5  Clearly A(x) is a compact base of N a(x), for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Moreover, since the values of A are all contained  in the compact ball B∗, the norm-to-weak∗ upper semicontinuity of A is equivalent to prove that  the graph of A is closed with respect to the s × w∗ topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Assume that the net {xα} converges  to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Since all the λi are continuous, it is not restrictive to assume that I(x) ⊆ I(xα) for all α and  we get:  A(xα) =  �  i∈I(x)  λi(xα)Ai(xα) +  �  i∈I(xα)\\I(x)  λi(xα)Ai(xα)  Moreover, from the continuity of the functions λi, we deduce  lim  α  �  i∈I(xα)\\I(x)  λi(xα) = 1 − lim  α  �  i∈I(x)  λi(xα) = 0  (1)  Now, let {x∗  α} be a net which weakly∗ converges to x∗ and such that x∗  α ∈ A(xα) for any α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Then,  there exist x∗  i,α ∈ Ai(xα) for every i ∈ I(xα) such that  x∗  α =  �  i∈I(x)  λi(xα)x∗  i,α +  �  i∈I(xα)\\I(x)  λi(xα)x∗  i,α  (2)  The second addend of (2) weakly∗ converges to zero since, thanks to (1), it converges to zero in  norm  ������  �  i∈I(xα)\\I(x)  λi(xα)x∗  i,α  ������  ∗  ≤  �  i∈I(xα)\\I(x)  λi(xα)∥x∗  i,α∥∗ ≤  �  i∈I(xα)\\I(x)  λi(xα)  On the other hand, without loss of generality, we may assume that {x∗  i,α} weakly∗ converges to  some x∗  i , for every i ∈ I(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Since Ai has closed graph, we obtain x∗  i ∈ Ai(x) and x∗ ∈ A(x) follows  from (2) taking the weak∗ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  ✷  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' An application  In this section, our aim is to consider a special optimization problem, called quasioptimization  problem, and to provide an existence result for this problem through the study of an associated  generalized quasivariational inequality where Theorem 3 plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' We start establishing a  new existence result for a generalized quasivariational inequality without requiring any assumption  of monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Let C be a nonempty subset of X and T : C ⇒ X∗ and K : C ⇒ C be two set-valued maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  the generalized quasivariational inequality GQV I(T, K) consists in finding  x ∈ K(x) such that ∃x∗ ∈ T (x) with ⟨x∗, y − x⟩ ≥ 0,  ∀y ∈ K(x)  6  One of the most classic existence results for GQV I(T, K) in the infinite dimensional setting is due  to Tan and it was originally stated for locally convex topological vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' We recall that the  set-valued map K : C ⇒ C is said to be lower semicontinuous if for every open set Ω the lower  inverse image {x ∈ C : K(x) ∩ Ω ̸= ∅} is open in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Moreover K is called compact if K(C) is  contained in a compact subset of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Theorem 4 (Theorem 1 in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let C be compact and convex and K be closed and lower semi-  continuous with nonempty convex values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Assume that T is norm-to-norm upper semicontinuous  with nonempty norm compact convex values, then GQV I(T, K) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  The existence of solutions for GQV I(T, K) can be obtained with a weaker continuity assumption  on T than in Theorem 4 if the space X is normed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' To this purpose, we need to recall the notion  of inside point of a convex set that appeared in 1956 in a paper by Michael [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The convex set  S ⊆ C is a face of C if x1, x2 ∈ C, t ∈ (0, 1) and tx1 + (1 − t)x2 ∈ S imply x1, x2 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let FC be  the (possibly empty) collection of all proper closed faces of cl C, which is the closure of C  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' A point x ∈ C is an inside point if it is not in any proper closed face of cl C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Denote by  I(C) = C \\  �  S∈FC  S  the set of the inside points of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  A comparison with other notions of relative interior is given in [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Thanks to this concept  of interior point, we can define the following family of convex sets  D(X) = {C ⊆ X : C is convex and I(cl C) ⊆ C}  It was proved [13] that D(X) contains all the convex sets which are either closed, or with nonempty  interior, or finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' In particular, when X is finite dimensional the class D(X) coincides  with the family of all convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Now we are in position to state and prove our existence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Let us denote by fix K the set of the fixed points of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let C be convex and K be a compact and lower semicontinuous set-valued map with  nonempty values in D(X), and fix K closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Assume that T is norm-to-weak∗ upper semicontinuous  with nonempty weak∗ compact convex values, then GQV I(T, K) has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Notice that K admits a continuous selection thanks to [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Hence the  Schauder fixed point theorem as formulated in [11, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='2] guarantees fix K ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Let us consider the set-valued map F : fix K ⇒ X defined as  F(x) =  �  x∗∈T (x)  {y ∈ X : ⟨x∗, y − x⟩ < 0} =  �  y ∈ X :  max  x∗∈T (x)⟨x∗, y − x⟩ < 0  �  Clearly, F has convex values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' To prove that F has open graph in fix K × X, it is sufficient to show  that the function m : fix K × X → R defined as  m(x, y) =  max  x∗∈T (x)⟨x∗, y − x⟩  is upper semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  First, fix K is compact since closed subset of the compact set which  contains K(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' From [2, Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='8], the subset T (fix K) is weak∗ compact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' hence, it is norm  bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Thanks to [2, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='40] the duality pairing ⟨·, ·⟩ restricted to T (fix K)× X is jointly  continuous, where X has its norm topology and X∗ has its weak∗ topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' hence, [2, Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='30]  guarantees the upper semicontinuity of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  By contradiction, assume that F(x) ∩ K(x) ̸= ∅ for all x ∈ fix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Fix (x0, y0) ∈ gph K and  define the map K0 : C ⇒ C as  K0(x) =  \uf8f1  \uf8f2  \uf8f3  K(x)  if x ̸= x0  {y0}  if x = x0  K0 is compact and lower semicontinuous, and K0(x) ∈ D(X) for every x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' From [10, Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='2] the map K0 admits a continuous selection, hence K is locally selectionable (see Definition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='1 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' From [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='4] we deduce that also F ∩ K is locally selectionable and  [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='2] guarantees that F ∩ K has a continuous selection f : fix K → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Therefore,  the set-valued map Υ : C ⇒ C defined as  Υ(x) =  \uf8f1  \uf8f2  \uf8f3  K(x)  if x /∈ fix K  {f(x)}  if x ∈ fix K  is lower semicontinuous [10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='3] with values in the class D(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Hence [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='2]  guarantees that f can be extended to a continuous selection ϕ for Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The Schauder fixed point  theorem guarantees that ϕ has a fixed point, that is, there exists x ∈ C such that x = ϕ(x) ∈ Υ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Clearly x ∈ fix K and this implies x = f(x) ∈ F(x) which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Therefore, there exists  8  x ∈ fix K such that F(x) ∩ K(x) = ∅, that is,  min  y∈K(x) max  x∗∈T (x)⟨x∗, y − x⟩ ≥ 0  Invoking the Sion’s minimax theorem we deduce that  max  x∗∈T (x) min  y∈K(x)⟨x∗, y − x⟩ ≥ 0  which means that x solves the generalized quasivariational inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  ✷  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let us compare our result with Theorem 4 due to Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' The first difference is about  the setting: Tan’s result works in a locally convex topological vector space instead Theorem 5 is  stated in a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Nevertheless, the other assumptions of Theorem 5 are rather weaker  than the ones in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Maybe, the most significant improvement consists in requiring the  norm-to-weak∗ upper semicontinuity of T instead of the stronger norm-to-norm upper semiconti-  nuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Moreover, the values of T are assumed weakly∗ compact instead of norm compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Also the  assumptions on K are weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' In Theorem 4 the map K is closed, which implies the closedness of  K(x), for all x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Conversely, in Theorem 5 we require only the closedness of fix K, that is necessary  for the closedness of K, and K(x) may not be closed but belonging to the class D(X) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Lastly,  we do not assume the compactness of C, not even its closedness, but only the fact that K(C) is  contained in a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Taking advantage of Theorem 5 and the good properties of the normal operator Na, our last  aim is to obtain an existence result for a quasioptimization problem through the study of a suitable  associated generalized quasivariational inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  A quasioptimization problem is an optimization problem in which the constraint set is subject  to modifications depending on the considered point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Given C ⊆ X nonempty, K : C ⇒ C and  f : C → R, a quasioptimization problem consists in finding  x ∈ K(x) such that f(x) ≤ f(y),  ∀y ∈ K(x)  Clearly, if K(x) = C for all x ∈ C, quasioptimization problem reduces to a classical optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let C be convex and K be a compact and lower semicontinuous set-valued map with  nonempty values in D(X), and fix K closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Assume that f is continuous and quasiconvex, then  the quasioptimization problem has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Let T : C ⇒ X∗ be defined as  T (x) =  \uf8f1  \uf8f2  \uf8f3  B∗  if x ∈ arg min f  A(x)  if x /∈ arg min f  where A is the norm-to-weak∗ upper semicontinuous set-valued map obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Since  arg min f is closed and A(x) ⊆ B∗, then T is norm-to-weak∗ upper semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' In this way,  thanks to Theorem 5, it follows that GQV I(T, K) has a solution x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Clearly, if x ∈ arg min f,  then f(x) ≤ f(y) for all y ∈ K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Instead, if x /∈ arg min f, then it results that  x∗ ∈ T (x) = A(x) ⊆ N a(x) \\ {0}  Hence, x is a solution to the generalized variational inequality associated to the operator N a \\ {0}  and the feasible set K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content=' Thanks to [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='2], the thesis follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='  ✷  Theorem 6 extends Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
+page_content='5 in [4] which is stated in a finite dimensional space and  requires also the compactness of C and the closedness of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dFKT4oBgHgl3EQf-S4-/content/2301.11957v1.pdf'}
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+JONES-WENZL IDEMPOTENTS IN THE TWISTED
+I-BUNDLE OF THE M¨OBIUS BAND
+DIONNE IBARRA
+Abstract. The Jones-Wenzl idempotent plays a vital role in quan-
+tum invariants of 3-manifolds and the colored Jones polynomial;
+it also serves as a useful tool for simplifying computations and
+proving theorems in knot theory. The relative Kauffman bracket
+skein module (RKBSM) for surface I-bundles and manifolds with
+marked boundaries have a well understood algebraic structure due
+to the work of J. H. Przytycki and T. T. Q. Lˆe.
+It has been
+well documented that the RKBSM of the I-bundle of the annulus
+and the twisted I-bundle of the M¨obius band have a distinct alge-
+braic structures even though the manifolds are homeomorphic. In
+this paper we will give various results on Jones-Wenzl idempotents
+in the twisted I-bundle of the M¨obius band when it is partially
+closed through the crosscap of the M¨obius band. In doing so we
+will uncover properties that differ from properties of Jones-Wenzl
+idempotents in Ann × I.
+Contents
+1.
+Introduction
+1
+1.1.
+Acknowledgements
+2
+2.
+Introduction to Jones-Wenzl idempotents
+2
+3.
+Crossingless connection in the M¨obius band
+9
+4.
+Jones-Wenzl idempotents in the M¨obius band
+10
+References
+15
+1. Introduction
+The Jones-Wenzl idempotent, discovered by V. F. R. Jones in [Jon],
+is an idempotent element in the Temperley-Lieb algebra. Originally,
+it was described as a certain symmetrizer using the Artin braid group
+Date: January 13, 2023.
+2020 Mathematics Subject Classification. Primary: 57K10. Secondary: 57K31.
+Key words and phrases. Jones-Wenzl idempotents, M¨obius band, twisted I-
+bundles, Kauffman bracket skein module, relative Kauffman bracket skein module.
+1
+arXiv:2301.04859v1  [math.GT]  12 Jan 2023
+
+2
+DIONNE IBARRA
+and the projection to the Temperley-Lieb algebra. In the late 1980’s,
+H. Wenzl in [Wen] discovered a recursive formula to the Jones-Wenzl
+idempotent.
+This formula is now widely used as the definition, see
+[Lic2].
+The Jones-Wenzl idempotent has played a significant role in defining
+quanum invariants of knots and 3-manifolds. For example, W. B. R.
+Lickorish’s Kauffman bracket skein theoretic approach to the Witten-
+Reshetikhin-Turaev 3-manifold invariants in [Lic1] uses a linear combi-
+nation of the trace (closure) of the idempotent elements along a framed
+knot or link. Similarly, the colored Jones polynomial quantum knot in-
+variant is defined by taking the trace of the nth Jones-Wenzl idempotent
+along a 0-framed knot in S3, see [Le, PBIMW]. These idempotent ele-
+ments are also used to decorate the edges of a tetrahedra to obtain the
+quantum 6j-symbols that are used in the definition of the Turaev-Viro
+quantum 3-manifold invariants, see [TV].
+The Jones-Wenzl idempotent has been a vital tool for simplifying
+computations and proving theorems in knot theory. An example of this
+is seen in X. Cai’s proof of a closed formula for the Gram determinant
+of type A in [Cai] and a closed formula for its generalization in [BIMP].
+In fact, this paper was conceived by needing properties of the Jones-
+Wenzl idempotents when it is closed in the twisted I-bundle of the
+M¨obius band in hopes to take a similar approach to [Cai] and [BIMP]
+to prove a closed formula for the Gram determinant of type Mb.
+In Section 2 we introduce the original definition of Jones-Wenzl idem-
+potents and also the RKBSM of the twisted I-bundle of the M¨obius
+band, then in Section 3 we give an illustration of the two different
+models of the M¨obius band as well as the antipodal properties of the
+crosscap. In Section 4 we prove many corollaries to the trace of Jones-
+Wenzl idempotents intersecting or surrounding the crosscap, then we
+end with a formula for when n − 1 curves from fn are closed around
+the crosscap and the last arc is closed through the crosscap.
+1.1. Acknowledgements. This work was supported by the Australian
+Research Council grant DP210103136.
+2. Introduction to Jones-Wenzl idempotents
+The first formal definition of the Temperley-Lieb algebra, denoted
+by TLn, was given by R. J. Baxter in [Bax] while describing the work
+of physicists N. Temperley and E. Lieb in [TL]. Jones independently
+introduced TLn in [Jon] while working on von Neumann algebras.
+
+JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND
+3
+Definition 2.1. Let R be a commutative ring with unity and d ∈ R.
+Let n ∈ N be fixed, then the nth Temperley-Lieb algebra, TLn,
+is defined to be the unital associative algebra over R with generators
+e1, . . . , en−1, identity element 1n, and relations
+(1) eiejei = ei for |i − j| = 1,
+(2) eiej = ejei for |i − j| > 1,
+(3) e2
+i = dei.
+L. H. Kauffman in [Kau], motivated by utilizing the Kauffman bracket,
+considered the Temperley-Lieb algebra over R = Z[A±1], where A is an
+indeterminate and d = −A2 − A−2. He then constructed a graphical
+interpretation using tangles.
+We will consider an n-tangle to be a rectangular shaped disk with
+n marked boundary points on the left (input points) and n marked
+boundary points on the right (output points). Kauffman’s graphical
+interpretation of the Temperley-Lieb algebra is obtained from the basis
+of crossingless tangles where the identity element corresponds to an n-
+tangle with n parallel arcs in which each ith input point is connected
+to the ith output point, and each ei corresponds to an n-tangle that
+has one input and one output cap on the ith and i + 1th position as
+illustrated in Figure 1.
+For simplicity we will label an arc by n to
+denote n parallel arcs as shown in Figure 1a.
+n
+(a) Identity element.
+n − i − 1
+i − 1
+(b) ei.
+Figure 1. The graphical interpretation of TLn.
+Definition 2.2. The n-tangle algebra is an R-module with basis
+elements consisting of n-tangles where multiplication of two n-tangles
+is defined by identifying the right side of the first n-tangle to the left
+side of the second n-tangle while respecting the boundary points and
+by letting any resulting trivial curve be denoted by d, see Figure 2 for
+an illustrative example. Kauffman’s diagrammatic interpretation of the
+Temperley-Lieb algebra, also known as the diagrammatic algebra,
+is a subalgebra of the n-tangle algebra. It is generated by tangles with
+no crossings where homotopically trivial curves are denoted by d ∈ R.
+
+4
+DIONNE IBARRA
+e3e3 =
+= d
+= de3.
+Figure 2. An illustration of multiplication.
+Theorem 2.3. [Kau] The diagrammatic algebra is isomorphic to TLn
+and can be thought of as a diagrammatic interpretation of it.
+We will give Jones’ constructive definition of the Jones-Wenzl idem-
+potent by using the relative Kauffman bracket skein module (RKBSM)
+and the Artin braid group before introducing Wenzl’s recursive formula.
+In doing so, we will first introduce the RKBSM and emphasize that the
+RKBSM of the twisted I-bundle of the M¨obius band and the RKBSM
+of Ann × I are different modules even though the two manifolds are
+homeomorphic. This will give us motivation to study the Jones-Wenzl
+idempotent in the twisted I-bundle of the M¨obius band. Furthermore,
+the corollaries and proposition in the last section will show that there
+are distinct differences when simple closed curves intersect the crosscap.
+Definition 2.4. Let M be an oriented 3-manifold and {xi}2n
+i=1 be the
+set of 2n framed points on ∂M. Let I = [−1, 1], and let Lfr(2n) be the
+set of all relative framed links (which consists of all framed links in M
+and all framed arcs, I×I, where I×∂I is connected to framed points on
+the boundary of M) up to ambient isotopy while keeping the boundary
+fixed in such a way that L ∩ ∂M = {xi}2n
+1 . Let R be a commutative
+ring with unity, A ∈ R be invertible, and let Ssub
+2,∞(2n) be the submodule
+of RLfr(2n) that is generated by the Kauffman bracket skein relations:
+(i) L+ − AL0 − A−1L∞, and
+(ii) L ⊔ ⃝
+⃝
+⃝ + (A2 + A−2)L,
+where ⃝
+⃝
+⃝ denotes the framed unknot and the skein triple (L+, L0, L∞)
+denotes three framed links in M that are identical except in a small
+3-ball in M where the difference is shown in Figure 3.
+Then, the relative Kauffman bracket skein module (RKBSM)
+of M is the quotient:
+S2,∞(M, {xi}2n
+1 ; R, A) = RLfr(2n)/Ssub
+2,∞(2n).
+Theorem 2.5. [Prz] Let F be a surface with ∂F ̸= ∅. If F is orientable
+then let M = F × I , otherwise let M = F ˆ×I. Let all {xi}2n
+1 be marked
+
+JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND
+5
+(a) L+.
+(b) L0.
+(c) L∞.
+Figure 3. The skein triple.
+points that lie on ∂F × {0}. Then S2,∞(M, {xi}2n
+1 ; R, A) is a free R-
+module whose basis is composed of relative links in F without trivial
+components. When n = 0, the empty link is also a generator.
+J. H. Przytycki’s corollary to Theorem 2.5 explicitly details the dif-
+ferences between the RKBSM of Ann × I and the RKBSM of Mbˆ×I
+even though both manifolds are homeomorphic to the solid torus.
+Corollary 2.6. [Prz]
+(1) S2,∞(Ann×I, {xi}2n
+1 ; R, A) where {xi}2n
+1 are located in the outer
+boundary component of the annulus is a free R[x]-module with
+Dn =
+�2n
+n
+�
+basis elements, where x denotes the homotopically
+nontrivial curve in the annulus and d = −A2 − A−2 denotes
+the homotopically trivial curve in the annulus. The basis is the
+set of all crossingless connections in the annulus with no trivial
+components or boundary parallel curves.
+(2) S2,∞(Mbˆ×I, {xi}2n
+1 ; R, A) is a free R-module. The standard ba-
+sis contains an infinite number of elements of the form bzi, bxzi
+for i ≥ 0, where x denotes the simple closed curve that intersects
+the M¨obius band once, z denotes the boundary parallel curve of
+the M¨obius band, and b is an element in the set of crossingless
+connections in the M¨obius band with no trivial components or
+boundary parallel curves for which the arcs do not intersect the
+crosscap.
+The rest of the elements in the standard basis are
+from a finite number of crossingless connections consisting of
+a collection of n − k arcs for 0 ≤ k < n that non-trivially in-
+tersects the crosscap. Among the finite collection there are
+�2n
+k
+�
+crossingless connections that intersect the crosscap n−k times.
+Definition 2.7. The Artin braid group is defined by the following
+group presentation:
+Bn =< σ1, . . . , σn−1; σiσj = σjσi for |i−j| > 1, σi±1σiσi±1 = σiσi±1σi > .
+The Artin braid group can be interpreted using n-tangles where el-
+ements of Bn are positive braids. More precisely, an element in Bn
+
+6
+DIONNE IBARRA
+can be represented as an n-tangle with positive crossings such that the
+boundary of each arc is attached to one input and one output point
+and when read from left to right each generator σi corresponds to the
+positive crossing of the ith and i + 1th arcs. That is, the ith generator
+element σi is a positive transposition of the ith and i + 1th arcs.
+Furthermore, there exists an epimorphism p : Bn → Sn from the
+Artin braid group to the permutation group that uniquely interprets a
+braid word. Let p be defined by sending generators of Bn, σi, to the
+transpositions in Sn; si = (i, i+1) for 1 ≤ i ≤ n−1. For a permutation
+π ∈ Sn, let bπ denote the unique minimal positive braid word such that
+p(bπ) = π.
+Definition 2.8. Let Z[A±1] denote the ring of Laurent polynomials
+in the variable A and Q(A) denote the field of rational functions in
+the variable A; whose elements are functions of the form P/Q where
+P, Q ∈ Z[A±1]. We define an unnormalized A-symmetrizer, Fn ∈
+Z[A±1]Bn, by the following
+Fn =
+�
+π∈Sn
+(A3)|π|bπ,
+and the normalized symmetrizer, also known as the A-symmetrizer
+and denoted by fn ∈ Q(A)Bn, by the formula
+fn =
+1
+[n]A4!Fn,
+where |π| denotes the minimal length of the permutation π written as
+elementary transposition generators, [n]A4 = 1+A4+A8+· · ·+A4(n−1) =
+A4n−1
+A4−1 and [n]A4! =
+n�
+i=1
+[i]A4.
+Fn evaluated in S2,∞(D2×I, {xi}2n
+1 ; R, A) is an element in the Temperley-
+Lieb algebra and the normalization is chosen so that fn is an idempotent
+element in TLn. The most recognized name for this A-symmetrizer is
+the Jones-Wenzl idempotent. We will denote this element as a
+square with n strands entering and n strands exiting, as shown in Fig-
+ure 4 and fn will denote the Jones-Wenzl idempotent.
+Wenzl’s recursive formula uses the Chebyshev polynomial of the sec-
+ond kind.
+Definition 2.9. The nth Chebyshev polynomial of the first kind
+is defined recursively by the initial conditions T0(d) = 2, T1(d) = d and
+Equation 2.1.
+(2.1)
+Tn(d) = dTn−1(d) − Tn−2(d).
+
+JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND
+7
+n
+Figure 4. The Jones-Wenzl idempotent.
+The nth Chebyshev polynomial of the second kind is defined
+recursively by the initial conditions S0(d) = 1, S1(d) = d and the same
+recursive relation as the first kind, Sn(d) = dSn−1(d) − Sn−2(d).
+When we substitute d = −A2 − A−2, the Chebyshev polynomial of
+the first kind has the following closed formula
+Tn(d) = (−1)n(A2n + A−2n),
+and the Chebyshev polynomial of the second kind has the following
+closed formula,denoted by ∆n,
+∆n = (−1)nA2n+2 − A−2n−2
+A2 − A−2
+= (−1)nA−2n[n + 1]A4.
+Theorem 2.10. [Wen] A recursion formula for the nth Jones-Wenzl
+idempotent, fn, is described in Equation 2.2:
+(2.2)
+fn =
+n − 1
+− ∆n−2
+∆n−1
+n − 1
+n − 1
+n − 2
+.
+The following lemma can be obtained from Wenzl’s recursive formula
+as discussed in [Lic2] or from the constructive definition of the Jones-
+Wenzl idempotent as detailed in [PBIMW].
+Lemma 2.11. [Lic2]
+(a) (fn − 1) is an element of the algebra generated by {ei}n−1
+i=1 .
+(b) eifn = fnei = 0 for 1 ≤ i ≤ n − 1.
+(c) fnfn = fn.
+A direct application of the next corollary will be given in Section 4.
+
+8
+DIONNE IBARRA
+Corollary 2.12. [Lic2] Let tr 1(fn) be obtained from fn by closing the
+top string in fn (see Figure 5). Then
+tr 1(fn) =
+∆n
+∆n−1
+fn−1.
+n − 1
+=
+∆n
+∆n−1
+n − 1
+Figure 5. Illustration of tr 1(fn).
+When defining the colored Jones polynomial, many authors tend to
+use the term “decorating a knot by the Chebyshev polynomial” when
+describing taking the trace of fn along a framed knot. This is because
+the trace of fn along the standard annulus S1×I where S1 is the trivial
+knot is equal to the Chebyshev polynomial of the second kind as stated
+in the next corollary.
+Corollary 2.13. [Lic2] Let Sn(z) denote the nth Chebyshev polynomial
+of the second kind and z denote the homotopically non-trivial curve in
+the annulus. Then
+tr Ann(fn) = Sn(z).
+Lemma 2.14. [Lic2]
+b
+a
+= ς
+b
+,
+where ς =
+(−1)a(A2(b+1)(a+1)−A−2(b+1)(a+1))
+A2(b+1)−A−2(b+1)
+.
+The following result is a well known corollary to Lemma 2.14, we will
+see similar corollaries in Section 4 for elements in the twisted I-bundle
+of the M¨obius band.
+Corollary 2.15. [Lic2]
+(2.3)
+m
+k
+= (−A2(k+1) − A−2(k+1))m∆k = ((−1)kTk+1)m∆k,
+where d = −A2 − A−2 and Tn(d) = (−1)n(A2n + A−2n) is the nth
+Chebyshev polynomial of the first kind.
+
+JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND
+9
+3. Crossingless connection in the M¨obius band
+Throughout this paper we will use the crosscap model of the M¨obius
+band where the boundary will be given in a rectangular form when
+marked points are included, as shown in Figure 6b, otherwise it will
+be displayed as a smooth circle as shown in Figure 7b.
+The three
+homotopically distinct arcs fixed on the boundary of the M¨obius band
+are given in Figure 6. In order to relate the arcs from the first and
+second model, a convention was chosen on the two distinct arcs fixed
+on the boundary that do not intersect the crosscap.
+(a) Formed from [0, 1] × [0, 1] by
+identifying {0} × [0, 1] with {1} ×
+[0, 1] as shown by the arrows.
+(b) Consists of a crosscap and
+highlights the boundary of the
+M¨obius band.
+Figure 6. Two models of the M¨obius band with 3 ho-
+motopically distinct arcs fixed on the boundary.
+d
+z
+x
+(a) First model.
+d
+z
+x
+(b) Second model.
+Figure 7. Two models of the M¨obius band with 3 ho-
+motopically distinct simple closed curves in M¨obius band
+denoted by d, x, and z, respectively.
+Figure 7 pictorially describes the three homotopically distinct simple
+closed curves in the M¨obius band. If a simple closed curve intersects
+the crosscap more than once then the number of intersection points can
+be reduced by two at a time. The following example will illustrate the
+process of removing two intersection points from the crosscap. Similar
+
+10
+DIONNE IBARRA
+moves can be applied to arcs attached to the boundary that intersect
+the crosscap more than once.
+Example 3.1. We will illustrate, in the first model then the second
+model, the removal of two intersection points of the crosscap from a
+simple closed curve. In the two examples, the curve will be multicolored
+in order to show which portion of the curve passes through the crosscap.
+Suppose we have a simple closed curve that is homotopically trivial
+and intersects the crosscap twice then, as shown in Equation 3.1, we
+may use a sequence of isotopy moves to remove the two intersection
+points.
+(3.1)
+∼
+∼
+.
+In Equation 3.2, we will illustrate the removal of the same intersec-
+tion points presented in the second model.
+(3.2)
+∼
+.
+Now, suppose we have a homotopically non-trivial curve that inter-
+sects the crosscap twice, for example the curve illustrated in Equation
+3.3. Then, we may remove the two intersection points by using one
+isotopy move as given below.
+(3.3)
+∼
+.
+Equation 3.4 gives an illustration of this move in the second model.
+(3.4)
+∼
+.
+4. Jones-Wenzl idempotents in the M¨obius band
+In this section we will introduce various properties associated to the
+Jones-Wenzl idempotents in the twisted I-bundle of the M¨obius band.
+
+JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND
+11
+n
+Figure 8. Illustration of the unique element in Mbn
+with n arcs intersecting the crosscap, denoted by 1Mbn.
+Lemma 4.1. Let (Mbn)k denote the set of elements of Mbn that in-
+tersect the crosscap k times, and let 1Mbn denote the unique element in
+Mbn that intersect the crosscap n times, then
+1Mbnei = en−i1Mbn ∈ (Mbn)n−2.
+Proof. This is a direct result from the antipodal property of the cross-
+cap that is explained in Section 3 and illustrated in Equations 4.1 and
+4.2.
+(4.1)
+1Mbnei =
+n − i − 1
+i − 1
+n − i − 1
+i − 1
+=
+n − i − 1
+i − 1
+n − i − 1
+i − 1
+∈ (Mbn)n−2.
+(4.2)
+en−i1Mbn =
+n − i − 1
+i − 1
+n − i − 1
+i − 1
+=
+n − i − 1
+i − 1
+n − i − 1
+i − 1
+∈ (Mbn)n−2.
+□
+Corollary 4.2. Sliding fn through the crosscap is achieved by the fol-
+lowing equations.
+n
+n
+=
+n
+n
+(4.3)
+=
+n
+n .
+(4.4)
+
+12
+DIONNE IBARRA
+Proof. By Lemma 2.11(a) and Lemma 4.1, all of the ei’s coming from
+the left fn from left hand side of Equation 4.3 can be pulled through
+the crosscap. Furthermore, for each ei this action results in a turn
+back on the second fn. Therefore, we obtain our desired result after
+applying Lemma 2.11(b). Equation 4.4 is obtained similarly.
+□
+We will now present the three direct corollaries to Lemma 2.14 that
+are obtained from closing fb through the crosscap.
+Corollary 4.3.
+n
+= x
+∆1
+(−1)nTn+1(d)∆n,
+where Tk(d) = (−1)k(A2k + A−2k) is the kth Chebyshev polynomial of
+the first kind.
+Proof. Let a = n and b = 1 in Lemma 2.14. Then by closing fb through
+the crosscap we have
+n
+= (−1)n(A4(n+1) − A−4(n+1))
+A4 − A−4
+.
+After simplification we have our desired result, where x denotes the
+simple closed curve that intersects the crosscap once.
+□
+Corollary 4.4.
+n
+m
+= (−1)n(A2(n+1)(m+1) − A−2(n+1)(m+1))
+A2(n+1) − A−2(n+1)
+m
+.
+Proof. This is obtained from directly applying Lemma 2.14 where a =
+n, b = m, and fb is closed through the crosscap.
+□
+
+JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND
+13
+Corollary 4.5.
+m
+n
+= ((−1)nTn+1(d))m
+n
+.
+Proof. We may remove one of the m meridional curves by applying
+Lemma 2.14 where a = 1 and b = n and closing fb through the crosscap.
+After simplification we have
+m
+n
+= ((−1)nTn+1(d))
+m − 1
+n
+.
+We obtain our desired result after repeating this argument m − 1 more
+times.
+□
+The following corollary is obtained from Corollary 2.13 by gluing a
+crosscap to the inner boundary of the annulus.
+Corollary 4.6. Let z denote the homotopically non-trivial curve in the
+M¨obius band that does not intersect the crosscap, then
+n
+= Sn(z).
+Corollary 4.7.
+m
+n
+=
+∆m+n
+∆n
+n
+.
+Proof. Apply Corollary 2.12 m times when closing n strands from fn+m
+through the crosscap then closing the rest away from the crosscap.
+□
+Let tr Mb1(fn) be obtained from closing one arc from fn through the
+crosscap and closing the rest of the arcs in such a way that it surrounds
+the crosscap, as shown in Figure 9. Then Lemma 2.12 can no longer
+
+14
+DIONNE IBARRA
+be directly applied. Instead we start with applying Wenzl’s recursion
+formula, Theorem 2.2, to obtain a recursive formula for tr Mb1(fn).
+n − 1
+Figure 9. Illustration of tr Mb1(fn).
+Lemma 4.8.
+tr Mb1(fn) = xSn−1(z) − ∆n−2
+∆n−1
+tr Mb1(fn−1).
+Proof. By applying Wenzl’s formula to the x-curve we have the follow-
+ing recursive formula.
+n − 1
+=
+n − 1
+− ∆n−2
+∆n−1
+n − 1
+.
+By Corollary 4.6 and Lemma 2.11(c),
+n − 1
+=
+xSn−1(z) − ∆n−2
+∆n−1
+n − 2
+.
+□
+Proposition 4.9.
+tr Mb1(fn) =
+x
+∆n−1
+n−1
+�
+k=0
+(−1)n−1+kSk(z)∆k.
+
+JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND
+15
+Proof. The base case is trivial,
+tr Mb1(f1) =
+= x.
+Suppose tr Mb1(fn−1) =
+x
+∆n−2
+�n−2
+k=0(−1)n−2+kSk(z)∆k. Then by Lemma
+4.8,
+tr Mb1(fn)
+=
+xSn−1(z) − ∆n−2
+∆n−1
+tr Mb1(fn−1)
+=
+xSn−1(z) −
+x
+∆n−1
+n−2
+�
+k=0
+(−1)n−2+kSk(z)∆k
+=
+x
+∆n−1
+Sn−1(z)∆n−1 +
+x
+∆n−1
+n−2
+�
+k=0
+(−1)n−1+kSk(z)∆k.
+□
+References
+[BIMP]
+R. P. Bakshi, D. Ibarra, S. Mukherjee, J. H. Przytycki, A generalization
+of the Gram determinant of type A, Topology Appl. 295 (2021), Paper
+No. 107663, 15 pp. e-print: arXiv:1905.07834 [math.GT].
+[Bax]
+R. J. Baxter, Exactly solved models in statistical mechanics. Academic
+Press, Inc., London (1982).
+[Cai]
+X. Cai, A Gram determinant of Lickorish’s bilinear form. Math. Proc.
+Cambridge Philos. Soc. 151 (2011), no. 1, 83–94. arXiv:1006.1297v3
+[math.GT].
+[Jon]
+V. F. R. Jones, Index for subfactors. Invent. Math. 72, 1983, 1-25.
+[Kau]
+L. H. Kauffman, An invariant of regular isotopy. Trans. Amer. Math.
+Soc. 318 (1990), no. 2, 417–471.
+[Le]
+T. T. Q. Lˆe, The colored Jones polynomial and the A-polynomial of
+knots. Adv. Math. 207 (2006), no. 2, 782–804. arXiv:math/0407521
+[math.GT].
+[Lic1]
+W. B. R. Lickorish, Invariants for 3-manifolds from the combinatorics
+of the Jones polynomial, Pacific Journ. Math.,149(2), 1991, 337-347.
+[Lic2]
+W. B. R. Lickorish, An introduction to knot theory. Graduate Texts in
+Mathematics, 175. Springer-Verlag, New York, 1997.
+[Prz]
+J. H. Przytycki, Fundamentals of Kauffman bracket skein modules. Kobe
+Math. J., 16(1), 1999, 45-66. arXiv:math/9809113 [math.GT].
+[PBIMW]
+J. H. Przytycki, R. P. Bakshi, D. Ibarra, G. Montoya-Vega, D. E. Weeks,
+Lectures on Knot Theory: An Exploration of Contemporary Topics,
+Springer Universitext (to appear).
+[TL]
+H. Temperley and E. Lieb, Relations Between the ‘Percolation’ and
+‘Colouring’ Problem and Other Graph-Theoretic Problems Associated
+
+16
+DIONNE IBARRA
+with Regular Plane Lattices: Some Exact Results for the ‘Percolation’
+Problem, Proceeds of the Royal Society of London 322 (1971), 251 - 280.
+[TV]
+V. G. Turaev, O. Ya. Viro, State sum invariants of 3-manifolds and
+quantum 6j-symbols. Topology 31 (1992), no. 4, 865–902.
+[Wen]
+H. Wenzl, On sequences of projections, C.R. Math. Rep. Acad. Sci., IX,
+1987, 5-9.
+School of Mathematics, 9 Rainforest Walk, Floor 4, Monash Uni-
+versity, VIC 3800, Australia
+Email address: [email protected]
+

7dE4T4oBgHgl3EQfCQuc/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf,len=426
+page_content='JONES-WENZL IDEMPOTENTS IN THE TWISTED  I-BUNDLE OF THE M¨OBIUS BAND  DIONNE IBARRA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The Jones-Wenzl idempotent plays a vital role in quan-  tum invariants of 3-manifolds and the colored Jones polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  it also serves as a useful tool for simplifying computations and  proving theorems in knot theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The relative Kauffman bracket  skein module (RKBSM) for surface I-bundles and manifolds with  marked boundaries have a well understood algebraic structure due  to the work of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Przytycki and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Lˆe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  It has been  well documented that the RKBSM of the I-bundle of the annulus  and the twisted I-bundle of the M¨obius band have a distinct alge-  braic structures even though the manifolds are homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' In  this paper we will give various results on Jones-Wenzl idempotents  in the twisted I-bundle of the M¨obius band when it is partially  closed through the crosscap of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' In doing so we  will uncover properties that differ from properties of Jones-Wenzl  idempotents in Ann × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Introduction  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Acknowledgements  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Introduction to Jones-Wenzl idempotents  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Crossingless connection in the M¨obius band  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Jones-Wenzl idempotents in the M¨obius band  10  References  15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Introduction  The Jones-Wenzl idempotent, discovered by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Jones in [Jon],  is an idempotent element in the Temperley-Lieb algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Originally,  it was described as a certain symmetrizer using the Artin braid group  Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Primary: 57K10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Secondary: 57K31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Jones-Wenzl idempotents, M¨obius band, twisted I-  bundles, Kauffman bracket skein module, relative Kauffman bracket skein module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='04859v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='GT]  12 Jan 2023  2  DIONNE IBARRA  and the projection to the Temperley-Lieb algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' In the late 1980’s,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Wenzl in [Wen] discovered a recursive formula to the Jones-Wenzl  idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  This formula is now widely used as the definition, see  [Lic2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  The Jones-Wenzl idempotent has played a significant role in defining  quanum invariants of knots and 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' For example, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Lickorish’s Kauffman bracket skein theoretic approach to the Witten-  Reshetikhin-Turaev 3-manifold invariants in [Lic1] uses a linear combi-  nation of the trace (closure) of the idempotent elements along a framed  knot or link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Similarly, the colored Jones polynomial quantum knot in-  variant is defined by taking the trace of the nth Jones-Wenzl idempotent  along a 0-framed knot in S3, see [Le, PBIMW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' These idempotent ele-  ments are also used to decorate the edges of a tetrahedra to obtain the  quantum 6j-symbols that are used in the definition of the Turaev-Viro  quantum 3-manifold invariants, see [TV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  The Jones-Wenzl idempotent has been a vital tool for simplifying  computations and proving theorems in knot theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' An example of this  is seen in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Cai’s proof of a closed formula for the Gram determinant  of type A in [Cai] and a closed formula for its generalization in [BIMP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  In fact, this paper was conceived by needing properties of the Jones-  Wenzl idempotents when it is closed in the twisted I-bundle of the  M¨obius band in hopes to take a similar approach to [Cai] and [BIMP]  to prove a closed formula for the Gram determinant of type Mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  In Section 2 we introduce the original definition of Jones-Wenzl idem-  potents and also the RKBSM of the twisted I-bundle of the M¨obius  band, then in Section 3 we give an illustration of the two different  models of the M¨obius band as well as the antipodal properties of the  crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' In Section 4 we prove many corollaries to the trace of Jones-  Wenzl idempotents intersecting or surrounding the crosscap, then we  end with a formula for when n − 1 curves from fn are closed around  the crosscap and the last arc is closed through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' This work was supported by the Australian  Research Council grant DP210103136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Introduction to Jones-Wenzl idempotents  The first formal definition of the Temperley-Lieb algebra, denoted  by TLn, was given by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Baxter in [Bax] while describing the work  of physicists N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Temperley and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Lieb in [TL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Jones independently  introduced TLn in [Jon] while working on von Neumann algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND  3  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let R be a commutative ring with unity and d ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Let n ∈ N be fixed, then the nth Temperley-Lieb algebra, TLn,  is defined to be the unital associative algebra over R with generators  e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' , en−1, identity element 1n, and relations  (1) eiejei = ei for |i − j| = 1,  (2) eiej = ejei for |i − j| > 1,  (3) e2  i = dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Kauffman in [Kau], motivated by utilizing the Kauffman bracket,  considered the Temperley-Lieb algebra over R = Z[A±1], where A is an  indeterminate and d = −A2 − A−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' He then constructed a graphical  interpretation using tangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  We will consider an n-tangle to be a rectangular shaped disk with  n marked boundary points on the left (input points) and n marked  boundary points on the right (output points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Kauffman’s graphical  interpretation of the Temperley-Lieb algebra is obtained from the basis  of crossingless tangles where the identity element corresponds to an n-  tangle with n parallel arcs in which each ith input point is connected  to the ith output point, and each ei corresponds to an n-tangle that  has one input and one output cap on the ith and i + 1th position as  illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  For simplicity we will label an arc by n to  denote n parallel arcs as shown in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  n  (a) Identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  n − i − 1  i − 1  (b) ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The graphical interpretation of TLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The n-tangle algebra is an R-module with basis  elements consisting of n-tangles where multiplication of two n-tangles  is defined by identifying the right side of the first n-tangle to the left  side of the second n-tangle while respecting the boundary points and  by letting any resulting trivial curve be denoted by d, see Figure 2 for  an illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Kauffman’s diagrammatic interpretation of the  Temperley-Lieb algebra, also known as the diagrammatic algebra,  is a subalgebra of the n-tangle algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' It is generated by tangles with  no crossings where homotopically trivial curves are denoted by d ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  4  DIONNE IBARRA  e3e3 =  = d  = de3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' An illustration of multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Kau] The diagrammatic algebra is isomorphic to TLn  and can be thought of as a diagrammatic interpretation of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  We will give Jones’ constructive definition of the Jones-Wenzl idem-  potent by using the relative Kauffman bracket skein module (RKBSM)  and the Artin braid group before introducing Wenzl’s recursive formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  In doing so, we will first introduce the RKBSM and emphasize that the  RKBSM of the twisted I-bundle of the M¨obius band and the RKBSM  of Ann × I are different modules even though the two manifolds are  homeomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' This will give us motivation to study the Jones-Wenzl  idempotent in the twisted I-bundle of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Furthermore,  the corollaries and proposition in the last section will show that there  are distinct differences when simple closed curves intersect the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let M be an oriented 3-manifold and {xi}2n  i=1 be the  set of 2n framed points on ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let I = [−1, 1], and let Lfr(2n) be the  set of all relative framed links (which consists of all framed links in M  and all framed arcs, I×I, where I×∂I is connected to framed points on  the boundary of M) up to ambient isotopy while keeping the boundary  fixed in such a way that L ∩ ∂M = {xi}2n  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let R be a commutative  ring with unity, A ∈ R be invertible, and let Ssub  2,∞(2n) be the submodule  of RLfr(2n) that is generated by the Kauffman bracket skein relations:  (i) L+ − AL0 − A−1L∞, and  (ii) L ⊔ ⃝  ⃝  ⃝ + (A2 + A−2)L,  where ⃝  ⃝  ⃝ denotes the framed unknot and the skein triple (L+, L0, L∞)  denotes three framed links in M that are identical except in a small  3-ball in M where the difference is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Then, the relative Kauffman bracket skein module (RKBSM)  of M is the quotient:  S2,∞(M, {xi}2n  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' R, A) = RLfr(2n)/Ssub  2,∞(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Prz] Let F be a surface with ∂F ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' If F is orientable  then let M = F × I , otherwise let M = F ˆ×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let all {xi}2n  1 be marked  JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND  5  (a) L+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (b) L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (c) L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The skein triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  points that lie on ∂F × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Then S2,∞(M, {xi}2n  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' R, A) is a free R-  module whose basis is composed of relative links in F without trivial  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' When n = 0, the empty link is also a generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Przytycki’s corollary to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='5 explicitly details the dif-  ferences between the RKBSM of Ann × I and the RKBSM of Mbˆ×I  even though both manifolds are homeomorphic to the solid torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Prz]  (1) S2,∞(Ann×I, {xi}2n  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' R, A) where {xi}2n  1 are located in the outer  boundary component of the annulus is a free R[x]-module with  Dn =  �2n  n  �  basis elements, where x denotes the homotopically  nontrivial curve in the annulus and d = −A2 − A−2 denotes  the homotopically trivial curve in the annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The basis is the  set of all crossingless connections in the annulus with no trivial  components or boundary parallel curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (2) S2,∞(Mbˆ×I, {xi}2n  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' R, A) is a free R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The standard ba-  sis contains an infinite number of elements of the form bzi, bxzi  for i ≥ 0, where x denotes the simple closed curve that intersects  the M¨obius band once, z denotes the boundary parallel curve of  the M¨obius band, and b is an element in the set of crossingless  connections in the M¨obius band with no trivial components or  boundary parallel curves for which the arcs do not intersect the  crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  The rest of the elements in the standard basis are  from a finite number of crossingless connections consisting of  a collection of n − k arcs for 0 ≤ k < n that non-trivially in-  tersects the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Among the finite collection there are  �2n  k  �  crossingless connections that intersect the crosscap n−k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The Artin braid group is defined by the following  group presentation:  Bn =< σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' , σn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' σiσj = σjσi for |i−j| > 1, σi±1σiσi±1 = σiσi±1σi > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  The Artin braid group can be interpreted using n-tangles where el-  ements of Bn are positive braids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' More precisely, an element in Bn  6  DIONNE IBARRA  can be represented as an n-tangle with positive crossings such that the  boundary of each arc is attached to one input and one output point  and when read from left to right each generator σi corresponds to the  positive crossing of the ith and i + 1th arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' That is, the ith generator  element σi is a positive transposition of the ith and i + 1th arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Furthermore, there exists an epimorphism p : Bn → Sn from the  Artin braid group to the permutation group that uniquely interprets a  braid word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let p be defined by sending generators of Bn, σi, to the  transpositions in Sn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' si = (i, i+1) for 1 ≤ i ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' For a permutation  π ∈ Sn, let bπ denote the unique minimal positive braid word such that  p(bπ) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let Z[A±1] denote the ring of Laurent polynomials  in the variable A and Q(A) denote the field of rational functions in  the variable A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' whose elements are functions of the form P/Q where  P, Q ∈ Z[A±1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' We define an unnormalized A-symmetrizer, Fn ∈  Z[A±1]Bn, by the following  Fn =  �  π∈Sn  (A3)|π|bπ,  and the normalized symmetrizer, also known as the A-symmetrizer  and denoted by fn ∈ Q(A)Bn, by the formula  fn =  1  [n]A4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='Fn,  where |π| denotes the minimal length of the permutation π written as  elementary transposition generators, [n]A4 = 1+A4+A8+· · ·+A4(n−1) =  A4n−1  A4−1 and [n]A4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' =  n�  i=1  [i]A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Fn evaluated in S2,∞(D2×I, {xi}2n  1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' R, A) is an element in the Temperley-  Lieb algebra and the normalization is chosen so that fn is an idempotent  element in TLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The most recognized name for this A-symmetrizer is  the Jones-Wenzl idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' We will denote this element as a  square with n strands entering and n strands exiting, as shown in Fig-  ure 4 and fn will denote the Jones-Wenzl idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Wenzl’s recursive formula uses the Chebyshev polynomial of the sec-  ond kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The nth Chebyshev polynomial of the first kind  is defined recursively by the initial conditions T0(d) = 2, T1(d) = d and  Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1)  Tn(d) = dTn−1(d) − Tn−2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND  7  n  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The Jones-Wenzl idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  The nth Chebyshev polynomial of the second kind is defined  recursively by the initial conditions S0(d) = 1, S1(d) = d and the same  recursive relation as the first kind, Sn(d) = dSn−1(d) − Sn−2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  When we substitute d = −A2 − A−2, the Chebyshev polynomial of  the first kind has the following closed formula  Tn(d) = (−1)n(A2n + A−2n),  and the Chebyshev polynomial of the second kind has the following  closed formula,denoted by ∆n,  ∆n = (−1)nA2n+2 − A−2n−2  A2 − A−2  = (−1)nA−2n[n + 1]A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Wen] A recursion formula for the nth Jones-Wenzl  idempotent, fn, is described in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2)  fn =  n − 1  − ∆n−2  ∆n−1  n − 1  n − 1  n − 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  The following lemma can be obtained from Wenzl’s recursive formula  as discussed in [Lic2] or from the constructive definition of the Jones-  Wenzl idempotent as detailed in [PBIMW].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Lic2]  (a) (fn − 1) is an element of the algebra generated by {ei}n−1  i=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (b) eifn = fnei = 0 for 1 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (c) fnfn = fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  A direct application of the next corollary will be given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  8  DIONNE IBARRA  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Lic2] Let tr 1(fn) be obtained from fn by closing the  top string in fn (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Then  tr 1(fn) =  ∆n  ∆n−1  fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  n − 1  =  ∆n  ∆n−1  n − 1  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Illustration of tr 1(fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  When defining the colored Jones polynomial, many authors tend to  use the term “decorating a knot by the Chebyshev polynomial” when  describing taking the trace of fn along a framed knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' This is because  the trace of fn along the standard annulus S1×I where S1 is the trivial  knot is equal to the Chebyshev polynomial of the second kind as stated  in the next corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Lic2] Let Sn(z) denote the nth Chebyshev polynomial  of the second kind and z denote the homotopically non-trivial curve in  the annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Then  tr Ann(fn) = Sn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Lic2]  b  a  = ς  b  ,  where ς =  (−1)a(A2(b+1)(a+1)−A−2(b+1)(a+1))  A2(b+1)−A−2(b+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  The following result is a well known corollary to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='14, we will  see similar corollaries in Section 4 for elements in the twisted I-bundle  of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' [Lic2]  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='3)  m  k  = (−A2(k+1) − A−2(k+1))m∆k = ((−1)kTk+1)m∆k,  where d = −A2 − A−2 and Tn(d) = (−1)n(A2n + A−2n) is the nth  Chebyshev polynomial of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Crossingless connection in the M¨obius band  Throughout this paper we will use the crosscap model of the M¨obius  band where the boundary will be given in a rectangular form when  marked points are included, as shown in Figure 6b, otherwise it will  be displayed as a smooth circle as shown in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  The three  homotopically distinct arcs fixed on the boundary of the M¨obius band  are given in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' In order to relate the arcs from the first and  second model, a convention was chosen on the two distinct arcs fixed  on the boundary that do not intersect the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (a) Formed from [0, 1] × [0, 1] by  identifying {0} × [0, 1] with {1} ×  [0, 1] as shown by the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (b) Consists of a crosscap and  highlights the boundary of the  M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Two models of the M¨obius band with 3 ho-  motopically distinct arcs fixed on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  d  z  x  (a) First model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  d  z  x  (b) Second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Two models of the M¨obius band with 3 ho-  motopically distinct simple closed curves in M¨obius band  denoted by d, x, and z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Figure 7 pictorially describes the three homotopically distinct simple  closed curves in the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' If a simple closed curve intersects  the crosscap more than once then the number of intersection points can  be reduced by two at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The following example will illustrate the  process of removing two intersection points from the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Similar  10  DIONNE IBARRA  moves can be applied to arcs attached to the boundary that intersect  the crosscap more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' We will illustrate, in the first model then the second  model, the removal of two intersection points of the crosscap from a  simple closed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' In the two examples, the curve will be multicolored  in order to show which portion of the curve passes through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Suppose we have a simple closed curve that is homotopically trivial  and intersects the crosscap twice then, as shown in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1, we  may use a sequence of isotopy moves to remove the two intersection  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1)  ∼  ∼  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  In Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2, we will illustrate the removal of the same intersec-  tion points presented in the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2)  ∼  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Now, suppose we have a homotopically non-trivial curve that inter-  sects the crosscap twice, for example the curve illustrated in Equation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Then, we may remove the two intersection points by using one  isotopy move as given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='3)  ∼  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='4 gives an illustration of this move in the second model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='4)  ∼  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Jones-Wenzl idempotents in the M¨obius band  In this section we will introduce various properties associated to the  Jones-Wenzl idempotents in the twisted I-bundle of the M¨obius band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND  11  n  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Illustration of the unique element in Mbn  with n arcs intersecting the crosscap, denoted by 1Mbn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let (Mbn)k denote the set of elements of Mbn that in-  tersect the crosscap k times, and let 1Mbn denote the unique element in  Mbn that intersect the crosscap n times, then  1Mbnei = en−i1Mbn ∈ (Mbn)n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' This is a direct result from the antipodal property of the cross-  cap that is explained in Section 3 and illustrated in Equations 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1)  1Mbnei =  n − i − 1  i − 1  n − i − 1  i − 1  =  n − i − 1  i − 1  n − i − 1  i − 1  ∈ (Mbn)n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2)  en−i1Mbn =  n − i − 1  i − 1  n − i − 1  i − 1  =  n − i − 1  i − 1  n − i − 1  i − 1  ∈ (Mbn)n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Sliding fn through the crosscap is achieved by the fol-  lowing equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  n  n  =  n  n  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='3)  =  n  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='4)  12  DIONNE IBARRA  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='11(a) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='1, all of the ei’s coming from  the left fn from left hand side of Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='3 can be pulled through  the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Furthermore, for each ei this action results in a turn  back on the second fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Therefore, we obtain our desired result after  applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='4 is obtained similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  □  We will now present the three direct corollaries to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='14 that  are obtained from closing fb through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  n  = x  ∆1  (−1)nTn+1(d)∆n,  where Tk(d) = (−1)k(A2k + A−2k) is the kth Chebyshev polynomial of  the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let a = n and b = 1 in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Then by closing fb through  the crosscap we have  n  = (−1)n(A4(n+1) − A−4(n+1))  A4 − A−4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  After simplification we have our desired result, where x denotes the  simple closed curve that intersects the crosscap once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  n  m  = (−1)n(A2(n+1)(m+1) − A−2(n+1)(m+1))  A2(n+1) − A−2(n+1)  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' This is obtained from directly applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='14 where a =  n, b = m, and fb is closed through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  □  JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND  13  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  m  n  = ((−1)nTn+1(d))m  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' We may remove one of the m meridional curves by applying  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='14 where a = 1 and b = n and closing fb through the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  After simplification we have  m  n  = ((−1)nTn+1(d))  m − 1  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  We obtain our desired result after repeating this argument m − 1 more  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  □  The following corollary is obtained from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='13 by gluing a  crosscap to the inner boundary of the annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Let z denote the homotopically non-trivial curve in the  M¨obius band that does not intersect the crosscap, then  n  = Sn(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  m  n  =  ∆m+n  ∆n  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Apply Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='12 m times when closing n strands from fn+m  through the crosscap then closing the rest away from the crosscap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  □  Let tr Mb1(fn) be obtained from closing one arc from fn through the  crosscap and closing the rest of the arcs in such a way that it surrounds  the crosscap, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='12 can no longer  14  DIONNE IBARRA  be directly applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Instead we start with applying Wenzl’s recursion  formula, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='2, to obtain a recursive formula for tr Mb1(fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  n − 1  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Illustration of tr Mb1(fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  tr Mb1(fn) = xSn−1(z) − ∆n−2  ∆n−1  tr Mb1(fn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' By applying Wenzl’s formula to the x-curve we have the follow-  ing recursive formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  n − 1  =  n − 1  − ∆n−2  ∆n−1  n − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='6 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='11(c),  n − 1  =  xSn−1(z) − ∆n−2  ∆n−1  n − 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  tr Mb1(fn) =  x  ∆n−1  n−1  �  k=0  (−1)n−1+kSk(z)∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  JONES-WENZL IDEMPOTENTS AND THE M¨OBIUS BAND  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' The base case is trivial,  tr Mb1(f1) =  = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  Suppose tr Mb1(fn−1) =  x  ∆n−2  �n−2  k=0(−1)n−2+kSk(z)∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Then by Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='8,  tr Mb1(fn)  =  xSn−1(z) − ∆n−2  ∆n−1  tr Mb1(fn−1)  =  xSn−1(z) −  x  ∆n−1  n−2  �  k=0  (−1)n−2+kSk(z)∆k  =  x  ∆n−1  Sn−1(z)∆n−1 +  x  ∆n−1  n−2  �  k=0  (−1)n−1+kSk(z)∆k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content='  □  References  [BIMP]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Bakshi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Ibarra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Mukherjee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
+page_content=' Przytycki, A generalization  of the Gram determinant of type A, Topology Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE4T4oBgHgl3EQfCQuc/content/2301.04859v1.pdf'}
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7tAyT4oBgHgl3EQf2_kh/content/tmp_files/2301.00759v1.pdf.txt

@@ -0,0 +1,1607 @@
+Avalanche scaling in large neural
+populations with distributed
+coupling to multiple dynamical
+latent variables
+Mia Morrell1, Ilya Nemenman2, Audrey J. Sederberg3*
+*For correspondence:
+[email protected] (AJS)
+1Department of Physics, New York University; 2Department of Physics, Department of
+Biology, Initiative in Theory and Modeling of Living Systems, Emory University;
+3Department of Neuroscience, University of Minnesota Medical School
+Abstract
+Observations of power laws in neural activity data have raised the intriguing notion
+that brains may operate in a critical state. One example of this critical state is “avalanche
+criticality,” which has been observed in a range of systems, including cultured neurons, zebrafish,
+and human EEG. More recently, power laws have also been observed in neural populations in the
+mouse under a coarse-graining procedure, and they were explained as a consequence of the
+neural activity being coupled to multiple latent dynamical variables. An intriguing possibility is
+that avalanche criticality emerges due to a similar mechanism. Here, we determine the
+conditions under which dynamical latent variables give rise to avalanche criticality. We find that a
+single, quasi-static latent variable can generate critical avalanches, but that multiple latent
+variables lead to critical behavior in a broader parameter range. We identify two regimes of
+avalanches, both of which are critical, but differ in the amount of information carried about the
+latent variable. Our results suggest that avalanche criticality arises in neural systems in which
+there is an emergent dynamical variable or shared inputs creating an effective latent dynamical
+variable, and when this variable can be inferred from the population activity.
+Introduction
+The neural criticality hypothesis – the idea that neural systems operate close to a phase transition,
+perhaps for optimal information processing – is at the same time ambitious and banal. Measure-
+ments from biological systems are limited in the range of spatial and temporal scales that can be
+sampled, not only because of limits of recording techniques but also due to fundamentally non-
+stationary behavior of most, if not all, biological systems. These limitations make proving that an
+observation indicates critical behavior difficult. At the same time, the idea that brain networks are
+critical echoes the anthropic principle: tuned another way, a network becomes quiescent or epilep-
+tic, and in either case seems unlikely to support perception, thought, or flexible behavior. Further
+muddying the water, researchers have reported multiple kinds of criticality in neural networks, in-
+cluding through analysis of avalanches (Beggs and Plenz, 2003; Plenz et al., 2021; O’Byrne and Jerbi,
+2022; Girardi-Schappo, 2021) and of coarse-grained activity (Meshulam et al., 2019), as well as of
+correlations (Dahmen et al., 2019). How these flavors of critical behavior relate to each other or to
+any functional network mechanism is not known.
+The phenomenon that we will refer to as “avalanche criticality” appears to be remarkably widespread.
+1 of 18
+arXiv:2301.00759v1  [q-bio.NC]  2 Jan 2023
+
+It was first observed in cultured neurons in a dish (Beggs and Plenz, 2003) and later studied in ze-
+brafish (Ponce-Alvarez et al., 2018), turtles (Shew et al., 2015), rodents (Ma et al., 2019), monkeys
+(Petermann et al., 2009), and even humans (Poil et al., 2008). The standard analysis, described
+thoroughly later, requires extracting power-law exponents from a fit to a distribution of avalanche
+size and duration and assessing the relationship between exponents. There is debate over whether
+these observations reflect true power laws, but within the resolution achievable from experiments,
+neural avalanches exhibit power laws with exponent relationships predicted from theory devel-
+oped in physical systems (Perković et al., 1995).
+Avalanche criticality is not the only form of criticality observed in neural systems. Zipf’s law (fre-
+quency of the network state being inversely proportional to its rank) appears in systems as diverse
+as fly motion estimation and salamander retina (Mora and Bialek, 2010; Schwab et al., 2014; Aitchi-
+son et al., 2016). More recently, Meshulam et al. (2019) measured various statistics of population
+activity in a mouse hippocampus, including the eigenvalue spectrum of the covariance matrix and
+the variance of activity. These were found to scale as populations were “coarse-grained” through
+a procedure in which neural activities were iteratively combined based on similarity. Neither the
+Zipf’s law nor the coarse-grained criticality can be explained by simple mechanistic models.
+Even though these three forms of criticality are observed through different analyses, it is pos-
+sible that they may originate from similar mechanisms. While avalanche power laws may result
+from critical dynamics, they can also appear due to quasi-static latent variables, which can pro-
+duce power laws, but not the relationships expected between the critical exponents (Priesemann
+and Shriki, 2018). We have previously shown that a dynamical latent variable (DLV) model, based
+on the coupling of neural populations to multiple dynamical latent variables, can reproduce scaling
+under coarse-graining analysis within experimental uncertainty (Morrell et al., 2021). The Zipf’s law
+has been explained by a similar mechanism (Schwab et al., 2014; Aitchison et al., 2016). However,
+it is not known under what conditions, if any, the DLV model generates avalanche criticality.
+In this paper, we systematically investigate avalanche statistics in the DLV model. We show that
+a system coupled to multiple dynamical latent variables can generate avalanche criticality, and we
+examine the requirements for the number and timescale of variables for this criticality to occur.
+We find that avalanche criticality is observed over a wide range of parameters, some of which may
+be optimal for information representation. Our results suggest that latent dynamical structure in
+large-scale neural recordings may be responsible for the observation of signatures of criticality
+across many systems.
+Results
+Critical exponents values and crackling noise
+We begin by defining the metrics used to quantify avalanche statistics and briefly summarize ex-
+perimental observations, which have been reviewed in detail elsewhere (Plenz et al., 2021; O’Byrne
+and Jerbi, 2022; Girardi-Schappo, 2021). Activity is recorded across a set of neurons and binned
+in time. Avalanches are then defined as contiguous time bins in which at least one neuron in the
+population is active. The duration of an avalanche is the number of contiguous time bins and the
+size is the summed activity during the avalanche. The distributions of avalanche size and duration
+are fit to power laws (푃(푆) ∼ 푆−휏 for size 푆, and 푃(퐷) ∼ 퐷−훼 for duration 퐷) using standard methods
+(Clauset et al., 2009).
+Power laws can be indicative of criticality, but they can also result from non-critical mechanisms
+(Touboul and Destexhe, 2017; Priesemann and Shriki, 2018). A more stringent test of criticality is
+the “crackling” relationship (Perković et al., 1995; Touboul and Destexhe, 2017), which involves
+fitting a third power-law relationship, ̄푆(퐷) ∼ 퐷훾fit, and comparing 훾fit to the predicted exponent
+훾pred, derived from the size and duration exponents, 휏 and 훼:
+훾fit
+?= 훾pred ≡ 훼 − 1
+휏 − 1 .
+(1)
+2 of 18
+
+Figure 1. Dynamical Latent Variable model produces avalanche criticality. A: Model structure. Latent
+dynamical variables ℎ휇(푡) are broadly coupled to neurons 푠푖(푡) in the recorded population. B: Raster plot of a
+sample of activity binned at 3-ms resolution across 128 neurons with five latent variables, each with
+correlation timescale 휏퐹 = 15 s. C: Projection of activity into a simulated field of view for illustration. D-F:
+Avalanche analysis in a network (parameters 푁퐹 = 5, 휏퐹 = 104, 휂 = 4 and 휖 = 12), showing size distribution (D),
+duration distribution (E), and size with duration scaling (F). Lower cutoffs used in fitting are shown with
+vertical lines and their values are indicated in the figures. There are 푁obs = 42725 avalanches of size 푆 ≥ 푆min in
+this simulated dataset. Estimated values of the critical exponents are shown in the titles of the panels.
+Previous work demonstrating approximate power laws in size and duration distributions through
+the mechanism of a slowly changing latent variable did not generate crackling (Touboul and Des-
+texhe, 2017; Priesemann and Shriki, 2018).
+Measuring power-laws in empirical data is challenging: it generally requires setting a lower cut-
+off in the size and duration, and the power-law behavior only has limited range due to the finite
+size and duration of the recording itself. Nonetheless, there is some consensus (Shew et al., 2015;
+Fontenele et al., 2019; Ma et al., 2019) that even if 휏 and 훼 vary over a wide range (1.5 to about 3)
+across recordings, the values of 훾fit and 훾pred stay in a relatively narrow range, from about 1.1 to 1.3.
+Avalanche scaling in the Dynamical Latent Variable (DLV) model
+We studied a population of neurons that are coupled to dynamical latent variables but not coupled
+to each other (Fig. 1A). We refer to this model as the Dynamical Latent Variable (DLV) model. The
+latent variables determine the inputs to the simulated population of neurons. We are agnostic as
+to the origin of these inputs: they may be externally driven from other brain areas, or they may
+arise from recurrent dynamics locally. We have previously shown that the DLV model with at least
+about five latent variables can produce power laws under the coarse-graining analysis (Morrell
+et al., 2021). In this paper, we examine avalanche criticality in the same model.
+Specifically, we model the neurons as binary units (푠푖) that are randomly (퐽푖휇 ∼ 푁(0, 1)) coupled
+to dynamical variables ℎ휇(푡). The probability of any pattern {푠푖}, given the current state of the latent
+3 of 18
+
+B
+h2(t)
+100
+1..
+neurons
+")
+50
+hm(t)
+S;(t)
+....
+time (s)
+100
+10°
+103
+Probability Density
+Probability Density
+Average Size
+10
+102
+10
+107
+10
+101
+100
+100
+102
+104
+106
+100
+102
+104
+100
+101
+102
+103
+Avalanche Size S
+Avalanche Duration D
+Duration Dvariables, is
+푃({푠푖}|ℎ휇(푡)) =
+1
+푍(ℎ휇(푡)) exp
+(
+−휂
+푁퐹
+∑
+휇=1
+푠푖퐽푖휇ℎ휇(푡) − 휖푠푖
+)
+,
+(2)
+where the parameter 휂 controls the scaling of the variables and 휖 controls the overall activity level.
+We modeled each latent variable as an Ornstein-Uhlenbeck process with the time scale 휏퐹 (see
+Methods). Thus our model has four parameters: 휂 (input scaling), 휖 (activity threshold), 휏퐹 (dynami-
+cal timescale), and 푁퐹 (number of neurons).
+Distributions of avalanche size and avalanche duration within this model followed approximate
+power laws (Fig. 1C; see Methods). In the example shown (푁퐹 = 5, 휏퐹 = 104, 휂 = 4 and 휖 = 12), we
+found exponents 휏 = 1.89 ± 0.02 (size) and 훼 = 2.11 ± 0.02 (duration). Further, the average size of
+avalanches with fixed duration scaled as 푆 ∼ 퐷훾, with the fitted 훾fit = 1.24 ± 0.02, in agreement with
+the predicted value 훾pred = 1.24±0.02. Thus, our model could generate avalanche scaling, at least for
+some parameter choices. In the following sections, we examine how avalanche scaling depends
+on model parameters (푁퐹 , 휏퐹 , 휂 and 휖; see Table 2). We first focus on two sets of simulations: one
+set with 푁퐹 = 1 latent variable, which does not generate scaling under coarse-graining (Morrell
+et al., 2021), and one set with 푁퐹 = 5 latent variables, which can generate such scaling for some
+values of parameters 휏퐹 , 휂, and 휖 (Morrell et al., 2021).
+Avalanche scaling depends on the number of latent variables
+We analyzed avalanches from one- and five-variable simulations, each with fixed latent dynamical
+timescale (휏퐹 = 5 × 103 time steps; see Table 2 for parameters). In the following sections, time is
+measured in simulation time steps, see Methods for converting time steps to seconds. We used es-
+tablished methods for measuring empirical power laws (Clauset et al., 2009). The minimum cutoffs
+for size (푆min) and duration (퐷min) are indicated by vertical lines in Fig. 2. For the population coupled
+to a single latent variable, the avalanche size distribution was not well fit by a power law (Fig. 2A).
+With a sufficiently high minimum cut-off (퐷min), the duration distribution was approximately power-
+law (Fig. 2B).
+We next assessed whether the simulation produced crackling. If so, the value 훾fit obtained by
+fitting ̄푆(퐷) ∼ 퐷훾fit would be similar to 훾pred = 훼−1
+휏−1 . In many cases, such as the one-variable example
+shown in Fig. 2C, the full range of avalanche durations were not fit by a single power law. There-
+fore, we determined the largest range over which a power law was a good fit to the simulated
+observations. In this case, slightly over two decades of apparent scaling were observed starting
+from avalanches with minimum duration slightly less than 100 time steps (Fig. 2C), with a best-fit
+value of 훾푓푖푡 ∈ [1.69, 1.74]. As we did not find a power-law in the size distribution, calculating 훾pred is
+meaningless. If we do it anyway, we obtain 훾푝푟푒푑 = 0.83 ± 0.03 (yellow line in Fig. 2C), which clearly
+deviates from the fitted value of 훾. Thus, for the single dynamical latent variable model (휏퐹 = 5000),
+power-law fits are poor, and there is no crackling.
+The five-variable model produces a different picture. We now find avalanches for which size and
+duration distributions are much better fit by power-law models starting from very low minimum
+cutoffs (Fig. 2D-E, Fig. 2-Supp. Fig. 2). Average size scaled with duration, again over more than
+two decades, with 훾fit = 1.27 ± 0.03, which was in close agreement with 훾pred = 1.25 ± 0.02 (Fig. 2F).
+Holding other parameters constant, we thus found that scaling relationships and crackling arise in
+the multi-variable model but not the single-variable model.
+Avalanche scaling depends on the time scale of latent variables
+Based on simulations in the previous section, we surmised that the five-variable simulation gen-
+erated scaling more readily due to creating an “effective” latent variable that had slower dynam-
+ics than any individual latent variable. We reasoned that at any moment in time, the latent vari-
+able state ℎ휇(푡) is a vector in the latent space. Because coupling to the latent variables is random
+throughout the population, only the length (∼
+√
+푁퐹 ) and not the direction of this vector matters,
+4 of 18
+
+Figure 2. Multiple latent variables generate avalanche scaling at shorter timescales than a single latent
+variable. Parameters used for simulations for this figure are found in Table 2. A-C: Scaling analysis for one
+variable models where the dynamic timescale is equal to 5 × 103 time steps. A: Distribution of avalanche sizes.
+MLE value of exponent for best-fit power law is 휏 = 1.98 (0.02 SE), with lower cutoff indicated by the vertical
+line. B: Distribution of avalanche duration. MLE value of 훼 is 1.81 (0.02 SE). C: Average size plotted against
+avalanche duration (blue points), with power-law fit (black line) and predicted relationship (yellow line) from
+MLE values for exponents in A and B. Gray bar on the horizontal axis indicates range over which a power law
+with 훾 = 1.72 fits the data (see Methods). D-F: Analysis of avalanches from a simulation of a population coupled
+to five independent latent variables where the dynamic timescale is equal to 5 × 103 time steps. G: Exponents
+휏 for avalanche size distributions across timescales for one-variable (blue) and five-variable (red) simulations.
+Each circle is a simulation with independently drawn coupling parameters. Simulations had to show scaling
+over at least two decades to be included in panels (G-J). H: Exponents 훼 for avalanche duration distributions
+for simulations in G. I: Fitted values of 훾 for simulations in G. J: Difference between fitted and predicted 훾
+values. Five-variable simulations produce crackling over a wider range of timescales than single-variable
+simulations. Figure 2–Figure supplement 1. Methods, power law distribution fits, one variable example.
+Figure 2–Figure supplement 2. Methods, power law distribution fits, five variable example.
+Figure 2–Figure supplement 3. Methods, gamma fit and range, one variable example.
+Figure 2–Figure supplement 4. Methods, gamma fit and range, five variables example.
+5 of 18
+
+5
+average size
+4
+2
+2
+601)
+３
+4
+pdf
+4
+¥2
+¥6
+-6
+S
+D
+min
+8
+8
+avalanche size
+avalanche duration
+duration
+5
+(10g10)
+average size
+4
+2
+¥３
+4
+pdf
+4
+2
+¥6
+OZIs
+6
+D
+min
+8
+8
+0
+avalanche size
+avalanche duration
+duration
+2.2
+1.8
+CDAD
+D
+團
+④
+2.4
+D OD)
+0
+2
+1.6
+0.5
+CD
+CRDD
+2.2
+KED
+CTXD
+(
+.8
+1.4
+00
+0
+BXD
+ED
+1.8
+1.6
+.2
+0.5
+10000
+00
+1000
+1000
+10000
+10000
+10000
+30000
+30000
+30000
+100000
+100000
+100000
+100000
+dynamical timescale
+dynamical timescale
+dynamical timescale
+dynamical timescale
+TEand the timescale of changes in this length would be much slower than 휏퐹 , the timescale of each
+of the components ℎ휇(푡). We therefore speculated that increasing the timescale of dynamics of the
+latent variables should eventually lead to scaling and crackling in the single-variable model as well
+as the five-variable one. To examine the dependence of avalanche scaling on this timescale, we
+simulated one-variable and five-variable networks at fixed 휂 and 휖 coupled to latent variables with
+the correlation time of their Ornstein-Uhlenbeck dynamics of 휏퐹 ∈ [103, 105] time steps, spanning
+from a factor of 10 faster to a factor of 10 slower than the original 휏퐹 in Fig. 1. Simulations were
+replicated five times at each combination of parameters by drawing new latent variable coupling
+values (퐽푖휇), as well as new latent variable dynamics and instances of neural firing. For simulations
+that passed the criteria to be fitted by power laws, we plot the fitted values of 휏 , 훼, 훾fit and 훾fit − 훾pred
+(Fig. 2G-J). Missing points are those for which distributions did not pass the power law fit criteria.
+In the single-variable model, best-fit exponents changed abruptly for latent variable timescale
+around 휏퐹 = 104 (Fig. 2G, H), while in the five-variable model, exponents tended to increase grad-
+ually (Fig. 2G, H, red). The discontinuity in the single-variable case reflected a change in the lower
+cutoff values in the power-law fits: size and duration distributions generated with faster latent
+dynamics could be fit reasonably well to a power law by using a high value of the lower cutoff
+(Fig. 2-Supp. Fig. 3). For time scales greater than ∼ 104, the minimum cutoffs dropped, and the
+single-variable model generated power-law distributed avalanches and crackling (Fig. 2J), similar
+to the five-variable model. In summary, in the DLV model, introducing multiple variables gener-
+ated scaling at faster timescales. However, by slowing the timescale of the latent dynamics, the
+DLV model generated signatures of critical avalanche scaling for both multi- and single-variable
+simulations.
+Avalanche criticality, input scaling, and firing threshold
+In the previous section, we found that a very slow single DLV model generated scaling. Thus, from
+now on, we simplify the model in order to characterize avalanche statistics across values of input
+scaling 휂 and firing threshold 휖. Specifically, we modeled a population of 푁 = 128 neurons coupled
+to a single quasi-static latent variable. We simulated 103 segments of 104 steps each and drew a
+new value of the latent variable (ℎ ∼ 푁(0, 1)) for each segment. Ten replicates of the simulation
+were generated at each of the combinations of 휂 and 휖 (see Methods).
+Almost independent of 휂 and 휖, we found quality power law fits and crackling. Fig. 3 shows
+the average (across 푛 = 10 network realizations) of the exponents extracted from size (휏, Fig. 3A)
+and duration (훼, Fig. 3C) distributions. At small firing threshold (휖 = 2), we do not observe scaling
+because the system is always active, and all avalanches merge into one. At large firing threshold 휖
+and low input scaling 휂, we do not observe scaling because activity is so sparse that all avalanches
+are small. At intermediate values of the parameters, the simulations generated plausible scaling
+relationships in size and duration. The difference between 훾fit and 훾pred was typically less than 0.1
+(Fig. 4D-F) which was consistent with previously reported differences between fit and predicted
+exponents (Ma et al., 2019). Thus, there appears to be no need for fine-tuning to generate apparent
+scaling in this model, at least in the limit of (near) infinite observation time. Wherever 휂 and 휖
+generate avalanches, there are approximate power-law distributions and crackling.
+To determine where avalanches occur, we derive the avalanche rate across values of the latent
+variable ℎ. In the quasi-static model, the probability of an avalanche initiation is the probability of a
+transition from the quiet to an active state. Because all neurons are conditionally independent, this
+is 푃ava = 푃silence(1 − 푃silence). Then the expected number of avalanches ̂푁ava is obtained by integrating
+푃ava over ℎ at each value of 휂 and 휖:
+̂푁ava = ∫ 푃ava(휖, 휂, ℎ; 퐽푖, 푁)푝(ℎ)푑ℎ = ∫
+∏
+푖
+(
+1
+1 + 푒휂퐽푖ℎ+휖
+) (
+1 −
+∏
+푖
+(
+1
+1 + 푒휂퐽푖ℎ+휖
+))
+푝(ℎ)푑ℎ,
+(3)
+where 푝(ℎ) is the standard normal distribution. This probability tracks the observed number of
+avalanches across simulations, Fig. 4A.
+6 of 18
+
+Figure 3. Exponents across network simulations. Each parameter combination 휂, 휖 was simulated for ten
+replicates, each time drawing randomly the couplings 퐽푖, the latent variable values, and the neural activities.
+A: Average across replicates for the size exponent 휏. B: Scatter plot of 훼 vs. 휏 for each network replicate for
+parameter combinations indicated in A. Linear relationships between 휏 and 훼, corresponding to the minimum
+and maximum values of 훾fit from panel E, are shown to guide the eye. C: Same as A, for duration exponent 훼.
+D: Predicted exponent, 훾pred, derived from A and C. E: Value of 훾fit from fit to ̄푆퐷 ∼ 퐷훾. F: Difference between
+훾pred and 훾fit.
+7 of 18
+
+Average Exponents: Size
+.8
+口
+0
+2.6
+las
+口
+2.4
+2.8
+米
+8
+2.2
+*
+(duration exponent)
+2.6
+2
+4
+6
+10
+米
+米
+n (gain)
+米
+柔
+2
+米
+Average Exponents: Duration
+2.8
+2
+2.6
+米
+(sel
+2.4
+8
+24680
+2.5
+3
+n (gain)
+(sizeexponent)
+Crackling Exponents
+Predicted
+Difference
+2
+0.2
+2
+1.3
+0.
+as
+Dias
+.2
+8
+1.
+-0.1
+0.2
+10
+810
+4
+aairTo gain an intuition for the conditions under which avalanches occur, we show two slices of
+the avalanche probability, at fixed 휂 (Fig. 4B) and at fixed 휖 (Fig. 4C). Black regions indicate where
+avalanches are likely to occur. If, for a given value of 휖 and 휂, there is no overlap between high
+avalanche probability regions and the distribution of ℎ, then there will be no avalanches. For large
+휖, avalanches occur because neurons with large coupling to the latent variable (휂|퐽푖| >> 1, recall
+퐽푖 ∼ 푁(0, 1)) are occasionally activated by a value of the latent variable ℎ that is sufficient to exceed
+휖 (Fig. 4B). Thus, the scaling parameter 휂 controls the value of ℎ for which avalanches occur most
+frequently (Fig. 4C). As 휖 decreases, avalanches occur for smaller and smaller ℎ until avalanches
+primarily occur when ℎ = 0.
+To calculate the probability of avalanches, we must integrate over all values of ℎ, but we can
+gain a qualitative understanding of which avalanche regime the system is in by examining the
+probability of avalanches at ℎ = 0. At ℎ = 0, the avalanche probability (see Methods) is
+푃ava(휖, 휂, ℎ = 0; 퐽푖, 푁) =
+(
+1
+1 + 푒휖
+)푁 (
+1 −
+(
+1
+1 + 푒휖
+)푁)
+,
+(4)
+which is maximized at 휖0 = − log(21∕푁 − 1), independent of 퐽푖 and 휂. The dependence on 푁 re-
+flects that a larger threshold is required for larger networks: large networks (푁 → ∞) are unlikely
+to achieve complete network silence, therefore preventing avalanches from occurring. Similarly,
+small networks (푁 ∼ 1) are unlikely to fire consecutively and thus are unlikely to avalanche.
+We plot 푃ava(휖, 휂; 퐽푖, 푁, ℎ = 0) as a function of 휖 in Fig. 4B. The peak at 휖0 divides the space into
+two regions. For 휖 < 휖0, a power-law is only observed in the large-size avalanches, which are rare
+(Fig. 4E, green). By contrast, when 휖 > 휖0, minimum size cutoffs are low (Fig. 4F, orange). Both
+regions, 휖 < 휖0 and 휖 > 휖0, exhibit crackling noise scaling. If observation times are not sufficiently
+long (estimated in Fig. 4-Supp. Fig. 1), then scaling will not be observed in the 휖 < 휖0 region, whose
+scaling relations consist of rare events. Insufficient observation times may explain experiments
+and simulations where avalanche scaling was not found.
+Inferring the latent variable
+Our analysis of 푃ava(휖, 휂, ℎ) at ℎ = 0 suggested that there are two types of avalanche regimes: one
+with high activity and high minimum cutoffs in the power law fit (Type 1), and the other with lower
+activity and size cutoffs (Type 2). Further, when 푃ava drops to zero, avalanches disappear because
+the activity is too high or too low. We now examine how information about the value of the latent
+variables represented in the network activity relates to the activity type. To delineate these types,
+we calculated numerically 휖∗(휂), the value of 휖 for which the probability of avalanches is maximized,
+and the contours of 푃ava (Fig. 5A). Curves for 휖∗(휂) and 휖0 and 푃ava = 10−3 are shown in Fig. 5A and B.
+We expect that the more cells fire, the more information they would convey, until the firing
+rate saturates, and inferring the value of the latent variable becomes impossible. Fig. 5B supports
+the prediction: generally, information is higher in regions with more activity (lower 휖, higher 휂), but
+only up to a limit: as 휖 → 0, information decreases. This decrease begins approximately where
+the probability of avalanches drops to nearly zero (dashed black lines, Fig. 5B-E) because all of
+the activity merges into a few very large avalanches. In other words, the Type-1 avalanche region
+coincides with the highest information about the latent variable.
+The critical brain hypothesis suggests that the brain operates in a critical state, and its func-
+tional role may be in optimizing information processing (Beggs, 2008; Chialvo, 2010). Under this
+hypothesis, we would expect the information conveyed by the network to be maximized in the
+regions we observe avalanche criticality. However, we see that critical regions do not always have
+optimal information transmission. In Fig. 5, the region that displays crackling noise is that where
+avalanches exist (푃ava > 0.001), which corresponds to any 휂 value and 휖 ≳ 3. This avalanche re-
+gion encompasses both networks with high information transmission and networks with low in-
+formation transmission. In summary, observing avalanche criticality in a system does not imply a
+high-information processing network state. However, the scaling can be seen at smaller cutoffs,
+8 of 18
+
+Figure 4. Avalanches in the DLV model with a single quasistatic variable. A: Number of avalanches in
+simulations as a function of the calculated probability of avalanches at fixed 휂 across values of 휖 and latent
+variable ℎ. Line indicates equality. B: Analytically calculated probability of avalanches with 휂 = 2 across values
+of 휖 and ℎ. The latent variable ℎ is normally distributed with mean 0 and variance 1. Where the distribution of
+ℎ overlaps with regions of high probability (black), avalanches occur. C: Analytically calculated probability of
+avalanches at 휖 = 8 across values of 휂 and ℎ. Increasing 휂 narrows the range of ℎ that generates avalanches. D:
+Analytically calculated probability of avalanches at ℎ = 0 for a populations of 128 neurons (black line) and for a
+varying 휖. Size distributions corresponding to simulations marked by the green and orange crosses are in E, F.
+E: Example of size distribution with 휖 < 휖0 (orange marker in D). Size cutoff is close to 100. F: Example of size
+distribution with 휖 > 휖0 (green marker in D). Size cutoff is < 10.
+Figure 4–Figure supplement 1. Estimated simulation time to observe avalanche criticality.
+and hence with shorter recordings, in the high-information state. This parallels the discussion by
+Schwab et al. (2014), who noticed that the Zipf’s law always emerges in neural populations driven
+by quasi-stationary latent fields, but it emerges at smaller system sizes when the information about
+the latent variable is high.
+Discussion
+Here we studied systems with distributed, random coupling to Dynamical Latent Variables (DLV)
+and we found that avalanche criticality is nearly always observed, with no fine-tuning required.
+Avalanche criticality was surprisingly robust to changes in input gain and firing rate threshold. Loss
+of avalanche criticality could occur if the latent process was not well-sampled, either because the
+simulation was not long enough or the dynamics of the latent variables were too fast. Finally, while
+information about the latent variables in the network activity was higher where avalanches were
+generated compared to when they were not, there was a range of information values across the
+critical avalanche regime. Thus, avalanche criticality alone was not a predictor of optimal informa-
+tion transmission.
+Explaining experimental exponents
+A wide range of critical exponents have been found in ex vivo and in vivo recordings from various
+systems. For instance, the seminal work on avalanche statistics in cultured neuronal networks
+by Beggs and Plenz (2003) found size and duration exponents of 1.5 and 2.0 respectively, along
+with 훾 = 2, when time was discretized with a time bin equal to the average inter-event interval in
+the system. These values are predicted by a theoretical model of a critical branching process. By
+contrast, a survey of many in vivo and ex vivo recordings found power-law size distributions with
+exponents ranging from 1 to 3 depending on the system (Fontenele et al., 2019). Separately, Ma
+9 of 18
+
+A
+B
+c
+n= 2
+E=8
+0.25
+10
+0.25
+Avalanche count (Na)
+2
+2
+8
+1.5
+(seIc
+a
+6
+1
+8
+m
+4
+0.5
+2
+0
+14
+0
+2
+-5
+0
+5
+-5
+0
+5
+7
+P
+(1-P
+h
+h
+D
+silence
+silence
+E
+F
+T = 2.24 (0.01 SE)
+T = 2.00 (0.01 SE)
+0.25
+0
+0.2
+-2
+-2
+0.15
+-4
+-4
+0.1
+-6
+-6
+P°0.05
+0
+-8
+-8 
+10
+15
+0
+E (bias)
+avalanche size
+avalanche sizeFigure 5. Information in the neural activity about the latent variable is higher in the low-휖 avalanche region,
+compared to high-휖 avalanche or high-rate avalanche-free activity. A: Probability of avalanche per time step
+across values of 휂 and 휖. Solid magenta curve follows 휖∗(휂), the value of 휖 maximizing the probability of
+avalanches at fixed 휂. Dashed magenta line indicates 휖0, calculated analytically, which matches 휖∗ at 휂 = 0. B:
+Information about latent variable, calculated from maximum likelihood estimate of ℎ using population
+activity. MLE approximation is invalid in the dark-blue region bounded by gray curve. Magenta line marks the
+maximum values of 푃ava, reproduced from A. Dashed black curve indicates 푃ava = 0.001. The highest
+information region falls between 휖∗(휂) and the contour for 푃ava = 0.001. C - E: Slices of B, showing 퐼MLE(휖) for
+휂 = {2, 5, 9}. Magenta and dashed black lines again indicate 휖∗ and 푃ava = 0.001, respectively, as in B. Black
+dashed line marks the approximate boundary between the high-activity/no avalanche and the high-cutoff
+avalanche, and magenta line marks boundary between high-cutoff and low-cutoff avalanche regions.
+10 of 18
+
+0
+0.25
+B
+0
+2.5
+2
+2
+ cells (MLE)
+0.2
+2
+0.001
+Probability of Avalanche
+三
+4 
+4
+0.02
+0.15
+1.5
+(bias)
+6
+(bias)
+9
+128
+8
+8
+0.1
+for N
+0.08
+1
+0
+10
+0.04
+10
+D
+=0.001
+ava
+0.05
+12
+12
+0
+e(n)
+14
+0
+14
+0
+0
+2
+4
+6
+8
+10
+2
+4
+6
+8
+10
+n (input scaling)
+n (input scaling)
+c
+D
+n=2
+n=5
+E
+n=9
+2
+2.5
+128
+128
+128
+2
+1.5
+=
+z
+z
+h*),
+h*),
+1
+0.5
+≤ 0.5
+0
+0
+0
+0
+5
+10
+0
+5
+10
+0
+5
+10
+E(
+(bias)
+E (bias)
+E (bias)et al. (2019) reported recordings in freely moving rats with size exponents ranging from 1.5 to 2.7.
+In all of the these recordings, when the crackling relationship held, the reported value of 훾 was
+near 1.2 (Fontenele et al., 2019; Ma et al., 2019).
+Our DLV model, across the parameters we tested that produced exponents consistent with the
+scaling relationship, generated 휏 values that ranged from 1.9 to about 2.5. Across those simulations,
+we found values 훾 within a narrow band from 1.1 to 1.3 (see Fig. 2I, J and Fig. 3H). While the exponent
+values our model produces are inconsistent with a critical branching process (훾 = 2), they match
+very closely the ranges of exponents reported by Fontenele et al. (2019).
+One possible resolution to the discrepancy in exponents derives from how the system is sub-
+sampled in space or coarse-grained in time, both of which systematically change exponents 휏 and
+훼 (Beggs and Plenz, 2003; Shew et al., 2015). Were we to change the time bin, our modeling results
+would exhibit different exponent values. However, neither manipulations of the latent variable
+timescale (휏퐹 or 푁퐹 ), nor of the overall activity level (휂, 휖) produced exponents close to 1.5 and 2.0,
+despite maintaining the crackling relationship across many different choices of parameters.
+A second possibility is that different experiments study similar, but distinct biological phenom-
+ena. In other words, the underlying biology can differ between networks that were cultured in vitro
+and those that were not, whether they are in vivo or ex vivo (i.e., brain slices). This could happen
+if cultured networks develop connections between neurons such that they truly do produce dy-
+namics that approximate a critical branching process, while brain networks that develop in a living
+brain have different structure and resulting dynamics and can be better understood as a system
+coupled to latent dynamical variables. This is especially true in sensory systems, where coupling
+to (latent) external stimuli in a way that the neural activity can be used to infer the stimuli is the
+reason for the networks’ existence (Schwab et al., 2014).
+Relationship to past modeling work
+Our model is not the first to produce approximate power-law size and duration distributions for
+avalanches from a latent variable process (Touboul and Destexhe, 2017; Priesemann and Shriki,
+2018). In particular, Priesemann and Shriki (2018) derived the conditions under which an inhomo-
+geneous Poisson process could produce such approximate scaling. The basic idea is to generate a
+weighted sum of exponentially distributed event sizes, each of which are generated from a homo-
+geneous Poisson process. How each process is weighted in this sum determines the approximate
+power-law exponent, allowing one to tune the system to obtain the critical values of 1.5 and 2. In-
+terestingly, this model did not generate non-trivial scaling of size with duration (푆 ∼ 퐷훾). Instead,
+they found 훾 = 1, not the predicted 훾 = 2. Our results differ significantly, in that 훾 was typically
+between 1.1 and 1.3 and it was nearly always close to the prediction from 훼 and 휏. We speculate
+that this is due to nonlinearity in the mapping from latent variable to spiking activity, as doubling
+the latent field ℎ does not double the population activity, but doubling the rate of a homogeneous
+Poisson process does double the expected spike count. As biological networks are likely to have
+such nonlinearities in their responses to common inputs, this scenario may be more applicable to
+certain kinds of recordings.
+Summary
+Latent variables – whether they are emergent from network dynamics (Clark et al., 2022; Seder-
+berg and Nemenman, 2020) or derived from shared inputs – are ubiquitous in large-scale neural
+population recordings. This fact is reflected most directly in the relatively low-dimensional struc-
+ture in large-scale population recordings (Stringer et al., 2019; Pandarinath et al., 2018; Nieh et al.,
+2021). We previously used a model based on this observation to examine signatures of neural crit-
+icality under a coarse-graining analysis and found that coarse-grained criticality is generated by
+systems driven by many latent variables (Morrell et al., 2021). Here we showed that the same
+model also generates avalanche criticality, and that when information about the latent variables
+can be inferred from the network, avalanche criticality is also observed. Crucially, finding signa-
+11 of 18
+
+tures of avalanche criticality required long observation times, such that the latent variable was
+well-sampled. Previous studies showed that Zipf’s law appears generically in systems coupled to a
+latent variable that changes slowly relative to the sampling time, and that the Zipf’s behavior is eas-
+ier to observe in the higher information regime (Schwab et al., 2014; Aitchison et al., 2016). How-
+ever, this also suggests that observation of either scaling at modest data set sizes indeed points
+to some fine-tuning — namely to the increase of the information in the individual neurons (and,
+since neurons in these models are conditionally independent, also in the entire network) about
+the value of the latent variables. In other words, one would expect a sensory part of the brain, if
+adapted to the statistics of the external stimuli, to exhibit all of these critical signatures at relatively
+modest data set sizes. In monocular deprivation experiments, when the activity in the visual cor-
+tex is transiently not adapted to its inputs, scaling disappears, at least for recordings of a typical
+duration, and is restored as the system adapts to the new stimulus (Ma et al., 2019). We speculate
+that the observed recovery of criticality by Ma et al. (2019) could be driven by neurons adapting
+to the reduced stimuli state, for instance, by adjusting 휂 (input scaling) and 휖 (firing rate threshold).
+Taken together, these results suggest that critical behavior in neural systems – whether based on
+the Zipf’s law, avalanches, or coarse-graining analysis – is expected whenever neural recordings ex-
+hibit some latent structure in population dynamics and this latent structure can be inferred from
+observations of the population activity.
+Methods and Materials
+Simulation of Dynamic Latent Variable (DLV) model
+We study a model from Morrell et al. (2021), incorporating only latent variables (no place variables),
+and assuming that every cell is coupled to every latent variable with some randomly drawn coupling
+strength.
+The probability of observing a certain population state {푠푖} given latent variables {ℎ휇(푡)} at time
+푡 is
+푃({푠푖}|{ℎ휇}) =
+1
+푍({ℎ휇})푒퐻({푠푖},{ℎ휇}),
+(5)
+where 푍 is the normalization, and 퐻 is the “energy”:
+퐻 =
+푁,푁f
+∑
+푖,푚=1
+휂ℎ휇(푡)퐽푖휇푠푖 + 휖푠푖.
+(6)
+The latent variables {ℎ휇(푡)} are Ornstein-Uhlenbeck processes with zero mean, unit variance, and
+time constant 휏푚. Couplings 퐽푖휇 are drawn from the standard normal distribution.
+The parameters 휂, 휖, and 휏푚 are constants, and we simulate 푁 = 1024 cells. For the infinite time
+constant simulation, we reset ℎ푛 ∼  (0, 1) (for each of 푛 = 1..푁푛) and simulate for 10000 time steps,
+then repeat for 1000 draws of ℎ푛.
+Time step units
+Most results were presented using arbitrary time units: all times (i.e., 휏퐹 and avalanche duration 퐷)
+are measured in units of an unspecified time step. Specifying a time bin converts the probability
+Table 1. Simulation parameters for Fig. 1.
+Parameter
+Description
+Value
+휖
+bias towards silence
+휖 = 12
+휂
+variance multiplier
+휂 = 4.0
+푁F
+number of latent fields
+푁F = 5
+휏퐹
+latent field time constant
+휏 = 104
+푁
+number of cells
+푁 = 1024
+12 of 18
+
+of firing into actual firing rates, in spikes per second, and this choice determines which part of the
+휂-휖 phase space is most relevant to a given experiment.
+The time step is the temporal resolution at which activity is discretized, which varies from sev-
+eral to hundreds of milliseconds across different experimental studies (Beggs and Plenz, 2003;
+Fontenele et al., 2019; Ma et al., 2019). In physical units and assuming a bin size of 3 ms to 10 ms,
+our choice of 휂 and 휖 in Fig. 2 would yield physiologically realistic firing rate ranges (Hengen et al.,
+2016), with high-firing neurons reaching averages rates of 20 − 50 spikes/second and median firing-
+rate neurons around 1 − 2 spikes/second. The timescales of latent variables examined range from
+about 3 seconds to 3000 seconds, assuming 3-ms bins. Simulations were carried out for the same
+number of time steps (2 × 106), which would be approximately 1 to 2 “hours,” which is a reasonable
+duration for in vivo neural recordings. Note that at large values of 휏퐹 , the latent variable space is
+not well sampled during this time period.
+Analysis of avalanche statistics
+Setting the threshold for observing avalanches
+In our model, we count avalanches as periods of continuous activity (in any subset of neurons)
+that is book-ended by time bins with no activity in the entire simulated neural network. For real
+neural populations of modest size, this method fails because there are no periods of quiescence.
+The typical solution is to set a threshold, and to only count avalanches when the population activity
+exceeds that threshold, with the hope that results are relatively robust to that choice. In our model,
+this operation is equivalent to changing 휖, which shifts the probability of firing up or down by a
+constant amount across all cells independent of inputs. Our results in Fig. 3 show that 훼 and 휏
+decrease as the threshold for detection is increased (equivalent to large |휖|), but that the scaling
+relationship is maintained. The model predicts that 훾pred − 훾fit would initially increase slightly with
+the detection threshold before decreasing back to near zero.
+Following the algorithm laid out in Clauset et al. (2009), we fit power laws to the size and dura-
+tion distributions from simulations generating avalanches. We use least-squares fitting to estimate
+훾fit, the scaling exponent for size with duration, assessing the consistency of the fit across decades.
+Reading power laws from data
+We want, from each simulation, a quantification of the quality of scaling (how many decades, min-
+imally) and an estimate of the scaling exponents (휏 for the size distribution, 훼 for the duration
+distribution). Following the steps outlined by Clauset et al. (2009), we use the maximum-likelihood
+estimator to determine the scaling exponent. This is the solution to the transcendental equation
+휁′(̂훼, 푥min)
+휁′(̂훼, 푥min) = −1
+푛
+푛
+∑
+푖=1
+ln 푥푖
+(7)
+where 휁(훼, 푥min) is the Hurwitz zeta function. For values of 푥min < 6, a numerical look-up table based
+on the built-in Hurwitz zeta function in the symbolic math toolbox was used (MATLAB2019b). For
+Table 2. Simulation parameters for Fig. 2.
+Parameter
+Description
+Value
+휖
+bias towards silence
+휖 = 8 (for 푁퐹 = 1) or
+휖 = 12 (for 푁퐹 = 5)
+휂
+variance multiplier
+휂 = 4.0
+푁F
+number of latent fields
+푁F = 1 or 5
+휏퐹
+latent field time constant
+휏 ∈  [log 103, log 105]
+푁
+number of cells
+푁 = 1024
+13 of 18
+
+푥min > 6 we use an approximation (Clauset et al. (2009)),
+̂훼 = 1 + 푛
+(
+∑
+푖
+ln
+푥푖
+푥min − 1
+2
+)−1
+.
+(8)
+To determine 푥min, we computed the maximum absolute difference between the empirical cu-
+mulative density (푆(푥)) function and model’s cumulative density function 푃(푥) (the Kolmogorov-
+Smirnov (KS) statistic; 퐷 = max푥≥푥min|푆(푥)−푃(푥)|). The KS statistic was computed between for power-
+law models with scaling parameter ̂훼 and cutoffs 푥min. The value of 푥min that minimizes the KS statis-
+tic was chosen. Occasionally the KS statistic had two local minima (as in Figure 2-Supplemental
+Figure 1), indicating two different power-laws. In these cases, the minimum size and duration cut-
+offs were the smallest values that were within 10% of the absolute minimum of the KS statistic.
+Note that the statistic is computed for each model only on the power-law portion of the CDF (i.e.
+푥푖 ≥ 푥min). We do not attempt to determine an upper cut-off value.
+To assess the quality of the power-law fit, Clauset et al. (2009) compared the empirical observa-
+tions to surrogate data generated from a semi-parametric power-law model. The semi-parametric
+model sets the value of the CDF equal to the empirical CDF values up to 푥 = 푥min and then according
+to the power-law model for 푥 > 푥min. If the KS statistic for the real data (relative to its fitted model)
+is within the distribution of the KS statistics for surrogate datasets relative to their respective fitted
+models, the power-law model was considered a reasonable fit.
+Strict application of this methodology could give misleading results. Much of this is due to the
+loss of statistical power when the minimum cutoff is so high that the number of observations drops.
+For instance, in the simulations shown in Fig. 2, the one-variable duration distribution passed the
+Clauset et al. (2009) criterion, with a minimum KS statistic of 0.03 when the duration cutoff was
+18 time steps. However, for the five-variable simulation in Fig. 2, a power-law would be narrowly
+rejected for both size and duration, despite having much smaller KS statistics: for 휏, the KS statistic
+was 0.0087 (simulation range: 0.0008 to 0.0082; number of avalanches observed: 58, 787) and for 훼 it
+was 0.0084 (simulation range: 0.0011 to 0.0075). Below we discuss this problem in more detail.
+Determining range over which avalanche size scales with duration
+For fitting 훾, our aim was to find the longest sampled range over which we have apparent power-
+law scaling of size with duration. Because our sampled duration values have linear spacing, error
+estimates are skewed if a naive goodness of fit criterion is used. We devised the following algorithm.
+First, the simulation must have at least one avalanche of size 500. We fit 푆 = 푐퐷훾 over one decade
+at a time. We chose as the lower duration cutoff the value of minimum duration for which the
+largest number of subsequent (longer-duration) fits produced consistent fit parameters (Figure
+2-Supp. Fig. 3 and 4, top row). Next, with the minimum duration set, we gradually increased the
+maximum duration cut-off, and we determined whether there was a significant bias in the residual
+over the first decade of the fit. We selected the highest duration cutoff for which there was no bias.
+Finally, over this range, we re-fit the power law relationship and extracted confidence intervals.
+Our analysis focused on finding the apparent power-law relationship that held over the largest
+log-scale range. A common feature across simulation parameters (휏퐹 , 푁퐹 ) was the existence of
+Table 3. Simulation parameters for Fig. 3 and 4.
+Parameter
+Description
+Value
+휖
+bias towards silence
+휖 ∈ {2, 4, ...14}
+휂
+variance multiplier
+휂 ∈ {1, 2, ...10}
+푁F
+number of latent fields
+푁F = 1
+휏퐹
+latent field time constant
+quasistatic
+푁
+number of cells
+푁 = 128
+14 of 18
+
+two distinct power-law regimes. This is apparent in Fig. 2I, which shows that when 푁퐹 = 1 at small
+휏퐹 , the best-fit 훾 (that showing the largest range with power-law-consistent scaling) is much larger
+(> 1.7), and then above 휏퐹 ∼ 3000, the best-fit 훾 drops to around 1.3.
+Statistical power of power-law tests
+In several cases, we found examples of power-law fits that passed the rejection criteria commonly
+used to determine avalanche scaling relationships because of limited number of observations. A
+key example is that of the single latent variable simulation shown in Fig. 2B, where we could not
+reject a power law for the duration distribution. Conversely, strict application of the surrogate cri-
+teria would reject a power law for distributions that were quantitatively much closer to a power-law
+(i. e., lower KS statistic), but for which we had many more observations and thus a much stronger
+surrogate test (Fig. 2). This points to the difficulty of applying a single criterion to determining a
+power-law fit. In this work, we adhere to the criteria set forth in Clauset et al. (2009), with a mod-
+ification to control for the unreasonably high statistical power of simulated data. Specifically, the
+number of avalanches used for fitting and for surrogate analysis was capped at 500, 000, drawn
+randomly from the entire pool of avalanches.
+Additionally, we found examples in which a short simulation was rejected, but increasing the
+simulation time by a factor of five yielded excellent power-law fits. We speculate that this arises
+due to insufficient sampling of the latent space. These observations raise an important biological
+point. Simulations provide the luxury of assuming the network is unchanging for as long as the
+simulator cares to keep drawing samples. In a biological network, this is not the case. Over the
+course of hours, the effective latent degrees of freedom could change drastically (e. g., due to
+circadian effects (Aton et al., 2009), changes in behavioral state (Fu et al., 2014), plasticity (Hooks
+and Chen, 2020), etc.), and the network itself (synaptic scaling, firing thresholds, etc.) could be
+plastic (Hengen et al., 2016). All of these factors can be modeled in our framework by determining
+appropriate cutoffs (in duration of recording, in time step sizes, for activity distributions) based on
+specific experimental timescales.
+Calculation of avalanche regimes
+In the quasistatic model, we derive the dependence of the avalanche rate on 휂, 휖 and number of
+neurons 푁, finding that there are two distinct regimes in which avalanches occur. Each time bin is
+independent, conditioned on the value of ℎ. For an avalanche to occur, the probability of silence
+in the population (i.e., all 푠푖 = 0) must not be too close to 0 (or there are no breaks in activity) or too
+close to 1 (or there is no activity). At fixed ℎ, the probability of silence is
+푃silence(휖, 휂; 퐽푖, 푁, ℎ) =
+∏
+푖
+1
+1 + exp(−휂퐽푖ℎ + 휖).
+(9)
+An avalanche occurs when a silent time bin is followed by an active bin, which has probability
+푃ava(휖, 휂; 퐽푖, 푁, ℎ) = 푃silence(1 − 푃silence).
+Information calculation
+Maximum-likelihood decoding
+For large populations coupled to a single latent variable, we estimate the information between pop-
+ulation spiking activity and the latent variable as the information between the maximum-likelihood
+estimator ℎ∗ of the latent variable ℎ and the latent variable itself. This approximation fails at ex-
+tremes of network activity levels (low or high).
+Specifically, we approximate the log-likelihood of ℎ∗ given ℎtrue near ℎ∗ by log 퐿(ℎ−ℎ∗) ≈ log 퐿푚푎푥−
+1
+2
+(ℎ−ℎ∗)2
+휎2
+ℎ∗
+, so we assume that ℎ∗ is normally distributed about ℎtrue with variance 휎2(ℎtrue). The variance
+is then derived from the curvature of the log-likelihood at the maximum. The information between
+15 of 18
+
+two Gaussian variables, here 푃(ℎ∗|ℎ) = 푁(ℎ, 휎2
+ℎ∗) and 푝(ℎ) = 푁(0, 1), is
+퐼(ℎ; ̄푠푖,푇 ) ≈ 1
+2
+⟨
+log
+푇
+휎2
+ℎtrue
+⟩
+ℎtrue
+,
+(10)
+where the average is taken over ℎtrue ∼ 푁(0, 1).
+Given a set of 푇 observations of the neurons {푠푖}, the likelihood is
+푃({푠푖}푡|ℎ) =
+푁,푇
+∏
+푖,푡
+푃(푠푖|ℎ) =
+푁,푇
+∏
+푖,푡
+푒−휂푠푖퐽푖ℎ−휖푠푖
+1 + 푒−(휂퐽푖ℎ+휖) .
+(11)
+Maximizing the log likelihood gives the following condition:
+0 = 휕(log 푃)
+휕ℎ
+||ℎ∗ = 휕
+휕ℎ
+(
+∑
+푖,푡
+((−휂푠푖퐽푖ℎ − 휖푠푖) − log(1 + 푒−(휂퐽푖ℎ+휖))
+)
+||ℎ∗
+(12)
+=
+∑
+푖
+−휂 ̄푠푖퐽푖푇 +
+푇 퐽푖휂
+1 + 푒휂퐽푖ℎ∗+휖 ,
+(13)
+where ̄푠푖 = 1
+푇
+∑
+푡 푠푖푡 is the average over observations 푡. The uncertainty in ℎ∗ is 휎ℎ, which was calcu-
+lated from the second derivative of the log likelihood:
+1
+휎2
+ℎ∗
+= −휕2(log 푃)
+휕ℎ2
+(14)
+= − 휕
+휕ℎ
+(
+∑
+푖
+−휂 ̄푠푖퐽푖푇 +
+푇 퐽푖휂
+1 + 푒휂퐽푖ℎ+휖
+)
+||ℎ∗
+(15)
+=
+∑
+푖
+푇 (휂퐽푖)2푒휂퐽푖ℎ∗+휖
+(1 + 푒휂퐽푖ℎ∗+휖)2
+(16)
+=
+∑
+푖
+푇 (휂퐽푖)2
+4 cosh2( 휂퐽푖ℎ∗+휖
+2
+)
+.
+(17)
+This expression depends on the observations ̄푠푖 only through the maximum-likelihood estimate ℎ∗.
+When ℎ∗ → ℎtrue, then the variance is
+1
+휎2
+ℎ∗
+=
+∑
+푖
+푇 (휂퐽푖)2
+4 cosh2( 휂퐽푖ℎtrue+휖
+2
+)
+≡
+푇
+휎2
+ℎtrue
+.
+(18)
+To generate Figure 5, we evaluated Eqn. 10 using Eqn. 18.
+Acknowledgments
+IN was supported in part by the Simons Foundation Investigator program, the Simons-Emory Con-
+sortium on Motor Control, NSF grant BCS/1822677 and NIH grant 2R01NS084844. AS was sup-
+ported in part by NIH grant 1RF1MH130413-01 and by startup funds from the University of Min-
+nesota Medical School.
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+18 of 18
+
+
+Figure 2–Figure supplement 1. Illustration of algorithm for determining 휏 and 훼, using one vari-
+able example in Fig. 2. A-B: Probability density function for avalanche size (A) and duration (B) on
+a log-log scale, with the best power law fit (red). C-D: In blue: Maximum likelihood exponent of a
+power-law model as a function of the minimum (lower cutoff) size (C) and duration (D). In red: KS
+statistics (see Methods) for each fit. “Best fit” is the power law with the minimum KS statistic. E-F:
+Surrogate data procedure. To generate each surrogate, samples were drawn from a power law
+with size / duration cutoff indicated (E, 푆푚푖푛 = 3; F, 퐷푚푖푛 = 18) and the KS statistic was computed.
+Histograms illustrate KS statistic across surrogates (blue), while values derived from data are in
+red. Because the red line does not fall within the blue histogram, the hypothesis that the data is
+fitted well by a power law fit was rejected in E. At the same time, since the red line falls within the
+blue histogram in F, the hypothesis was accepted.
+
+B
+A
+-10
+8
+15
+10
+0
+2
+og1
+size
+0.07
+0.07
+0.06
+0.06
+KS
+KS
+0.05
+0.05
+statistic
+statistic
+0.04
+0.04
+王王
+0.03
+0.03
+0.02
+0.02
+0.5
+1.5
+2
+0.5
+1.5
+2
+log10
+log
+E
+min
+250
+counts (2000 surrogates)
+300
+200
+150
+00
+50
+50
+3
+2.5
+1.5
+1.8
+-1.6
+1.4
+og
+KS statistic
+log1
+KS statisticFigure 2–Figure supplement 2. Illustration of algorithm for determining 휏 and 훼, using example
+in Fig. 2, five latent variables. Notation the same as in Fig. 2-Fig. Supplement 3.
+
+B
+2
+6
+8
+10
+0.04
+0.05
+0.04
+0.03
+KS
+0.03
+statistic
+stat
+0.02
+tist
+0.02
+0.01
+0.01
+0.5
+1.5
+0.5
+1.5
+log10
+log
+E
+min
+counts (2000 surrogates)
+300
+counts (2000 surrogates)
+250
+250
+200
+200
+150
+100
+100
+50
+50
+3
+-2.8-2.6-2.4-2.2
+2.8
+-2.6
+-2.4-2.2
+2
+log1
+KSstatistic
+log10
+KS statisticFigure 2–Figure supplement 3. Illustration of algorithm for fitting the exponent 훾 and determining
+the range, over which power law scaling of average size with duration is observed, using example in
+Fig. 2(A-D). A-C: Determining the lower bound, the minimum duration 퐷푚푖푛. A: The relation log 푆 =
+푏 + 훾 log 퐷 was fit using linear least-squares, restricted to (overlapping) 1-decade ranges (blue, red:
+example decades). B: Confidence intervals for fit parameters (훾, 푏 for fits starting at each value
+of 퐷푚푖푛. C: Best value of 퐷min was selected based on how many subsequent start points yielded
+consistent slope/intercept values. D-F: Determining the upper bound, maximum duration 퐷푚푎푥. D:
+Keeping 퐷푚푖푛 fixed based of value obtained in C, we test values of 퐷푚푎푥 up to the maximum duration
+event, and fit over the range [퐷min, 퐷max]. E: Average residual over the fit range [퐷min, 퐷min + 1],
+calculated for each fit and plotted against the value of 퐷max used for that fit. The largest value of
+퐷max without evidence of bias in the residual was then selected. F: Final fit and range.
+
+alldurations
+fit to dec. 1
+Estimate of  value and power-law range
+ex.decade 1
+fit to dec. 2
+ex. decade 2
+fit parameters for single-decade fits
+fit range starting from each d.
+min
+8
+1.8
+0.5
+scale)
+1.7
+t (b, 95% CI)
+10
+6
+average size (log s
+% 1.6
+8
+4
+0.5
+9
+intercept 
+2
+0
+1.3
+number 
+2
+-2 
+1.2
+-1.5
+0
+0
+1
+2
+3
+4
+0
+1
+2
+3
+0
+2
+3
+duration (log scale)
+d
+(lower bound on fitted decade)
+d.
+min
+min
+all durations
+fit to (i)
+(i) dmax = 3.95
+fit to (i)
+all end points
+(i) dmax = 2.8 
+all
+final fit
+ within 95% CI
+final fit range
+Part 2: select upper cutoff (d,
+)forS~D
+SE)
+residuals for fits ending at d.
+max
+= 1.72, range: 2.15
+8 
+0.01
+max
++/- 1 
+8
+ scale)
+ scale)
+6
+0.005
+2.8]
+average size (log
+6o) azis 
+4
+4
+2
+80. -0.005
+average s
+2
+0
+-0.01
+0
+error
+-2 
+-0.015
+-2
+0
+1
+2
+4
+3
+3.5
+4
+0
+1
+2
+3
+4
+duration (log scale)
+d.
+duration (log scale)
+maxFigure 2–Figure supplement 4. Illustration of algorithm for fitting the exponent 훾 and determining
+the range over which power-law scaling of average size with duration is observed, using example
+in Fig. 2 E-H. See Fig. 2-Fig. Supplement 3 for caption. In this example, a lower value of 퐷min was
+selected. Panel E, which was flat for Fig. 2-Fig. Supplement 3, now shows how extending the range
+to high values of 퐷max can generate systematic errors at the low range of the fit, even while having
+a high overall goodness of fit metric.
+Figure 4–Figure supplement 1. Estimate of how long it takes to observe avalanche criticality at
+each combination of 휂 and 휖. We took a parameter combination with a low rate of avalanches but
+good apparent scaling (휂 = 4 and 휖 = −14) and assumed that this is a reasonable estimate of the
+minimum number of observations (approximately 106 avalanches) required to observe scaling. To
+translate to observation length (in hours), we divided the number of avalanches observed in each
+full-length simulation by this minimum count and converted to a time using a time bin of 10 ms.
+Simulations were for a recorded population of 128 neurons. For this size of population, 휖0 = 5.2.
+
+Reguired Sim Time (est.)
+300h
+6
+0
+30 h
+-10
+14
+3
+4
+6
+8
+10
+n (gain)all durations
+fit to dec. 1
+Estimate of  value and power-law range
+ex. decade 1
+fit to dec. 2
+ex. decade 2
+Part 1: select lower cutoff (d,
+fit parameters for single-decade fits
+fit range starting from each d.
+min
+6
+1.6
+0.5
+intercept (b, 95% ClI)
+%S6
+10
+0.5
+5
+number of 
+1
+1.2
+0
+0
+1
+2
+3
+4
+0
+2
+3
+0
+1
+3
+2
+duration (log scale)
+d
+(lower bound on fitted decade)
+d.
+min
+min
+all durations
+fit to (i)
+fit to (i)
+all end points
+selectedd
+dmin
+max
+all
+final fit 
+ within 95% CI
+final fit range
+Part 2: select upper cutoff (d,
+)forS~D
+SE)
+residuals for fits ending at d.
+= 1.26, range: 2.15
+6
+6
+( +/- 1
+0.1
+50.08
+0.06
+uo
+0.02
+ave error
+1
+0
+0
+0
+2
+1
+4
+2
+3
+4
+0
+1
+2
+3
+4
+duration (log scale)
+d
+duration (log scale)
+max

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+Draft version January 4, 2023
+Typeset using LATEX default style in AASTeX63
+Gaussian Process Modeling Blazar Multiwavelength Variability: Indirectly Resolving Jet Structure
+Haiyun Zhang (张海云),1 Dahai Yan (闫大海),1 and Li Zhang (张力)1
+1Department of Astronomy, Key Laboratory of Astroparticle Physics of Yunnan Province, Yunnan University, Kunming 650091, China
+ABSTRACT
+Blazar jet structure can be indirectly resolved by analyzing the multiwavelength variability. In this
+work, we analyze the long-term variability of blazars in radio, optical and X-ray energies with the
+Gaussian process (GP) method. The multiwavelength variability can be successfully characterized by
+the damped-random walk (DRW) model. The nonthermal optical characteristic timescales of 38 blazars
+are statistically consistent with the γ-ray characteristic timescales of 22 blazars. For three individuals
+(3C 273, PKS 1510-089, and BL Lac), the nonthermal optical, X-ray, and γ-ray characteristic timescales
+are also consistent within the measured 95% errors, but the radio timescale of 3C 273 is too large to be
+constrained by the decade-long light curve. The synchrotron and inverse-Compton emissions have the
+same power spectral density, suggesting that the long-term jet variability is irrelevant to the emission
+mechanism. In the plot of the rest-frame timescale versus black hole mass, the optical-γ-ray timescales
+of the jet variability occupy almost the same space with the timescales of accretion disk emission from
+normal quasars, which may imply that the long-term variabilities of the jet and accretion disk are
+driven by the same physical process. It is suggested that the nonthermal optical-X-ray and γ-ray
+emissions are produced in the same region, while the radio core which can be resolved by very-long-
+baseline interferometry locates at a far more distant region from the black hole. Our study suggests
+a new methodology for comparing thermal and nonthermal emissions, which is achieved by using the
+standard GP method.
+Keywords: Blazars (164), Jets (870), Light curves (918), Time series analysis (1916)
+1. INTRODUCTION
+Flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs) are classed into a special class of active
+galactic nuclei (AGNs) called blazars, whose jets nearly point to the Earth. Blazars are highly variable over the entire
+electromagnetic bands. One popular scenario is that the accretion onto a supermassive black hole is the central engine,
+driving relativistic jet. But the detailed process is still unclear. Thanks to the high variability of blazars, one can
+investigate the physical process close to the central engine (e.g., Rieger 2019), such as the location of the emitting
+region and the jet-disk connection (e.g., Ackermann et al. 2016; Meyer et al. 2019; Zhang et al. 2022).
+Using advanced interferometric instruments, blazar radio jet can be directly resovled on ∼parsec-scale (see Hovatta
+& Lindfors 2019, for a recent review). This provides a calibrator for multi-band variability analysis. There have been
+lots of works attempting to investigate the underlying physical process of blazar jet with multi-band variability (e.g.,
+Chatterjee et al. 2012; Nakagawa & Mori 2013; Xiong et al. 2017; Goyal et al. 2018, 2022). Max-Moerbeck et al. (2014)
+investigated the time-domain relationship between radio and γ-ray emission of blazars, and found the correlations
+only exist in a minority of the sources over a 4 yr period. They found radio variations lagging the γ-ray variations,
+suggesting that the γ-ray emission originates upstream of the radio emission. This result is further verified by Liodakis
+et al. (2018) who concluded that the radio variation is usually substantially delayed to the other wavelengths for
+blazars. Bhatta (2021) analyzed the correlation between optical (V -band) and γ-ray variabilities for blazars and found
+that the optical variability is highly correlated with the γ-ray variability except for 3C 273, however, no significant
+Corresponding author: Dahai Yan
+[email protected]
+Corresponding author: Li Zhang
+[email protected]
+arXiv:2301.01025v1  [astro-ph.HE]  3 Jan 2023
+
+2
+lagging is found. The multi-band variability analysis can be considered as an indirect approach for resolving blazar
+jet.
+The GP method becomes popular in modern time-domain astronomy (e.g., Ryan et al. 2019; Burke et al. 2021; Yang
+et al. 2021; Griffiths et al. 2021; Covino et al. 2022; Rueda et al. 2022; Stone et al. 2022; Zhang et al. 2022). The GP
+method enables us to effectively extract information from astronomical variability. For example, Zhang et al. (2022)
+used a GP method to characterize the γ-ray variability of AGNs with stochastic process. It is found that the DRW
+model can successfully fit the γ-ray variability, which is similar with the optical variability of AGN accretion disk
+(Kelly et al. 2009; Li & Wang 2018; Burke et al. 2021). Moreover, Zhang et al. (2022) suggested a connection between
+the jet and the accretion disk by comparing the rest-frame γ-ray timescales of blazars with the optical accretion disk
+timescales of quasars.
+In this work, we analyze the multi-band variability of blazars with the GP method, which is independent of the
+temporal correlation analysis. We hope to extract additional information from the variability. Using the data from
+Fermi-Large Area Telescope (Fermi-LAT), we carried out systematic research of γ-ray variability of AGNs recently
+(Zhang et al. 2022). So far, the Small and Moderate Aperture Research Telescope System (SMARTS) monitoring
+program1 (Bonning et al. 2012) and the Steward Observatory (SO) spectropolarimetric monitoring project2 (Smith
+et al. 2009) can provide almost ten years’ (from 2008 to 2018) optical data of Fermi blazars. RXTE AGN Timing
+& Spectral Database3 (Rivers et al. 2013) provides long-term X-ray data, and the Owens Valley Radio Observatory
+(OVRO) 40 m program (Richards et al. 2011) provides radio light curves (LCs) from 2008 to 20204. Using these public
+data, we analyze the radio, optical and X-ray variability of three individual blazars, as well as optical variability for a
+sample including 38 Fermi blazars. The format of this paper is as follows. In Section 2, we describe the data as well
+as the GP method. The modeling results of the three individual sources and 38 blazars are shown in Section 3. We
+give discussions and physical interpretations of the results in Section 4. In Section 5, we conclude the paper with a
+brief summary.
+2. DATA AND GAUSSIAN PROCESS METHOD
+2.1. Data and Sources
+We use photometric data of blazars from the SMARTS and SO monitoring projects. The SMARTS program gives
+photometric data at five wavelength bands (B, V, R, J, K), which were taken from the 1.3 m telescope at the Cerro
+Tololo Inter-American Observatory. SO is a long-term optical program to support the Fermi Telescope, utilizing both
+the 2.3 m Bok Telescope on Kitt Peak and the 1.54 m Kuiper Telescope on Mt.Bigelow in Arizona. The campaign
+of the SO program spanned almost a decade from 2008 November to 2018 July. The X-ray data can be gained from
+RXTE observation which provided us with 16 yr (1996-2012) data in 2-10 keV. OVRO 40 m program gives radio data
+of blazars from 2008 to 2020, which is a large-scale, fast-cadence 15 GHz radio monitoring program. We select sources
+having long-term continuous observations and a good sampling. For the source with a large gap in the LC, we only
+use the data covering a longer period before or after the gap for analysis. Finally, We have 38 blazars in the optical
+band, including 23 FSRQs and 15 BL Lacs. Three blazars (3C 273, BL Lac, and PKS 1510-089) have long-term RXTE
+X-ray data. They are also in the sample of selected optical sources. Unfortunately, among the three sources, only 3C
+273 has the OVRO LC. Table 1 gives the general information of these targets.
+2.2. Gaussian Process Method
+GPs are a class of statistical models, which are widely applied for modeling stochastic processes. For the one who is
+interested in the stochastic behavior of astronomical variability, GP provides a flexible method to model the LC with
+stochastic processes. The application of GPs for astronomical time-series is discussed in a recent review (Aigrain &
+Foreman-Mackey 2022). Considering a data set of yn at coordinates xn, the GP model consists of a mean function
+µθ(x) parameterized by θ and a kernel function (covariance function) kα(xn, xm) parameterized by parameters α
+(Foreman-Mackey et al. 2017). For time-series data, the GP is one-dimensional, and the coordinates are time, xn=tn.
+After obtaining the likelihood function with the above information, one can use Bayesian inference to estimate the
+posterior distribution over the parameter space.
+1 http://www.astro.yale.edu/smarts/glast/home.php
+2 http://james.as.arizona.edu/∼psmith/Fermi/datause.html
+3 https://cass.ucsd.edu/∼rxteagn/
+4 http://astro.caltech.edu/ovroblazars/
+
+3
+Table 1. Information of 38 blazars.
+Object
+z
+Type
+logMBH/M⊙
+Ref.
+(1)
+(2)
+(3)
+(4)
+(5)
+1ES 1959+650
+0.048
+BLL
+8.2 ± 0.17
+1
+1ES 2344+514
+0.044
+BLL
+8.7 ± 0.18
+2
+3C 66A
+0.37
+BLL
+8.570.03
+0.6
+3,4
+3C 454.3
+0.859
+FSRQ
+9.1 ± 0.5
+6
+PKS 0235+164
+0.94
+BLL
+9.0
+7
+4C +38.41
+1.81396
+FSRQ
+9.5 ± 0.5
+7
+CTA 102
+1.032
+FSRQ
+8.7
+7
+Mkn 421
+0.03002
+BLL
+8.3 ± 0.2
+6
+Mkn 501
+0.03298
+BLL
+9.2 ± 0.2
+6
+OJ 287
+0.3056
+BLL
+8.8 ± 0.5
+6
+4C +21.35
+0.43383
+FSRQ
+8.9 ± 0.15
+8
+PKS 2155-304
+0.1167
+BLL
+8.9
+9
+S5 0716+714
+0.31
+BLL
+8.7
+10
+W Com
+1.25813
+BLL
+8.5
+14
+4C +01.02
+2.099
+FSRQ
+9.5
+16
+PKS 0208-512
+1.003
+FSRQ
+9.2
+7
+PKS 0235-618
+0.46657
+FSRQ
+9.0
+14
+PKS 0402-362
+1.42284
+FSRQ
+9.0
+14
+PKS 0426-380
+1.105
+BLL
+8.6
+7
+PKS 0458-02
+2.286
+FSRQ
+8.7
+11
+PKS 0502+049
+0.954
+FSRQ
+8.9 ± 0.5
+12
+PKS 0528+134
+2.06
+FSRQ
+9.0
+13
+PMN J0531-4827
+0.8116
+BLL
+· · ·
+· · ·
+PMN J0850-1213
+0.566
+FSRQ
+8.7
+14
+PKS 1144-379
+1.048
+BLL
+8.5
+7
+PKS 1244-255
+0.633
+FSRQ
+8.3
+14
+PKS B1406-076
+1.494
+FSRQ
+9.4
+17
+PKS 1730-130
+0.902
+FSRQ
+8.7
+14
+PKS 1954-388
+0.63
+FSRQ
+8.0 ± 0.5
+5
+PKS 2142-75
+1.139
+FSRQ
+9.7
+15
+PKS 2233-148
+0.33
+BLL
+8.7
+14
+PKS 2326-502
+0.518
+FSRQ
+9.3
+14
+PMN J2345-1555
+0.621
+FSRQ
+8.2 ± 0.17
+8
+Ton 599
+0.72474
+FSRQ
+8.5 ± 0.5
+5
+PKS 2052-47
+1.489
+FSRQ
+8.9
+14
+3C 273
+0.15834
+FSRQ
+8.9 ± 0.5
+5
+BL Lac
+0.0686
+BLL
+8.5 ± 0.2
+6
+PKS 1510-089
+0.36
+FSRQ
+7.8+0.05
+−0.04
+18
+Note—(1) source name, (2) redshift, (3) source type, (4) black hole mass
+(in solar mass) collected from the references in the last column.
+X-ray
+variability analysis is performed for the last three sources. References: (1)
+Falomo et al. (2003), (2) Woo et al. (2005), (3) Kaur et al. (2017), (4) Gupta
+et al. (2012), (5) Liu et al. (2006), (6) Wang et al. (2004), (7) Sbarrato et al.
+(2012), (8) Shaw et al. (2012), (9) Ghisellini et al. (2010), (10) Kaur et al.
+(2018), (11) Fan & Cao (2004), (12) Oshlack et al. (2002), (13) Palma et al.
+(2011), (14) Paliya et al. (2017), (15) Dutka et al. (2013), (16) Schutte et al.
+(2022), (17) Xue et al. (2016), (18) Rakshit (2020).
+
+4
+In practical application, the key point is choosing the kernel function. The DRW process (called Ornstein-Uhlenbeck
+process in physics) is widely used to describe the variability of AGNs (e.g., Burke et al. 2021), and it is defined by an
+exponential covariance function (e.g., Kelly et al. 2009; Zu et al. 2013),
+k(tnm) = 2σ2
+DRW · exp(−tnm/τDRW) ,
+(1)
+where tnm = |tn − tm| is the time lag between measurements m and n. The amplitude term (σDRW) represents the
+amplitude of the random disturbance, and the damping (characteristic) timescale (τDRW) represents the timescale that
+the system returns to the stability after experiencing a disturbance. Sometimes, an excess white noise term (σ2
+nδnm
+where σn is the excess white noise amplitude and δnm is the Kronecker δ function) is needed in the situation that there
+is a white noise in the LC in addition to the quoted measurement errors (Foreman-Mackey 2018; Burke et al. 2021).
+A more complex kernel is the stochastically driven damped simple harmonic oscillator (SHO), which is described by
+a second-order differential equation (Foreman-Mackey et al. 2017). The SHO kernel has been used to model the AGN
+accretion disk (Yu et al. 2022) and jet (Zhang et al. 2022) variability.
+Celerite software package is a GP tool for a stationary process (Foreman-Mackey et al. 2017). It uses the semi-
+separated structure of a special covariance matrix to directly analyze and compute the GP likelihood for large data
+sets. Yang et al. (2021) and Zhang et al. (2021, 2022) have tested the efficiency of this method for the study of AGN
+jet variability, and suggested that celerite has a strong potentiality for studying AGN variability (also see Burke et al.
+2021). Here, we use the DRW model implemented in celerite package to model the multi-band variability of blazars.
+The Markov Chain Monte Carlo (MCMC) sampler emcee5 is adopted to perform posterior analysis. We assume
+log-uniform priors on each of the parameters. The MCMC sampler is run for 50,000 iterations with 32 parallel walkers.
+The first 20,000 steps are taken as burn-in. After modeling the LCs, we should estimate the fitting quality for assessing
+whether the fitting results are reliable, e.g., whether the standardized residuals follow a Gaussian white-noise sequence.
+The power spectral density (PSD) can be constructed by using the fitting results. The DRW PSD is in the form of
+S(ω) =
+�
+8
+π σ2
+DRWτDRW
+1
+1 + (ωτDRW)2 .
+(2)
+It is a broken power-law form with slope 0 below the broken frequency (fb) and slope -2 above the broken frequency.
+The conversion between the timescale τDRW and fb is τDRW = 1/(2πfb).
+The LC with large cadence or insufficient length leads to a large bias on the characteristic timescale derived from
+modeling.
+If the timescale is larger than the mean cadence of LC and less than 1/10 of the length of LC, the
+measurement of the damping timescale from the LC is reliable (Burke et al. 2021).
+3. RESULTS
+3.1. Results of 3C 273, PKS 1510-089 and BL Lac
+We first analyze the multi-band variability of the three individual sources, 3C 273, PKS 1510-089, and BL Lac. We
+present the celerite fitting results of the LC for each source in the following. The measured timescales given in the
+main text are with errors in 95% confidence intervals.
+For 3C 273, the optical data in both B and V bands are available. We show the modeling results in Figure 1, in
+which the left panel is for B-band LC and the right is for V -band LC. The DRW model can agree well with both
+LCs. Looking at the distribution of standardized residuals and the auto-correlation function (ACF) of standardized
+residuals (see details in Zhang et al. 2022), we believe the characteristic of each LC has been captured successfully.
+Through MCMC sampling, we get the posterior probability density distributions of two parameters (σDRW and τDRW)
+and show them in Figure 2. The values are listed in Table 2. The results are different between the two bands. The
+parameters can be constrained by the B-band data but with large uncertainties, e.g., τDRW = 59+41
+−28 days. Comparing
+the timescale with the cadence and the length of the LC, we believe that the B-band timescale is reliable. A broken
+frequency corresponding to the characteristic timescale is shown in the B-band PSD (Figure 3). While the V -band
+timescale is ≈ 3200 days, very close to the length of the LC. This means that the V -band timescale is unreliable, which
+is also confirmed by the single power-law PSD (Figure 3). We show the modeling results of X-ray LC, the posterior
+probability density distribution of parameters, and the PSD in the right panel in Figure 4, Figure 5 and Figure 6
+5 https://github.com/dfm/emcee
+
+5
+12.8
+12.9
+13.0
+13.1
+13.2
+13.3
+Magnitude
+optical B-band
+0
+500
+1000
+1500
+2000
+2500
+3000
+time-2454677.5 (JD)
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+Standardized Residuals
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+Standardized Residuals
+0.0
+0.2
+0.4
+0.6
+0.8
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.5
+0.0
+0.5
+1.0
+ACF of Residuals
+optical V-band
+0
+500
+1000
+1500
+2000
+2500
+3000
+time-2454795.0 (JD)
+Standardized Residuals
+5
+0
+5
+Standardized Residuals
+0.0
+0.2
+0.4
+0.6
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.00
+0.25
+0.50
+0.75
+1.00
+ACF of Residuals
+3C 273
+Figure 1. DRW fitting results of 3C 273 in B-band (left panel) and V -band (right panel). For each column, the top panel
+presents the observed LC (black points) and the modeled LC (orange/blue line). We show the standardized residuals (black
+points) in the middle panel. In the bottom panel, there are two parts. The probability density of standardized residuals (black
+ladder diagram) as well as the best-fit normal distribution (orange/blue solid line) are shown in the left part. The ACF of
+residuals with the 95% confidence limits of the white noise (the gray region) are shown in the right part.
+ln
+DRW = 
+2.50+0.15
+0.12
+2.4
+1.6
+0.8
+0.0
+ln
+DRW
+4.5
+6.0
+7.5
+9.0
+ln
+DRW(day)
+4.5
+6.0
+7.5
+9.0
+ln
+DRW(day)
+ln
+DRW(day) = 4.08+0.31
+0.26
+3C 273
+ln
+DRW = 
+1.93+0.64
+0.46
+3.0
+2.4
+1.8
+1.2
+0.6
+ln
+DRW
+6
+7
+8
+9
+10
+ln
+DRW(day)
+6
+7
+8
+9
+10
+ln
+DRW(day)
+ln
+DRW(day) = 8.06+1.29
+0.93
+3C 273
+Figure 2. Posterior probability densities of model parameters for 3C 273 in B-band (left) and V -band (right). The vertical
+dotted lines mark the median value and 68% confidence intervals of the distribution of the parameter.
+respectively. The values of the parameters can be found in Table 3. It is shown that the DRW model can describe the
+X-ray variability of 3C 273. The parameters are well constrained. The X-ray PSD presents a broken frequency that
+corresponds to a timescale of τDRW = 28+7
+−6 days. We give the radio results together with the X-ray results. The radio
+PSD (the left panel of Figure 6) of 3C 273 is a single power law. The radio timescale is too large to be reliable.
+
+6
+10
+3
+10
+2
+10
+1
+Frequency (day
+1)
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Power(Magnitude2 day)
+3C 273
+PSD obtained from B-band
+PSD obtained from V-band
+y = v
+2
+Figure 3. B-band and V -band PSDs of 3C 273 constructed from the modeling results with DRW model. The orange line is
+B-band PSD, and the blue line is V -band PSD. The corresponding color region denotes the 1σ confidence interval. The dashed
+black line is a reference line with a slope of -2.
+16
+18
+20
+22
+24
+26
+28
+30
+32
+Jy
+radio
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+time-2454677.5 (JD)
+6
+4
+2
+0
+2
+4
+Standardized Residuals
+5
+0
+5
+Standardized Residuals
+0.0
+0.1
+0.2
+0.3
+0.4
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.00
+0.25
+0.50
+0.75
+1.00
+ACF of Residuals
+5
+10
+15
+20
+25
+30
+flux (×10
+11 erg cm
+2 s
+1)
+X-ray
+0
+1000
+2000
+3000
+4000
+5000
+time-2454795.0 (JD)
+Standardized Residuals
+2.5
+0.0
+2.5
+5.0
+7.5
+Standardized Residuals
+0.0
+0.2
+0.4
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.00
+0.25
+0.50
+0.75
+1.00
+ACF of Residuals
+3C 273
+Figure 4. DRW fitting results of 3C 273 for radio (left panel) and X-ray (right panel) data. The symbols and lines are the
+same as those in Figure 1.
+For PKS 1510-089, the V and B-band LCs can be described by the DRW model (Figure 7 and Figure 8). The
+V -band τDRW of 39+18
+−14 days is larger than the B-band τDRW of 11 ± 3 days (Table 4). As expected, we get a smaller
+value of fb in V -band PSD (Figure 9). The X-ray LC of PKS 1510-089 also can be fitted well by the DRW model
+(Figure 10). The parameters are well constrained (Table 3), and the PSD is in the form of typical DRW PSD. A
+trusted timescale τDRW = 26 ± 3 days is obtained.
+For BL Lac, only the V -band and X-ray data are available. For the X-ray data, there are two large gaps in the first
+2800 days of LC, we then take the following 2500 days of LC for analysis. When modeling the two LCs of BL Lac,
+we get poor fitting (ACF of residuals deviating from the white noise) with the two-parameter DRW model. An excess
+white noise term is then added to the DRW model, and we model the LCs with the three-parameter DRW model
+again. The modeling of LC, the posterior distribution of parameters, and the broken power-law PSDs are shown in
+Figure 11, Figure 12, and Figure 13, respectively. Optical results are shown in the left panels and the X-ray results
+
+7
+ln
+DRW = 1.54+0.42
+0.41
+0.4
+0.8
+1.2
+1.6
+2.0
+ln
+DRW
+7
+8
+9
+10
+ln
+DRW(day)
+7
+8
+9
+10
+ln
+DRW(day)
+ln
+DRW(day) = 8.95+0.84
+0.83
+3C 273
+ln
+DRW = 0.91+0.06
+0.05
+0.75
+0.90
+1.05
+1.20
+ln
+DRW
+3.00
+3.25
+3.50
+3.75
+4.00
+ln
+DRW(day)
+3.00
+3.25
+3.50
+3.75
+4.00
+ln
+DRW(day)
+ln
+DRW(day) = 3.34+0.12
+0.11
+3C 273
+Figure 5. Posterior probability densities of model parameters for 3C 273 in radio (left) and X-ray (right) energies. The vertical
+dotted lines mark the median value and 68% confidence intervals of the distribution of the parameter.
+10
+3
+10
+2
+10
+1
+Frequency (day
+1)
+10
+3
+10
+2
+10
+1
+100
+101
+102
+103
+Power(flux2 day)
+3C 273
+PSD obtained from radio band
+y = v
+2
+10
+3
+10
+2
+10
+1
+Frequency (day
+1)
+10
+3
+10
+2
+10
+1
+100
+101
+102
+103
+Power(flux2 day)
+3C 273
+PSD obtained from X-ray
+y = v
+2
+Figure 6. Radio and X-ray PSDs constructed from modeling LCs of 3C 273 with DRW model. The symbols and lines are the
+same as those in Figure 3.
+are shown in the right panels. One can see that the three-parameter DRW can fit the LCs well. Note that the highest
+flux point in the LC is poorly fitted. After removing the highest flux point, the modeling results are unchanged. We
+obtain the X-ray timescale of τDRW = 63+49
+−30 days and V -band τDRW of 47+26
+−19 days (Table 4).
+We applied the SHO model to the optical and X-ray data of BL Lac. The fitting is not improved significantly, and
+the parameters cannot be constrained. This suggests that the SHO model is not a good choice. The DRW including
+an additional white noise can describe the variability behavior. The value of σ2
+n (0.01 for the V -band LC; 0.04 for
+the X-ray LC) is larger than the squared of the mean size of the light curve uncertainties (σy2) where σy2=0.0001 for
+V -band and 0.0036 for X-ray data. We have σ2
+DRW >σ2
+n+σy2 for both the V -band and X-ray data, which ensures that
+the fitted DRW amplitude is reasonable (Burke et al. 2021). It is possible that the quoted measurement errors are
+underestimated, and the excess white noise can account for excess measurement noise.
+The γ-ray variability of the three sources has been analyzed in our previous work (Zhang et al. 2022) with the same
+method. We give the multi-band timescales with the errors in 95% confidence intervals of the three sources in Table 4.
+For 3C 273, the B-band, X-ray, and γ-ray timescales are consistent within the errors. The V -band and radio PSDs
+
+8
+14
+15
+16
+17
+18
+Magnitude
+optical B-band
+0
+500
+1000
+1500
+2000
+2500
+3000
+time-2454677.5 (JD)
+5
+0
+5
+Standardized Residuals
+5
+0
+5
+Standardized Residuals
+0.0
+0.2
+0.4
+0.6
+0.8
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.00
+0.25
+0.50
+0.75
+1.00
+ACF of Residuals
+optical V-band
+0
+500
+1000
+1500
+2000
+2500
+3000
+time-2454795.0 (JD)
+Standardized Residuals
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+Standardized Residuals
+0.0
+0.2
+0.4
+0.6
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.0
+0.5
+1.0
+ACF of Residuals
+PKS 1510-089
+Figure 7. DRW fitting results of B-band (left panel) and V -band (right panel) LCs for PKS 1510-089. The symbols and lines
+are the same as those in Figure 1.
+ln
+DRW = 
+1.26+0.06
+0.05
+1.35
+1.20
+1.05
+0.90
+ln
+DRW
+2.1
+2.4
+2.7
+3.0
+ln
+DRW(day)
+2.1
+2.4
+2.7
+3.0
+ln
+DRW(day)
+ln
+DRW(day) = 2.40+0.13
+0.12
+PKS 1510-089
+ln
+DRW = 
+1.16+0.10
+0.09
+1.25
+1.00
+0.75
+0.50
+0.25
+ln
+DRW
+3.0
+3.6
+4.2
+4.8
+5.4
+ln
+DRW(day)
+3.0
+3.6
+4.2
+4.8
+5.4
+ln
+DRW(day)
+ln
+DRW(day) = 3.67+0.22
+0.19
+PKS 1510-089
+Figure 8. Posterior probability densities of model parameters of B-band (left) and V -band (right) LCs for PKS 1510-089. The
+symbols and lines are the same as those in Figure 2.
+are single power-law, having no corresponding characteristic timescales. For PKS 1510-089, the V -band, X-ray and
+γ-ray timescales are consistent within the errors but the B-band one has a smaller value. For BL Lac, the V -band,
+X-ray and γ-ray timescales are also consistent within the errors.
+3.2. Optical Results of 38 Blazars
+The DRW model can successfully fit the long-term optical LCs of the 38 blazars. Based on the criteria of selecting
+reliable measurements of the damping timescale, we get reliable optical timescales for the 38 blazars.
+The basic
+information of the 38 blazars and the modeling results are given in Table 1 and Table 2, respectively. Except for
+
+9
+Table 2. Modeling results of optical data for 38 blazars.
+Object
+Data sources
+Waveband
+Parameter of DRW
+Damping timescale
+Cadence
+Length
+ln σDRW
+ln τDRW
+(days)
+(days)
+(days)
+(1)
+(2)
+(3)
+(4)
+(5)
+(6)
+(7)
+(8)
+1ES 1959+650
+SO
+V
+−1.36+0.25
+−0.17
+5.13+0.53
+−0.39
+169+90
+−66
+23.5
+3548.2
+1ES 2344+514
+SO
+V
+−3.14+0.20
+−0.16
+4.92+0.58
+−0.46
+137+79
+−63
+17.35
+3539.1
+3C 66A
+SO
+V
+−1.15+0.28
+−0.17
+5.35+0.57
+−0.36
+210+120
+−75
+9.04
+3217.9
+3C 454.3
+SO
+V
+−0.81+0.10
+−0.09
+3.82+0.21
+−0.18
+46+10
+−8
+5.97
+3563.3
+PKS 0235+164
+SO
+V
+−0.41+0.13
+−0.11
+3.93+0.28
+−0.24
+51+14
+−12
+14.0
+3417.9
+4C +38.41
+SO
+V
+−0.98+0.09
+−0.08
+3.51+0.19
+−0.17
+33+6
+−6
+9.7
+3561.2
+CTA 102
+SO
+V
+−0.10+0.15
+−0.12
+4.26+0.32
+−0.26
+71+23
+−18
+10.13
+3182.2
+Mkn 421
+SO
+V
+−1.19+0.19
+−0.14
+4.97+0.39
+−0.29
+144+56
+−42
+5.95
+3562.7
+Mkn 501
+SO
+V
+−3.13+0.10
+−0.09
+3.92+0.24
+−0.21
+50+12
+−11
+7.31
+3561.0
+OJ 287
+SO
+B
+−1.01+0.11
+−0.09
+3.57+0.23
+−0.19
+36+8
+−7
+5.32
+3079.7
+4C +21.35
+SO
+V
+−1.46+0.18
+−0.13
+4.79+0.37
+−0.28
+120+45
+−34
+7.70
+3357.9
+PKS 2155-304
+SO
+V
+−1.28+0.15
+−0.12
+4.42+0.32
+−0.26
+83+27
+−22
+11.03
+3561.2
+S5 0716+714
+SO
+V
+−0.96+0.11
+−0.09
+2.99+0.24
+−0.21
+20+5
+−4
+14.98
+3414.9
+W Com
+SO
+V
+−1.13+0.18
+−0.13
+4.74+0.37
+−0.29
+114+42
+−33
+9.94
+3538.7
+4C +01.02
+S
+B
+−1.53+0.18
+−0.13
+3.27+0.40
+−0.32
+26+11
+−8
+10.35
+1408.2
+PKS 0208-512
+S
+V
+−0.56+0.15
+−0.12
+4.35+0.30
+−0.24
+77+23
+−19
+5.22
+3301.9
+PKS 0235-618
+S
+R
+−0.97+0.12
+−0.10
+2.73+0.29
+−0.25
+15+4
+−4
+6.76
+966.6
+PKS 0402-362
+S
+V
+−1.18+0.20
+−0.15
+3.91+0.42
+−0.32
+50+21
+−16
+8.39
+1459.0
+PKS 0426-380
+S
+R
+−0.34+0.31
+−0.19
+4.85+0.64
+−0.39
+128+82
+−50
+2.63
+1509.0
+PKS 0458-02
+S
+R
+−1.35+0.17
+−0.13
+3.20+0.38
+−0.31
+25+9
+−8
+10.53
+1148.1
+PKS 0502+049
+S
+B
+−0.56+0.16
+−0.13
+2.80+0.36
+−0.28
+16+6
+−5
+5.87
+769
+PKS 0528+134
+S
+V
+−1.38+0.20
+−0.15
+3.38+0.48
+−0.38
+29+14
+−11
+6.60
+956.6
+PMN J0531-4827
+S
+V
+0.02+0.23
+−0.17
+4.03+0.49
+−0.37
+56+28
+−21
+9.27
+1808.1
+PMN J0850-1213
+S
+R
+−0.79+0.14
+−0.11
+3.39+0.31
+−0.27
+30+9
+−8
+11.86
+1696.6
+PKS 1144-379
+S
+B
+−0.62+0.30
+−0.19
+4.72+0.62
+−0.41
+112+70
+−46
+8.99
+1590.7
+PKS 1244-255
+S
+R
+−0.99+0.12
+−0.10
+2.85+0.28
+−0.25
+17+5
+−4
+7.90
+1515.9
+PKS B1406-076
+S
+R
+−1.19+0.10
+−0.09
+3.24+0.22
+−0.20
+26+6
+−5
+130.9
+3365.9
+PKS 1730-130
+S
+V
+−1.59+0.14
+−0.11
+2.91+0.30
+−0.25
+18+6
+−5
+4.48
+1008.2
+PKS 1954-388
+S
+R
+−1.19+0.16
+−0.13
+3.57+0.39
+−0.34
+36+14
+−12
+20.12
+1851.0
+PKS 2142-75
+S
+V
+−2.03+0.11
+−0.10
+3.04+0.26
+−0.22
+21+5
+−5
+11.24
+1843.1
+PKS 2233-148
+S
+R
+−0.30+0.17
+−0.13
+3.37+0.36
+−0.28
+29+10
+−8
+6.34
+1110.2
+PKS 2326-502
+S
+B
+−0.38+0.22
+−0.16
+3.11+0.46
+−0.34
+22+10
+−8
+4.29
+720.1
+PMN J2345-1555
+S
+R
+−0.48+0.18
+−0.14
+3.44+0.39
+−0.30
+31+12
+−9
+6.59
+1424.1
+Ton 599
+SO
+V
+−0.39+0.20
+−0.15
+3.78+0.42
+−0.32
+44+18
+−14
+6.85
+1143.9
+PKS 2052-47
+S
+V
+−0.91+0.22
+−0.16
+4.41+0.54
+−0.40
+82+44
+−33
+10.25
+2121.2
+3C 273
+SO
+B
+−2.50+0.15
+−0.12
+4.08+0.31
+−0.26
+59+18
+−15
+8.9
+3179.12
+SO
+V
+−1.93+0.64
+−0.46
+8.06+1.29
+−0.93
+· · ·
+4.9
+3423.6
+PKS 1510-089
+SO
+V
+−1.16+0.10
+−0.09
+3.67+0.22
+−0.19
+39+9
+−7
+8.91
+3476.7
+SO
+B
+−1.26+0.06
+−0.05
+2.40+0.13
+−0.12
+11+1
+−1
+4.92
+3315.9
+BL Lac
+SO
+V
+−0.91+0.10
+−0.08
+3.86+0.25
+−0.22
+47+12
+−10
+4.78
+3569.2
+Note— (1) source name, (2) data source, S is for SMARTS and SO is for Steward Observatory blazar data archive,
+(3) wavebands of optical data, including B, V, R-band here, (4)(5) posterior parameters of modeling LC with DRW
+model, (6) damping timescale in the observed frame. The uncertainties of model parameters and damping timescales
+represent 1σ confidence intervals, (7) the mean cadence of the LC, and (8) the length of LC.
+
+10
+10
+3
+10
+2
+10
+1
+Frequency (day
+1)
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Power(Magnitude2 day)
+PKS 1510-089
+PSD obtained from B-band
+PSD obtained from V-band
+y = v
+2
+Figure 9. B and V -band PSDs of PKS 1510-089 constructed from the modeling results with the DRW model. The symbols
+and lines are the same as those in Figure 3.
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Flux (×10
+11 ph cm
+2 s
+1)
+X-ray
+0
+1000
+2000
+3000
+4000
+5000
+time-50115.1 (MJD)
+4
+2
+0
+2
+4
+Standardized Residuals
+4
+2
+0
+2
+4
+Standardized Residuals
+0.0
+0.1
+0.2
+0.3
+0.4
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.00
+0.25
+0.50
+0.75
+1.00
+ACF of Residuals
+PKS 1510-089
+ln
+DRW = 
+1.87+0.05
+0.05
+1.95
+1.80
+1.65
+ln
+DRW
+3.00
+3.25
+3.50
+3.75
+ln
+DRW(day)
+3.00
+3.25
+3.50
+3.75
+ln
+DRW(day)
+ln
+DRW(day) = 3.25+0.12
+0.12
+PKS 1510-089
+10
+3
+10
+2
+10
+1
+Frequency (day
+1)
+10
+5
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+Power(flux2 day)
+PKS 1510-089
+PSD obtained from X-ray
+y = v
+2
+Figure 10. Modeling results of the X-ray LC (left), the posterior probability density distribution of parameters (middle), and
+the X-ray PSD (right) for PKS 1510-089.
+Table 3. Modeling results of X-ray data for the 3 blazars.
+Object
+Parameter of DRW
+Damping timescale
+Cadence
+Length
+ln σDRW
+ln τDRW
+ln σn
+(days)
+(days)
+(days)
+(1)
+(2)
+(3)
+(4)
+(5)
+(6)
+(7)
+3C 273
+0.91+0.06
+−0.05
+3.34+0.12
+−0.11
+· · ·
+28+3
+−3
+2.97
+5811.5
+PKS 1510-089
+−1.87+0.05
+−0.05
+3.25 ± 0.12
+· · ·
+26+3
+−3
+4.16
+5495.3
+BL Lac
+−1.18+0.14
+−0.11
+4.15+0.33
+−0.26
+−1.64+0.03
+−0.04
+63+21
+−16
+2.19
+2493.1
+Note— (1) source name, (2)(3)(4) posterior parameters of modeling LC with DRW model, and (5) damping timescale in
+the observed frame. The uncertainties of model parameters and damping timescales represent 1σ confidence intervals,
+(6) the mean cadence of the LC, and (7) the length of LC.
+
+11
+13.5
+14.0
+14.5
+15.0
+15.5
+16.0
+Magnitude
+optical V-band
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+time-2454677.5 (JD)
+5
+0
+5
+10
+15
+Standardized Residuals
+5
+0
+5
+Standardized Residuals
+0.0
+0.2
+0.4
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.00
+0.25
+0.50
+0.75
+1.00
+ACF of Residuals
+1
+2
+3
+4
+5
+flux (×10
+11 erg cm
+2 s
+1)
+X-ray
+0
+500
+1000
+1500
+2000
+time-2454795.0 (JD)
+Standardized Residuals
+5
+0
+5
+10
+Standardized Residuals
+0.0
+0.2
+0.4
+Normalized Counts [a.u]
+0
+10
+20
+30
+40
+50
+Time Lag
+0.00
+0.25
+0.50
+0.75
+1.00
+ACF of Residuals
+BL Lac
+Figure 11. DRW fitting results of V -band (left panel) and X-ray data (right panel) for BL Lac. The symbols and lines are the
+same as those in Figure 1.
+ln
+DRW = 
+0.91+0.10
+0.08
+3.2
+4.0
+4.8
+5.6
+ln
+DRW(day)
+ln
+DRW(day) = 3.86+0.25
+0.22
+1.2
+0.9
+0.6
+0.3
+0.0
+ln
+DRW
+2.6
+2.4
+2.2
+2.0
+ln
+n
+3.2
+4.0
+4.8
+5.6
+ln
+DRW(day)
+2.6
+2.4
+2.2
+2.0
+ln
+n
+ln
+n = 
+2.32+0.08
+0.09
+BL Lac
+ln
+DRW = 
+1.18+0.14
+0.11
+4.5
+6.0
+7.5
+9.0
+ln
+DRW(day)
+ln
+DRW(day) = 4.15+0.33
+0.26
+0.8
+0.0
+0.8
+ln
+DRW
+1.76
+1.68
+1.60
+1.52
+ln
+n
+4.5
+6.0
+7.5
+9.0
+ln
+DRW(day)
+1.76
+1.68
+1.60
+1.52
+ln
+n
+ln
+n = 
+1.64+0.03
+0.04
+BL Lac
+Figure 12. V -band (left) and X-ray (right) posterior probability densities of model parameters for BL Lac. The symbols and
+lines are the same as those in Figure 2.
+3C 273 and PKS 1510-089 which are analyzed in Section 3.1, the timescales for different optical bands are consistent
+for the other 36 sources. This indicates that the optical emission of the 36 blazars has the same origin, i.e., the jet
+emission. In Table 2, we only list one optical band result for these sources. The timescale is between 10 days and 200
+days.
+Some notes should be given on PKS 2052-47 and Ton 599. The fitting to the LC of PKS 2052-47 needs an additional
+white noise, and the relation σ2
+DRW(0.16) > σ2
+n(0.026) + σy2(0.0016) still holds. Ton 599 has big gaps and few data in
+the first half of its V -band LC, and we select the second half of the LC to analyze.
+
+12
+10
+3
+10
+2
+10
+1
+Frequency (day
+1)
+10
+2
+10
+1
+100
+101
+102
+103
+Power(Magnitude2 day)
+BL Lac
+PSD obtained from V-band
+y = v
+2
+10
+3
+10
+2
+10
+1
+Frequency (day
+1)
+10
+4
+10
+3
+10
+2
+10
+1
+100
+101
+102
+Power(flux2 day)
+BL Lac
+PSD obtained from X-ray
+y = v
+2
+Figure 13. V -band (left) and X-ray (right) PSDs for BL Lac. The symbols and lines are the same as those in Figure 3.
+Table 4. Damping timescale of 3C 273, PKS 1510-089 and BL Lac.
+Object
+B-band timescale
+V -band timescale
+X-ray timescale
+γ-ray timescale
+(days)
+(days)
+(days)
+(days)
+(1)
+(2)
+(3)
+(4)
+(5)
+3C 273
+59+41
+−28
+unreliable
+28+7
+−6
+31+12
+−10
+PKS 1510-089
+11+3
+−3
+39+18
+−14
+26+7
+−6
+40+14
+−12
+BL Lac
+no data
+47+26
+−19
+63+49
+−30
+69+36
+−25
+Note— (1) source name, (2)(3)(4)(5) multi-band damping timescales in the observed frame. The uncertainties
+of the damping timescales represent 95% confidence intervals of the distribution of the parameter.
+3.3. Origin of the optical emission from 3C 273 and PKS 1510-089
+The optical emission of 3C 273 and PKS 1510-089 is complicated. Blue bump can be seen in their multi-band spectral
+energy distributions (SEDs; e.g., Abdo et al. 2010; Nalewajko et al. 2012; Castignani et al. 2017). SED modeling results
+showed that the accretion disk has a significant contribution to the optical emissions of 3C 273 and PKS 1510-089
+(e.g., Nalewajko et al. 2012; Yan et al. 2012; Castignani et al. 2017). In addition, Zhang et al. (2019) found that a
+long-term variation trend in the optical continuum LC of 3C 273 does not appear in the emission-line variation. This
+suggests that the long-term variation trend is not contributed by the accretion disk, and it could originate from the
+jet. Li et al. (2020) quantitatively decoupled the optical emissions from the jet and accretion disk in 3C 273 and found
+that the jet emission accounts for 10%-40% of the total optical emission. Pandey et al. (2022) studied the correlation
+between V -band flux and polarization degree (PD) variations using SO observation during 2008-2018. They found a
+significant positive correlation only in two of the ten observing cycles. Note that the PD is quite small, and it changes
+from 0.04% to 1.58% during 2008-2018. The V -band single power-law PSD we obtained here is different from the
+typical PSD of the accretion disk (Suberlak et al. 2021; Burke et al. 2021) and jet variability (Zhang et al. 2022). The
+complicated mixture of the jet and accretion disk emissions at the V -band may result in the single power-law PSD.
+The mixed emission also results in the weak correlation between V -band and Fermi γ-ray variabilities reported by
+Bhatta (2021). We find no significant correlation between B-band variability and γ-ray variability for 3C 273 and PKS
+1510-089. Looking at the location of the blue bump in SED (Roy et al. 2021), we suggest that the B-band emission
+of 3C 273 is dominated by the accretion disk photons.
+For PKS 1510-089, the V and B-band timescales are clearly different, indicating different origins for the two bands’
+emissions. The V -band polarization of PSK 1510-089 is averagely greater than that of 3C 273, varying from 0.2%
+to 25.82% (Pandey et al. 2022). Among the ten observing cycles during 2008-2018, a significant positive correlation
+
+13
+Table 5. Mean timescales (redshift-corrected) of blazars
+in γ-ray and optical energies.
+Waveband
+logMBH/M⊙
+Mean timescale
+(1)
+(2)
+(3)
+γ-ray
+8 − 9
+58+21
+−16
+9 − 10
+32+10
+−8
+8 − 10
+53+18
+−14
+optical
+8 − 9
+51+23
+−11
+9 − 10
+19+6
+−5
+8 − 10
+42+18
+−13
+Note— (1) waveband, (2) the range of black hole
+mass in solar mass, (3) the mean damping timescale
+(redshift-corrected) with unit day.
+The uncertainties
+of timescales represent 1σ confidence intervals.
+between V -band flux and PD variations is found in 5 cycles. Moreover, Castignani et al. (2017) found a good correlation
+between the long-term SO V -band and γ-ray LCs. These results suggest that the V -band emission is dominated by
+jet contribution. Also looking at the location of the blue bump in SED (Nalewajko et al. 2012), the B-band emission
+with a smaller timescale of 11 days is suggested as the accretion disk contribution.
+3.4. Comparing Optical and γ-ray results
+Long-term Fermi γ-ray LCs of 22 blazars have been analyzed by Zhang et al. (2022) with the same GP method. The
+optical timescale in this work is generally consistent with the γ-ray timescale (Figure 14). We examine the consistency
+of the timescales in the two energy-bands by using a statistical significance test (T-test). We get t-statistic=1.1 and
+p-value=0.28 (>0.05), which means that in statistic there is little difference between the two groups of timescales. The
+optical amplitude term σDRW is less than one, and the γ-ray σDRW can be greater than 10. This means that γ-ray
+variability can be more energetic than optical variability.
+We separated the sources into two groups with MBH < 109M⊙ and MBH > 109M⊙. The mean timescales (redshift-
+corrected) in different ranges of black hole mass are listed in Table 5. It is found that the mean timescale of the sources
+in the mass range of 109-1010M⊙ is smaller in both γ-ray and optical energies. However, we have a few sources with
+the mass of 109-1010M⊙, therefore this result may be tentative.
+In Figure 15, we plot the relationship between the damping timescale in the rest frame (τ rest
+damping) and the black hole
+mass of blazars along with the results of normal quasars from Burke et al. (2021). The timescales should be modified
+into the rest frame with the following formula:
+τ rest
+damping = τDRW δD
+1 + z
+.
+(3)
+An average Doppler factor of δD=10 is used here and the redshift z for each source is given in table 1. We show the
+optical, X-ray, and γ-ray results in the plot. It is found that the nonthermal optical τ rest
+damping of blazars and the thermal
+optical timescale of normal quasars occupy the same space in the plot of τ rest
+damping − MBH.
+The X-ray results for the three individual blazars are also in the same area as the optical results. The B-band
+timescale of 3C 273 is a typical value of accretion disk timescale.
+The B-band timescale of PKS 1510-089 is an
+outlier value among the accretion disk timescales. This value significantly deviates from the relation between damping
+timescale and black hole mass reported by Burke et al. (2021).
+4. DISCUSSION
+
+14
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+log(
+DRW)
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+2.50
+log(
+damping/days)
+Optical data
+Gamma-ray data
+0.0
+0.5
+1.0
+1.5
+2.0
+Normolized Counts [a,u]
+0
+2
+4
+Normolized Counts [a,u]
+Figure 14. Plot of the redshift-corrected timescale τDRW versus the amplitude σDRW. The red and blue points represent the
+optical and γ-ray results, respectively. The side panels show the normalized histograms of the distributions of redshift-corrected
+τDRW (right) and σDRW (top) for blazars.
+104
+105
+106
+107
+108
+109
+1010
+MBH (M
+)
+100
+101
+102
+103
+rest
+damping(days)
+optical normal quasars
+gamma-ray blazars
+optical blazars
+x-ray blazars
+B-band PKS 1510-089
+Figure 15.
+Plot of the rest-frame timescale versus black hole mass.
+The gray data, lines, and area represent the optical
+accretion disk results for normal quasars taken from Burke et al. (2021). Red data are γ-ray results for blazars taken from
+Zhang et al. (2022), and the purple and blue data respectively represent the optical and X-ray results for blazars obtained in
+this work.
+
+15
+It is difficult to directly resolve the inner jet structure of the blazar6. Especially, the location of the high-energy
+emission region is still a hot open question (e.g., Madejski & Sikora 2016; B¨ottcher 2019). Multi-band variability
+analysis provides an indirect approach to resolve the emission regions. The cross-correlation method is frequently used
+in multi-band variability analysis (e.g., Liodakis et al. 2018; Bhatta 2021).
+GP method has been wildly used to characterize the AGN accretion disk variability (Kelly et al. 2009; Zhang et al.
+2018; Lu et al. 2019; Burke et al. 2021). In blazar science, it becomes popular in recent several years (e.g., Goyal et al.
+2018; Ryan et al. 2019; Covino et al. 2020; Tarnopolski et al. 2020; Yang et al. 2021; Zhang et al. 2022). In this work,
+we use the GP method to study the multi-band variability of the blazar. This provides results independent of the
+cross-correlation method.
+The γ-ray variability of the blazar has been studied by Zhang et al. (2022) with the GP method. Here we focus on
+the X-ray and optical variability of the blazar. Multi-band emission from the blazar is dominated by the nonthermal
+jet contribution. Two special blazars are 3C 273 and PKS 1510-089. An optical-ultraviolet bump appears in their
+SED, which is associated with their thermal accretion disk emission (e.g., Nalewajko et al. 2012; Yan et al. 2012;
+Castignani et al. 2017).
+We fit the long-term optical LCs from the database of SO and SMARTS with the DRW model. Finally, 38 blazars
+with a reliable characteristic timescale are selected. Except for 3C 273 and PKS 1510-089, the timescales in different
+optical colors agree with each other for the remaining 36 blazars. This indicates that the emissions in different optical
+colors of the 36 blazars have the same origin, i.e., the jet emission.
+Ruan et al. (2012) modeled the optical LCs covering from 2002 December through 2008 March of 51 blazars using
+the DRW model. They found that the observed damping timescale peaks at ∼80 days, and the intrinsic timescale
+τ rest
+damping peaks at ∼800 days7. The distribution of the optical timescale obtained in this work is flat (Figure 14), and
+the average optical τ rest
+damping is ∼400 days, which is smaller than the result of Ruan et al. (2012). All blazars in our
+sample are Fermi-detected γ-ray sources. While the sample studied by Ruan et al. (2012) would be dominated by the
+blazars of non-Fermi detection. Therefore, the results indicate that the optical timescale of the blazar of non-Fermi
+detection may be longer than that of the blazar of Fermi detection. Xiong et al. (2015) found that the two population
+blazars indeed have different physical properties, for example, the blazar of non-Fermi detection has a smaller Doppler
+factor (Paliya et al. 2017).
+In the reverberation mapping studies of 3C 273 and PSK 1510-089, a nonechoed long-term trend is found in the
+optical continuum LC (Zhang et al. 2019; Li et al. 2020; Rakshit 2020). This reveals the mixed origin of their optical
+emission. New clues on the origin of the optical emission can be found in our results. The V and B-band timescales of
+PSK 1510-089 are different. Its long-term V -band variability is correlated with the γ-ray variability (Castignani et al.
+2017), suggesting that the V -band emission is dominated by jet contribution. The long-term polarization variation
+(Pandey et al. 2022) also supports that the nonthermal component is dominated at V -band. The V -band emission of
+3C 273 seems to be more complicated. The jet contribution to V -band emission may be strongly time-dependent and
+may vary in a large range. This complicated mixture of jet and accretion disk emission results in a single power-law
+PSD. For the two sources, no significant correlation is found between B-band and γ-ray variabilities in our analysis.
+The B-band emission is naturally considered as the accretion disk contribution. For 3C 273, the B-band timescale of
+≈ 60 days is a typical value for the accretion disk emission of normal quasars. While the B-band timescale of ≈ 11
+days of PKS 1510-089 is significantly smaller, and it deviates from the τ rest
+damping − MBH relation of Burke et al. (2021)
+(Figure 15). This short timescale may imply special properties of its accretion disk.
+The nonthermal optical, X-ray and γ-ray variabilities all have the typical DRW PSD. Namely, the PSD of synchrotron
+emission is the same as that of inverse-Compton (IC) emission, consistent with the simulations with a time-dependent
+one-zone leptonic blazar emission model (Thiersen et al. 2022). In other words, the long-term jet variability is irrelevant
+to the underlying emission mechanism.
+Burke et al. (2021) suggested that the DRW damping timescale measured from the accretion disk variability of
+normal quasars could be associated with the thermal instability timescale expected in the AGN standard accretion
+disk theory. Zhang et al. (2022) measured the γ-ray DRW damping timescale of AGNs from the Fermi-LAT data, and
+found that the γ-ray timescales of 23 AGNs occupy almost the same space with the optical variability timescales of
+normal quasars in the plot of τ rest
+damping − MBH. In this work, we add the nonthermal optical timescale of blazars in this
+6 The inner parsec jet of the blazar J19242914 has been resolved by the Event Horizon Telescope (Issaoun et al. 2022).
+7 They also used δD = 10 for the Doppler effect correction.
+
+16
+plot. The nonthermal optical timescale of blazars also locates at the same region with the thermal optical timescale
+of normal quasars in the plot (Figure 15). This implies that the jet variability is relevant to the accretion disk. The
+thermal instability in accretion disk may not only cause the accretion disk variability but also the jet multi-band
+variability.
+Statistically, the nonthermal optical τ rest
+damping of 38 blazars are consistent with the γ-ray τ rest
+damping of 22 blazars.
+Individually (3C 273, PKS 1510-089, and BL Lac), the damping timescales of the jet variability in optical, X-ray,
+and γ-ray energies are consistent within the measured errors. Our results indicate that multi-band jet emissions are
+produced in the same region. However, we still cannot know the distance from the emission region to the central black
+hole. The radio observation is helpful to constrain this distance (Max-Moerbeck et al. 2014). We modeled the OVRO
+radio LCs covering over ∼ten years, and we obtain a single power-law PSD. In this work, we only show the radio result
+for 3C 273 as an example. We also modeled the 30-yr radio LCs of 3C 279 and 3C 454.3 obtained from Aalto University
+Mets¨ahovi Radio Observatory, and we still get an unconstrained timescale. The results indicate the radio timescale is
+very large and may be larger than 10 years. Through the very long baseline interferometry (VLBI) observation, one
+can determine the distance from the radio core to the central black hole. Comparing the optical/X-ray/γ-ray timescale
+and the radio timescale, we can infer that the optical/X-ray/γ-ray emission region is far upstream from the radio core.
+5. SUMMARY
+We analyze the blazar’s radio, optical, and X-ray variabilities using the GP tool celerite. The DRW model can
+successfully fit the jet multi-band variabilities. The multi-band characteristic timescale is used to probe the structure
+of the emission region in the blazar jet. Our main results are as follows.
+(i) The synchrotron and IC emissions have the same PSD, i.e., the typical DRW PSD. This indicates that the jet’s
+long-term variability is irrelevant to the underlying emission processes. In the plot of τ rest
+damping−MBH, the jet timescales
+locate at almost the same space as the accretion disk timescales of normal quasars, implying that the jet and accretion
+disk variability is driven by the same physical process (Zhang et al. 2022).
+(ii) The nonthermal optical, X-ray, and γ-ray variability has a consistent characteristic timescale. The radio char-
+acteristic timescale is very long which cannot be constrained by decades-long LC. The results indicate that the non-
+thermal optical-X-ray-γ-ray emission is produced in the same region, which is upstream and far from the radio core.
+This supports the basic hypothesis of the standard Synchrotron-Self-Compton jet model.
+The GP method provides a flexible approach to understand the variability pattern of AGN in the framework of
+stochastic process. Adopting the standard GP tool (Foreman-Mackey et al. 2017), we build the link between accretion
+disk (thermal emission) and the jet (nonthermal emission), i.e., Figure 15. This is a new methodology for comparing
+thermal and nonthermal emissions, additional to the comparison between the thermal and nonthermal luminosities
+(e.g, Ghisellini et al. 2011; Sbarrato et al. 2012; Ghisellini et al. 2014).
+ACKNOWLEDGMENTS
+We thank the referees’ valuable report. This work is partially supported by the National Key R & D Program of
+China under grant No. 2018YFA0404204. H. Y. Zhang acknowledges the financial support from the Scientific Research
+Fund project of Yunnan Education Department (2022Y053) and the Graduate Research innovation project of Yunnan
+University (2021Y034). The work of D. H. Yan is also supported by the CAS Youth Innovation Promotion Association
+and Basic research Program of Yunnan Province (202001AW070013).
+Data from the Steward Observatory spectropolarimetric monitoring project were used. This program is supported
+by Fermi Guest Investigator grants NNX08AW56G, NNX09AU10G, NNX12AO93G, and NNX15AU81G. This re-
+search has made use of up-to-date SMARTS optical/nearinfrared light curves.
+This research has made use of
+data from the OVRO 40-m monitoring program, which is supported by private funding from the California In-
+situte of Technology and the Max Planck Institute for Radio Astronomy, and by NASA grants NNX08AW31G,
+NNX11A043G, and NNX14AQ89G and NSF grants AST-0808050 and AST- 1109911.
+This work also has made
+use of {lightcurves} {spectral files} provided by the University of California, San Diego Center for Astrophysics and
+Space Sciences, X-ray Group (R.E. Rothschild, A.G. Markowitz, E.S. Rivers, and B.A. McKim).
+Facility: SMARTS.
+
+17
+Software: corner.py (Foreman-Mackey 2016), celerite (Foreman-Mackey et al. 2017), emcee (Foreman-Mackey et al.
+2013), NumPy (Harris et al. 2020), Matplotlib (Hunter 2007), Astropy (Astropy Collaboration et al. 2013, 2018), SciPy
+(Virtanen et al. 2020).
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+Springer Nature 2021 LATEX template
+Heterogeneous Tri-stream Clustering Network
+Xiaozhi Deng1, Dong Huang1,2* and Chang-Dong Wang3
+1*College of Mathematics and Informatics, South China
+Agricultural University, Guangzhou, China.
+2Key Laboratory of Smart Agricultural Technology in Tropical
+South China, Ministry of Agriculture and Rural Affairs, China.
+3School of Computer Science and Engineering, Sun Yat-sen
+University, Guangzhou, China.
+*Corresponding author(s). E-mail(s): [email protected];
+Contributing authors: [email protected];
+[email protected];
+Abstract
+Contrastive deep clustering has recently gained significant attention
+with its ability of joint contrastive learning and clustering via deep
+neural networks. Despite the rapid progress, previous works mostly
+require both positive and negative sample pairs for contrastive cluster-
+ing, which rely on a relative large batch-size. Moreover, they typically
+adopt a two-stream architecture with two augmented views, which over-
+look the possibility and potential benefits of multi-stream architectures
+(especially with heterogeneous or hybrid networks). In light of this,
+this paper presents a new end-to-end deep clustering approach termed
+Heterogeneous Tri-stream Clustering Network (HTCN). The tri-stream
+architecture in HTCN consists of three main components, including two
+weight-sharing online networks and a target network, where the param-
+eters of the target network are the exponential moving average of that
+of the online networks. Notably, the two online networks are trained by
+simultaneously (i) predicting the instance representations of the target
+network and (ii) enforcing the consistency between the cluster repre-
+sentations of the target network and that of the two online networks.
+Experimental results on four challenging image datasets demonstrate the
+superiority of HTCN over the state-of-the-art deep clustering approaches.
+The code is available at https://github.com/dengxiaozhi/HTCN.
+Keywords: Data clustering, Image clustering, Deep clustering, Deep neural
+network, Contrastive learning
+1
+arXiv:2301.04451v1  [cs.LG]  11 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Heterogeneous Tri-stream Clustering Network
+1 Introduction
+Data clustering is the process of grouping data samples into multiple clus-
+ters in an unsupervised manner, which is a fundamental task in a variety
+of applications [1–3]. The traditional clustering algorithms typically focus
+on some low-level information and lack the representation learning ability,
+which may lead to sub-optimal performance when dealing with some complex
+high-dimensional data like images.
+In recent years, the deep learning has gained tremendous progress [4–6],
+which has also been exploited for tackling the clustering task, giving rise to the
+rapid development of the deep clustering algorithms [7–11]. For example, Xie
+et al. [7] presented a deep clustering method called Deep Embedded Clustering
+(DEC), which simultaneously learns representations and cluster assignments
+with an objective loss based on Kullback-Leibler (KL) divergence. Guo et al.
+[8] extended DEC by incorporating the reconstruction loss (via autoencoder)
+to preserve local structures. Ji et al. [10] sought to learn invariant information
+of data by maximizing the mutual information between paired samples. More
+recently, the contrastive learning has emerged as a promising technique for
+exploiting sample-wise (or augmentation-wise) contrastiveness to improve the
+deep clustering performance. Van Gansbeke et al. [12] presented the Semantic
+Clustering by Adopting Nearest neighbors (SCAN) method, which first adopts
+contrastive learning to learn discriminant features and then performs semantic
+clustering with the K-nearest neighbors exploited. Dang et al. [13] matched
+local-level and global-level nearest neighbors to further improve clustering per-
+formance. Li et al. [14] presented the Contrastive Clustering (CC) method to
+perform feature learning and clustering with simultaneous instance-level and
+cluster-level contrastive learning.
+Despite significant success, these contrastive deep clustering methods [12–
+14] are mostly faced with two limitations. On the one hand, they typically
+requires both positive sample pairs and negative sample pairs during their
+contrastive learning process, which rely on a relatively large batch-size (for
+sufficient negative pairs) and may bring in a heavier computational burden.
+On the other hand, these prior works generally adopt a two-stream architec-
+ture (with two weight-sharing augmented views), which neglect the possibility
+of going beyond the two-stream architecture to utilize three or even more
+streams of networks (with heterogeneous or hybrid structures). Recently Grill
+et al. [15] presented the Bootstrap Your Own Latent (BYOL) method, which
+adopts an asymmetric two-stream architecture (with an online network and a
+target network) and conducts the contrastive learning without negative pairs,
+where the online network is trained by predicting the feature representations
+of the target network. Though the requirement for negative sample pairs is
+remedied, BYOL still complies with the two-stream architecture and also lacks
+the ability of directly learning the clustering structure. It remains a challeng-
+ing problem how to incorporate contrastive learning into multiple streams of
+heterogeneous networks while alleviating the dependence on negative sample
+pairs for strengthened deep clustering performance.
+
+Springer Nature 2021 LATEX template
+Heterogeneous Tri-stream Clustering Network
+3
+In light of this, this paper presents a novel deep clustering approach
+termed Heterogeneous Tri-stream Clustering Network (HTCN), which lever-
+ages three streams of heterogeneous networks for simultaneous cluster-level
+and instance-level contrastive learning without requiring negative sample pairs
+(as illustrated in Fig. 1). Inspired by BYOL [15], we design a novel tri-stream
+architecture with three augmented views, corresponding to two online net-
+works and a target network, respectively. Note that the online network and
+the target network are heterogeneous, which differ from each other in the net-
+work structure and the updating mechanism. The two online networks share
+the same parameters, while the parameters of the target network are the
+exponential moving average of that of the online networks. Here, the exponen-
+tial moving average is a type of moving average that places a greater weight
+and significance on the most recent data samples [15]. Each online network
+is associated with an instance predictor and a cluster predictor, which pro-
+duce the instance-level representations and the cluster-level representations,
+respectively. Different from the online networks, the target network utilizes a
+cluster predictor to generate the cluster-level representations while producing
+the instance-level representations by the projector directly. The incorporation
+of an instance predictor in the online networks is meant to prevent the potential
+collapse where the networks produce the same feature representations for most
+samples. Then we train the two online networks by (i) predicting the target
+network’s representation of the same image via the mean squared error (MSE)
+loss (for the instance-level contrastive learning) and (ii) enforcing the consis-
+tency between the predicted cluster distributions of the two online networks
+and that of the target network via the information noise contrastive estimation
+(InfoNCE) [16] loss (for the cluster-level contrastive learning). Experiments
+conducted on four image datasets demonstrate the superiority of our approach
+over the state-of-the-art deep clustering approaches.
+For clarity, the contributions of this work are summarized below.
+• A heterogeneous tri-stream architecture is designed, where two online net-
+works and a target network are jointly leveraged for instance-level and
+cluster-level contrastive learning.
+• A novel deep clustering approach termed HTCN is proposed, which utilizes
+three augmented views for contrastive learning without requiring negative
+sample pairs.
+• Experimental results on four image datasets confirm the advantegeous clus-
+tering performance of our HTCN approach over the state-of-the-art deep
+clustering approaches.
+The rest of the paper is organized as follows. The related works on deep
+clustering and self-supervised learning are reviewed in Section 2. The proposed
+HTCN framework is described in Section 3. The experiments are reported in
+Section 4. Finally, Section 5 concludes the paper.
+
+Springer Nature 2021 LATEX template
+4
+Heterogeneous Tri-stream Clustering Network
+2 Related Work
+In this section, we will introduce the related works on deep clustering and
+self-supervised learning.
+2.1 Deep Clustering
+Traditional clustering methods such as K-means [17] and spectral cluster-
+ing (SC) [18] have achieved promising results in handling low-dimensional
+data, but they may result in sub-optimal performance when faced with high-
+dimensional data (e.g., images and videos) due to the lack of the representation
+learning ability. To address this, the deep learning based clustering methods,
+referred to as the deep clustering methods, have recently achieved significant
+success, which leverage the power of feature learning of deep neural networks
+for the clustering task [7–10, 12–14, 19–27].
+Previous deep clustering methods can be divided into two main categories,
+namely, the one-stage methods and the two-stage methods. The goal of the
+one-stage approach is to perform feature representation learning and clustering
+assignment simultaneously. Xie et al. [7] proposed a Deep Embedding Cluster-
+ing (DEC) method, which jointly optimizes feature learning and clustering with
+a KL-divergence loss. Caron et al. [21] iteratively clustered the learned features
+with K-means and regarded the cluster assignments as supervisory signals to
+optimize the network. Li et al.[14] presented a Contrastive Clustering (CC)
+method that performs contrastive learning at instance-level and cluster-level
+for deep clustering. Besides the one-stage methods, some researchers have also
+made considerable efforts to the two-stage clustering methods. Van Gansbeke
+et al.[12] proposed a two-stage clustering method called Semantic Clustering
+by Adopting Nearest neighbors (SCAN), which first learns the semantic fea-
+tures via contrastive learning and then utilizes the features for clustering in
+the next stage. To extend SCAN, Dang et al. [13] designed a Nearest Neighbor
+Matching (NNM) method, which selects both local and global nearest neigh-
+bors to optimize the network, where the neighbors are forced to be close to
+each other.
+2.2 Self-supervised Learning
+Self-supervised learning has recently emerged as a powerful technique with the
+ability to learn representation from raw data without human supervision, in
+which the contrastive learning methods [28–31] have been a representative and
+promising category.
+The goal of contrastive learning is to minimize the distance between posi-
+tive sample pairs while maximizing the distance between negative sample pairs
+in a self-supervised manner, where positive pairs and negative pairs are defined
+through data augmentations. In particular, some researchers maintained a
+memory bank [28, 29] that contains large amounts of representations of nega-
+tive samples to achieve high performance. However, these methods that utilize
+memory banks to store and update representations may be computationally
+
+Springer Nature 2021 LATEX template
+Heterogeneous Tri-stream Clustering Network
+5
+expensive. To address the problems with memory banks, He et al. [30] pro-
+posed a Momentum Contrast (MoCo) method that trains an encoder by the
+momentum update mechanism maintaining a long queue of negative examples.
+Following the MoCo method, Chen et al. [31] proposed a Simple framework
+for Contrastive LeaRning (SimCLR) method which carefully designs the strat-
+egy of data augmentation and a non-linear transformation head. In addition,
+the clustering based methods [32, 33] adopt a clustering approach to group
+similar features together, which address the issue that every sample is consid-
+ered as a discrete class in previous works. More recently, some self-supervised
+learning methods that only rely on positive pairs and directly predict the out-
+put of one augmented view from another augmented view [15, 34, 35] have
+been developed, among which a representative method is the BYOL method
+[15]. The BYOL method [15] adopts an asymmetric two-stream architecture,
+which, however, lacks the ability to learn the clustering structure directly and
+also overlooks the opportunities and potential benefits of going beyond the
+two-stream architecture to three or more streams of networks (even with het-
+erogeneous or hybrid structures) to further enhance the contrastive learning
+and clustering performance.
+3 Proposed Framework
+3.1 Framework Overview
+This paper presents a heterogeneous tri-stream network architecture termed
+HTCN for contrastive deep clustering (as illustrated in Fig. 1), which goes
+beyond the traditional two-stream architecture to explore the constrastive net-
+work in a multi-stream manner. Also, HTCN doesn’t require negative sample
+pairs, which makes it more resilient to different batch-size. Specifically, HTCN
+consists of three main components, including two online networks and a target
+network. The online networks and the target network are respectively parame-
+terized by different sets of weights, where the parameters of the target network
+are an exponential moving average of that of the online networks.
+Given a batch of N images, we perform three types of augmentations on
+each image, denoted as xi with i ∈ [1, N], to generate 3 · N augmented (or
+distorted) images, denoted as {xa
+1, . . . , xa
+N, xb
+1, . . . , xb
+N, xc
+1, . . . , xc
+N}. The back-
+bones (i.e., fθ and fξ) and projectors (i.e., gθ and gξ) are adopted to extract
+features from the distorted images via za
+i = gθ(fθ(xa
+i )), zb
+i = gξ(fξ(xb
+i)) and
+zc
+i = gθ(fθ(xc
+i)). Then the instance predictors transform za
+i and zc
+i to ya
+i
+and yc
+i , respectively, while the cluster predictors transform za
+i , zb
+i and zc
+i to
+˜qa
+i , ˜qb
+i and ˜qc
+i , respectively. Note that, similar to the asymmetric architecture
+of BYOL, the target network is not associated with an instance predictor,
+and the representations generated by its projector are used to guide the
+instance-level learning of the two online networks. The row space of the feature
+matrix learned by the projector or the instance predictor is expressed as the
+instance-level representations, while the column space of the feature matrix
+learned by the cluster predictor is expressed as the cluster-level representations.
+
+Springer Nature 2021 LATEX template
+6
+Heterogeneous Tri-stream Clustering Network
+𝑓!
+𝑓"
+𝑔!
+𝑔"
+ℎ"
+ℎ!
+𝑝!
+𝑥#
+𝑥$
+𝑧#
+𝑧$
+𝑦#
+𝑞#
+𝑞$
+𝑥
+𝑞$
+𝑞#
+Minimize InfoNCE Loss
+Distorted images
+Backbone
+Cluster predictor
+Projector
+Online network
+Target network
+Instance predictor
+𝑦#
+𝑧$
+Minimize MSE Loss
+𝑦%
+𝑧$
+Input images
+𝑞$
+𝑞%
+Minimize InfoNCE Loss
+Cluster predictor
+Distorted images
+Backbone
+Projector
+𝑇#
+𝑇$
+𝑇%
+𝑓!
+𝑔!
+ℎ!
+𝑝!
+𝑥%
+𝑧%
+𝑦%
+𝑞%
+Cluster predictor
+Distorted images
+Backbone
+Projector
+Instance predictor
+sg
+Online network
+Target network
+Fig. 1 Illustration of the proposed HTCN framework. The tri-stream network consists of
+two weight-sharing online networks and a target network, where the parameters of the target
+network is an exponential moving average of that of the online networks. Instance predictors
+and cluster predictors are incorporated in the three networks, after which the MSE loss
+and the InfoNCE loss are ultilized for instance-level contrastive learning and cluster-level
+contrastive learning, respectively. The network architecture can be trained in an end-to-end
+manner, where the final clustering is obtained via the cluster predictor of the target network.
+The instance-level representations are utilized to enforce the instance-level
+contrastive learning with an MSE loss optimized, while the cluster-level repre-
+sentations are utilized to enforce the cluster-level contrastive learning with an
+InfoNCE loss optimized. Finally, the instance-level and cluster-level contrastive
+losses are simultaneously utilized to optimize the tri-stream network.
+3.2 Instance-level Contrastiveness
+Our HTCN approach simultaneously performs feature learning and clustering
+without requiring negative sample pairs. In instance-level contrastive learn-
+ing, we aim to train the two online networks by predicting the instance
+representations of target network. Specifically, let ya
+i and zb
+i be the instance
+representations of xi in the first online network and the target network,
+respectively. The instance-level contrastive loss between them is defined as
+La,b,i = ∥ya
+i − zb
+i ∥2
+2 = 2 − 2 ·
+⟨ya
+i , zb
+i ⟩
+∥ya
+i ∥2 · ∥zb
+i ∥2
+,
+(1)
+
+Springer Nature 2021 LATEX template
+Heterogeneous Tri-stream Clustering Network
+7
+where ya
+i and zb
+i are the normalized representations. Thus the loss between the
+first and second views can be expressed as
+La,b = 1
+N
+N
+�
+i=1
+La,b,i
+(2)
+Similar to BYOL [15], the exchange of the online and target views is performed
+during each training step. Also, we utilize another online network to predict
+the representations produced by the target network, whose loss is defined as
+Lb,c,i = ∥yc
+i − zb
+i ∥2
+2 =2 − 2 ·
+⟨yc
+i , zb
+i ⟩
+∥yc
+i ∥2 · ∥zb
+i ∥2
+, i ∈ [1, N],
+(3)
+Lb,c = 1
+N
+N
+�
+i=1
+Lb,c,i,
+(4)
+Therefore, the instance-level contrastive loss among the three streams of
+networks is defined as
+Linstance = La,b + Lb,c.
+(5)
+3.3 Cluster-level Contrastiveness
+The cluster predictor maps the representations produced by the projector to
+M-dimensional probability vectors, where M is the number of clusters. These
+probability vectors, whose i-th element denotes how likely the image belongs to
+the i-th cluster, can be interpreted as the soft label. Let qa, qb, qc ∈ RN×M be
+the feature matrices produced by the cluster predictors of the three networks,
+respectively. Each column of the feature matrix denotes an N-dimensional clus-
+ter representation, denoted as qk
+i , while the each row denotes a M-dimensional
+probability vector, denoted as ˜qk
+i (for k ∈ {a, b, c}).
+For a cluster representation qa
+i , we regard qa
+i and qb
+i as a positive cluster
+pair, and the other 2·M −2 pairs (in the first and second views) as the negative
+cluster pairs. The pair-wise similarity is defined as
+s(qa
+i , qb
+j) = ⟨qa
+i , qb
+j⟩
+∥qa
+i ∥∥qb
+j∥,
+i, j ∈ [1, M]
+(6)
+Then the InfoNCE loss for qa
+i is computed by
+ℓa
+i = − log
+exp(s(qa
+i , qb
+j)/τ)
+�M
+j=1[exp(s(qa
+i , qa
+j )/τ) + exp(s(qa
+i , qb
+j)/τ)]
+,
+(7)
+where τ is the temperature parameter. After traversing all cluster representa-
+tions, the cluster-level contrastive loss between the first and second augmented
+
+Springer Nature 2021 LATEX template
+8
+Heterogeneous Tri-stream Clustering Network
+views can be obtained as
+ˆLa,b =
+1
+2M
+M
+�
+i=1
+(ℓa
+i + ℓb
+i) − H(Q),
+(8)
+where H(Q) is the entropy of the cluster-assignment probability, which helps
+to avoid a degenerate solution that most images fall into the same cluster and
+is computed as
+H(Q) = −
+M
+�
+i=1
+[P(qa
+i ) log P(qa
+i ) + P(qb
+i ) log P(qb
+i )],
+(9)
+P(qk
+i ) =
+N
+�
+j=1
+qk
+ji
+∥q∥1
+,
+k ∈ {a, b}
+(10)
+For each batch of images, a view pair is formed between each online network
+and the target network, leading to a total of two view pairs for the cluster-
+level contrastive learning. Therefore, the cluster-level contrastive loss can be
+defined as
+Lcluster = ˆLa,b + ˆLb,c.
+(11)
+3.4 Overall Loss Function
+The tri-stream network of HTCN is trained by simultaneously considering the
+instance-level contrastiveness and the cluster-level contrastiveness. The overall
+loss function is defined as
+L = Linstance + Lcluster.
+(12)
+At each training step, we optimize the overall loss function w.r.t. the online
+networks’ parameters θ only, but not the target network’s parameters ξ. The
+parameters of the target is updated as an exponential moving average of that
+of the online networks. That is
+θ ← optimizer(θ, ∇θL, η),
+(13)
+ξ ← αξ + (1 − α)θ.
+(14)
+where η is the learning rate and α is the momentum coefficient. After the
+training, we only keep the target network to perform clustering, which can be
+obtained in the cluster predictor.
+
+Springer Nature 2021 LATEX template
+Heterogeneous Tri-stream Clustering Network
+9
+Table 1 Description of the benchmark image datasets.
+Dataset
+#Images
+#Classes
+CIFAR-100
+60,000
+20
+ImageNet-10
+13,000
+10
+ImageNet-Dogs
+19,500
+15
+Tiny-ImageNet
+100,000
+200
+(a) CIFAR-100
+(b) ImageNet-10
+(c) ImageNet-Dogs
+(d) Tiny-ImageNet
+Fig. 2 Some examples of the four image datasets.
+3.5 Implementation Details
+In HTCN, we use the ResNet34 [36] as the backbone. The projectors and the
+instance predictors have the same network structure, each of which is a multi-
+layer perceptron (MLP) with 256-dimensional output units. Each of the cluster
+predictors is a two-layer MLP, whose output dimension is equal to the desired
+number of clusters.
+Three augmented (or distorted) views are generated by applying a family of
+transformations to each input image. Five types of augmentations are utilized,
+including ResizedCrop, HorizontalFlip, ColorJitter, Grayscale and Gaussian-
+Blur [14]. As each transformation has a probability of being adopted, the
+distortions of the three streams can thus be randomly decided. During opti-
+mization, we use the Adam optimizer and train the model for 1000 epochs.
+The learning rate is set to 0.0003.The batch size is set to 128.
+4 Experiments
+4.1 Datasets and Evaluation Metrics
+The experiments are conducted on four widely-used image datasets, namely,
+CIFAR-100 [37], ImageNet-10 [38], ImageNet-Dogs [38], and Tiny-ImageNet
+[39]. The statistics of these benchmark datasets are given in Table 1, and some
+sample images of these datasets are illustrated in Fig. 2.
+To compare the clustering results of different clustering methods, three
+evaluation metrics are adopted, including normalized mutual information
+
+Springer Nature 2021 LATEX template
+10
+Heterogeneous Tri-stream Clustering Network
+Table 2 The NMI(%) scores by different clustering methods (The best score in each
+column is in bold).
+Dataset
+CIFAR-100
+ImageNet-10
+ImageNet-Dogs
+Tiny-ImageNet
+K-means [17]
+8.4
+11.9
+5.5
+6.5
+SC [18]
+9.0
+15.1
+3.8
+6.3
+AC [43]
+9.8
+13.8
+3.7
+6.9
+NMF [44]
+7.9
+13.2
+4.4
+7.2
+AE [45]
+10.0
+21.0
+10.4
+13.1
+DAE [46]
+11.1
+20.6
+10.4
+12.7
+DCGAN [47]
+12.0
+22.5
+12.1
+13.5
+DeCNN [48]
+9.2
+18.6
+9.8
+11.1
+VAE [49]
+10.8
+19.3
+10.7
+11.3
+JULE [22]
+10.3
+17.5
+5.4
+10.2
+DEC [7]
+13.6
+28.2
+12.2
+11.5
+DAC [38]
+18.5
+39.4
+21.9
+19.0
+DCCM [50]
+28.5
+60.8
+32.1
+22.4
+GATC [51]
+28.5
+59.4
+28.1
+-
+PICA [19]
+31.0
+80.2
+35.2
+27.7
+DRC [26]
+35.6
+83.0
+38.4
+32.1
+CC [14]
+43.1
+85.9
+44.5
+34.0
+HTCN
+46.5
+87.5
+49.4
+35.6
+(NMI) [40], clustering accuracy (ACC) [41], and adjusted rand index (ARI)
+[42].
+4.2 Comparison with State-of-the-Art
+In this section, we compare the proposed method against four non-deep
+clustering methods, namely, K-means [17], Spectral Clustering (SC) [18],
+Agglomerative Clustering (AC) [43], and Nonnegative Matrix Factorization
+(NMF) [44], and thirteen deep clustering methods, namely, Auto-Encoder
+(AE) [45], Denoising Auto-Encoder (DAE) [46], Deep Convolutional Genera-
+tive Adversarial Networks (DCGAN) [47], DeConvolutional Neural Networks
+(DeCNN) [48], Aariational Auto-Encoder (VAE) [49], Joint Unsupervised
+LEarning (JULE) [22], Deep Embedded Clustering (DEC) [7], Deep Adap-
+tive Clustering (DAC) [38], Deep Comprehensive Correlation Mining (DCCM)
+[50], Gaussian ATtention Network for image Clustering (GATC) [51], PartI-
+tion Confidence mAximization (PICA) [19], Deep Robust Clustering (DRC)
+[26] and Contrastive Clustering (CC) [14].
+As shown in Table 2, 3 and 4, our HTCN method achieves the best scores
+on all the four benchmark datasets w.r.t. NMI, ACC, and ARI. Notably, on
+the ImageNet-Dogs dataset, our HTCN method obtains NMI(%),ACC(%) and
+ARI(%) scores of 49.4, 49.3, and 35.2, respectively, which significantly outper-
+forms the second best method (i.e., CC) that obtains NMI(%),ACC(%) and
+ARI(%) scores of 44.5, 42.9, and 27.4. The experimental results in Table 2,
+
+Springer Nature 2021 LATEX template
+Heterogeneous Tri-stream Clustering Network
+11
+Table 3 The ACC(%) scores by different clustering methods (The best score in each
+column is in bold).
+Dataset
+CIFAR-100
+ImageNet-10
+ImageNet-Dogs
+Tiny-ImageNet
+K-means [17]
+13.0
+24.1
+10.5
+2.5
+SC [18]
+13.6
+27.4
+11.1
+2.2
+AC [43]
+13.8
+24.2
+13.9
+2.7
+NMF [44]
+11.8
+23.0
+11.8
+2.9
+AE [45]
+16.5
+31.7
+18.5
+4.1
+DAE [46]
+15.1
+30.4
+19.0
+3.9
+DCGAN [47]
+15.3
+34.6
+17.4
+4.1
+DeCNN [48]
+13.3
+31.3
+17.5
+3.5
+VAE [49]
+15.2
+33.4
+17.9
+3.6
+JULE [22]
+13.7
+30.0
+13.8
+3.3
+DEC [7]
+18.5
+38.1
+19.5
+3.7
+DAC [38]
+23.8
+52.7
+27.5
+6.6
+DCCM [50]
+32.7
+71.0
+38.3
+10.8
+GATC [51]
+32.7
+73.9
+32.2
+-
+PICA [19]
+33.7
+87.0
+35.2
+9.8
+DRC [26]
+36.7
+88.4
+38.9
+13.9
+CC [14]
+42.9
+89.3
+42.9
+14.0
+HTCN
+47.2
+90.5
+49.3
+16.0
+Table 4 The ARI(%) scores by different clustering methods (The best score in each
+column is in bold).
+Dataset
+CIFAR-100
+ImageNet-10
+ImageNet-Dogs
+Tiny-ImageNet
+K-means [17]
+2.8
+5.7
+2.0
+0.5
+SC [18]
+2.2
+7.6
+1.3
+0.4
+AC [43]
+3.4
+6.7
+2.1
+0.5
+NMF [44]
+2.6
+6.5
+1.6
+0.5
+AE [45]
+4.8
+15.2
+7.3
+0.7
+DAE [46]
+4.6
+13.8
+7.8
+0.7
+DCGAN [47]
+4.5
+15.7
+7.8
+0.7
+DeCNN [48]
+3.8
+14.2
+7.3
+0.6
+VAE [49]
+4.0
+16.8
+7.9
+0.6
+JULE [22]
+3.3
+13.8
+2.8
+0.6
+DEC [7]
+5.0
+20.3
+7.9
+0.7
+DAC [38]
+8.8
+30.2
+11.1
+1.7
+DCCM [50]
+17.3
+55.5
+18.2
+3.8
+GATC [51]
+17.3
+55.2
+16.3
+-
+PICA [19]
+17.1
+76.1
+20.1
+4.0
+DRC [26]
+20.8
+79.8
+23.3
+5.6
+CC [14]
+26.6
+82.2
+27.4
+7.1
+HTCN
+30.5
+83.9
+35.2
+7.6
+
+Springer Nature 2021 LATEX template
+12
+Heterogeneous Tri-stream Clustering Network
+Table 5 The clustering performance of HTCN using different combinations of network
+architectures.
+Architecture
+NMI
+ACC
+ARI
+Tri-stream architecture
+46.5
+47.2
+30.5
+Dual-stream (Online+Target)
+42.2
+42.5
+26.8
+Dual-stream (Online+Online)
+39.9
+40.2
+24.4
+Table 6 The clustering performance of HTCN using different loss functions.
+Loss function
+NMI
+ACC
+ARI
+With instance and cluster losses
+46.5
+47.2
+30.5
+With only instance loss
+43.3
+35.6
+14.6
+With only cluster loss
+38.4
+36.4
+22.7
+3 and 4 confirm the advantageous clustering performance of HTCN over the
+baseline methods.
+4.3 Influence of the Tri-stream Architecture
+In the proposed framework, we present a tri-stream architecture which consists
+of two online networks and a target network. In this section, we test the influ-
+ence of the three streams of networks. As shown in Table 5, using an online
+network and a target network leads to better clustering results than using
+two online networks, while using three streams of networks outperforms both
+variants of using two streams, which shows the benefits of the heterogeneous
+tri-stream architecture.
+4.4 Influence of Two Types of Contrastive losses
+In the section, we test the influence of the two types of contrastive losses,
+i.e., the instance-level contrastive loss and the cluster-level contrastive loss. As
+shown in Table 6, training with both types of losses can lead to better clustering
+performance than training with only one of them, which confirm the joint
+contribution of the instance-level and cluster-level losses in the self-supervised
+training.
+4.5 Influence of the Asymmetric Settings
+Two symmetry-breaking mechanisms are enforced between the online and tar-
+get networks [15]. First, an instance predictor is incorporated in each online
+network, which does not exist in the target network. Second, the so-called
+stop-gradient is incorporated in the target network, which indicates that this
+network is not updated using backpropagation. We test the influence of the
+
+Springer Nature 2021 LATEX template
+Heterogeneous Tri-stream Clustering Network
+13
+Table 7 The NMI(%), ACC(%), and ARI(%) by HTCN removing different asymmetric
+settings.
+Asymmetric settings
+NMI
+ACC
+ARI
+HTCN
+46.5
+47.2
+30.5
+No predictor
+40.9
+41.2
+25.0
+No stop-gradient
+39.2
+39.3
+23.7
+0
+200
+400
+600
+800
+1000
+Epochs
+0
+10
+20
+30
+40
+50
+55
+NMI
+(a) CIFAR-100
+0
+200
+400
+600
+800
+1000
+Epochs
+0
+20
+40
+60
+80
+100
+NMI
+(b) ImageNet-10
+0
+200
+400
+600
+800
+1000
+Epochs
+0
+10
+20
+30
+40
+50
+55
+NMI
+(c) ImageNet-dogs
+0
+200
+400
+600
+800
+1000
+Epochs
+0
+5
+10
+15
+20
+25
+30
+35
+40
+NMI
+(d) Tiny-ImageNet
+Fig. 3 Illustration of the convergence of HTCN (w.r.t. its NMI performance) on the four
+benchmark datasets.
+asymmetric settings by removing one of the instance predictor and the stop-
+gradient. As shown in Table 7, training with both asymmetric settings leads
+to better performance than training with only one of them.
+4.6 Convergence Analysis
+In this section, we test the convergence of the proposed HTCN method as
+the number of epochs increases. As shown in Fig. 3, the clustering scores
+(w.r.t. NMI) of the proposed HTCN method rapidly increase during the first
+200 epochs on the benchmark datasets. When going beyond 200 epochs, the
+increase of epochs still benefits the clustering performance consistently. In this
+paper, the number of epochs is set to 1000 on all benchmark datasets.
+5 Conclusion and Future Work
+The paper develops a new deep clustering approach termed HTCN, which
+breaks through the conventional two-stream contrastive architecture to explore
+the rich possibilities in heterogeneous multi-stream contrastive learning and
+clustering. In HTCN, the two weight-sharing online networks are trained by
+predicting the instance representations of the target network and enforcing
+the consistency between the cluster representations of the target and online
+networks. Thus the tri-stream network architecture can be optimized in an
+end-to-end manner via simultaneous instance-level and cluster-level contrastive
+learning. Experimental results on four challenging image datasets have shown
+the superior performance of our HTCN approach over the state-of-the-art deep
+
+Springer Nature 2021 LATEX template
+14
+Heterogeneous Tri-stream Clustering Network
+clustering approaches. In this paper, we mainly focus on the deep clustering
+task for images. In the future work, a possible direction is to extend the pro-
+posed framework to the deep clustering tasks for more complex data types,
+such as time series data and document data.
+Declarations
+• Funding.
+This
+work
+was
+supported
+by
+the
+NSFC
+(61976097
+&
+61876193) and the Natural Science Foundation of Guangdong Province
+(2021A1515012203).
+• Conflict of interest. The authors declare that they have no conflict of
+interest.
+• Ethical approval. This article does not contain any studies with human
+participants or animals performed by any of the authors.
+• Consent to participate. Informed consent to participate was obtained
+from all individual participants included in the study.
+• Consent for publication. Informed consent for publication was obtained
+from all individual participants included in the study.
+• Availability of data and materials. All datasets used in this paper are
+publicly-available datasets.
+• Code
+availability.
+The
+code
+is
+available
+at
+https://github.com/
+dengxiaozhi/HTCN.
+• Authors’ contributions. XD: Conceptualization, Methodology, Writing–
+Original Draft. DH: Conceptualization, Writing–Review & Editing. CDW:
+Optimization, Writing–Review & Editing.
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+

E9AyT4oBgHgl3EQf4_pw/content/tmp_files/2301.00796v1.pdf.txt

@@ -0,0 +1,779 @@
+Direct lattice calculation of inclusive hadronic decay rates
+of the 𝝉 lepton
+A. Evangelista,𝑎,∗ R. Frezzotti,𝑎 G. Gagliardi,𝑏 V. Lubicz,𝑐 F. Sanfilippo,𝑏 S. Simula𝑏
+and N. Tantalo𝑎
+𝑎Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata", Via della Ricerca Scientifica 1,
+I-00133 Rome, Italy
+𝑏Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome,
+Italy
+𝑐Dipartimento di Matematica e Fisica, Università di Roma Tre and INFN, Sezione di Roma Tre, Via della
+Vasca Navale 84, I-00146 Rome, Italy
+E-mail: [email protected]
+The inclusive hadronic decay–rates of the 𝜏 lepton are particularly interesting from the phe-
+nomenological point of view since they give access to the CKM matrix elements 𝑉𝑢𝑑 and 𝑉𝑢𝑠. In
+this talk, we discuss how a recent method for the extraction of smeared spectral densities from Eu-
+clidean lattice correlators can be used to obtain a direct lattice determination of inclusive hadronic
+𝜏 decay rates. We also present preliminary numerical results obtained by applying this method
+to correlators measured on two gauge ensembles produced by the ETMC with 𝑁 𝑓 = 2 + 1 + 1
+dynamical flavours at physical pion masses, lattice spacing 𝑎 ≃ 0.08 fm and volumes 𝐿 ≃ 5.1 fm
+and 𝐿 ≃ 7.6 fm.
+The 39th International Symposium on Lattice Field Theory (Lattice2022),
+8-13 August, 2022
+Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.00796v1  [hep-lat]  2 Jan 2023
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+𝜏
+𝜈𝜏
+𝑓
+¯𝑔
+𝑝𝜏
+𝑝𝜈
+𝑋 𝑓 𝑔
+Figure 1: The 𝜏 → 𝜈𝜏𝑋 𝑓 𝑔 Feynman diagram (with no gluons). The hadronic final state 𝑋 𝑓 𝑔, with flavour
+quantum numbers 𝑓 and 𝑔, has fixed 4-momentum 𝑝𝑋 = 𝑝𝜏 − 𝑝𝜈.
+1.
+Introduction
+Inclusive hadronic 𝜏 decays are particularly interesting from the phenomenological viewpoint since
+they give access to the CKM matrix elements 𝑉𝑢𝑑 and 𝑉𝑢𝑠. The determinations of 𝑉𝑢𝑠 from leptonic
+and semileptonic kaon decays [1] are in fairly good agreement with the one of Ref. [2] but, for many
+years, a puzzling tension with other determinations obtained from inclusive hadronic 𝜏 decays has
+been observed and debated [1, 3]. On the lattice, hadronic 𝜏 decays have been studied by using
+dispersion relations and by combining non-perturbative lattice inputs with perturbative and/or OPE
+calculations (see for example [4]).
+Here we present a method to perform a fully non-perturbative direct lattice calculation of the
+𝜏 ↦→ 𝑋𝜈𝜏 decay rate. In our method, the decay rate is extracted from the two-point Euclidean
+correlators of the hadronic weak currents that mediate the decay. This is done by using the algorithm
+of Ref. [5] that allows to extract smeared spectral densities from Euclidean lattice correlators and,
+building on Refs. [6, 7], by using as smearing kernels smoothed versions of the step-functions that
+define the physical phase-space integration domain.
+We also present preliminary numerical results obtained by applying this method to the relevant cor-
+relators measured on two gauge ensembles produced by the Extended Twisted–Mass Collaboration
+(ETMC) with 𝑁 𝑓 = 2 + 1 + 1 dynamical flavours with physical pion mass. The two ensembles,
+corresponding to the cB211.072.64 (B64 in short) and cB211.072.96 (B96 in short) entries in
+TABLE V of Ref. [8], have the same lattice spacing, 𝑎 = 0.07957(13) fm, and differ only for the
+physical volumes that are 𝐿 = 5.09 fm and 𝐿 = 7.64 fm respectively.
+2.
+Reconstruction of the inclusive rate using the HLT method
+By relying on the Fermi effective theory for the weak interactions and by neglecting long–distance
+QED radiative corrections, the ratio 𝑅 𝑓 𝑔 of the inclusive hadronic decay rate Γ[𝜏 ↦→ 𝑋 𝑓 𝑔𝜈𝜏] with
+the leptonic decay rate Γ[𝜏 ↦→ 𝑒 ¯𝜈𝑒𝜈𝜏] can be expressed as
+𝑅 𝑓 𝑔 = 12𝜋 𝑆𝐸𝑊
+��𝑉𝑓 𝑔
+��2 ∫ 1
+𝑟 𝑓 𝑔
+d𝜔 𝜔
+�
+1 − 𝜔2�2 �
+𝜌𝐿
+𝑓 𝑔(𝜔) + 𝜌𝑇
+𝑓 𝑔(𝜔)
+�
+1 + 2𝜔2��
+.
+(1)
+In the previous formula, 𝑓 and 𝑔 label the flavour quantum numbers of the final hadronic states
+𝑋 𝑓 𝑔 having four–momentum 𝑝𝑋, 𝑟 𝑓 𝑔 = 𝑚 𝑓 𝑔/𝑚𝜏 is the ratio of the mass of the lightest hadronic
+state and the 𝜏-mass, 𝑆𝐸𝑊 = 1.0201(3) is the short–distance electroweak correction [9]. The
+2
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+longitudinal and transverse form factors 𝜌𝐿
+𝑓 𝑔(𝜔) and 𝜌𝑇
+𝑓 𝑔(𝜔) parametrize the hadronic spectral
+density
+H 𝜇𝜈
+𝑓 𝑔(𝑝𝑋) = (2𝜋)4 ⟨0| 𝐻𝜇
+𝑓 𝑔(0) 𝛿(4) (P − 𝑝𝑋) 𝐻𝜈†
+𝑓 𝑔(0) |0⟩
+= 𝑝𝜇
+𝑋 𝑝𝜈
+𝑋 𝜌𝐿
+𝑓 𝑔(𝜔) +
+�
+𝑝𝜇
+𝑋 𝑝𝜈
+𝑋 − 𝑔𝜇𝜈𝑝2
+𝑋
+�
+𝜌𝑇
+𝑓 𝑔(𝜔) ,
+𝜔2 = 𝑝2
+𝑋
+𝑚2𝜏
+,
+(2)
+where P = (H, �P) is the QCD four–momentum operator and 𝐻𝜇
+𝑓 𝑔 = 𝑉 𝜇
+𝑓 𝑔 − 𝐴𝜇
+𝑓 𝑔 is the hadronic weak
+current that mediates the decay.
+In the following, we concentrate on the 𝑢𝑑-flavour channel and omit the 𝑓 𝑔 flavour indexes in
+intermediate expressions. Moreover, we study separately the longitudinal (𝐿) and transverse (𝑇)
+contributions to 𝑅𝑢𝑑 and also the contributions coming from the vector (𝑉 𝜇) and axial-vector (𝐴𝜇)
+currents. To this end we introduce the indexes
+𝐼 = {𝐿,𝑇} ,
+𝐽 = {𝑉, 𝐴} ,
+(3)
+and the different components of the spectral density H 𝜇𝜈(𝑝𝑋) according to
+H 𝐿
+𝐽 (𝜔) ≡ H00
+𝐽 (𝜔) ,
+H𝑇
+𝐽 (𝜔) ≡ 1
+3
+3
+∑︁
+𝑖=1
+H𝑖𝑖
+𝐽 (𝜔) ,
+H 𝐼 (𝜔) ≡ H 𝐼
+𝑉 (𝜔) + H 𝐼
+𝐴(𝜔) ,
+(4)
+with
+H 𝜇𝜈
+𝐽 (𝑝𝑋) ≡ (2𝜋)4 ⟨0| 𝐽𝜇(0) 𝛿(4) (P − 𝑝𝑋) 𝐽𝜈†(0) |0⟩ .
+(5)
+By working in the reference frame where the final hadronic state is at rest,
+𝑝𝑋 = (𝑚𝜏𝜔, �0) ,
+(6)
+we have
+𝑅𝐼
+𝐽 (𝜎) = 12𝜋 𝑆𝐸𝑊 |𝑉𝑢𝑑|2
+∫ ∞
+𝑟𝑢𝑑
+d𝜔 H 𝐼
+𝐽 (𝜔) 𝐾𝐼
+𝜎(𝜔) ,
+𝑅𝑢𝑑 = lim
+𝜎→0 𝑅𝑢𝑑(𝜎) = lim
+𝜎→0
+�
+𝑅𝐿
+𝑉 (𝜎) + 𝑅𝐿
+𝐴(𝜎) + 𝑅𝑇
+𝑉 (𝜎) + 𝑅𝑇
+𝐴(𝜎)
+�
+,
+(7)
+where, in analogy to Refs. [6, 7], we have introduced the longitudinal (𝐾 𝐿
+𝜎) and transverse (𝐾𝑇
+𝜎)
+smearing kernels
+𝐾 𝐿
+𝜎(𝜔) = (1 − 𝜔2)2
+𝜔
+Θ𝜎(1 − 𝜔) ,
+𝐾𝑇
+𝜎(𝜔) = (1 − 𝜔2)2(1 + 2𝜔2)
+𝜔
+Θ𝜎(1 − 𝜔) .
+(8)
+The function Θ𝜎(𝑥) appearing in the previous formula can be any 𝐶∞ smoothed version of the
+step-function 𝜃(𝑥) such that lim𝜎↦→0 Θ𝜎(𝑥) = 𝜃(𝑥). In the following, we will consider the three
+different choices given by
+Θ(1)
+𝜎 (𝑥) =
+1
+1 + 𝑒−𝑥/𝜎 ,
+Θ(2)
+𝜎 (𝑥) =
+1
+1 + 𝑒− sinh(𝑥/𝜎) ,
+Θ(3)
+𝜎 (𝑥) = 1 + Erf (𝑥/𝜎)
+2
+.
+(9)
+3
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+Under the assumption that the spectral densities H 𝐼 (𝜔) are regular at the end-point of the phase-
+space, i.e. 𝜔 = 1, an analytical calculation shows that the corrections to the 𝜎 ↦→ 0 limit are even
+functions of 𝜎, starting at O�𝜎4�, i.e.
+∫ ∞
+𝑟𝑢𝑑
+d𝜔 H 𝐼 (𝜔)
+�
+𝐾𝐼
+𝜎(𝜔) − 𝐾𝐼
+0 (𝜔)
+�
+= O(𝜎4) ,
+(10)
+This assumption is of course not valid on a finite volume where the spectral densities are not regular.
+Indeed, because of the quantization of the spectrum, the finite–volume spectral densities H 𝐼 (𝜔)
+are sums of Dirac 𝛿-functions localized in correspondence of the eigenvalues of the finite–volume
+Hamiltonian. However, precisely for this reason and as emphasized in Ref. [5], the 𝜎 → 0 limit in
+Eq. (7) has to be taken after performing the necessary 𝐿 → ∞ extrapolation of the lattice data. A
+detailed numerical investigation of the dependence upon the volume of our results is postponed to a
+future publication. Here, see below, we simply check that the results obtained on the two ensembles
+with volumes 𝐿 ≃ 5.1 fm and 𝐿 ≃ 7.6 fm are compatible within the statistical uncertainties and
+then attempt a 𝜎 → 0 extrapolation by relying on Eq. (10).
+The representation of 𝑅𝑢𝑑 given in Eq. (7) allows for a straightforward application of the method
+developed in Ref. [5] along the lines of Ref. [7]. The starting point is the relation between the
+hadronic spectral density H 𝜇𝜈
+𝐽 (𝑝𝑋) and the Euclidean two-point correlator 𝐶𝜇𝜈
+𝐽
+at vanishing three-
+momentum (our lattice input), i.e.
+𝐶𝜇𝜈
+𝐽 (𝑡) ≡
+∫
+d3𝑥 T ⟨0| 𝐽𝜇(𝑎𝑡, �𝑥)𝐽𝜈†(0) |0⟩ = 𝑚𝜏
+2𝜋
+∫ ∞
+𝑟𝑢𝑑
+𝑑𝜔 H 𝜇𝜈
+𝐽 (𝑝𝑋) 𝑒−𝑎𝑚𝜏 𝜔𝑡,
+𝑝𝑋 = (𝑚𝜏𝜔, �0),
+(11)
+where 𝑡 is the Euclidean time in units of the lattice spacing 𝑎.1 The main idea is then to express the
+smeared-kernels 𝐾 𝐿
+𝜎(𝜔) and 𝐾𝑇
+𝜎(𝜔) in terms of the basis function {𝑒−𝑎𝑚𝜏 𝜔𝑡}𝑡=1,...,∞, i.e.
+𝐾𝐼
+𝜎(𝜔) =
+∞
+∑︁
+𝑡=1
+𝑔𝐼 (𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 .
+(12)
+In this way, once the coefficients 𝑔𝐼 (𝑡, 𝜎) are known, the longitudinal (𝑅𝐿
+𝐽 ) and transverse (𝑅𝑇
+𝐽 )
+contributions to 𝑅𝑢𝑑 can be computed from the knowledge of
+𝐶𝐿
+𝐽 (𝑡) = − 2𝜋
+𝑚𝜏
+𝐶00
+𝐽 (𝑡) ,
+𝐶𝑇
+𝐽 (𝑡) = 2𝜋
+3𝑚𝜏
+3
+∑︁
+𝑖=1
+𝐶𝑖𝑖
+𝐽 (𝑡) ,
+(13)
+by using
+∞
+∑︁
+𝑡=1
+𝑔𝐼 (𝑡, 𝜎)𝐶𝐼
+𝐽 (𝑡) =
+∫ ∞
+𝑟𝑢𝑑
+d𝜔 H 𝐼
+𝐽 (𝜔) 𝐾𝐼
+𝜎(𝜔) ,
+(14)
+and inserting the result in Eqs. (7). However, as discussed thoroughly in Refs. [5], the problem
+of finding the coefficients 𝑔𝐼 (𝑡, 𝜎) presents a certain number of technical difficulties. First of all,
+1On a lattice having a finite temporal extent 𝑇, Eq. (11) must be modified replacing in the r.h.s. 𝑒−𝑎𝑚𝜏 𝜔𝑡 with
+𝑒−𝑎𝑚𝜏 𝜔𝑡 + 𝑒−𝑎𝑚𝜏 𝜔(𝑇 −𝑡).
+4
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+the sums appearing on the r.h.s. of Eqs. (12) need necessarily to be truncated at a finite value
+𝑡 = 𝑡𝑚𝑎𝑥, hence the goal is to find a finite set of coefficients 𝑔𝐼 (𝑡, 𝜎), with 𝑡 ∈ {1, . . . , 𝑡𝑚𝑎𝑥},
+such that both the statistical (due to the fluctuation of 𝐶𝐼
+𝐽 (𝑡)) and the systematic errors (due to the
+inexact reconstruction of the kernels) in the resulting determination of 𝑅𝐼
+𝐽 are under control. If
+we were only concerned with systematic errors, the best coefficients 𝑔𝐼 (𝑡, 𝜎) could be obtained by
+minimizing the quadratic form
+𝐴𝐼
+𝛼[𝒈] =
+∫ ∞
+𝐸0
+d𝜔 𝑒𝑎𝑚𝜏 𝜔𝛼�� 𝑓 (𝜔; 𝒈) − 𝐾𝐼
+𝜎(𝜔)
+��2 ,
+(15)
+with
+𝑓 (𝜔; 𝒈) ≡
+𝑡max
+∑︁
+𝑡=1
+𝑔(𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 .
+(16)
+Indeed, for any 𝛼 < 2 and 0 < 𝐸0 < 𝑟𝑢𝑑, the functional in Eq. (15) corresponds to a weighted
+𝐿2-norm in the functional space defined in the interval [𝐸0, ∞]. However, for small values of
+𝜎, the coefficients 𝑔𝐼 (𝑡, 𝜎) resulting from the minimization of 𝐴𝐼
+𝛼[𝒈] turn out to be very large
+in magnitude and oscillating in sign, strongly amplifying the statistical errors of 𝐶𝐼
+𝐽 (𝑡) when the
+𝑡max-truncated version of the sum in Eq. (14) is evaluated (see Ref. [5] for more details on this
+point).
+The method of Ref. [5], provides a regularization mechanism to this problem, enabling to find an
+optimal balance between statistical and systematic errors. This is achieved by minimizing a linear
+combination
+𝑊 𝐼 𝐽
+𝛼 [𝒈] ≡ 𝐴𝐼
+𝛼[𝒈]
+𝐴𝐼𝛼[0] + 𝜆𝐵𝐼 𝐽 [𝒈] ,
+(17)
+of the norm-functional 𝐴𝐼
+𝛼[𝒈] and of the error-functional
+𝐵𝐼 𝐽 [𝒈] =
+1
+(𝐶𝐼
+𝐽 (0))2
+𝑡𝑚𝑎𝑥
+∑︁
+𝑡1,𝑡2=1
+𝑔(𝑡1, 𝜎) 𝑔(𝑡2, 𝜎) CovI
+J(𝑡1, 𝑡2) ,
+(18)
+where CovI
+J(𝑡1, 𝑡2) is the covariance matrix of the lattice correlator 𝐶𝐽
+𝐼 (𝑡), and 𝜆 is the so-called
+trade-off parameter [5]. For any fixed value of the algorithmic parameters 𝒑 ≡ {𝛼, 𝐸0, 𝜆, 𝑡𝑚𝑎𝑥}, the
+minimization
+𝜕𝑊 𝐼 𝐽
+𝛼 [𝒈]
+𝜕𝑔(𝑡, 𝜎)
+����
+𝒈=𝒈𝐼 𝐽
+𝒑
+= 0 ,
+(19)
+defines the coefficients 𝒈𝐼 𝐽
+𝒑 . The systematic error associated to the inexact reconstruction of the
+smeared kernel,
+𝐾𝐼 𝐽
+𝒑 (𝜔) ≡ 𝑓 (𝜔; 𝒈𝐼 𝐽
+𝒑 ) =
+𝑡max
+∑︁
+𝑡=1
+𝑔𝐼 𝐽
+𝒑 (𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 ,
+(20)
+5
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+0.005
+0.006
+0.007
+0.008
+0.009
+0.010
+dT(gTV
+p )
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+RT
+V( ) |Vud|2
+ensemble=B64
+= 0
+= 1
+= 2
+0.005
+0.006
+0.007
+0.008
+0.009
+0.010
+dT(gTV
+p )
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+RT
+V( ) |Vud|2
+ensemble=B96
+= 0
+= 1
+= 2
+0
+2
+4
+6
+8
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+KT(
+)
+KT(
+)
+= 0
+= 1
+= 2
+0
+2
+4
+6
+8
+10
+0.04
+0.02
+0.00
+0.02
+0.04
+(KT(
+)
+KT, V
+*
+(
+))
+= 0
+= 1
+= 2
+reg=TM   
+= 0.0500
+Figure 2: Top: the contribution 𝑅𝑇
+𝑉 /|𝑉𝑢𝑑|2 obtained using 𝛼 = 0 (green), 𝛼 = 1 (yellow) and 𝛼 = 2−
+(blue), is plotted against 𝑑𝑇 (𝒈𝑇 𝑉
+𝒑
+) for 𝜎 = 0.05. For 𝛼 = 2−, the rightmost (leftmost) vertical dashed line
+indicates the point satisfying Eq. (23) with 𝑟 = 104 (103), while the horizontal blue band corresponds to our
+final determination obtained combining in quadrature the statistical and the systematic errors. The results
+are shown in the TM lattice regularization for both the B64 (top-left figure) and the B96 (top-right figure)
+ensembles at 𝜎 = 0.05. Bottom: the reconstructed smearing kernels 𝐾𝑇 𝑉
+∗
+(𝜔), obtained using the coefficients
+𝒈𝑇 𝑉
+∗
+of Eq. (23) are compared, for 𝛼 = 0, 1, 2−, with the target one 𝐾𝑇
+𝜎(𝜔) for 𝜎 = 0.05 (bottom-left figure).
+In the bottom-right figure we show 𝜔 · (𝐾𝑇
+𝜎(𝜔) − 𝐾𝑇 𝑉
+∗
+(𝜔)).
+can be quantified through the quantity
+𝑑𝐼 (𝒈) =
+�
+�
+𝐴𝐼
+0 [𝒈]
+𝐴𝐼
+0 [0] .
+(21)
+In the following, we will quote our best estimate for the four contributions 𝑅𝐿,𝑇
+𝑉 ,𝐴(𝜎), see Eq. (7),
+performing the so-called stability analysis (see Ref. [10] and also the Supplementary Material of
+Ref. [11]), which amounts to select the algorithmic parameters 𝒑 in such a way that the corresponding
+𝑑𝐼 (𝒈𝐼 𝐽
+𝒑 ) is sufficiently small and the results stable, within statistical errors, under variations of 𝒑
+(the so-called statistically dominated regime).2
+3.
+Numerical results
+In this section, we present our preliminary results for 𝑅𝑢𝑑. These have been obtained by using
+the Euclidean lattice correlators 𝐶𝐽
+𝐼 (𝑡) produced by the ETMC on the two ensembles B64 and
+B96. We have considered two different discretized versions of the local weak current, peculiar to
+our twisted-mass LQCD setup, that in the following will be indicated as twisted-mass (TM) and
+Osterwalder-Seiler (OS) [12]. The results obtained using the two discretizations only differ by O(𝑎2)
+cut-off effects, enabling us to approach the continuum limit in two different ways. Furthermore, we
+2More numerical details on this point will be given in a forthcoming publication.
+6
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0
+1
+2
+3
+4
+Rud( ) |Vud|2
+RL
+A( ) |Vud|2
+RT
+A( ) |Vud|2
+RL
+V( ) |Vud|2
+RT
+V( ) |Vud|2
+ens=B64   reg=TM
+(1)
+(2)
+(3)
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0
+1
+2
+3
+4
+Rud( ) |Vud|2
+RL
+A( ) |Vud|2
+RT
+A( ) |Vud|2
+RL
+V( ) |Vud|2
+RT
+V( ) |Vud|2
+ens=B96   reg=TM
+(1)
+(2)
+(3)
+Figure 3: The decay rate 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 as a function of 𝜎 in the range [0.0044, 0.1]. The results have
+been obtained in the TM regularization and are shown for both the volumes (B64 top, B96 bottom) and for
+the three choices of Θ(𝜔) in Eqs. (9). In the case Θ(1)
+𝜎 (𝜔) we also show, separately, the four contributions
+𝑅𝐿,𝑇
+𝑉 ,𝐴(𝜎)/|𝑉𝑢𝑑|2.
+considered three different values,
+𝛼 = {0, 1, 2−} ,
+(22)
+for the parameter 𝛼 appearing in Eq. (15), where 𝛼 = 2− in practice means 𝛼 = 1.99. We set the
+parameter 𝐸0 in Eq. (15) to 𝐸0 = 0.05 ≃ 0.6 𝑚 𝜋/𝑚𝜏 and use 𝑡𝑚𝑎𝑥 = 64, 96 respectively for the
+ensembles B64 and B96.
+In Figure 2 we show our determination of 𝑅𝑇
+𝑉 (𝜎)/|𝑉𝑢𝑑|2 in the TM regularization and at 𝜎 = 0.05,
+obtained employing the three values of 𝛼 and the smeared kernel Θ(1)
+𝜎 , see Eqs. (9). The results are
+shown as a function of the parameter 𝑑𝑇 (𝒈𝑇 𝑉
+𝒑
+) defined in Eq. (21) and provide an illustrative example
+of our stability analysis. For large values of 𝑑𝑇 (𝒈𝑇 𝑉
+𝒑
+) the results corresponding to different values
+of 𝛼 are substantially different because in this regime the reconstruction of the smearing kernel
+is very bad. At very small values of 𝑑𝑇 (𝒈𝑇 𝑉
+𝒑
+), where the quality of the reconstruction becomes
+excellent, the results corresponding to the different values of 𝛼 become compatible because the
+statistical errors are quite large. We observe that the results corresponding to 𝛼 = 1, 2− stabilize at
+much larger values of 𝑑𝑇 (𝒈𝑇 𝑉
+𝒑
+) than the 𝛼 = 0 ones. This behaviour, already observed in Ref. [11]
+where the same 𝐿2-norms have been used, can be explained by noticing that for 𝛼 > 0 the presence
+7
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+5
+10
+15
+20
+25
+Rud( ) |Vud|2
+2 dof = 0.065
+Rud(
+= 0) |Vud|2 = 3.675 ± 0.072
+ens=B64   reg=TM
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+3.50
+3.75
+4.00
+4.25
+4.50
+(1)
+(2)
+(3)
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+5
+10
+15
+20
+25
+Rud( ) |Vud|2
+2 dof = 0.058
+Rud(
+= 0) |Vud|2 = 3.562 ± 0.057
+ens=B96   reg=TM
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+3.50
+3.75
+4.00
+4.25
+4.50
+(1)
+(2)
+(3)
+Figure 4: Combined 𝜎 → 0 extrapolations of our results for 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 obtained in the TM regulariza-
+tion for both volumes. The datasets corresponding to the three choices of Θ(𝜔) appearing in Eqs. (9) have
+different colours. Assuming negligible finite-volume effects, these are expected to have the same 𝜎 → 0
+limit and to differ at finite 𝜎 with leading corrections of 𝑂(𝜎4). The data have been fitted using the ansatz of
+Eq. (25). The green point is the result of the extrapolation while the solid curves are the fitted curves 𝑅𝑘 (𝜎)
+for 𝑘 = 1 (red), 𝑘 = 2 (blue) and 𝑘 = 3 (yellow).
+of the exponential 𝑒𝑎𝑚𝜏 𝜔𝛼 in Eq. (15) improves the quality of the reconstruction in the large-𝜔
+region. Indeed, the errors in the reconstruction of the smearing kernels (e.g. 𝐾𝑇
+𝜎(𝜔)) for large
+values of 𝜔 get amplified in the corresponding smeared quantities (e.g. 𝑅𝑇
+𝐽 (𝜎)) because, in general,
+spectral densities grow asymptotically with the energy (e.g. H𝑇
+𝐽 (𝜔) ∝ 𝜔2).
+For 𝛼 = 1, 2−, we found that the results obtained at the point 𝒈𝐼 𝐽
+∗
+such that the condition
+𝐴𝐼
+𝛼[𝒈𝐼 𝐽
+∗ ]
+𝐴𝐼𝛼[0]
+= 𝑟𝐵𝐼 𝐽 [𝒈𝐼 𝐽
+∗ ] ,
+𝑟 = 104 ,
+(23)
+holds true, are in the statistically dominated regime. In what follows, the central values of the
+four contributions to 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 in Eq. (7) are estimated by using the 𝛼 = 2− results (that are
+remarkably stable) and the coefficients 𝒈𝐼 𝐽
+∗ . Residual systematic errors are instead evaluated by
+re-performing the analysis using 𝑟 = 103 (see the vertical lines in Figure 2). Any variation of the
+result corresponding to the choice 𝑟 = 103 w.r.t. the result corresponding to 𝑟 = 104 that goes
+beyond a mere statistical fluctuation is added in quadrature to the statistical error.
+In Figure 3 we show our preliminary results for 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 obtained in the TM regularization
+8
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+𝐿
+TM (𝑎 = 0.08 fm)
+OS (𝑎 = 0.08 fm)
+HFLAV+HT (𝑎 = 0)
+5.1 fm
+3.675(72)
+3.550(60)
+7.6 fm
+3.562(57)
+3.676(236)
+∞
+3.6615(78)
+Table 1: Preliminary results for 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 obtained in this work at fixed lattice spacing 𝑎 ≃ 0.08 fm in
+both the TM and OS lattice regularizations on the volumes 𝐿 ≃ 5.1 fm (ensemble B64) and 𝐿 ≃ 7.6 fm
+(ensemble B96). For comparison, we also show the result obtained by taking 𝑅𝑢𝑑 form Ref. [13] (HFLAV)
+and 𝑉𝑢𝑑 from Ref. [14] (HT).
+by using the three different smearing kernels of Eq. (9) and 23 values of 𝜎 in the range
+𝜎 ∈ [0.0044, 0.2] .
+(24)
+We observe a remarkably flat behaviour for 𝜎 < 0.05 3. Moreover, the results corresponding to the
+two volumes 𝐿 = 5.1 fm and 𝐿 = 7.6 fm are compatible at all values of 𝜎 within less than 1.5 standard
+deviations. This implies that finite-volume effects are negligible within the quoted errors, even at
+the smallest value of 𝜎 that we have considered. In the light of these observations, we attempted a
+combined 𝜎 → 0 extrapolation of our results by relying on the infinite-volume asymptotic formula
+of Eq. (10). On each ensemble and for each regularization, the results corresponding to the three
+smearing kernels Θ(𝑘)
+𝜎 (𝑘 = 1, 2, 3) have been fitted by using the following ansatz
+𝑅𝑘(𝜎) = 𝑅 + 𝑐1,𝑘 · 𝜎4 + 𝑐2,𝑘 · 𝜎6 ,
+(25)
+where 𝑐1,𝑘 and 𝑐2,𝑘 are free fit parameters which depend on the smearing kernel while 𝑅 ≡
+𝑅𝑢𝑑/|𝑉𝑢𝑑|2 is the common 𝜎 = 0 extrapolation. The quality of the fits is excellent on both volumes
+and for both regularizations. In the case of the TM regularization, the results of these extrapolations
+are shown in Figure. 4, again for the two volumes.
+In Table. 1, we report our final determination, for the two considered volumes and for both the
+TM and OS regularization. For comparison, we also reported in the table the result for 𝑅𝑢𝑑/|𝑉𝑢𝑑|2
+obtained by taking 𝑅𝑢𝑑 form Ref. [13] (HFLAV) and 𝑉𝑢𝑑 from Ref. [14] (HT). Although our OS
+results on the B96 ensemble are still affected by a quite large uncertainty, the spread between the TM
+and OS results on the B64 ensemble, where the accuracy is less than 2%, gives a first encouraging
+indication about the size of the cut-off effects. We are currently performing a more detailed analysis
+of all systematic effects and plan to extend the analysis to all the physical point ETMC ensembles
+in order to carry out a reliable continuum limit extrapolation.
+4.
+Conclusion and Outlooks
+We illustrated a method that allows to compute on the lattice the inclusive hadronic decay rates of
+the 𝜏 lepton without the need of perturbative and/or OPE inputs. We also presented preliminary
+3According to our experience, see e.g. Refs. [7, 10, 11], the numerical reconstruction of smearing kernels corre-
+sponding to 𝜃-functions in the 𝜎 → 0 limit is much easier than in the case of kernels corresponding to Dirac 𝛿-functions.
+9
+
+Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton
+A. Evangelista
+results for the inclusive decay rate in the 𝑢𝑑-flavour channel. These have been obtained by applying
+this method on two 𝑁 𝑓 = 2 + 1 + 1 QCD gauge ensembles produced by the ETMC with physical
+pion masses, at fixed cutoff 𝑎 = 0.07957(13) fm and with volumes 𝐿 = 5.09 fm and 𝐿 = 7.64 fm.
+In our method, as originally proposed in Refs. [6, 7], the step-functions that define the physical
+phase-space integration domain are smoothed and used as smearing kernels in the algorithm of
+Ref. [5]. Controlling the limit in which the smearing radius goes to zero is a crucial step of the
+method, to be performed after the necessary infinite-volume extrapolations. Our numerical results
+provide a rather solid evidence that this limit can be taken with controlled theoretical and numerical
+errors.
+We postpone to future work a more detailed illustration of the theoretical analysis of the vanishing
+smearing width limit and a thorough investigation of all systematic uncertainties, including the
+required continuum extrapolations.
+We also plan to extend our computation to the more phe-
+nomenologically relevant 𝑢𝑠-flavour channel.
+References
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+(2022) 869 [2111.09849].
+[2] R.J. Hudspith, R. Lewis, K. Maltman and J. Zanotti, A resolution of the inclusive flavor-breaking 𝜏
+|𝑉𝑢𝑠| puzzle, Phys. Lett. B 781 (2018) 206 [1702.01767].
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+1 (2019) 006.
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+Hadronic Vacuum Polarization Functions, Phys. Rev. Lett. 121 (2018) [1803.07228].
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+[hep-ph/0211345].
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+spectral densities in the two-dimensional O(3) non-linear 𝜎-model, 2111.12774.
+[11] C. Alexandrou et al., Probing the R-ratio on the lattice, 2212.08467.
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+implications for V𝑢𝑑 and CKM unitarity, Phys. Rev. C 102 (2020) 045501.
+10
+

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+page_content='Direct lattice calculation of inclusive hadronic decay rates  of the 𝝉 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista,𝑎,∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Frezzotti,𝑎 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Gagliardi,𝑏 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Lubicz,𝑐 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Sanfilippo,𝑏 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Simula𝑏  and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Tantalo𝑎  𝑎Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata", Via della Ricerca Scientifica 1,  I-00133 Rome, Italy  𝑏Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome,  Italy  𝑐Dipartimento di Matematica e Fisica, Università di Roma Tre and INFN, Sezione di Roma Tre, Via della  Vasca Navale 84, I-00146 Rome, Italy  E-mail: antonio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='evangelista@roma2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='it  The inclusive hadronic decay–rates of the 𝜏 lepton are particularly interesting from the phe-  nomenological point of view since they give access to the CKM matrix elements 𝑉𝑢𝑑 and 𝑉𝑢𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' In  this talk, we discuss how a recent method for the extraction of smeared spectral densities from Eu-  clidean lattice correlators can be used to obtain a direct lattice determination of inclusive hadronic  𝜏 decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' We also present preliminary numerical results obtained by applying this method  to correlators measured on two gauge ensembles produced by the ETMC with 𝑁 𝑓 = 2 + 1 + 1  dynamical flavours at physical pion masses, lattice spacing 𝑎 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='08 fm and volumes 𝐿 ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='1 fm  and 𝐿 ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  The 39th International Symposium on Lattice Field Theory (Lattice2022),  8-13 August, 2022  Bonn, Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='00796v1  [hep-lat]  2 Jan 2023  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  𝜏  𝜈𝜏  𝑓  ¯𝑔  𝑝𝜏  𝑝𝜈  𝑋 𝑓 𝑔  Figure 1: The 𝜏 → 𝜈𝜏𝑋 𝑓 𝑔 Feynman diagram (with no gluons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The hadronic final state 𝑋 𝑓 𝑔, with flavour  quantum numbers 𝑓 and 𝑔, has fixed 4-momentum 𝑝𝑋 = 𝑝𝜏 − 𝑝𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  Introduction  Inclusive hadronic 𝜏 decays are particularly interesting from the phenomenological viewpoint since  they give access to the CKM matrix elements 𝑉𝑢𝑑 and 𝑉𝑢𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The determinations of 𝑉𝑢𝑠 from leptonic  and semileptonic kaon decays [1] are in fairly good agreement with the one of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [2] but, for many  years, a puzzling tension with other determinations obtained from inclusive hadronic 𝜏 decays has  been observed and debated [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' On the lattice, hadronic 𝜏 decays have been studied by using  dispersion relations and by combining non-perturbative lattice inputs with perturbative and/or OPE  calculations (see for example [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  Here we present a method to perform a fully non-perturbative direct lattice calculation of the  𝜏 ↦→ 𝑋𝜈𝜏 decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' In our method, the decay rate is extracted from the two-point Euclidean  correlators of the hadronic weak currents that mediate the decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' This is done by using the algorithm  of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [5] that allows to extract smeared spectral densities from Euclidean lattice correlators and,  building on Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [6, 7], by using as smearing kernels smoothed versions of the step-functions that  define the physical phase-space integration domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  We also present preliminary numerical results obtained by applying this method to the relevant cor-  relators measured on two gauge ensembles produced by the Extended Twisted–Mass Collaboration  (ETMC) with 𝑁 𝑓 = 2 + 1 + 1 dynamical flavours with physical pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The two ensembles,  corresponding to the cB211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='64 (B64 in short) and cB211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='96 (B96 in short) entries in  TABLE V of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [8], have the same lattice spacing, 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='07957(13) fm, and differ only for the  physical volumes that are 𝐿 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='09 fm and 𝐿 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='64 fm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  Reconstruction of the inclusive rate using the HLT method  By relying on the Fermi effective theory for the weak interactions and by neglecting long–distance  QED radiative corrections, the ratio 𝑅 𝑓 𝑔 of the inclusive hadronic decay rate Γ[𝜏 ↦→ 𝑋 𝑓 𝑔𝜈𝜏] with  the leptonic decay rate Γ[𝜏 ↦→ 𝑒 ¯𝜈𝑒𝜈𝜏] can be expressed as  𝑅 𝑓 𝑔 = 12𝜋 𝑆𝐸𝑊  ��𝑉𝑓 𝑔  ��2 ∫ 1  𝑟 𝑓 𝑔  d𝜔 𝜔  �  1 − 𝜔2�2 �  𝜌𝐿  𝑓 𝑔(𝜔) + 𝜌𝑇  𝑓 𝑔(𝜔)  �  1 + 2𝜔2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  (1)  In the previous formula, 𝑓 and 𝑔 label the flavour quantum numbers of the final hadronic states  𝑋 𝑓 𝑔 having four–momentum 𝑝𝑋, 𝑟 𝑓 𝑔 = 𝑚 𝑓 𝑔/𝑚𝜏 is the ratio of the mass of the lightest hadronic  state and the 𝜏-mass, 𝑆𝐸𝑊 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0201(3) is the short–distance electroweak correction [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The  2  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  longitudinal and transverse form factors 𝜌𝐿  𝑓 𝑔(𝜔) and 𝜌𝑇  𝑓 𝑔(𝜔) parametrize the hadronic spectral  density  H 𝜇𝜈  𝑓 𝑔(𝑝𝑋) = (2𝜋)4 ⟨0| 𝐻𝜇  𝑓 𝑔(0) 𝛿(4) (P − 𝑝𝑋) 𝐻𝜈†  𝑓 𝑔(0) |0⟩  = 𝑝𝜇  𝑋 𝑝𝜈  𝑋 𝜌𝐿  𝑓 𝑔(𝜔) +  �  𝑝𝜇  𝑋 𝑝𝜈  𝑋 − 𝑔𝜇𝜈𝑝2  𝑋  �  𝜌𝑇  𝑓 𝑔(𝜔) ,  𝜔2 = 𝑝2  𝑋  𝑚2𝜏  ,  (2)  where P = (H, �P) is the QCD four–momentum operator and 𝐻𝜇  𝑓 𝑔 = 𝑉 𝜇  𝑓 𝑔 − 𝐴𝜇  𝑓 𝑔 is the hadronic weak  current that mediates the decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  In the following, we concentrate on the 𝑢𝑑-flavour channel and omit the 𝑓 𝑔 flavour indexes in  intermediate expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Moreover, we study separately the longitudinal (𝐿) and transverse (𝑇)  contributions to 𝑅𝑢𝑑 and also the contributions coming from the vector (𝑉 𝜇) and axial-vector (𝐴𝜇)  currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' To this end we introduce the indexes  𝐼 = {𝐿,𝑇} ,  𝐽 = {𝑉, 𝐴} ,  (3)  and the different components of the spectral density H 𝜇𝜈(𝑝𝑋) according to  H 𝐿  𝐽 (𝜔) ≡ H00  𝐽 (𝜔) ,  H𝑇  𝐽 (𝜔) ≡ 1  3  3  ∑︁  𝑖=1  H𝑖𝑖  𝐽 (𝜔) ,  H 𝐼 (𝜔) ≡ H 𝐼  𝑉 (𝜔) + H 𝐼  𝐴(𝜔) ,  (4)  with  H 𝜇𝜈  𝐽 (𝑝𝑋) ≡ (2𝜋)4 ⟨0| 𝐽𝜇(0) 𝛿(4) (P − 𝑝𝑋) 𝐽𝜈†(0) |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  (5)  By working in the reference frame where the final hadronic state is at rest,  𝑝𝑋 = (𝑚𝜏𝜔, �0) ,  (6)  we have  𝑅𝐼  𝐽 (𝜎) = 12𝜋 𝑆𝐸𝑊 |𝑉𝑢𝑑|2  ∫ ∞  𝑟𝑢𝑑  d𝜔 H 𝐼  𝐽 (𝜔) 𝐾𝐼  𝜎(𝜔) ,  𝑅𝑢𝑑 = lim  𝜎→0 𝑅𝑢𝑑(𝜎) = lim  𝜎→0  �  𝑅𝐿  𝑉 (𝜎) + 𝑅𝐿  𝐴(𝜎) + 𝑅𝑇  𝑉 (𝜎) + 𝑅𝑇  𝐴(𝜎)  �  ,  (7)  where, in analogy to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [6, 7], we have introduced the longitudinal (𝐾 𝐿  𝜎) and transverse (𝐾𝑇  𝜎)  smearing kernels  𝐾 𝐿  𝜎(𝜔) = (1 − 𝜔2)2  𝜔  Θ𝜎(1 − 𝜔) ,  𝐾𝑇  𝜎(𝜔) = (1 − 𝜔2)2(1 + 2𝜔2)  𝜔  Θ𝜎(1 − 𝜔) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  (8)  The function Θ𝜎(𝑥) appearing in the previous formula can be any 𝐶∞ smoothed version of the  step-function 𝜃(𝑥) such that lim𝜎↦→0 Θ𝜎(𝑥) = 𝜃(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' In the following, we will consider the three  different choices given by  Θ(1)  𝜎 (𝑥) =  1  1 + 𝑒−𝑥/𝜎 ,  Θ(2)  𝜎 (𝑥) =  1  1 + 𝑒− sinh(𝑥/𝜎) ,  Θ(3)  𝜎 (𝑥) = 1 + Erf (𝑥/𝜎)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  (9)  3  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  Under the assumption that the spectral densities H 𝐼 (𝜔) are regular at the end-point of the phase-  space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 𝜔 = 1, an analytical calculation shows that the corrections to the 𝜎 ↦→ 0 limit are even  functions of 𝜎, starting at O�𝜎4�, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  ∫ ∞  𝑟𝑢𝑑  d𝜔 H 𝐼 (𝜔)  �  𝐾𝐼  𝜎(𝜔) − 𝐾𝐼  0 (𝜔)  �  = O(𝜎4) ,  (10)  This assumption is of course not valid on a finite volume where the spectral densities are not regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  Indeed, because of the quantization of the spectrum, the finite–volume spectral densities H 𝐼 (𝜔)  are sums of Dirac 𝛿-functions localized in correspondence of the eigenvalues of the finite–volume  Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' However, precisely for this reason and as emphasized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [5], the 𝜎 → 0 limit in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (7) has to be taken after performing the necessary 𝐿 → ∞ extrapolation of the lattice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' A  detailed numerical investigation of the dependence upon the volume of our results is postponed to a  future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Here, see below, we simply check that the results obtained on the two ensembles  with volumes 𝐿 ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='1 fm and 𝐿 ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6 fm are compatible within the statistical uncertainties and  then attempt a 𝜎 → 0 extrapolation by relying on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  The representation of 𝑅𝑢𝑑 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (7) allows for a straightforward application of the method  developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [5] along the lines of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The starting point is the relation between the  hadronic spectral density H 𝜇𝜈  𝐽 (𝑝𝑋) and the Euclidean two-point correlator 𝐶𝜇𝜈  𝐽  at vanishing three-  momentum (our lattice input), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  𝐶𝜇𝜈  𝐽 (𝑡) ≡  ∫  d3𝑥 T ⟨0| 𝐽𝜇(𝑎𝑡, �𝑥)𝐽𝜈†(0) |0⟩ = 𝑚𝜏  2𝜋  ∫ ∞  𝑟𝑢𝑑  𝑑𝜔 H 𝜇𝜈  𝐽 (𝑝𝑋) 𝑒−𝑎𝑚𝜏 𝜔𝑡,  𝑝𝑋 = (𝑚𝜏𝜔, �0),  (11)  where 𝑡 is the Euclidean time in units of the lattice spacing 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='1 The main idea is then to express the  smeared-kernels 𝐾 𝐿  𝜎(𝜔) and 𝐾𝑇  𝜎(𝜔) in terms of the basis function {𝑒−𝑎𝑚𝜏 𝜔𝑡}𝑡=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=',∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  𝐾𝐼  𝜎(𝜔) =  ∞  ∑︁  𝑡=1  𝑔𝐼 (𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  (12)  In this way, once the coefficients 𝑔𝐼 (𝑡, 𝜎) are known, the longitudinal (𝑅𝐿  𝐽 ) and transverse (𝑅𝑇  𝐽 )  contributions to 𝑅𝑢𝑑 can be computed from the knowledge of  𝐶𝐿  𝐽 (𝑡) = − 2𝜋  𝑚𝜏  𝐶00  𝐽 (𝑡) ,  𝐶𝑇  𝐽 (𝑡) = 2𝜋  3𝑚𝜏  3  ∑︁  𝑖=1  𝐶𝑖𝑖  𝐽 (𝑡) ,  (13)  by using  ∞  ∑︁  𝑡=1  𝑔𝐼 (𝑡, 𝜎)𝐶𝐼  𝐽 (𝑡) =  ∫ ∞  𝑟𝑢𝑑  d𝜔 H 𝐼  𝐽 (𝜔) 𝐾𝐼  𝜎(𝜔) ,  (14)  and inserting the result in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' However, as discussed thoroughly in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [5], the problem  of finding the coefficients 𝑔𝐼 (𝑡, 𝜎) presents a certain number of technical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' First of all,  1On a lattice having a finite temporal extent 𝑇, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (11) must be modified replacing in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 𝑒−𝑎𝑚𝜏 𝜔𝑡 with  𝑒−𝑎𝑚𝜏 𝜔𝑡 + 𝑒−𝑎𝑚𝜏 𝜔(𝑇 −𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  4  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  the sums appearing on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (12) need necessarily to be truncated at a finite value  𝑡 = 𝑡𝑚𝑎𝑥, hence the goal is to find a finite set of coefficients 𝑔𝐼 (𝑡, 𝜎), with 𝑡 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' , 𝑡𝑚𝑎𝑥},  such that both the statistical (due to the fluctuation of 𝐶𝐼  𝐽 (𝑡)) and the systematic errors (due to the  inexact reconstruction of the kernels) in the resulting determination of 𝑅𝐼  𝐽 are under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' If  we were only concerned with systematic errors, the best coefficients 𝑔𝐼 (𝑡, 𝜎) could be obtained by  minimizing the quadratic form  𝐴𝐼  𝛼[𝒈] =  ∫ ∞  𝐸0  d𝜔 𝑒𝑎𝑚𝜏 𝜔𝛼�� 𝑓 (𝜔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 𝒈) − 𝐾𝐼  𝜎(𝜔)  ��2 ,  (15)  with  𝑓 (𝜔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 𝒈) ≡  𝑡max  ∑︁  𝑡=1  𝑔(𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  (16)  Indeed, for any 𝛼 < 2 and 0 < 𝐸0 < 𝑟𝑢𝑑, the functional in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (15) corresponds to a weighted  𝐿2-norm in the functional space defined in the interval [𝐸0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' However, for small values of  𝜎, the coefficients 𝑔𝐼 (𝑡, 𝜎) resulting from the minimization of 𝐴𝐼  𝛼[𝒈] turn out to be very large  in magnitude and oscillating in sign, strongly amplifying the statistical errors of 𝐶𝐼  𝐽 (𝑡) when the  𝑡max-truncated version of the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (14) is evaluated (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [5] for more details on this  point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  The method of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [5], provides a regularization mechanism to this problem, enabling to find an  optimal balance between statistical and systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' This is achieved by minimizing a linear  combination  𝑊 𝐼 𝐽  𝛼 [𝒈] ≡ 𝐴𝐼  𝛼[𝒈]  𝐴𝐼𝛼[0] + 𝜆𝐵𝐼 𝐽 [𝒈] ,  (17)  of the norm-functional 𝐴𝐼  𝛼[𝒈] and of the error-functional  𝐵𝐼 𝐽 [𝒈] =  1  (𝐶𝐼  𝐽 (0))2  𝑡𝑚𝑎𝑥  ∑︁  𝑡1,𝑡2=1  𝑔(𝑡1, 𝜎) 𝑔(𝑡2, 𝜎) CovI  J(𝑡1, 𝑡2) ,  (18)  where CovI  J(𝑡1, 𝑡2) is the covariance matrix of the lattice correlator 𝐶𝐽  𝐼 (𝑡), and 𝜆 is the so-called  trade-off parameter [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' For any fixed value of the algorithmic parameters 𝒑 ≡ {𝛼, 𝐸0, 𝜆, 𝑡𝑚𝑎𝑥}, the  minimization  𝜕𝑊 𝐼 𝐽  𝛼 [𝒈]  𝜕𝑔(𝑡, 𝜎)  ����  𝒈=𝒈𝐼 𝐽  𝒑  = 0 ,  (19)  defines the coefficients 𝒈𝐼 𝐽  𝒑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The systematic error associated to the inexact reconstruction of the  smeared kernel,  𝐾𝐼 𝐽  𝒑 (𝜔) ≡ 𝑓 (𝜔;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 𝒈𝐼 𝐽  𝒑 ) =  𝑡max  ∑︁  𝑡=1  𝑔𝐼 𝐽  𝒑 (𝑡, 𝜎)𝑒−𝑎𝑚𝜏 𝜔𝑡 ,  (20)  5  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='009  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='010  dT(gTV  p )  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='2  RT  V( ) |Vud|2  ensemble=B64  = 0  = 1  = 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='009  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='010  dT(gTV  p )  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='2  RT  V( ) |Vud|2  ensemble=B96  = 0  = 1  = 2  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0  KT(  )  KT(  )  = 0  = 1  = 2  0  2  4  6  8  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='04  (KT(  )  KT, V (  ))  = 0  = 1  = 2  reg=TM  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0500  Figure 2: Top: the contribution 𝑅𝑇  𝑉 /|𝑉𝑢𝑑|2 obtained using 𝛼 = 0 (green), 𝛼 = 1 (yellow) and 𝛼 = 2−  (blue), is plotted against 𝑑𝑇 (𝒈𝑇 𝑉  𝒑  ) for 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' For 𝛼 = 2−, the rightmost (leftmost) vertical dashed line  indicates the point satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (23) with 𝑟 = 104 (103), while the horizontal blue band corresponds to our  final determination obtained combining in quadrature the statistical and the systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The results  are shown in the TM lattice regularization for both the B64 (top-left figure) and the B96 (top-right figure)  ensembles at 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Bottom: the reconstructed smearing kernels 𝐾𝑇 𝑉  ∗  (𝜔), obtained using the coefficients  𝒈𝑇 𝑉  ∗  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (23) are compared, for 𝛼 = 0, 1, 2−, with the target one 𝐾𝑇  𝜎(𝜔) for 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='05 (bottom-left figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  In the bottom-right figure we show 𝜔 · (𝐾𝑇  𝜎(𝜔) − 𝐾𝑇 𝑉  ∗  (𝜔)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  can be quantified through the quantity  𝑑𝐼 (𝒈) =  �  �  𝐴𝐼  0 [𝒈]  𝐴𝐼  0 [0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  (21)  In the following, we will quote our best estimate for the four contributions 𝑅𝐿,𝑇  𝑉 ,𝐴(𝜎), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (7),  performing the so-called stability analysis (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [10] and also the Supplementary Material of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [11]), which amounts to select the algorithmic parameters 𝒑 in such a way that the corresponding  𝑑𝐼 (𝒈𝐼 𝐽  𝒑 ) is sufficiently small and the results stable, within statistical errors, under variations of 𝒑  (the so-called statistically dominated regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  Numerical results  In this section, we present our preliminary results for 𝑅𝑢𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' These have been obtained by using  the Euclidean lattice correlators 𝐶𝐽  𝐼 (𝑡) produced by the ETMC on the two ensembles B64 and  B96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' We have considered two different discretized versions of the local weak current, peculiar to  our twisted-mass LQCD setup, that in the following will be indicated as twisted-mass (TM) and  Osterwalder-Seiler (OS) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The results obtained using the two discretizations only differ by O(𝑎2)  cut-off effects, enabling us to approach the continuum limit in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Furthermore, we  2More numerical details on this point will be given in a forthcoming publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  6  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='10  0  1  2  3  4  Rud( ) |Vud|2  RL  A( ) |Vud|2  RT  A( ) |Vud|2  RL  V( ) |Vud|2  RT  V( ) |Vud|2  ens=B64   reg=TM  (1)  (2)  (3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='10  0  1  2  3  4  Rud( ) |Vud|2  RL  A( ) |Vud|2  RT  A( ) |Vud|2  RL  V( ) |Vud|2  RT  V( ) |Vud|2  ens=B96   reg=TM  (1)  (2)  (3)  Figure 3: The decay rate 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 as a function of 𝜎 in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0044, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The results have  been obtained in the TM regularization and are shown for both the volumes (B64 top, B96 bottom) and for  the three choices of Θ(𝜔) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' In the case Θ(1)  𝜎 (𝜔) we also show, separately, the four contributions  𝑅𝐿,𝑇  𝑉 ,𝐴(𝜎)/|𝑉𝑢𝑑|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  considered three different values,  𝛼 = {0, 1, 2−} ,  (22)  for the parameter 𝛼 appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (15), where 𝛼 = 2− in practice means 𝛼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' We set the  parameter 𝐸0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (15) to 𝐸0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='05 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6 𝑚 𝜋/𝑚𝜏 and use 𝑡𝑚𝑎𝑥 = 64, 96 respectively for the  ensembles B64 and B96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  In Figure 2 we show our determination of 𝑅𝑇  𝑉 (𝜎)/|𝑉𝑢𝑑|2 in the TM regularization and at 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='05,  obtained employing the three values of 𝛼 and the smeared kernel Θ(1)  𝜎 , see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The results are  shown as a function of the parameter 𝑑𝑇 (𝒈𝑇 𝑉  𝒑  ) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (21) and provide an illustrative example  of our stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' For large values of 𝑑𝑇 (𝒈𝑇 𝑉  𝒑  ) the results corresponding to different values  of 𝛼 are substantially different because in this regime the reconstruction of the smearing kernel  is very bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' At very small values of 𝑑𝑇 (𝒈𝑇 𝑉  𝒑  ), where the quality of the reconstruction becomes  excellent, the results corresponding to the different values of 𝛼 become compatible because the  statistical errors are quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' We observe that the results corresponding to 𝛼 = 1, 2− stabilize at  much larger values of 𝑑𝑇 (𝒈𝑇 𝑉  𝒑  ) than the 𝛼 = 0 ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' This behaviour, already observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [11]  where the same 𝐿2-norms have been used, can be explained by noticing that for 𝛼 > 0 the presence  7  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='200  5  10  15  20  25  Rud( ) |Vud|2  2 dof = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='065  Rud(  = 0) |Vud|2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='675 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='072  ens=B64   reg=TM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
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+page_content='10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='50  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='75  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='50  (1)  (2)  (3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
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+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
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+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='200  5  10  15  20  25  Rud( ) |Vud|2  2 dof = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='058  Rud(  = 0) |Vud|2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='562 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='057  ens=B96   reg=TM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='50  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='75  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='50  (1)  (2)  (3)  Figure 4: Combined 𝜎 → 0 extrapolations of our results for 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 obtained in the TM regulariza-  tion for both volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The datasets corresponding to the three choices of Θ(𝜔) appearing in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (9) have  different colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Assuming negligible finite-volume effects, these are expected to have the same 𝜎 → 0  limit and to differ at finite 𝜎 with leading corrections of 𝑂(𝜎4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The data have been fitted using the ansatz of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The green point is the result of the extrapolation while the solid curves are the fitted curves 𝑅𝑘 (𝜎)  for 𝑘 = 1 (red), 𝑘 = 2 (blue) and 𝑘 = 3 (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  of the exponential 𝑒𝑎𝑚𝜏 𝜔𝛼 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (15) improves the quality of the reconstruction in the large-𝜔  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Indeed, the errors in the reconstruction of the smearing kernels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 𝐾𝑇  𝜎(𝜔)) for large  values of 𝜔 get amplified in the corresponding smeared quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 𝑅𝑇  𝐽 (𝜎)) because, in general,  spectral densities grow asymptotically with the energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' H𝑇  𝐽 (𝜔) ∝ 𝜔2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  For 𝛼 = 1, 2−, we found that the results obtained at the point 𝒈𝐼 𝐽  ∗  such that the condition  𝐴𝐼  𝛼[𝒈𝐼 𝐽  ∗ ]  𝐴𝐼𝛼[0]  = 𝑟𝐵𝐼 𝐽 [𝒈𝐼 𝐽  ∗ ] ,  𝑟 = 104 ,  (23)  holds true, are in the statistically dominated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' In what follows, the central values of the  four contributions to 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (7) are estimated by using the 𝛼 = 2− results (that are  remarkably stable) and the coefficients 𝒈𝐼 𝐽  ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Residual systematic errors are instead evaluated by  re-performing the analysis using 𝑟 = 103 (see the vertical lines in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Any variation of the  result corresponding to the choice 𝑟 = 103 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' the result corresponding to 𝑟 = 104 that goes  beyond a mere statistical fluctuation is added in quadrature to the statistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  In Figure 3 we show our preliminary results for 𝑅𝑢𝑑(𝜎)/|𝑉𝑢𝑑|2 obtained in the TM regularization  8  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  𝐿  TM (𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='08 fm)  OS (𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='08 fm)  HFLAV+HT (𝑎 = 0)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='1 fm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='675(72)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='550(60)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6 fm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='562(57)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='676(236)  ∞  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6615(78)  Table 1: Preliminary results for 𝑅𝑢𝑑/|𝑉𝑢𝑑|2 obtained in this work at fixed lattice spacing 𝑎 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='08 fm in  both the TM and OS lattice regularizations on the volumes 𝐿 ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='1 fm (ensemble B64) and 𝐿 ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6 fm  (ensemble B96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' For comparison, we also show the result obtained by taking 𝑅𝑢𝑑 form Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [13] (HFLAV)  and 𝑉𝑢𝑑 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [14] (HT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  by using the three different smearing kernels of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (9) and 23 values of 𝜎 in the range  𝜎 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='0044, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  (24)  We observe a remarkably flat behaviour for 𝜎 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Moreover, the results corresponding to the  two volumes 𝐿 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='1 fm and 𝐿 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='6 fm are compatible at all values of 𝜎 within less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='5 standard  deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' This implies that finite-volume effects are negligible within the quoted errors, even at  the smallest value of 𝜎 that we have considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' In the light of these observations, we attempted a  combined 𝜎 → 0 extrapolation of our results by relying on the infinite-volume asymptotic formula  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' On each ensemble and for each regularization, the results corresponding to the three  smearing kernels Θ(𝑘)  𝜎 (𝑘 = 1, 2, 3) have been fitted by using the following ansatz  𝑅𝑘(𝜎) = 𝑅 + 𝑐1,𝑘 · 𝜎4 + 𝑐2,𝑘 · 𝜎6 ,  (25)  where 𝑐1,𝑘 and 𝑐2,𝑘 are free fit parameters which depend on the smearing kernel while 𝑅 ≡  𝑅𝑢𝑑/|𝑉𝑢𝑑|2 is the common 𝜎 = 0 extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' The quality of the fits is excellent on both volumes  and for both regularizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' In the case of the TM regularization, the results of these extrapolations  are shown in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 4, again for the two volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  In Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' 1, we report our final determination, for the two considered volumes and for both the  TM and OS regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' For comparison, we also reported in the table the result for 𝑅𝑢𝑑/|𝑉𝑢𝑑|2  obtained by taking 𝑅𝑢𝑑 form Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [13] (HFLAV) and 𝑉𝑢𝑑 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [14] (HT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Although our OS  results on the B96 ensemble are still affected by a quite large uncertainty, the spread between the TM  and OS results on the B64 ensemble, where the accuracy is less than 2%, gives a first encouraging  indication about the size of the cut-off effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' We are currently performing a more detailed analysis  of all systematic effects and plan to extend the analysis to all the physical point ETMC ensembles  in order to carry out a reliable continuum limit extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  Conclusion and Outlooks  We illustrated a method that allows to compute on the lattice the inclusive hadronic decay rates of  the 𝜏 lepton without the need of perturbative and/or OPE inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' We also presented preliminary  3According to our experience, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [7, 10, 11], the numerical reconstruction of smearing kernels corre-  sponding to 𝜃-functions in the 𝜎 → 0 limit is much easier than in the case of kernels corresponding to Dirac 𝛿-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  9  Direct lattice calculation of inclusive hadronic decay rates of the 𝜏 lepton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Evangelista  results for the inclusive decay rate in the 𝑢𝑑-flavour channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' These have been obtained by applying  this method on two 𝑁 𝑓 = 2 + 1 + 1 QCD gauge ensembles produced by the ETMC with physical  pion masses, at fixed cutoff 𝑎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='07957(13) fm and with volumes 𝐿 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='09 fm and 𝐿 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='64 fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  In our method, as originally proposed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [6, 7], the step-functions that define the physical  phase-space integration domain are smoothed and used as smearing kernels in the algorithm of  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Controlling the limit in which the smearing radius goes to zero is a crucial step of the  method, to be performed after the necessary infinite-volume extrapolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content=' Our numerical results  provide a rather solid evidence that this limit can be taken with controlled theoretical and numerical  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
+page_content='  We postpone to future work a more detailed illustration of the theoretical analysis of the vanishing  smearing width limit and a thorough investigation of all systematic uncertainties, including the  required continuum extrapolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9AyT4oBgHgl3EQf4_pw/content/2301.00796v1.pdf'}
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ENA0T4oBgHgl3EQfA_81/content/tmp_files/2301.01969v1.pdf.txt

@@ -0,0 +1,1530 @@
+Prepared for submission to JINST
+Imaging of muon track in CsI(Tl) crystal with
+single photon sensitive camera
+Zhimin Wang,a,b,c,1 Min Li,a,b Diru Wu,a,b Jinchang Liu,a,c Yongpeng Zhang,a,c
+Xiangcheng Meng,a Caimei Liu,a,b Changgen Yanga,b
+aInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
+bUniversity of Chinese Academy of Sciences, Beijing 100049, China
+cState Key Laboratory of Particle Detection and Electronics, Beijing 100049, China
+E-mail: [email protected]
+Abstract: As a novel approach on visual photon imaging by a single photon sensitive
+camera and PMTs, this work is trying to measure and identify muon tracks from the 2-D
+images of CsI(Tl) crystal (scintillator detectors). It is possible that muon tracks can be
+seen directly with a good signal-to-noise ratio neither with further amplification nor external
+light, which provides an evolution method for particle measurement in the photon-starved
+regime of scintillation detectors. The setup of the crystal and camera testing system and
+the identification algorithm of muon track will be discussed in detail including the system
+calibration, identification model, signal-to-noise ratio, muon track confirmation, and an
+expectation on further improvements and applications.
+Keywords: photon detectors, scintillator detector, imaging, single photon, camera, muon
+track
+ArXiv ePrint: 1234.56789
+1Corresponding author.
+arXiv:2301.01969v1  [physics.ins-det]  5 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+CsI(Tl) crystal with camera
+2
+2.1
+Setup
+2
+2.2
+Calibration
+3
+3
+Muon track
+5
+3.1
+Signal vs. Noise
+5
+3.2
+Image track survey
+8
+3.3
+Muon identification
+10
+4
+Discussion
+12
+4.1
+Muon tracks or not?
+12
+4.2
+Possible system optimization
+13
+4.3
+Further applications
+13
+5
+Summary
+14
+1
+Introduction
+Vertex and track reconstruction are critical for most particle physics experiments, such as
+studies on neutrino, dark matter, and others. There is a long list of related technologies
+including but not limited to emulsion film[1–3], cloud chamber[4], bubble chamber[5], spark
+chamber[6], multi-wire proportional chamber[7], TPC[8], Si strip[9] and Si pixel[10] etc. In
+the case of photon-based detection, in particular, PMT or SiPM is the commonly used sensor
+for timing, intensity, and crude spatial reconstruction, such as JUNO[11, 12], Darkside[13],
+JUNO-TAO[14], SNO+[15, 16], and DUNE[17] etc., where computer algorithms are further
+used to have a better reconstruction on the vertex or track[18–20].
+Recently, many efforts are focusing on photon imaging-related projects following the
+new development of sensors, where the critical challenges are the need for high spatial
+resolution over large volumes[21] and better effective signal-to-noise ratio under the photon-
+starved regime.
+For many years classical emulsion film radiography is being replaced by digital detec-
+tor imaging, especially in medical applications due to faster and more reliable diagnostics
+and computed tomography and tomosynthesis capabilities[22]. The single photon count-
+ing X-ray CCD camera spectrometer is used in laser-plasma interaction experiments as a
+simple tool to study the K-shell X-ray generation. A CCD detector enables the spectrum
+of the impinging X-ray radiation to be obtained without further dispersive devices[23].
+– 1 –
+
+Among the imaging systems used for thermal neutron imaging worldwide, the most preva-
+lent configuration is CCD camera based[24].
+Single-photon light detection and ranging
+(lidar) offers single-photon sensitivity and picosecond timing resolution, which is desirable
+for high-precision three-dimensional (3D) imaging over long distances[25]. Single image 3D
+photography enables viewers to view a still image from novel viewpoints[26]. Some good
+sensors are developed too, such as SPC3[27], a single photon counting camera based on a
+2-D imaging array. A small, high resolution, high signal-to-noise GEM-based TPC with a
+2-D CCD readout designed to provide a benchmark for background discrimination and di-
+rectional sensitivity that could be used for future optimization studies for directional dark
+matter experiments [28, 29]. A skipper CCD was also developed for very low noise and
+directly measured a muon track through ionization inside the sensor[30].
+But, generally, it is not suitable to directly image of vertex or track in case of a starved-
+photon regime and uniform angular distribution of the photons[21, 31]. Photography by
+CCD or other technologies, in particular single photon imaging, provides another new pos-
+sibility, such as our previous study for particle imaging by event[32].
+In this article, we will try to have a further detailed check on the imaging of muon track
+in CsI(Tl) crystal with a single photon sensitive camera and PMTs. Sec.2 will introduce
+the system setup and calibration. Sec.3 will discuss the expected features of muon tracks,
+measurements, and track surveys.
+Sec.4 will provide further discussions on the results,
+possible improvements of the system, and further expectations. And a short summary is in
+Sec.5.
+2
+CsI(Tl) crystal with camera
+2.1
+Setup
+An imaging system, as in [32], is set up with a single photon sensitive camera of ORCA-
+Quest qCMOS C15550-20UP, which is a new product of Hamamatsu Photonics [33]. The
+detailed layout of the system is shown in Fig. 1. The output of the camera will save in tif
+format with 16 bits of each of the 4096(H)×2304(V) pixels and the volume of each photo is
+around 16 MB. The CsI(Tl) crystal (7.5×7.5×15 cm3) is located in front of the camera and
+the two 3-inch PMTs, where the distance between them can be adjusted. An alpha source
+of 241Am is used and put on the top surface of the crystal. The two 3-inch PMTs are used
+to calibrate and monitor the signal intensity of the crystal, the coincidence of which is used
+as a trigger of the CAEN DT5751 (1 GS/s with 1 V p-p dynamic range, [34]) for waveform
+data taking. The threshold of each 3-inch PMT is set to around 1 p.e. (photon-electron).
+The maximum rate of the data-taking system is limited by the DT5751, which is generally
+lower than 100 Hz with data saving of four channels and 10000 samples of each channel.
+Here the window length of the waveform recording is set to 6 µs (6000 samples/waveform),
+and the maximum data-taking rate is around 70-80 Hz.
+In order to increase the acceptance of the emitted photons from the crystal, a lens with
+a much short focal length and small number aperture (1/2”, C type, 6-∞ mm, f/1.4) is used.
+The images of the crystal with different distances are shown in Fig. 2, which are taken with
+illumination before the dark box is closed. The field of view with the used lens is in a circle
+– 2 –
+
+Figure 1: Layout of the imaging measurement system.
+and much smaller than the full size of the camera sensor. The circle shape and its outside
+of the field of view will be considered in the following measurement and analysis. Please
+note that there is a clear distortion around the edge of the field of view (crystal region)
+when the object distance is too small as in Fig. 2b.
+(a) 15 cm
+(b) 4 cm
+Figure 2: Crystal photos when the dark box is open with natural illumination and different
+object distances.
+2.2
+Calibration
+Fig. 3a shows the measured charge spectra with the alpha source located when the object
+distance is around 15 cm: the two 3-inch PMTs and the sum of them. A long tail can be
+found at the right of the spectrum, which is known as cosmic muons. A 400 p.e. cut to the
+sum spectrum is used to select the events of muons. A factor of particle identification (PID)
+is calculated by each waveform of each PMT, and it is defined by the ratio of the charge in
+the first 300 ns to the whole window, as shown in Fig. 3b. A 2-D cut on the PID is used to
+identify the events from the alpha source, the red dash line (PID_pmt_0+PID_pmt_1)
+as shown in Fig. 3c.
+The selected spectra are shown in Fig. 3d, where the blue curve
+is selected as the alpha-like events by (PID_pmt_0+PID_pmt_1>0.5), the Magenta
+– 3 –
+
+couple
+Source
+Camera
+Lens
+Power
+USB cable
+Crystal
+Dark Boxcurve is selected for the muon events candidates by (PID_pmt_0+PID_pmt_1<0.5 and
+charge_pmt_0+charge_pmt_1>400 p.e.), and the green curve is assumed as the gamma-
+like events by (PID_pmt_0+PID_pmt_1<0.5).
+(a) Charge spectra of eac PMT and sum
+(b) PID of each event of each PMT
+(c) 2-D distribution of PID of the two PMTs
+(d) Selected events by PID and charge
+Figure 3: Measured charge spectra by the 3-inch PMTs, and the PID distrbution of the
+waveform.
+Taking into account the dead time of the data-taking system, the actual event rate of
+each measurement is re-normalized according to the selected muon rate, where the reference
+muon rate of the crystal is from the measurement and selection of without source with object
+distance of 15 cm.
+It is around 3 Hz for selected muons, 2.6 Hz for gamma-like events,
+and 2.2 Hz for alpha-like events of the measurement without source and object distance of
+15 cm. The data-taking rate and the re-normalized rate are listed in Table 1 for different
+configurations. The rate of alpha-like events is around 100 Hz and increases from 93 Hz to
+188 Hz following the shortening of the object distance from 15 cm to 4 cm. The contribution
+from the alpha source is much higher than that from the background of around 2.2 Hz.
+Following the classification of the events, the mean charge of each kind of event is
+calculated too. The mean charge of the alpha-like events is 130 p.e. of 15 cm, 150 p.e. of
+10 cm, and 159 p.e. of 4 cm, respectively. The signal intensity is not following the solid angle
+simply, for the 4 cm, in particular, which is because the distance to the 3-inch PMTs is
+rather small than the object distance of the camera to the crystal according to the layout
+of the system. The mean intensity of the muon events is around 2100 p.e., which suffers
+from statistic uncertainty and solid angle issues too.
+The images of the crystal with the alpha source are taken by the camera with di���erent
+exposure times and different object distances. The region of the source is selected and shown
+in Fig. 4, where the selected image dimension of the sensor is around 0.23 mm ×0.28 mm
+– 4 –
+
+Count
+Sum single_S
+sPMT single 0
+102
+sPMT single1
+10
+10
+102
+103
+Charge/peCount
+700
+h1D_pidpmto
+600
+h1D pid pmt1
+500
+400
+300
+200
+100
+0.2
+0.4
+0.6
+0.8
+PIDPMT_
+16
+14
+0.8
+12
+0.6
+10
+8
+0.4
+6
+4
+0.2
+0.2
+0.4
+0.6
+0.8
+C
+0
+PMT 0Count
+10
+Sum singleS
+Sum_single_S_high
+Sum_single_S_low
+Sumsingle_S_low_gamma
+102
+SumsingleSlowmuon
+10
+10
+102
+103
+Charge/peTable 1: Event rate and charge intensity of crystal with two 3-inch PMTs. The distance
+is between the camera and the crystal front surface.
+The events are measured by the
+coincidence by the two 3-inch PMTs, and the charge is from the sum of the two PMTs
+of each event. The alpha-like events are selected by the sum of PID, and the separation
+between muon and gamma-like is by a charge cut after the PID cut.
+Type
+DAQ Rate
+Normalized
+Rate (Hz)
+Mean Charge (p.e.)
+(Distance)
+(Hz)
+Rate (Hz)
+Muon
+Gamma-like
+Alpha-like
+Muon
+Gamma-like
+Alpha-like
+w/o source 15 cm
+∼8
+7.8
+3.0
+2.6
+2.2
+2099.0
+181.3
+124.5
+w/ source 15 cm
+∼80
+93.1
+3.0
+6.2
+83.9
+2108.9
+184.0
+130.2
+w/ source 10 cm
+∼70
+160.1
+3.0
+18.2
+138.9
+2057.4
+204.0
+150.4
+w/ source 4 cm
+∼70
+188.9
+3.0
+21.3
+164.6
+2062.8
+207.1
+159.5
+with 50 pixel (V) ×60 pixels (H) and 4.6 µm × 4.6 µm per pixel. The regime of the alpha
+source can be identified as around 3 mm scale of an object distance of 15 cm, 2 mm scale
+of 10 cm, and 1 mm scale of 4 cm, and the light intensity gradually dims when shortening
+the exposure time. It is almost identified to event level with 0.05 s exposure time but on a
+higher noise background, where only a few alphas occur during the time. The dimension of
+the image of the source is enlarging when the object distance reduces as expected.
+The intensity of the source region is integrated and converted into p.e. as shown in
+Fig. 5, where the noise (baseline is around 200 ADC) of the camera is subtracted according
+to a parallel region of the source with equal area [32].
+The conversion factor is around
+7.8 ADC/p.e. The diameter of the source region is 21 pixels for an object distance of 15 cm,
+22 pixels for 10 cm, and 33 pixels for 4 cm. The fitted intensity per second by a linear curve
+is around 314 p.e. of an object distance of 15 cm, 862 p.e. of 10 cm, and 2255 p.e. of 4 cm,
+respectively.
+Considering the rate of the alpha source measured under different object
+distances as in Tab. 1, the ratio of measured charge intensity of the camera and the PMTs
+is around 3% of an object distance of 15 cm, 4% of 10 cm, and 8% of 4 cm, respectively.
+The expected typical charge intensity of alpha-like event viewed by the camera is around
+3.9 p.e. of an object distance of 15 cm, 6.0 p.e. of 10 cm, and 12.8 p.e. of 4 cm, respectively.
+The expected typical charge intensity of each muon viewed by the camera is around 60 p.e. of
+an object distance of 15 cm, 85 p.e. of 10 cm, and 177 p.e. of 4 cm, respectively.
+3
+Muon track
+3.1
+Signal vs. Noise
+As stated in [32], the noise of the camera is still much higher than the traditional used
+PMT or SiPM, which is much worse when we are trying to use many pixels for imaging
+measurement. It can be expected that it will help to identify the target by a smaller area
+and stronger intensity of the same object, as seen in the left plateau of the curves in Fig. 5,
+where the difference of the plateau level (noise) is mainly from the dimension of the imaging
+area. The minimum of the plateau is from the object distance of 15 cm configuration, which
+is proportional to the ratio squared of the focal length to object distance, even the final
+– 5 –
+
+Figure 4: 2-D images of alpha source with object distances of 15 cm (50 pixel (V) ×60 pixels
+(H), left), 10 cm (50 pixel (V) ×60 pixels (H), middle) and 4 cm (50 pixel (V) ×60 pixels (H),
+right) versus exposure time of 60 s, 1 s, 100 ms, 50 ms and 20 ms. The pixel size is 4.6 µm
+× 4.6 µm. The z-axis is in the unit of ADC which is the camera output per pixel directly
+relative to the light intensity. The baseline of each pixel is around 200 ADC, and the gain
+factor is around 7.8 ADC/p.e.
+signal-to-noise (SN) ratio is also proportional to the square of the reciprocal of focal length
+except for the effective aperture.
+A calculation is further executed to compare the effect of the imaging shape under the
+same noise level. Here a pure statistic model is used to check the mean of each pixel and
+total sum with an assumption of 0.3 p.e. noise level per pixel (sigma of Gaussian) in a circle
+(diameter) or line (in length and in width of one pixel) shape. The uncertainty of the mean
+of each pixel is inversely proportional to the square root of total pixel numbers as shown in
+Fig. 6a, where more pixels will have smaller fluctuation comparing circles to lines. While
+the total sum of all pixels is proportional to the square root of pixel numbers shown in
+Fig. 6b, where the required sum in lines is much smaller than that of in circles to identify
+a signal. If we aim to identify a signal by a three-times signal-to-noise ratio with similar
+– 6 –
+
+15cm
+10cm
+183
+1695
+-
+1515
+1825
+4cm
+1400
+169
+1510
+60s
+1820
+60s
+1200
+1685
+1815
+168
+1810
+1000
+1675
+1805
+800
+1670
+0081
+1665
+1795
+600
+1660
+1790
+400
+1655
+1785
+1968
+1780
+1830
+4cm
+212
+1695
+15cm
+1515
+10cm
+1825
+211
+1690
+1s
+1510
+1820
+1s
+1685
+210
+1505
+1815
+1680
+1500
+1810
+209
+1675
+1495
+208
+1670
+1490E
+1800
+207
+1665
+1485E
+1795
+206
+1660
+1480
+1790
+1655
+1475
+1785
+205
+160
+1790
+15cm
+1520
+10cm
+-
+1695
+:
+4cm
+1515
+211
+1690
+100ms
+100ms
+100ms
+1685
+1505
+1815
+1810月
+209
+1680
+1500
+1675
+208
+1670
+1490
+1800E
+1665E
+1485
+1795
+:
+206
+1660
+1480
+1790
+1785
+205
+1655
+1475
+190
+204
+1520
+-
+212
+10cm
+-4cm
+1695
+15cm
+1515
+211
+1510
+50ms
+1820
+50ms
+50ms
+210
+1685
+:
+1505
+-
+1815
+1500
+1810
+209
+1680
+1675
+-
+1495
+:
+1805
+208
+1670
+1490
+1800
+207
+1665E
+1485
+1795
+206
+1660
+1480
+1790
+1655
+1475
+1785
+205
+1R50E
+1470元
+-
+AEA
+AA7A
+178
+70
+04
+1700
+1520
+1830
+212
+15cm
+-
+1695
+1515
+10cm
+：
+1825
+211
+1690
+20ms
+1510
+20ms
+1820
+20ms
+210
+1685
+1505
+1815
+1680
+1500
+1810
+209
+1675
+1805
+208
+1670E
+1490
+1800E
+207
+1665
+1485
+-
+1795
+206
+1660
+■
+1480
+1790
+1655E
+1475
+1785
+-
+:
+1479020
+165010
+2030
+2040
+2020
+2030
+2040
+2050
+2060
+2060
+2070
+179640
+2050
+2060
+2070
+2080
+Hpixel
+Hpie
+2090
+2100Figure 5: Measured intensity of alpha source by camera versus exposure time under dif-
+ferent object distances.
+total intensity, the signal in a line is much more effective than that in a circle.
+(a) Uncertainty of the mean of each pixel: in
+p.e.
+(b) Sum of noise vs. dimension: in p.e.
+Figure 6: Pure noise versus image dimension in circles or lines.
+Following this strategy, an alpha-like event in a circle is difficult to be identified directly
+according to their aimed imaging area and captured light intensity if we only can locate
+a range of an image by one camera as in [32]. While, in another hand, a muon track is
+a good candidate to check if we assume its image follows a straight line with tiny width.
+With the configurations of the crystal system (20000 photons/MeV, 2 MeV/cm, focal length
+6 mm, sensor quantum efficiency 30%, a 4 cm track, around 8 p.e./cm and 330 pixels/cm at
+an object distance of 4 cm), we can anticipate the averaged intensity of each pixel along the
+muon track imaging versus object distance and effective numerical aperture (ap), as shown
+in Fig. 7a. A smaller object distance and effective numerical aperture mean more light is
+collected with the same focal length and lens diameter. The expected signal-to-noise ratio
+with the assumption of 0.3 p.e. noise, and 4 cm track length can be further checked as shown
+in Fig. 7b, and it is around three times SN when the object distance is 4 cm.
+– 7 –
+
+Average inpeperpixel
+average_circle_pixel
+10
+average_line_pixel
+10-2
+10-3
+10
+0
+100
+200
+300
+400
+500
+DimensioninpixelTotalsuminpe
+102
+10
+ sum_pe circle
+sum pe line
+0
+100
+200
+300
+400
+500
+DimensioninpixelIntensity
+105
+1 2inch 6mm 4cm
+104
+ 1 2inch 6mm 10cm
+1 2inch 6mm 15cm
+103
+102
+10
+10-3
+10-2
+10-1
+10
+Exposuretime/s(a) Averaged intensity of muon track in pixel
+(b) S/N expectation following the model
+Figure 7:
+Intensity and signal-to-noise ratio of a muon track:
+20000 photons/MeV,
+2 MeV/cm, focal length 6 mm, sensor quantum efficiency 30%, a 4 cm track, around
+8 p.e./cm and 330 pixels/cm at object distance of 4 cm.
+The measured total intensity of a track will increase following the track length as shown
+in Fig. 8a, where a length of 10 mm track means around 330 pixels with 6 mm focal length
+and 4 cm object distance. The signal-to-noise ratio also will improve following the track
+length increasing as in Fig. 8b. The effect of the noise level in each pixel is further evaluated
+in Fig. 8c. It will reach around three times the signal-to-noise level with 0.3 p.e. noise level,
+4 cm track length, 6 mm focal length, 1.4 aperture, and 4 cm object distance. A smaller
+noise per pixel and smaller aperture will help improve the signal-to-noise ratio as well as
+larger optical lens dimensions.
+3.2
+Image track survey
+The images of the crystal system with the alpha source and 1 s exposure time are taken.
+Fig. 9 shows an example of the image, where the location of the alpha source can be identified
+clearly, and it will be used as an online anchor for light intensity and location.
+While
+according to the calculation in Sec. 3.1, the object distance of 4 cm configuration is run
+continuously for 30 s to check out possible muon racks with better signal-to-noise ratio.
+With the images (object distance of 4 cm as an example), a survey was done to each
+assumed line in aimed pixel range (from vertical line 1700 to vertical line 1600, the minimum
+track length is around 100 pixels) following the strategy, where the averaged intensity of each
+pixel is calculated among each assumed lines as shown in Fig. 10a and Fig. 10c. In Fig. 10a, a
+green dotted line is also plotted, which is a used cut of five times of noise uncertainty which
+is relative to the pixel number in a model (0.3*7.8 ADC/sqrt(pixel number)). Some tracks
+can be identified as tagged in red and shown in Fig. 10c. The sum of each assumed track
+is also plotted in Fig. 10b and Fig. 10d including the selected possible track candidates. An
+offset of the pixel average is related to the baseline of each pixel which is further corrected
+during the sum calculation. The five-times cut selects the possible tracks efficiently from
+noise, which is better for long track and higher than the calculated three-times to avoid
+more noise as seen.
+The identified candidate of tracks by the survey can be found in Fig. 11, while it seems
+too many than the expectation which is around three muon per second hitting the crystal.
+– 8 –
+
+Average inpeperpixel
+0.
+0.09
+— ap_1.0
+0.08
+0.07
+ap_1.4
+0.06
+0.05
+0.04
+0.03
+0.02
+0.01
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Objectdistanceinmm3
+12
+10
+sn_ap_1.0
+8
+sn_ap _1.4
+6
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+Objectdistanceinmm(a) Total intensity versus track length
+(b) S/N versus track length
+(c) S/N versus noise level per pixel
+Figure 8: Total intensity and signal-to-noise ratio versus track length, and signal-to-
+noise ratio versus noise level per pixel and effective numerical aperture with a 4 cm track:
+20000 photons/MeV, 2 MeV/cm, focal length 6 mm, sensor quantum efficiency 30%, around
+8 p.e./cm and 330 pixels/cm at an object distance of 4 cm.
+(a) Object distance of 15 cm
+(b) Object distance of 10 cm
+(c) Object distance of 4 cm
+Figure 9: Full image of the crystal with alpha source under 1 s exposure time.
+– 9 –
+
+Total light intensity in pe
+180
+160
+sum line ap1.0
+140
+sum line ap1.4
+120
+100
+80
+60
+40
+20
+0
+20
+40
+60
+80
+100
+Tracklengthinmm3
+10
+-SNap1.0
+SN ap1.4
+20
+40
+60
+80
+100
+Track length inmm3
+20
+18
+16
+- SN ap1.0
+14
+SN_ap1.4
+12
+10
+8
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Noiselevelperpixelinpepixel
+212
+2200
+>
+2000
+211
+1800
+210
+1600
+1400
+209
+1200
+208
+1000
+207
+800
+600
+206
+400
+205
+200
+3500
+204
+500
+1000
+1500
+2000
+2500
+3000
+4000
+H pixelpixel
+212
+2200
+>
+2000
+211
+1800
+210
+1600
+1400
+209
+1200
+208
+1000
+207
+800
+600
+206
+400
+205
+200
+204
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+H pixelpixel
+212
+2200
+>
+2000
+211
+1800
+210
+1600
+1400
+209
+1200
+208
+1000
+207
+800
+600
+206
+400
+205
+200
+204
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+H pixel(a) Averaged intensity of each pixel 2-D
+(b) Intensity sum 2-D
+(c) Averaged intensity of each pixel 1-D
+(d) Intensity sum 1-D
+Figure 10: Averaged intensity of each pixel and the sum of the assumed tracks.
+The direction of the candidate tracks and locations also exceed the range of the view field
+of the lens in Fig. 2. It still needs further checking on the quality of the identification even
+if a five-times cut is used.
+Figure 11: Selected tracks under 1 s exposure time.
+3.3
+Muon identification
+The identified track candidates need to be further checked as muon track candidates. Check-
+ing the uniform trend along the tracking candidate is a good way to avoid the noise effect.
+It is extended to another 2000 pixels along the track in maximum (Fig. 12a) to check the
+summed intensity (Fig. 12b) and its averaged intensity per pixel versus the track’s length
+– 10 –
+
+h2Dtracklengthintensity
+2.8
+1200
+2.6
+1000
+2.4
+2.2
+800
+2
+600
+1.8
+1.6
+400
+1.4
+200
+1.2
+500
+1500
+3000
+0
+1000
+2000
+2500
+tracklengthinpixelh2Dtrack_sumlength
+track intensityinADC
+800
+4000
+700
+3500
+600
+3000
+500
+2500
+400
+2000
+300
+1500
+200
+1000
+100
+500
+500
+1000
+1500
+2000
+2500
+3000
+tracklengthinpixelh1Dtrackpixel_average
+Count
+h1D_track_pixel_average
+105
+h1D_track_pixel_average_select
+104
+103
+102
+10
+0.5
+1.5
+2.5
+3
+3.5
+Pixel average ADCh1Dtracksum
+Count
+104
+h1Dtrack_sum
+h1Dtracksumselect
+103
+102
+10
+400
+200
+0
+200
+400
+600
+800
+1000
+tracksuminADCh2Dtrack
+/pixel
+230
+2200
+>
+2000
+225
+1800
+1600
+220
+1400
+1200
+215
+1000
+800
+210
+600
+400
+205
+200
+00
+200
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+H pixel(Fig. 12c). As shown in Fig. 12b, most of the candidates are excluded after the extension,
+and the sum of a few candidates keeps increasing versus the length of a true track candidate
+as expected in Fig. 8a. As shown in Fig. 12c, most of the candidates are excluded too, and
+the average of a few candidates keeps increasing or stable versus the length which is over
+the cut.
+(a) Extended tracks
+(b) Extended sum
+(c) Extended average
+Figure 12: Track candidate checking by extension.
+The tracks after further checking are drawn in the 3-D plots as shown in Fig. 13 and
+Fig. 14. Fig. 13 shows a candidate with a short length, the direction (Fig. 13a), and intensity
+(Fig. 13b) distribution is reasonable.
+Fig. 14 shows a candidate with a long length, the
+direction (Fig. 14a) and intensity (Fig. 14b) distribution is reasonable too. They are good
+candidates for muon tracks.
+(a) Extended track 1-D
+(b) Extended track in 3-D
+Figure 13: Track candidates one in 3-D
+(a) Extended track: 1-D
+(b) extended track: 3-D
+Figure 14: Track candidates two in 3-D
+– 11 –
+
+h2D trackext only
+pixel
+230
+2200
+>
+2000
+225
+1800
+1600
+220
+1400
+1200
+215
+1000
+800
+210
+600
+400
+205
+200
+200
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+Hpixelh2D_track_sumlength
+track intensity in ADC
+1000
+4000
+900
+3500
+800
+700
+3000
+600
+2500
+500
+2000
+400
+1500
+300
+1000
+200
+100
+500
+500
+1000
+1500
+2000
+2500
+3000
+-
+track length in pixelh2D_tracklength_intensity
+track intensity in ADC
+:
+2.8
+1200
+2.6
+1000
+2.4
+2.2
+800
+600
+1.8
+1.6
+400
+1.4
+200
+1.2
+500
+1000
+1500
+2000
+2500
+3000
+track length in pixelpixelV:pixelH(trackNum==50)
+2000
+1900
+1800
+1700
+1600
+1500
+1400
+1300
+1200
+1100
+100600
+1500
+2000
+2500
+3000
+3500
+4000pixe/Value:pixe/V:pixelH{trackNum==50
+214
+212
+210
+208
+206
+204
+202
+200
+198
+196
+4000
+3500
+1650
+1640
+3000
+1630
+1620
+2500
+1610
+1600
+2000
+1590
+15801500pixelV:pixelH (trackNum==2)
+2000
+1800
+1600
+1400
+1200
+1000
+800
+E
+600
+400E
+-
+2500
+2600
+2700
+2800
+3000
+3100
+3200
+3300
+3400
+3500pixelValue:pixelV:pixelH(trackNum==2)
+214
+212
+210
+208
+206
+204
+202
+200
+198
+196
+2890
+2900
+2870
+2880
+2850
+2860
+02830
+28404
+Discussion
+4.1
+Muon tracks or not?
+The angle distribution of the selected muon track candidates is further plotted as in Fig. 15
+after the track extension check as shown in Fig. 15b, where the theta angle is defined in
+the range [-90,+90]◦ to distinguish left and right relative to the camera. There are still too
+many abnormal tracks around 0 or |90|◦ as seen in Fig. 15a and Fig. 15c, which are related to
+some kind of systematic readout noise of the camera. After excluding the abnormal tracks
+around 0 or |90|◦, the theta distribution of the track candidates is basically consistent with
+expectation and a peak around 40◦ (cos(θ)∼0.7) as a hint (Fig. 15c and Fig. 15d). But the
+statistics are still not enough to have a good check by the muon angle distribution, even the
+selected muon candidates are much more than the expectation during the 30 s data-taking
+period (around 3 Hz × 30 s).
+(a) All selected tracks
+(b) Total intensity in ADC vs. track length
+(c) Theta in degree
+(d) Cos(theta)
+Figure 15: Distribution of selected tracks
+According to [35, 36], the quenching factor of the alpha of 241Am in CsI(Tl) is taken
+as 0.5 with an energy of 5.4 MeV, then the light intensity viewed by the camera is around
+1.4 p.e./MeV at an object distance of 15 cm, 2.2 p.e./MeV of 10 cm, and 4.7 p.e./MeV of 4 cm
+following the measurement in Sec. 2.2, respectively. Assuming the energy deposit of muon
+is around 2 MeV per cm, it means 2.8 p.e./400µm at an object distance of 15 cm (around
+80 pixels, 0.035 p.e./pixel) on camera, 4.4 p.e./625µm at an object distance of 10 cm (around
+105 pixels, 0.042 p.e./pixel) on camera, and 9.4 p.e./1300µm at an object distance of 4 cm
+(around 260 pixels, 0.036 p.e./pixel) on camera. With the data in Fig.15b, the intensity per
+pixel of the selected muon track candidates is from 0.1 to 1 ADC or 0.01 to 0.1 p.e., which
+– 12 –
+
+pixel
+230
+2200
+2000
+225
+1800
+1600
+220
+1400
+1200
+215
+1000
+800
+210
+600
+400
+205
+200
+200
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+H pixelSum
+1200
+1000
+800
+600
+400
+200
+500
+1000
+1500
+2000
+2500
+3000
+Track length in pixelCount
+10°
+102
+10
+80
+60
+40
+-20
+20
+40
+60
+80
+Theta/degreeCount
+103
+102
+10
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Cos(theta)is wide than the expectation from the measurement in Sec. 2.2 and could be related the
+variation of muon location.
+4.2
+Possible system optimization
+Taking more data to accumulate the muon tracks is an effective solution for more precise
+angle checking to find out more features of the noise, but the data volume will increase
+too. Following the issues of muon track identification, shorter exposure time is one of the
+effective methods to reduce the camera noise as known, while the data volume will increase
+too for similar statistics.
+To realize a good muon tagging and reconstruction method by visual photons directly,
+except to find a camera with a further lower noise level as the skipper CCD[30], it is
+possible to further improve the system by updating crystals with higher light yield, and
+better apertures, and to reduce the distortion of the image by the lens.
+Improving the track identification algorithms including distortion identification is also
+a valuable solution for track identification and better data compression. The coincidence
+with more than one camera is also another effective way to reduce the noise and improve
+the measurement as suggested in [32].
+4.3
+Further applications
+With the novel method to measure the muon track or realize a similar topology of a particle
+measurement in a scintillation detector, it can be used in a huge LS detector to measure the
+track of muons precisely as shown in Fig. 16, including to identify possible showers along
+the tracks as in JUNO, which is valuable to further study relevant background and suppress
+their contribution to neutrino measurement.
+Figure 16: Simulated 1 GeV Muon in liquid scintilator
+– 13 –
+
+4000
+2000
+0
+1000
+y/m500
+0
+1000
+500
+x/mm
+-500
+0
+-500
+-1000-1000With further improved sensitivity of the system, it is possible to be used to tag the
+topology of different particles to do particle identification as simulated in Fig. 17, where
+electron, positron, gamma, alpha, proton, pion of 1 GeV in liquid scintillator can be further
+identified through their topology for further direction or physics study.
+(a) 1 GeV Gamma
+(b) 1 GeV proton
+(c) 1 GeV π0
+(d) 1 GeV π+
+Figure 17: Simulated 1 GeV particle in liquid scintillator.
+5
+Summary
+With the crystal system viewed by the single photon sensitive camera and PMTs, system
+calibration was further discussed. The muon track imaging was tested further, and the
+data were analyzed according to the understanding of the characteristic expectation. Some
+possible tracks are identified with the averaged signal intensity cuts. But there still are a few
+critical items that need to be finalized. Some improvements are proposed and suggested.
+The realization of muon track direct measurement is valuable for future experiments and
+applications.
+Acknowledgments
+This work was supported by the National Natural Science Foundation of China (NSFC)
+No. 11875282 and 11475205, the State Key Laboratory of Particle Detection and Electronics,
+SKLPDE-ZZ-202208.
+– 14 –
+
+N
+4000
+2000
+0
+1000
+y/m200
+0
+1000
+500
+-500
+0
+x/mm
+-1000-1000
+500/mm
+1000
+N
+500
+0
+1000
+y/mm
+0
+1000
+-1000
+500
+x/mm
+0
+-2000-1000
+-5006000
+mm
+N
+4000
+2000
+0
+1500
+yx900
+him
+500
+1000
+0
+500
+x/mm
+-500
+0
+-1000-1000
+5004000
+N3000
+2000
+1000
+0
+1000
+2000
+-2000
+1000
+x/mm
+-3000
+0
+-4000-2000
+-1000References
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+– 17 –
+

IdE0T4oBgHgl3EQfiAEY/content/tmp_files/2301.02438v1.pdf.txt

@@ -0,0 +1,971 @@
+Experimental demonstration of position-controllable topological interface states
+in high-frequency topological integrated circuits
+Tetsuya Iizuka,∗ Haochen Yuan,† Yoshio Mita,† Akio Higo,‡ Shun Yasunaga,† and Motohiko Ezawa§
+(Dated: January 9, 2023)
+Topological integrated circuits are integrated circuit realizations of topological systems. We perform an ex-
+perimental demonstration by taking instances of the Su-Schrieffer-Heeger model and the Kitaev topological
+superconductor model. They are found to realize high frequency resonances around 17GHz. We explicitly
+observe the spatial profile of a topological edge state. In particular, the topological interface state between a
+topological segment and a trivial segment is the Majorana-like state in the Kitaev model. We construct a switch-
+able structure in the integrated circuit, which enables us to control the position of a Majorana-like interface state
+arbitrarily along a chain. Our results will open topological electronics with high frequency integrated circuits.
+Topological insulators and superconductors are fascinat-
+ing new states of matter[1, 2].
+The Su-Schrieffer-Heeger
+(SSH) model[3] and the Kitaev topological superconduc-
+tor model[4] are simplest one-dimensional (1D) systems re-
+alizing topological insulators and superconductors, respec-
+tively.
+Especially, topological superconductors host Ma-
+jorana edge states[5–8], which are the key elements of a
+topological quantum computer[9, 10].
+The area of topo-
+logical physics is expanded nowadays to photonic[11–16],
+acoustic[17–21], mechanical[22–29] and electronic-circuit
+systems[30–37].
+They are called artificial topological sys-
+tems.
+There are several merits which are difficult to be
+achieved in inorganic crystals: 1) It is possible to make a fine
+tuning of the system, which is crucial for observing topologi-
+cal edge states. 2) It is possible to construct a few site systems.
+3) It is possible to directly measure the site dependent infor-
+mation.
+It is relatively easy to materialize the SSH model because it
+involves only real hoppings. On the other hand, this is not the
+case with respect to the Kitaev model because it is a p-wave
+topological superconductor, although the Majorana edge state
+itself can be generated in a s-wave superconductor with the
+aid of a topological insulator nanowire[38, 39].
+Electronic circuits present an ideal platform to realize var-
+ious topological phases[30–37, 40–46].
+The emergence of
+topological edge states are observed by means of impedance
+resonance. However, experimental demonstrations have so far
+been restricted to printed circuit boards with discrete com-
+ponents. Indeed, to the best of our knowledge, there is no
+integrated-circuit realization working at high resonant fre-
+quency, although a simulation of the SSH model was done
+recently[47]. The integrated circuit realization is an important
+step toward industrial applications of topological electronics.
+In order to generate Majorana-like states, it is necessary to
+simulate electron and hole bands in electronic circuits. Al-
+though there is a proposal with the use of chains of capacitors
+∗ Systems Design Lab., School of Engineering, The University of Tokyo.
+[email protected] (corresponding author)
+† Department of Electrical Engineering and Information Systems, The Uni-
+versity of Tokyo.
+‡ Systems Design Lab., School of Engineering, The University of Tokyo.
+§ Department of Applied Physics, The University of Tokyo. [email protected]
+tokyo.ac.jp (corresponding author)
+and inductors[44, 45], there is so far no experimental demon-
+stration of this theoretical proposal.
+Most of previous experiments were carried out based on
+patterned structures, where it is impossible to control the topo-
+logical and trivial phases once the sample is manufactured.
+Actually, it is very hard to introduce switch structures in in-
+organic materials, photonic crystals and acoustic systems. On
+the other hand, transistors act as switches in electronic cir-
+cuits and hence, there is a possibility to construct a switchable
+topological system based on electronic circuits.
+In this paper, we perform for the first time an experimen-
+tal demonstration of topological integrated circuits, which are
+integrated circuit realizations of topological systems, by tak-
+ing instances of the SSH model and the Kitaev model. An
+integrated circuit implementation enables us to realize very
+high resonant frequency as large as 17GHz. We explicitly ob-
+serve the spatial profile of a topological edge state and deter-
+mine its penetration length. The system may contain several
+topological and trivial segments simultaneously along a chain.
+In particular, we observe the signal of a Majorana-like state
+emerging at the interface of a topological segment and a triv-
+ial segment. It is topologically protected since it necessarily
+emerges between the topological and trivial segments. These
+two topologically different segments are interchangeable sim-
+ply by exchanging inductors and capacitors.
+SSH chain
+The SSH chain is the basic model of a topological insula-
+tor. It was implemented in a printed circuit board with dis-
+crete components[31] a few years ago. The electronic circuit
+is illustrated in Fig.1a. Capacitances are alternating along the
+chain, and each node is grounded via an inductor.
+We have implemented the SSH model in two 32-unit cell
+chains on chips in an integrated circuit as shown in Fig.1b.
+Both capacitors and inductors are implemented by metallic
+wires. An inductor is materialized by a swirling structure of
+wire as shown in Fig.1c, while a capacitor is materialized by
+a comb-teeth structure as shown in Fig.1d. The comb-teeth
+structure is prepared in order to increase the capacitance with
+small occupation area. The explicit values of the capacitance
+and the inductance are shown in Table.1, which is very tiny
+compared with printed circuit boards with discrete compo-
+nents. The small capacitance and inductance lead to a high
+resonant frequency ωresonant = 1/
+√
+LC.
+The impedance as a function of the input frequency is
+arXiv:2301.02438v1  [cond-mat.mes-hall]  6 Jan 2023
+
+2
+FIG. 1. SSH chain. a, An illustration of an electronic-circuit representing the SSH chain. b, A picture of a 32-unit cell integrated circuit for
+the SSH chain. c, A zoom-in view of its unit cell layout. d, A picture of the comb teeth capacitor. Frequency dependence of the impedance
+measured from the left edge of the 17.2 GHz chain for e, all-topological and f, all-trivial setups. g, The spatial profile of the impedance values
+for all-topological mode measured from the left edge at the resonant frequency of 13.1 GHz. In e and f, solid and dashed lines show the
+measured and simulated results of the SSH chain, respectively.
+shown in Fig.1e and f.
+The impedance is evaluated with
+the two-point impedance between two nodes. The solid and
+dashed lines show measurement and simulation results, re-
+spectively. A peak emerges at the characteristic frequency for
+a topological phase as shown in Fig.1e. The measured res-
+onant frequency is 10.7GHz. On the other hand, there is no
+peak for a trivial phase as shown in Fig.1f.
+The node-dependent impedance is shown in Fig.1g. The
+impedance decays exponentially. The penetration length of
+the topological edge state is estimated as 0.414, which is in
+good agreement with the theoretical value 1/ log(C2/C1) =
+0.449. See Supplementary Information IV for details.
+We have also carried out measurements on the SSH chain
+with 8.8 GHz. See Supplementary Information I for details.
+Kitaev chain
+The Kitaev chain model is the basic model of a topologi-
+cal superconductor. Our main result is its implementation in
+an integrated electronic circuit. To realize a Cooper pair it
+is necessary to prepare an electron band and a hole band to-
+gether with cross terms between these two bands, as shown in
+TABLE I. Parameters used for the SSH chain (left table) and the
+Kitaev chain (right table).
+8.8 GHz 17.2 GHz
+C1
+42 fF
+22 fF
+C2
+414 fF
+204 fF
+L
+721 pH
+378 pH
+8.8 GHz 17.2 GHz
+C
+440 fF
+220 fF
+L
+747 pH
+384 pH
+Cx
+396 fF
+204 fF
+Lx 830 pH
+427 pH
+C0
+880 fF
+440 fF
+L0 374 pH
+192 pH
+Methods.
+We first illustrate an electronic circuit for the Kitaev
+chain [44, 45] in Fig. 2a, b and c. The capacitor channel (in-
+dicated in red) corresponds to the electron band, while the
+inductor channel (in blue) corresponds to the hole band. The
+two main channels are crosslinked through Cx and Lx. Each
+node is connected to the ground via an inductor L0 or a ca-
+pacitor C0 to realize a topological state or a trivial state, re-
+spectively, as shown in Figs. 2a and b. The topological phase
+
+400μm
+c
+Unit cell
+Unit cell
+C2
+C1
+forProbing Test
+H&}
+ContactPads
+区HH区
+HH区
+240μm
+000
+GNDLine
+b
+Unit cell
+d
+480μm
+3300μm
+e
+f
+Topological phase
+Trivial phase
+g
+1000
+1000
+Two-Point Impedance [Q]
+Two-Point Impedance [2]
+a
+100
+10.7GHz
+Two-Point Impedance
+100
+100
+Node
+10
+Node0
+10
+10
+Node
+Node.2
+1
+0.1
+0.1
+0.1
+Node
+Penetrationlength=0.414
+Node6
+0.01
+Node7
+0.01
+1
+2
+3456
+10
+20
+1
+2
+3456
+10
+20
+0
+1
+2
+3
+4
+5
+6
+Frequency [GHz]
+Frequency [GHz]
+Node Number3
+FIG. 2. Kitaev chain. a, b and c, The electronic-circuit representation of the Kitaev chain. a, All-topological configuration where the
+topological edge state emerges at both the left and right edges of the chain. b, All-trivial configuration that does not have a topological edge
+state. c, The implemented state-configurable Kitaev chain circuit. d, By using two SPDT switches with inverters in the unit cell, the connection
+of L0 and C0 can be swapped to change its topological/trivial state. The SPDT switch is realized by two CMOS transmission gate switches. e,
+A picture of an 16-unit cell integrated circuit for the SSH chain. f, A picture of a unit cell. g, A zoom of SPDT switches in Fig.2f. Each SPDT
+switch is composed of an inverter and two transmission gates with n-type and p-type MOS FETs as in Fig.2d. h and i, Frequency dependence
+of the impedance measured from the right edge of the electronic circuit with the characteristic frequency ωresonant =17.2 GHz Kitaev chain
+for all-topological and all-trivial setups, respectively. Solid and dashed lines show the measured and simulated results of the Kitaev chain,
+respectively. j, The spatial profile of the impedance values for all-topological mode measured from both left and right edges at the resonant
+frequency of 13.1 GHz.
+is realized by the configuration shown in Fig. 2a, while the
+trivial phase is realized by the configuration shown in Fig. 2b.
+A single Kitaev chain may accommodate several segments
+which are either topological or trivial. A Majorana-like state
+emerges at an interface between the two phases. We introduce
+two single-pole double-throw (SPDT) switches in each unit
+cell as illustrated in Fig. 2c. The electric circuit for the SPDT
+switch is shown in Fig. 2d. The switching is done by swapping
+the connection of L0 and C0, by way of which the position of
+a Kitaev interface state is controlled. In the integrated circuits,
+the SPDT switch is simply implemented with an inverter and
+two CMOS transmission gates, composed of n-type and p-
+type metal oxide semiconductor (MOS) field-effect transistors
+(FETs) as shown in Fig. 2f and g.
+The Kitaev chain circuit shown in Fig. 2c is implemented
+onto the chip using 180 nm CMOS technology as shown in
+
+a
+Topological phase
+b
+Trivial phase
+Unit cell
+g
+Switches
+220μm
+GNDLine
+Transmission
+SPDTSwitch
+Gate
+forProbingTest
+C
+ContactPads
+Transmission
+Gate
+000
+000
+000
+400μm
+Inverter
+000
+000
+Switches
+C
+d
+SPDT switch
+Probingpoints
+SPDTSwitch
+IN
+000
+区
+OUT1
+OUT2
+Co
+SPDT
+switch
+GNDLine
+000
+SW
+OUT1
+OUT2
+e
+Unit cell
+Unit cell
+400μm
+3800μm
+h
+Topological phase
+Trivial phase
+J
+100
+100
+Two-Point Impedance [Q]
+Two-Point Impedance [Ω]
+Two-Point Impedance [Ω]
+100
+13.1GHZ
+D..
+From-leftedge
+Node 16
+Penetration length=0.660
+10
+10
+10
+Node16
+Node 15
+Fromirightedge
+Penetration length=0.666
+Node.15
+Node 14
+Node.
+1
+lode
+0.1
+0.1
+0.1
+Nodel
+Node 12
+0.01
+0.01
+0.01
+2
+3
+456
+10
+20
+2
+3
+456
+10
+20
+0
+2
+4
+6
+8
+10
+12
+14
+16
+Frequency [GHz]
+Freguency[GHz]
+Node Number4
+ 0
+ 5
+ 10
+ 15
+ 20
+ 25
+ 30
+ 35
+ 40
+ 45
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+Impedance [Ω]
+Node Number
+3 trivial
+3 trivial
+2 trivial
+4 topological
+4 topological
+8.8GHz chain
+17.2GHz chain
+From node 4
+From node 7
+From node 10 From node 13
+From node 4
+From node 7
+From node 10 From node 13
+ 0
+ 5
+ 10
+ 15
+ 20
+ 25
+ 30
+ 35
+ 40
+ 45
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+Impedance [Ω]
+Node Number
+4 trivial
+3 trivial
+1 trivial
+4 topological
+4 topological
+8.8GHz chain
+17.2GHz chain
+From node 5
+From node 8
+From node 10
+From node 13
+From node 5
+From node 8
+From node 10
+From node 13
+ 0
+ 5
+ 10
+ 15
+ 20
+ 25
+ 30
+ 35
+ 40
+ 45
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+Impedance [Ω]
+Node Number
+4 trivial
+4 trivial
+8 topological
+8.8GHz chain
+17.2GHz chain
+From node 5
+From node 12
+From node 5
+From node 12
+a
+b
+c
+FIG. 3. Topological interface states. Measurement results of the
+topological edge state locations depending on different Kitaev chain
+configurations.
+"n trivial (topological)" indicates that the trivial
+(topological) segment contains n unit cells. Blue (red) data points
+are for 8.8 GHz (17.2 GHz) chain. a, The topological edge states
+emerge at 4th, 7th, 10th and 11th nodes. b, The locations of the edge
+states move to the 5th, 8th, 10th and 11th nodes. c, When two topo-
+logical segments combine to one segment, the edge states emerge
+only at 5th and 12th nodes.
+Fig. 2e. On a 5 mm×5 mm chip, two 16-unit cell Kitaev chain
+circuits were integrated for two different target resonant fre-
+quencies, 8.8 GHz and 17.2 GHz. We show a zoom-in view of
+the unit cell layout in Fig. 2f, which shows that it includes 3
+inductors L, Lx and L0, 3 capacitors C, Cx and C0, 2 SPDT
+switches, and a contact pad at each node for direct probing
+measurement with GSG (Ground, Signal, Ground) probes. A
+photo of the SPDT switches is shown in Fig. 2g. Two trans-
+mission gates and an inverter are integrated for each SPDT
+switch. The values for the capacitors and inductors are sum-
+marized in TABLE I.
+Fig. 2h, i and j summarizes the impedance measurement re-
+sults of the Kitaev chain designed for 17.2 GHz resonant fre-
+quency. Figs. 2h and i shows the frequency dependence of the
+impedance measured from the right edge of the chain for topo-
+logical and trivial setups, respectively. The solid and dashed
+lines show measurement and simulation results, respectively.
+We have also carried out a measurement for the Kitaev
+chain with 8.8 GHz. See Supplementary Information II for
+details.
+As we can see from the impedance peak of the rightmost
+edge in Fig. 2h, the measured resonant frequency is shifted
+down from the calculated value of 17.2 GHz to 13.1 GHz.
+This is mainly caused by the parasitic inductance of the metal
+wires in the unit cell to connect the circuit elements. With-
+out considering the wires, the simulated resonant frequency is
+16.4 GHz, which is much closer to the theoretical value. For
+both topological and trivial setups, the measurement results
+agree well with the simulation especially around the resonant
+frequency.
+Fig. 2j summarizes the two-point impedance values at the
+measured resonant frequency 13.1 GHz.
+The blue and red
+lines show the impedance measured from the left and the right
+edges, respectively. The leftmost (0-th) and rightmost (16-
+th) node impedance correspond to Z11 value of the 2 × 2
+impedance matrix.
+In the topological setup, the impedance peaks are observed
+at both the edges. The penetration length of the topological
+edge state is 0.660 unit cell for the left edge and 0.666 unit
+cell for the right edge, which show a good agreement with the
+theoretical value 0.610 unit cell. See Supplementary Informa-
+tion V for details.
+We have so far observed the topological edge states. There
+is also a topological interface state between topological and
+trivial phases.
+It is possible to switch the topological and
+trivial phases for each segment. Fig. 3 summarizes the 2-
+point impedance at the resonant frequency with 3 different
+switch configurations for the Kitaev chains with 8.8 GHz and
+17.2 GHz designs, where one point is fixed at the topologi-
+cal/trivial interface and the other point is moved from 1 to
+16. In Fig. 3a we divided the chain into 4 segments as shown
+Fig. 3a. The impedance peak that corresponds to the topo-
+logical interface state emerges at the edges of the topological
+segments. When we move the left topological segment to the
+right by one unit, the location of the edge states moves ac-
+cordingly as shown in Fig. 3b. Then if the two separated topo-
+logical segments are combined into one segment as shown in
+Fig. 3c, we observe only two impedance peaks at the left and
+right edges of the single topological segments. This clearly
+demonstrates the movement of the topological interface state
+that emerges on the electronic-circuit realization of the Kitaev
+chain implemented onto the integrated circuit. We also ob-
+serve the same behavior for two chains with different resonant
+frequencies, which proves that the topological interface state
+emerges independent of the designed resonant frequency.
+Conclusion
+We have materialized the SSH model and the Kitaev model
+in integrated circuits. These models have topological and triv-
+ial phases. It is possible to create several segments which
+are either topological or trivial in a single chain. Topological
+edge states emerge at both the edges of a topological segment,
+which are observable by mean of the impedance resonance.
+We have demonstrated that the segment size can be as small
+as one unit cell because the penetration length can be made
+smaller than one unit cell: See Fig.3b. Furthermore, we have
+equipped our integrated circuit with a switchable structure,
+which enables us to control the position of a topological in-
+terface state arbitrarily along a chain. Such a possibility is a
+great merit of topological electric circuits over other artificial
+
+5
+FIG. 4. Setup. a, A microphotograph of the chip that integrates the SSH model and the Kitaev chain, where A (B) shows the circuits for the
+32-stage SSH model with 17.2GHz (8.8GHz), while C (D) shows the circuits for the 16-stage Kitaev model with 8.8GHz(17.2GHz). b, A
+photo of the measurement setup. c, A block diagram of the measurement setup.
+topological systems, where an integrated topological pattern
+is printed once and for all.
+We have observed that the resonant frequency is lower than
+the theoretical value estimated from ωresonant = 1/
+√
+LC. This
+is due to the parasitic inductance present in the wires. Details
+are shown in Supplementary Information III.
+The integrated circuit has small inductance and capaci-
+tance, which leads to high frequency operation. The size of
+the unit cell is 200µm and hence, largely integrated circuits
+are possible. Furthermore, mass production is possible in in-
+tegrated circuits, which will benefit for future industrial appli-
+cations of topological electronics.
+Methods
+Measurements. A block diagram and a photo of the mea-
+surement setup are shown in Fig.4. We observed the topolog-
+ical edge state based on two-point impedance measurement.
+We observe two-point impedance with a vector network
+analyzer (VNA), Keysight N5222B. The chip measurement
+is done on the probe station, Formfactor Summit11000. A
+2×2 Z-matrix is derived from the 2×2 S-parameter measured
+by the VNA. The chain configuration (the state of the SPDT
+switches) is controlled by the serial-parallel interface (SPI) in-
+tegrated on the same chip, whose configuration data are writ-
+ten from an external PC.
+Simulation is done with a circuit simulator, Cadence Spec-
+tre. The S-parameters of the passive components such as ca-
+pacitors and inductors are extracted for circuit simulation with
+Cadence EMX, which is a planar 3D electromagnetic simula-
+tor based on the Fast Multipole Method (FMM) designed for
+high-frequency integrated circuits.
+SSH model.
+The SSH is defined by the following 1D
+Hamiltonian,
+H =
+N
+�
+x=1
+tA
+�
+c†
+2x−1c2x + c†
+2xc2x−1
+�
++tB
+�
+c†
+2xc2x+1 + c†
+2x+1c2x
+�
+.
+(1)
+It is realized by an LC circuit as shown in Fig.1a. When we ap-
+ply an AC source with frequency ω, with Kirchhoff’s current
+law, the sum of currents from all adjacent nodes m flowing
+into node n leads to the following formula,
+In(ω) =
+�
+m
+Jnm(ω)Vm(ω),
+(2)
+where Jnm(ω) is the circuit Laplacian. By Fourier transform-
+ing from the node x to the momentum k, it is summarized
+as
+�
+IA (k)
+IB (k)
+�
+= JAB(ω)
+�
+VA (k)
+VB (k)
+�
+,
+(3)
+where
+JAB(ω) = iω
+�
+1
+ω2L − (C1 + C2)
+C1 + C2e−ik
+C1 + C2eik
+1
+ω2L − (C1 + C2)
+�
+(4)
+is the circuit Laplacian. The condition for the impedance reso-
+nance is determined by the condition where the diagonal term
+is zero at the resonant frequency and the resonant frequency
+is determined as
+ωresonant = 1/
+�
+L (C1 + C2)
+(5)
+for the topological phase.
+On the other hand, there is no
+impedance resonance for the trivial phase.
+1D p-wave Kitaev topological superconductor model.
+The original Kitaev p-wave superconductor model is defined
+on the 1D lattice as
+H = −µ
+�
+x
+c†
+xcx − t
+2
+�
+x
+�
+c†
+xcx+1 + c†
+x+1cx
+�
+−1
+2
+�
+x
+�
+∆eiφcxcx+1 + ∆e−iφc†
+x+1c†
+x
+�
+,
+(6)
+where µ is the chemical potential, t > 0 is the nearest-
+neighbor hopping strength and ∆ > 0 is the p-wave pairing
+amplitude of the superconductor.
+By introducing the Nambu representation Ψ†
+k =
+�
+c†
+k, c−k
+�
+and Ψk =
+�
+ck, c†
+−k
+�T
+one can write the Hamiltonian in the
+
+a
+b
+c
+5mm
+NetworkAnalyzer
+Vecto Network Analyzer
+Screen
+（KeysightN5222B）
+Port1
+Port2
+ABCD
+Network Analyzer
+Chip
+(Behindthe Microscope)
+UnderTest
+5mm
+A
+GPLLLLLL
+Microscopic View
+oftheChip
+Powersupplyand
+Serial-Parallel Interface
+Chipon the Probe Station
+(Tocontrolswitches)
+SPI control signals6
+Bogoliubov-de Gennes form
+H = 1
+2
+�
+k
+Ψ†
+kH(k)Ψk,
+(7)
+with a 2×2 form Hamiltonian
+H(k) = 1
+2
+�
+−t cos k − µ
+i∆0 sin k
+−i∆0 sin k
+t cos k + µ
+�
+.
+(8)
+The zero-energy state of the Bogoliubov-de Gennes Hamilto-
+nian is a Majorana state, and hence, there appear Majorana
+edge states in the topological phase of the Kitaev model.
+Here, t, µ, σi and ∆i represent the hopping amplitude, the
+chemical potential, the spin degree of freedom, and the su-
+perconducting gap parameter, respectively. It is well known
+that the system is topological for |µ| < |2t| and trivial for
+|µ| > |2t| irrespective of ∆i provided ∆i ̸= 0.
+We then realize this p-wave Kitaev model by way of an
+electronic circuit. As shown in Fig.2a, this circuit chain con-
+tains two main lines, one connected by a series of capacitors C
+implementing the electrons band, while another connected by
+a series of inductors L implementing the holes band, respec-
+tively. Pairing interaction between the two bands is simulated
+by bridging capacitors Cx and inductors Lx. Each electron
+node and each hole node is connected to the ground via a ca-
+pacitor C0 and inductors L0, respectively. The hopping am-
+plitudes t realized in the electrons band and holes band are op-
+posite since the capacitors C contained in the electrons band
+contribute the terms iωC while the inductors L contained in
+the holes band contribute the terms 1/(iωL).
+The circuit Laplacian is given by
+Jab(ω) =
+�
+f1 g1
+g2 f2
+�
+,
+(9)
+where
+f1 = −2C cos k + 2C −
+�
+ω2L0
+�−1
+f2 = 2
+�
+ω2L
+�−1 cos k − 2
+�
+ω2L
+�−1 + C0
+g1 = −Cxeik +
+�
+ω2Lx
+�−1 e−ik
+g2 =
+�
+ω2Lx
+�−1 eik − Cxe−ik,
+(10)
+for topological phase and
+f1 = −2C cos k + 2C + C0
+f2 = 2
+�
+ω2L
+�−1 cos k − 2
+�
+ω2L
+�−1 −
+�
+ω2L0
+�−1
+g1 = −Cxeik +
+�
+ω2Lx
+�−1 e−ik
+g2 =
+�
+ω2Lx
+�−1 eik − Cxe−ik,
+(11)
+for trivial phase.
+The essence to realize the 1D model in circuit form is
+to make the circuit Laplacian equal to the system Hamilto-
+nian.
+Clearly, to make it possible, particle-hole symmetry
+(PHS) must be respected, which requires these three pairs
+of LC resonators shares the same resonant frequency, that is,
+ωresonant ≡ 1/
+√
+LC = 1/√L0C0 = 1/√LxCx. Once PHS
+is respected, the relationship between circuit components and
+Hamiltonian parameters could be induced and expressed as
+follows:
+�
+�
+�
+t = −C,
+µ = −2C + C0,
+∆0 = −Cx.
+(12)
+To make the 1D circuit chain topological, we set µ to 0 to
+meet the topological mode requirements of |µ| < |2t|. This
+topological property is satisfied by the emergence of grounded
+capacitors C0 and inductors L0, since the system will be pre-
+cisely located at the critical point between the topological and
+trivial states. Therefore, by exchanging the connections of
+C0 and L0, we could perform transitions between these two
+states.
+Impedance resonance. The emergence of a topological
+edge states is observed via impedance resonance. The topo-
+logical edge state is a zero-energy eigenstate of the Hamilto-
+nian. It corresponds to the zero admittance, and hence, the
+emergence is observable by the divergence in the impedance.
+The two-point impedance between the a and b nodes is
+given by[32]
+Zab ≡ Va/Ib = Gab,
+(13)
+where G is the Green function defined by the inverse of the
+Laplacian J, G ≡ J−1.
+
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+Acknowledgments
+This work is supported by CREST, JST (Grants No. JPMJCR20T2).
+The authors would like to thank Z. Yang, S. Li and X. Chen for their
+measurement support. The on-chip passive components are designed
+based on RF cell library developed by Umeda laboratory, Tokyo Uni-
+versity of Science. The LSI chip in this study was designed and fab-
+ricated through the activities of VDEC, The University of Tokyo, in
+collaboration with Cadence Design Systems, Rohm Corporation and
+Toppan Printing Corporation.
+Author contributions
+M.E., Y.M. and T.I. planned the study.
+T.I. and H.Y. designed
+the topological circuits and performed the experiments. T.I., M.E.
+and H.Y. collected and analyzed the data. M.E. and T.I. wrote the
+manuscript with input from H.Y., Y.M., A.H. and S.Y. All the au-
+thors discussed the project and the results.
+Additional information
+Supplementary information is available.
+Competing financial interests
+The authors declare no competing financial interests.
+

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@@ -0,0 +1,799 @@
+Self-Supervised Object Segmentation with a Cut-and-Pasting GAN
+Kunal Chaturvedi, Ali Braytee, Jun Li, Mukesh Prasad
+aSchool of Computer Science, University of Technology Sydney, Ultimo, 2007, NSW, Australia
+Abstract
+This paper proposes a novel self-supervised based Cut-and-Paste GAN to perform foreground object seg-
+mentation and generate realistic composite images without manual annotations. We accomplish this goal by
+a simple yet effective self-supervised approach coupled with the U-Net based discriminator. The proposed
+method extends the ability of the standard discriminators to learn not only the global data representations
+via classification (real/fake) but also learn semantic and structural information through pseudo labels created
+using the self-supervised task. The proposed method empowers the generator to create meaningful masks
+by forcing it to learn informative per-pixel as well as global image feedback from the discriminator. Our
+experiments demonstrate that our proposed method significantly outperforms the state-of-the-art methods on
+the standard benchmark datasets.
+Keywords: Generative adversarial networks, Self-supervised learning, Cut-and-Paste, Segmentation
+1. Introduction
+Generative adversarial networks (GANs) [1] have
+become a popular class of image synthesis meth-
+ods due to their demonstrated ability to create high-
+dimensional samples with desired data distribution.
+The primary objective of GANs is to generate di-
+verse, high-quality images while also ensuring the
+stability of GAN training [2] [3]. GAN consists of
+generator and discriminator networks trained in an
+adversarial manner. The generator attempts to syn-
+thesize the real data distribution to fool the discrim-
+inator, whereas the discriminator’s goal is to distin-
+guish between the generator’s real and fake data. In
+image segmentation, several compositional genera-
+tive models have been proposed [4, 5, 6, 7] , where
+the generator creates a synthesized composite image
+by copying the object from one image and pasting
+it in another to fool the discriminator about think-
+ing the synthesized composite image is real. But,
+the generator may not perform any segmentation,
+and the background may look realistic. Therefore,
+Email address: Kunal.Chaturvedi, Ali.Braytee,
+Jun.Li, [email protected] (Mukesh Prasad)
+for effective training, the discriminator needs to pro-
+vide the generator with informative learning signals
+by learning relevant semantics and structures of the
+data that may result in more effective generators.
+However, the current state-of-the-art GANs employ
+discriminators based on the classification network,
+which learn only a single discriminative signal such
+as the difference between real and fake images. In
+such a non-stationary environment, the generator be-
+comes prone to catastrophic forgetting and may lead
+to training instability or mode collapse [8].
+To address the aforementioned issues, additional
+discriminatory signals are required to guide the train-
+ing mechanism and assist the generator in producing
+high-quality images. This can be accomplished by
+increasing the capacity of the discriminator with aux-
+iliary tasks and signals. These auxiliary tasks on the
+labeled datasets resist the forgetting issues and im-
+prove the training stability of GANs, but it suffers
+with unlabeled datasets.
+Recently, self-supervised
+learning has been explored on numerous GANs
+methods [8],[9],[10],[11]. The self-supervised tasks
+provide the learning environment with additional
+guidance to the standard training mechanism. Most
+of the recent self-supervision methods based GANs
+Preprint submitted to Nuclear Physics B
+January 3, 2023
+arXiv:2301.00366v1  [cs.CV]  1 Jan 2023
+
+use auxiliary tasks based on transformation. For ex-
+ample, SS-GAN developed by Chen et al. [8] uses
+rotation prediction as an auxiliary task. In FX-GAN,
+Huang et al.
+[10] use the pretext task of predic-
+tion on corrupted real images, and in LT-GAN [9],
+the authors use distinguishing GAN-induced trans-
+formation as a pretext task. However, the goals of
+these transformation-based self-supervised tasks are
+inconsistent with the GAN’s goal of mimicking the
+real data distribution. Moreover, this problem ampli-
+fies when the generator’s task is to construct segmen-
+tation masks from the foreground images.
+In order to maintain an enriched real data repre-
+sentation and improve the quality of generated seg-
+mentation masks, we propose a Self-Supervised Cut-
+and-Paste GAN (SS-CPGAN) based on U-net archi-
+tecture [12], which unifies cut-and-paste adversar-
+ial training with a self-supervised task.
+It allows
+the discriminator to learn both local and global dif-
+ferences between real and fake data.
+In contrast
+to the existing transformation based self-supervision
+methods, our self-supervision learning method cre-
+ates pseudo labels using unsupervised segmentation
+methods. Then, it simultaneously forces the discrim-
+inator to provide the generator with global feedback
+(real or fake) and the per-pixel feedback of the syn-
+thesized images with the help of pseudo labels.
+To sum up, our main contributions are:
+• We propose a novel Self-Supervised Cut-and-
+Paste GAN (SS-CPGAN), that unifies cut-and-
+paste adversarial training with a segmentation-
+based self-supervised task. SS-CPGAN lever-
+age unlabeled data to maximize segmentation
+performance and generate highly realistic com-
+posite images.
+• The proposed self-supervised task in SS-
+CPGAN improves the discriminator’s represen-
+tation ability by enhancing structure learning
+with global and local feedback. This enables the
+generator with additional discriminatory sig-
+nals to achieve superior results and stabilize the
+training process.
+• We perform a comprehensive analysis on the
+benchmark datasets and compare our proposed
+method with the baseline method.
+2. Related Works
+2.1. Unsupervised Object Segmentation via GANs
+Unsupervised segmentation using GANs is an im-
+portant topic in research. Several works [4, 5, 6, 7]
+investigate the use of compositional generative mod-
+els to obtain high quality segmentation masks. Copy-
+pasting GAN [7] performs unsupervised object dis-
+covery by extracting foreground objects and then
+copying and pasting them onto the different back-
+grounds.
+Similarly, PerturbGAN [5] generates a
+foreground mask along with a background and fore-
+ground image in an adversarial manner. Recently,
+Abdal et al.
+(2021) [6] propose a method to use
+an alpha network that includes two pretrained gen-
+erators and a discriminator based on the StyleGAN
+to generate high quality masks.
+These methods
+learn object segmentation without needing to use
+annotations. However, they are prone to degener-
+ate solutions or other trivial cases.
+For example,
+the generator may not perform any segmentation,
+and the background looks realistic or the generator
+may segment foreground masks consisting of all-
+ones. To avoid such problems, special care needs
+to be taken while training the compositional genera-
+tive models. Copy-pasting GAN uses anti-shortcut,
+border-zeroing, blur, and grounded fakes to prevent
+trivial solutions [7].
+PerturbGAN avoids such so-
+lutions by randomly shifting object segments rel-
+ative to the background [5].
+However, Abdal et
+al. (2021) [6] make several changes to the original
+StyleGAN and use truncation trick along with regu-
+larization to avoid degenerate solutions.
+2.2. Self-supervised learning
+Self-supervised learning learns useful feature rep-
+resentations from data with the help of pretext tasks.
+Recently, many pretext tasks coupled with adversar-
+ial training have been introduced [8]. The motiva-
+tion for using self-supervised learning is to: (1) pre-
+vent discriminator forgetting [13]; (2) improve train-
+ing stability [14]; (3) and ensure high quality of im-
+ages generated [15]. The self-supervision techniques
+rely on pretext tasks on geometric transformations
+(e.g., prediction on rotated images[8], corrupted im-
+ages [10], GAN-induced transformations [9], or a
+deshuffling task that predicts the shuffled orders [16])
+2
+
+to increase the discriminator’s representation power.
+Unlike the aforementioned methods, we incorporate
+segmentation-based self-supervised learning coupled
+with the Cut-and-Paste GAN to obtain high-quality
+segmentation masks.
+Most importantly, with our
+self-supervised approach, no extra care is needed to
+deal with the trivial solutions prevalent in composi-
+tional generative models.
+3. Method
+In this section, we first present the standard ter-
+minology of adversarial training and the encoder-
+decoder based discriminator.
+We then introduce
+our Self-Supervised Cut-and-Paste GAN built upon
+the cut-and-paste adversarial training.
+The uni-
+fied framework with the segmentation-based self-
+supervised task encourages the generator to empha-
+size local and global structures while synthesizing
+masks.
+3.1. Adversarial Training
+As shown in Fig. 2, we build a generative model
+in which the generator takes the foreground image
+as the input and generates a composite image using
+a combination of the predicted mask, source fore-
+ground image and the background image to fool the
+discriminator. Formally, we define the input fore-
+ground source image as If ∈ Pdata and background
+image as Ib ∈ Pdata where Pdata denotes the set of
+input images. Now, we define a generator (G) that
+is trained in an adversarial manner against the dis-
+criminator (D).
+During the training process, the
+generator predicts a segmentation mask defined by
+mg(If) = G(I f) where mg(I f) ∈ [0, 1]. Then, using
+the predicted mask mg(If), foreground source image
+If, and background image Ib, we define composite
+image as follows
+IC = mg(I f)I f + (1 − mg(I f))Ib
+(1)
+The discriminator’s objective is to classify the
+composite image as real or fake. As a result, the stan-
+dard objective of the discriminator and the generator
+of the CPGAN is defined as follows
+LD = max
+D
+E
+�
+log D(If) + log(1 − D(IC))
+�
+(2)
+LG = min
+G E �log D(IC)�
+(3)
+The discriminator works as a classification network
+that is restricted to learn only through the discrim-
+inative differences between the real and fake sam-
+ples.
+Thus, the discriminator fails to provide any
+useful information to the generator. Therefore, we
+use an encoder-decoder-based discriminator network
+with self-supervised learning to mitigate this prob-
+lem.
+3.2. Encoder-Decoder based Discriminator
+In this work, we replace standard classification-
+based discriminator with the U-net based discrimi-
+nator [12]. The U-net is an encoder-decoder-based
+architecture that consists of a network of convolu-
+tional layers, skip connections for semantic segmen-
+tation. It was initially proposed for biomedical im-
+age segmentation, which achieved precise segmen-
+tation results with few training images. Further, it
+demonstrates good results in other applications, in-
+cluding ArcGIS [17], remote sensing [18], and oth-
+ers. Its architecture (see Figure 1) is symmetric and
+consists of two paths, an Encoder that extracts spatial
+features from the input image (downscaling process),
+and a Decoder that constructs the segmentation map
+from the extracted feature maps (upscaling process).
+We use the encoder part of the U-net as the standard
+classification-based discriminator that performs the
+binary decision on real/fake composite images. And
+the decoder part of the U-net architecture is utilized
+by the self-supervised task to give per-pixel feedback
+of the synthesized images with the help of pseudo la-
+bels. This allows the discriminator to learn both rel-
+evant local and global differences between real/fake
+images.
+3.3. Self-Supervised
+Cut-and-Paste
+GAN
+(SS-
+CPGAN)
+To improve the representation learning ability of
+the CPGAN, the discriminator must be able to learn
+semantic as well as structural information from the
+synthesized images.
+Therefore, we focus our ap-
+proach on using self-supervised learning to build
+comprehensive representations for the CPGAN. In
+this work, we employ a segmentation based self-
+supervised task, with the primary goal of enabling
+3
+
+Figure 1: An overview of U-net architecture. The different ar-
+rows denote the different operations used in the encode-decoder
+based architecture.
+the discriminator with enhanced learned features that
+ultimately empower the generator to create consis-
+tent and structurally coherent masks.
+The pseudo
+segmentation masks mUS(I f) ∈ [0, 1] are created
+using graph-based unsupervised segmentation algo-
+rithm [16]. These masks obtained by the GrabCut
+technique acts as a good prior for the U-net based
+discriminator (see, Figure 2). Here, the discriminator
+performs two important tasks, i.e., 1) classification
+of real/fake compositing images and 2) performing
+per-pixel based classification on I f ∈ Pdata to gener-
+ate segmentation masks. Given the self-supervised
+pseudo labels, we train the discriminator for accu-
+rate pixel-level prediction. The introduction of self-
+supervisory signals empowers the discriminator by
+enhancing its localization ability and forces the dis-
+criminator it to learn useful semantic representations.
+This mechanism enables the generator to achieve op-
+timized results and makes the training process more
+stabilized.
+Formally, we define I f
+∈ Pdata as the source
+image containing foreground object, and Pdata de-
+notes the set of input images. Given a source fore-
+ground image I f, we create pseudo label denoted by
+mUS(I f) ∈ [0, 1], using an unsupervised segmenta-
+tion algorithm. Then, we define mw(IC) ∈ [0, 1] as
+the pixel-wise segmentation mask produced by the
+decoder of the discriminator. Hereafter, we optimize
+the overall discriminator loss function (Eq. 5) by
+augmenting a new self-supervision based loss (Eq. 4)
+Lsel f−supervised = L
+�
+mw(IC), 1 − mUS(If)
+�
+(4)
+L′
+D = LD + λLsel f−supervised
+(5)
+where L is the cross-entropy loss, and λ denotes the
+loss weight for the self-supervision based loss. This
+hyperparameter is updated based on the compari-
+son between mUS(I f) and mw(IC), using intersection-
+over-union (IoU). The details of the hyperparameter
+chosen are explained in the implementation details
+section. The framework of the self-supervised learn-
+ing is shown in Figure 2.
+4. Experimentation
+This section discusses the implementation details
+of the proposed method and an extensive set of ex-
+periments on various datasets.
+4.1. Datasets
+We utilize five different datasets for the foreground
+and background set to train our SS-CPGAN as de-
+scribed below:
+• Caltech-UCSD Birds (CUB) 200-2011 is a fre-
+quently used benchmark for unsupervised im-
+age segmentation. It consists of 11,788 images
+from 200 birds species.
+• Oxford 102 Flowers consists of 8,189 images
+from 102 flower classes.
+• FGVC Aircraft (Airplanes) contains 102 dif-
+ferent aircraft model variants with 100 images
+of each. This dataset was originally used for the
+purpose of fine-grained visual categorization.
+• MIT Places2 is a scene-centric dataset with
+more than 10 million images consisting of over
+400 unique scene classes. However, in the ex-
+periments, we use the classes: rainforest, for-
+est, sky, and swamp as a background set for
+the Caltech-UCSD Birds dataset, and the Ox-
+ford 102 Flowers, we use the class: herb garden
+as a background set.
+4
+
+64
+64
+128
+t9
+64
+Input
+indno
+Image
+abe
+128
+128
+256
+128
+256
+256
+512
+256
+Conv 3 X 3
+Max pool2x 2
+Up-conv 2 × 2
+512
+512
+1024512
+Concatenation
+1024
+Finalconv 1x 1Figure 2: The proposed Self-supervised cut-and-paste GAN (SS-CPGAN)
+• Singapore Whole-sky IMaging CATegories
+(SWIMCAT) contains 784 images of a total
+of five categories: patterned clouds, clear sky,
+thick dark clouds, veil clouds, and thick white
+clouds.
+We use the SWIMCAT dataset as a
+background set for the FGCV dataset.
+We chose background datasets similar to the back-
+ground of the images from the foreground dataset.
+For the foreground datasets, we use Caltech-UCSD
+Birds (CUB) 200-2011 [19], Oxford 102 Flowers
+[20], and FGCV Aircraft (Airplanes) [21]. During
+the training, we do not utilize the masks available
+with datasets, Caltech-UCSD Birds and Flowers-
+102.
+For the background datasets, we use MIT
+Places2 [22], and SWIMCAT [23].
+4.2. Implementation Details
+Our implementation is based on the PyTorch
+framework. For training our models, we deploy a
+batch size of 16 and the Adam optimizer with an
+initial learning rate of 2.10−4.
+We use the unsu-
+pervised segmentation algorithm based on the Grab-
+Cut technique [24] for the self-supervision task.
+Then, we set the weighting parameters for the self-
+supervised term in the loss function according to
+the Intersection-Over-Union (IoU) score between
+the pseudo label (mask) and the predicted mask
+by the discriminator.
+Initially, when IoU < 0.2,
+the hyperparameter value is set to 0.5 to boost the
+model’s ability to learn useful representations from
+the pseudo label. When the 0.2 < IoU < 0.8, we
+refine the predicted mask using the hyperparameter
+value λ of 0.1. To avoid the pseudo labels compro-
+mising the predicted masks, we restrict the value λ to
+0 when the IoU > 0.8.
+4.3. Results
+We utilized the Fr´echet Inception Distance (FID)
+score and mean Intersection over Union (mIoU) met-
+ric for quantitative evaluation of our methods.
+In
+this work, we use the FID score on the datasets
+CUB2011, Oxford 102 Flowers, and FGCV Aircraft
+(see Table 3) to compare the SS-CPGAN model with
+the CPGAN model images spatially scaled to 64×64,
+128 × 128, and 256 × 256. And for the datasets with
+available ground truth masks, including CUB2011,
+5
+
+Self-
+Supervised
+Loss (Eq. 4)
+SS-CPGAN
+Encodel
+Foreground
+U-Net
+Generator
+Predicted Mask
+Composite
+Real/Fake
+Background
+U-Net
+Discriminator
+Itask
+Unsupervised
+Self-Supervised 
+Segmentation
+using
+GrabCut
+Pseudo Label
+Predicted Mask
+byDiscriminator
+Similarity
+Update Loss
+Check
+WeightTable 1: FID comparison of the proposed method with the base-
+line CPGAN model
+FID ↓
+Methods
+Image size
+Caltech
+UCSD-
+Bird 200
+FGCV-
+Aircraft
+Oxford
+102
+Flowers
+CPGAN
+64 x 64
+27.724
+43.353
+81.724
+128 x 128
+23.125
+39.674
+44.825
+256 x 256
+22.846
+44.825
+51.218
+SS-CPGAN 64 x 64
+23.751
+39.578
+63.343
+128 x 128
+16.893
+37.756
+54.982
+256 x 256
+13.422
+33.149
+49.181
+Table 2: mIOU comparison of the proposed method with the
+baseline CPGAN model
+mIoU ↑
+Methods
+Image size
+Caltech
+UCSD-
+Bird 200
+Oxford
+102
+Flowers
+w/o Self-Supervision 64 x 64
+0.537
+0.632
+128 x 128
+0.492
+0.674
+256 x 256
+0.484
+0.779
+Self-Supervision
+64 x 64
+0.571
+0.625
+128 x 128
+0.543
+0.719
+256 x 256
+0.518
+0.791
+Oxford 102 Flowers, we use the mIoU metric as
+shown in Table 2.
+In Figure 2, we report the FID scores over the
+training iterations. We show that our method sta-
+bilizes GAN training across all the datasets by al-
+lowing GAN training to converge faster and con-
+sistently improve performance throughout the train-
+ing. According to Table 3 and Figure 2, our method,
+SS-CPGAN, utilizing self-supervision outperforms
+the baseline method, CPGAN, on each dataset used.
+Furthermore, as shown in Fig. 3, the generated masks
+and composite images of our proposed SS-CPGAN
+are of superior quality. The standard classification
+based discriminator of CPGAN does not provide ef-
+fective guidance to the generator. During the train-
+ing, the standard discriminator is not encouraged to
+learn a more robust data representation. The classi-
+fication task learns only the representation based on
+the discriminative differences between real/fake im-
+ages and fails to give information on why the synthe-
+sized image looks fake. Notedly, our self-supervision
+based task assigned to the U-net based discrimina-
+tor provides the generator with global feedback (real
+or fake) as well as per-pixel feedback of the masks
+with the help of pseudo labels. The self-supervisory
+signals prevent the two scenarios for the generator
+which the standard discriminator fails to do, i.e., cre-
+ating constant masks of only all-zeros pixel values or
+all-ones pixel values. The enhanced discriminator of
+SS-CPGAN influences the generator to create high
+quality masks that are devoid of any such anoma-
+lies. As shown in Figure 3, the qualitative analysis of
+the proposed SS-CPGAN shows that the generated
+masks and composite images are of superior quality.
+4.4. Comparison with the state-of-the-art
+We compare our self-supervision based Cut-and-
+Paste GAN (SS-CPGAN) with state-of-the-art. As
+shown in Table 3, we report and compare the
+FID score on the Caltech UCSD-Bird 200 dataset.
+Specifically, the FID scores of StackGANv2 [25],
+OneGAN [26], LR-GAN [27], ELGAN [28], and
+FineGAN [29] are listed.
+The results in Table 3
+show that our method delivers better performance
+and outperforms the existing methods.
+LR-GAN
+[27] performed the worst, followed by the other
+methods. The low performance of layer-wise GANs
+[27] [28] is attributed to the fact that these methods
+are prone to degenerate during the training phase,
+with all the pixels being assigned as one compo-
+nent. In Table 4, we compare the performance of our
+method to the recent methods using the mIoU met-
+ric on Caltech UCSD-Bird 200 and Oxford flowers-
+102 respectively. In comparison to PerturbGAN [5],
+ContraCAM [30], ReDO [4], UISB [31] and IIC-
+seg [32], our method outperforms by a large mar-
+gin on Caltech UCSD-Bird 200 dataset. On the Ox-
+ford flowers-102 dataset, we perform better than the
+methods, ReDO [4], Kyriazi et. al [33] and Voynov
+et. al. [34]. Here, ReDO and Kyriazi et. al (2021)
+are unsupervised approaches whereas Voynov et. al
+(2021) is a weakly supervised approach to create seg-
+mentation maps. The ability to leverage pseudo la-
+bels in the training of Cut-and-Paste GAN assists in
+creating foreground masks of superior quality.
+6
+
+Figure 3: Visualization results with the proposed SS-CPGAN on the datasets: Oxford 102 Flowers (left), FGVC Aircraft (center),
+and Caltech-UCSD Birds (CUB) 200-2011 (right).
+Table 3: FID comparison of our proposed method SS-CPGAN
+with the state-of-art on Caltech UCSD-Bird 200 dataset
+Method
+FID
+StackGANv2
+21.4
+FineGAN
+23.0
+OneGAN
+20.5
+LR-GAN
+34.91
+ELGAN
+15.7
+SS-CPGAN
+13.42
+Table 4: Quantitative comparison of the segmentation perfor-
+mance of our method SS-CPGAN with the state-of-art
+Dataset
+Method
+mIoU
+Caltech UCSD-Bird 200
+PerturbGAN
+0.380
+ContraCAM
+0.460
+ReDO
+0.426
+UISB
+0.442
+IIC-seg
+0.365
+SS-CPGAN
+0.571
+Oxford 102 flowers
+ReDO
+0.764
+Kyriazi et. al.
+0.541
+Voynov et al.
+0.540
+SS-CPGAN
+0.791
+5. Conclusion
+In this work, we proposed a novel Self-Supervised
+Cut-and-Paste GAN method to learn object seg-
+mentation. Specifically, we unify the cut-and-paste
+adversarial training with the proposed segmenta-
+tion based self-supervision learning.
+Unlike the
+existing transformation based self-supervised meth-
+ods, our method improves the discriminator’s rep-
+resentation ability by enhancing structure learning
+with global and local feedback from the synthesized
+masks. Furthermore, SS-CPGAN overcomes the is-
+sue of unwanted trivial solutions (generating con-
+stant masks of only all-zeros or all-ones pixel values)
+that plagues the generator. The experimental results
+show that our approach generates superior quality
+images and achieves promising results on the bench-
+mark datasets.
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+

ONAyT4oBgHgl3EQfgvjr/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf,len=463
+page_content='Self-Supervised Object Segmentation with a Cut-and-Pasting GAN  Kunal Chaturvedi, Ali Braytee, Jun Li, Mukesh Prasad  aSchool of Computer Science, University of Technology Sydney, Ultimo, 2007, NSW, Australia  Abstract  This paper proposes a novel self-supervised based Cut-and-Paste GAN to perform foreground object seg-  mentation and generate realistic composite images without manual annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' We accomplish this goal by  a simple yet effective self-supervised approach coupled with the U-Net based discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The proposed  method extends the ability of the standard discriminators to learn not only the global data representations  via classification (real/fake) but also learn semantic and structural information through pseudo labels created  using the self-supervised task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The proposed method empowers the generator to create meaningful masks  by forcing it to learn informative per-pixel as well as global image feedback from the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Our  experiments demonstrate that our proposed method significantly outperforms the state-of-the-art methods on  the standard benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Keywords: Generative adversarial networks, Self-supervised learning, Cut-and-Paste, Segmentation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Introduction  Generative adversarial networks (GANs) [1] have  become a popular class of image synthesis meth-  ods due to their demonstrated ability to create high-  dimensional samples with desired data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  The primary objective of GANs is to generate di-  verse, high-quality images while also ensuring the  stability of GAN training [2] [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' GAN consists of  generator and discriminator networks trained in an  adversarial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The generator attempts to syn-  thesize the real data distribution to fool the discrim-  inator, whereas the discriminator’s goal is to distin-  guish between the generator’s real and fake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' In  image segmentation, several compositional genera-  tive models have been proposed [4, 5, 6, 7] , where  the generator creates a synthesized composite image  by copying the object from one image and pasting  it in another to fool the discriminator about think-  ing the synthesized composite image is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' But,  the generator may not perform any segmentation,  and the background may look realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Therefore,  Email address: Kunal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='Chaturvedi, Ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='Braytee,  Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='Li, Mukesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='Prasad@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='au (Mukesh Prasad)  for effective training, the discriminator needs to pro-  vide the generator with informative learning signals  by learning relevant semantics and structures of the  data that may result in more effective generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  However, the current state-of-the-art GANs employ  discriminators based on the classification network,  which learn only a single discriminative signal such  as the difference between real and fake images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' In  such a non-stationary environment, the generator be-  comes prone to catastrophic forgetting and may lead  to training instability or mode collapse [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  To address the aforementioned issues, additional  discriminatory signals are required to guide the train-  ing mechanism and assist the generator in producing  high-quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' This can be accomplished by  increasing the capacity of the discriminator with aux-  iliary tasks and signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' These auxiliary tasks on the  labeled datasets resist the forgetting issues and im-  prove the training stability of GANs, but it suffers  with unlabeled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Recently, self-supervised  learning has been explored on numerous GANs  methods [8],[9],[10],[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The self-supervised tasks  provide the learning environment with additional  guidance to the standard training mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Most  of the recent self-supervision methods based GANs  Preprint submitted to Nuclear Physics B  January 3, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='00366v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='CV]  1 Jan 2023  use auxiliary tasks based on transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' For ex-  ample, SS-GAN developed by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' [8] uses  rotation prediction as an auxiliary task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' In FX-GAN,  Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  [10] use the pretext task of predic-  tion on corrupted real images, and in LT-GAN [9],  the authors use distinguishing GAN-induced trans-  formation as a pretext task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' However, the goals of  these transformation-based self-supervised tasks are  inconsistent with the GAN’s goal of mimicking the  real data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Moreover, this problem ampli-  fies when the generator’s task is to construct segmen-  tation masks from the foreground images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  In order to maintain an enriched real data repre-  sentation and improve the quality of generated seg-  mentation masks, we propose a Self-Supervised Cut-  and-Paste GAN (SS-CPGAN) based on U-net archi-  tecture [12], which unifies cut-and-paste adversar-  ial training with a self-supervised task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  It allows  the discriminator to learn both local and global dif-  ferences between real and fake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  In contrast  to the existing transformation based self-supervision  methods, our self-supervision learning method cre-  ates pseudo labels using unsupervised segmentation  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Then, it simultaneously forces the discrim-  inator to provide the generator with global feedback  (real or fake) and the per-pixel feedback of the syn-  thesized images with the help of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  To sum up, our main contributions are:  We propose a novel Self-Supervised Cut-and-  Paste GAN (SS-CPGAN), that unifies cut-and-  paste adversarial training with a segmentation-  based self-supervised task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' SS-CPGAN lever-  age unlabeled data to maximize segmentation  performance and generate highly realistic com-  posite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  The proposed self-supervised task in SS-  CPGAN improves the discriminator’s represen-  tation ability by enhancing structure learning  with global and local feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' This enables the  generator with additional discriminatory sig-  nals to achieve superior results and stabilize the  training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  We perform a comprehensive analysis on the  benchmark datasets and compare our proposed  method with the baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Related Works  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Unsupervised Object Segmentation via GANs  Unsupervised segmentation using GANs is an im-  portant topic in research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Several works [4, 5, 6, 7]  investigate the use of compositional generative mod-  els to obtain high quality segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Copy-  pasting GAN [7] performs unsupervised object dis-  covery by extracting foreground objects and then  copying and pasting them onto the different back-  grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Similarly, PerturbGAN [5] generates a  foreground mask along with a background and fore-  ground image in an adversarial manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Recently,  Abdal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  (2021) [6] propose a method to use  an alpha network that includes two pretrained gen-  erators and a discriminator based on the StyleGAN  to generate high quality masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  These methods  learn object segmentation without needing to use  annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' However, they are prone to degener-  ate solutions or other trivial cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  For example,  the generator may not perform any segmentation,  and the background looks realistic or the generator  may segment foreground masks consisting of all-  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' To avoid such problems, special care needs  to be taken while training the compositional genera-  tive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Copy-pasting GAN uses anti-shortcut,  border-zeroing, blur, and grounded fakes to prevent  trivial solutions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  PerturbGAN avoids such so-  lutions by randomly shifting object segments rel-  ative to the background [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  However, Abdal et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' (2021) [6] make several changes to the original  StyleGAN and use truncation trick along with regu-  larization to avoid degenerate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Self-supervised learning  Self-supervised learning learns useful feature rep-  resentations from data with the help of pretext tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Recently, many pretext tasks coupled with adversar-  ial training have been introduced [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The motiva-  tion for using self-supervised learning is to: (1) pre-  vent discriminator forgetting [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' (2) improve train-  ing stability [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' (3) and ensure high quality of im-  ages generated [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The self-supervision techniques  rely on pretext tasks on geometric transformations  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=', prediction on rotated images[8], corrupted im-  ages [10], GAN-induced transformations [9], or a  deshuffling task that predicts the shuffled orders [16])  2  to increase the discriminator’s representation power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Unlike the aforementioned methods, we incorporate  segmentation-based self-supervised learning coupled  with the Cut-and-Paste GAN to obtain high-quality  segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Most importantly, with our  self-supervised approach, no extra care is needed to  deal with the trivial solutions prevalent in composi-  tional generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Method  In this section, we first present the standard ter-  minology of adversarial training and the encoder-  decoder based discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  We then introduce  our Self-Supervised Cut-and-Paste GAN built upon  the cut-and-paste adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  The uni-  fied framework with the segmentation-based self-  supervised task encourages the generator to empha-  size local and global structures while synthesizing  masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Adversarial Training  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' 2, we build a generative model  in which the generator takes the foreground image  as the input and generates a composite image using  a combination of the predicted mask, source fore-  ground image and the background image to fool the  discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Formally, we define the input fore-  ground source image as If ∈ Pdata and background  image as Ib ∈ Pdata where Pdata denotes the set of  input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Now, we define a generator (G) that  is trained in an adversarial manner against the dis-  criminator (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  During the training process, the  generator predicts a segmentation mask defined by  mg(If) = G(I f) where mg(I f) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Then, using  the predicted mask mg(If), foreground source image  If, and background image Ib, we define composite  image as follows  IC = mg(I f)I f + (1 − mg(I f))Ib  (1)  The discriminator’s objective is to classify the  composite image as real or fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' As a result, the stan-  dard objective of the discriminator and the generator  of the CPGAN is defined as follows  LD = max  D  E  �  log D(If) + log(1 − D(IC))  �  (2)  LG = min  G E �log D(IC)�  (3)  The discriminator works as a classification network  that is restricted to learn only through the discrim-  inative differences between the real and fake sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Thus, the discriminator fails to provide any  useful information to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Therefore, we  use an encoder-decoder-based discriminator network  with self-supervised learning to mitigate this prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Encoder-Decoder based Discriminator  In this work, we replace standard classification-  based discriminator with the U-net based discrimi-  nator [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The U-net is an encoder-decoder-based  architecture that consists of a network of convolu-  tional layers, skip connections for semantic segmen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' It was initially proposed for biomedical im-  age segmentation, which achieved precise segmen-  tation results with few training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Further, it  demonstrates good results in other applications, in-  cluding ArcGIS [17], remote sensing [18], and oth-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Its architecture (see Figure 1) is symmetric and  consists of two paths, an Encoder that extracts spatial  features from the input image (downscaling process),  and a Decoder that constructs the segmentation map  from the extracted feature maps (upscaling process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  We use the encoder part of the U-net as the standard  classification-based discriminator that performs the  binary decision on real/fake composite images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' And  the decoder part of the U-net architecture is utilized  by the self-supervised task to give per-pixel feedback  of the synthesized images with the help of pseudo la-  bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' This allows the discriminator to learn both rel-  evant local and global differences between real/fake  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Self-Supervised  Cut-and-Paste  GAN  (SS-  CPGAN)  To improve the representation learning ability of  the CPGAN, the discriminator must be able to learn  semantic as well as structural information from the  synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Therefore, we focus our ap-  proach on using self-supervised learning to build  comprehensive representations for the CPGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' In  this work, we employ a segmentation based self-  supervised task, with the primary goal of enabling  3  Figure 1: An overview of U-net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The different ar-  rows denote the different operations used in the encode-decoder  based architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  the discriminator with enhanced learned features that  ultimately empower the generator to create consis-  tent and structurally coherent masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  The pseudo  segmentation masks mUS(I f) ∈ [0, 1] are created  using graph-based unsupervised segmentation algo-  rithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' These masks obtained by the GrabCut  technique acts as a good prior for the U-net based  discriminator (see, Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Here, the discriminator  performs two important tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=', 1) classification  of real/fake compositing images and 2) performing  per-pixel based classification on I f ∈ Pdata to gener-  ate segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Given the self-supervised  pseudo labels, we train the discriminator for accu-  rate pixel-level prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The introduction of self-  supervisory signals empowers the discriminator by  enhancing its localization ability and forces the dis-  criminator it to learn useful semantic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  This mechanism enables the generator to achieve op-  timized results and makes the training process more  stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Formally, we define I f  ∈ Pdata as the source  image containing foreground object, and Pdata de-  notes the set of input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Given a source fore-  ground image I f, we create pseudo label denoted by  mUS(I f) ∈ [0, 1], using an unsupervised segmenta-  tion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Then, we define mw(IC) ∈ [0, 1] as  the pixel-wise segmentation mask produced by the  decoder of the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Hereafter, we optimize  the overall discriminator loss function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' 5) by  augmenting a new self-supervision based loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' 4)  Lsel f−supervised = L  �  mw(IC), 1 − mUS(If)  �  (4)  L′  D = LD + λLsel f−supervised  (5)  where L is the cross-entropy loss, and λ denotes the  loss weight for the self-supervision based loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' This  hyperparameter is updated based on the compari-  son between mUS(I f) and mw(IC), using intersection-  over-union (IoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The details of the hyperparameter  chosen are explained in the implementation details  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The framework of the self-supervised learn-  ing is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Experimentation  This section discusses the implementation details  of the proposed method and an extensive set of ex-  periments on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Datasets  We utilize five different datasets for the foreground  and background set to train our SS-CPGAN as de-  scribed below:  Caltech-UCSD Birds (CUB) 200-2011 is a fre-  quently used benchmark for unsupervised im-  age segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' It consists of 11,788 images  from 200 birds species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Oxford 102 Flowers consists of 8,189 images  from 102 flower classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  FGVC Aircraft (Airplanes) contains 102 dif-  ferent aircraft model variants with 100 images  of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' This dataset was originally used for the  purpose of fine-grained visual categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  MIT Places2 is a scene-centric dataset with  more than 10 million images consisting of over  400 unique scene classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' However, in the ex-  periments, we use the classes: rainforest, for-  est, sky, and swamp as a background set for  the Caltech-UCSD Birds dataset, and the Ox-  ford 102 Flowers, we use the class: herb garden  as a background set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  4  64  64  128  t9  64  Input  indno  Image  abe  128  128  256  128  256  256  512  256  Conv 3 X 3  Max pool2x 2  Up-conv 2 × 2  512  512  1024512  Concatenation  1024  Finalconv 1x 1Figure 2: The proposed Self-supervised cut-and-paste GAN (SS-CPGAN)  Singapore Whole-sky IMaging CATegories  (SWIMCAT) contains 784 images of a total  of five categories: patterned clouds, clear sky,  thick dark clouds, veil clouds, and thick white  clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  We use the SWIMCAT dataset as a  background set for the FGCV dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  We chose background datasets similar to the back-  ground of the images from the foreground dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  For the foreground datasets, we use Caltech-UCSD  Birds (CUB) 200-2011 [19], Oxford 102 Flowers  [20], and FGCV Aircraft (Airplanes) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' During  the training, we do not utilize the masks available  with datasets, Caltech-UCSD Birds and Flowers-  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  For the background datasets, we use MIT  Places2 [22], and SWIMCAT [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Implementation Details  Our implementation is based on the PyTorch  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' For training our models, we deploy a  batch size of 16 and the Adam optimizer with an  initial learning rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  We use the unsu-  pervised segmentation algorithm based on the Grab-  Cut technique [24] for the self-supervision task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Then, we set the weighting parameters for the self-  supervised term in the loss function according to  the Intersection-Over-Union (IoU) score between  the pseudo label (mask) and the predicted mask  by the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Initially, when IoU < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='2,  the hyperparameter value is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='5 to boost the  model’s ability to learn useful representations from  the pseudo label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' When the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='2 < IoU < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='8, we  refine the predicted mask using the hyperparameter  value λ of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' To avoid the pseudo labels compro-  mising the predicted masks, we restrict the value λ to  0 when the IoU > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Results  We utilized the Fr´echet Inception Distance (FID)  score and mean Intersection over Union (mIoU) met-  ric for quantitative evaluation of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  In  this work, we use the FID score on the datasets  CUB2011, Oxford 102 Flowers, and FGCV Aircraft  (see Table 3) to compare the SS-CPGAN model with  the CPGAN model images spatially scaled to 64×64,  128 × 128, and 256 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' And for the datasets with  available ground truth masks, including CUB2011,  5  Self-  Supervised  Loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' 4)  SS-CPGAN  Encodel  Foreground  U-Net  Generator  Predicted Mask  Composite  Real/Fake  Background  U-Net  Discriminator  Itask  Unsupervised  Self-Supervised  Segmentation  using  GrabCut  Pseudo Label  Predicted Mask  byDiscriminator  Similarity  Update Loss  Check  WeightTable 1: FID comparison of the proposed method with the base-  line CPGAN model  FID ↓  Methods  Image size  Caltech  UCSD-  Bird 200  FGCV-  Aircraft  Oxford  102  Flowers  CPGAN  64 x 64  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='724  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='353  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='724  128 x 128  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='125  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='674  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='825  256 x 256  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='846  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='825  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='218  SS-CPGAN 64 x 64  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='751  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='578  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='343  128 x 128  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='893  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='756  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='982  256 x 256  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='422  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='149  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='181  Table 2: mIOU comparison of the proposed method with the  baseline CPGAN model  mIoU ↑  Methods  Image size  Caltech  UCSD-  Bird 200  Oxford  102  Flowers  w/o Self-Supervision 64 x 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='537  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='632  128 x 128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='492  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='674  256 x 256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='484  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='779  Self-Supervision  64 x 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='571  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='625  128 x 128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='543  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='719  256 x 256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='518  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='791  Oxford 102 Flowers, we use the mIoU metric as  shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  In Figure 2, we report the FID scores over the  training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' We show that our method sta-  bilizes GAN training across all the datasets by al-  lowing GAN training to converge faster and con-  sistently improve performance throughout the train-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' According to Table 3 and Figure 2, our method,  SS-CPGAN, utilizing self-supervision outperforms  the baseline method, CPGAN, on each dataset used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' 3, the generated masks  and composite images of our proposed SS-CPGAN  are of superior quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The standard classification  based discriminator of CPGAN does not provide ef-  fective guidance to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' During the train-  ing, the standard discriminator is not encouraged to  learn a more robust data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The classi-  fication task learns only the representation based on  the discriminative differences between real/fake im-  ages and fails to give information on why the synthe-  sized image looks fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Notedly, our self-supervision  based task assigned to the U-net based discrimina-  tor provides the generator with global feedback (real  or fake) as well as per-pixel feedback of the masks  with the help of pseudo labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The self-supervisory  signals prevent the two scenarios for the generator  which the standard discriminator fails to do, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=', cre-  ating constant masks of only all-zeros pixel values or  all-ones pixel values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The enhanced discriminator of  SS-CPGAN influences the generator to create high  quality masks that are devoid of any such anoma-  lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' As shown in Figure 3, the qualitative analysis of  the proposed SS-CPGAN shows that the generated  masks and composite images are of superior quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Comparison with the state-of-the-art  We compare our self-supervision based Cut-and-  Paste GAN (SS-CPGAN) with state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' As  shown in Table 3, we report and compare the  FID score on the Caltech UCSD-Bird 200 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Specifically, the FID scores of StackGANv2 [25],  OneGAN [26], LR-GAN [27], ELGAN [28], and  FineGAN [29] are listed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  The results in Table 3  show that our method delivers better performance  and outperforms the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  LR-GAN  [27] performed the worst, followed by the other  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The low performance of layer-wise GANs  [27] [28] is attributed to the fact that these methods  are prone to degenerate during the training phase,  with all the pixels being assigned as one compo-  nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' In Table 4, we compare the performance of our  method to the recent methods using the mIoU met-  ric on Caltech UCSD-Bird 200 and Oxford flowers-  102 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' In comparison to PerturbGAN [5],  ContraCAM [30], ReDO [4], UISB [31] and IIC-  seg [32], our method outperforms by a large mar-  gin on Caltech UCSD-Bird 200 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' On the Ox-  ford flowers-102 dataset, we perform better than the  methods, ReDO [4], Kyriazi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' al [33] and Voynov  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Here, ReDO and Kyriazi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' al (2021)  are unsupervised approaches whereas Voynov et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' al  (2021) is a weakly supervised approach to create seg-  mentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The ability to leverage pseudo la-  bels in the training of Cut-and-Paste GAN assists in  creating foreground masks of superior quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  6  Figure 3: Visualization results with the proposed SS-CPGAN on the datasets: Oxford 102 Flowers (left), FGVC Aircraft (center),  and Caltech-UCSD Birds (CUB) 200-2011 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Table 3: FID comparison of our proposed method SS-CPGAN  with the state-of-art on Caltech UCSD-Bird 200 dataset  Method  FID  StackGANv2  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='4  FineGAN  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='0  OneGAN  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='5  LR-GAN  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='91  ELGAN  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='7  SS-CPGAN  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='42  Table 4: Quantitative comparison of the segmentation perfor-  mance of our method SS-CPGAN with the state-of-art  Dataset  Method  mIoU  Caltech UCSD-Bird 200  PerturbGAN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='380  ContraCAM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='460  ReDO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='426  UISB  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='442  IIC-seg  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='365  SS-CPGAN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='571  Oxford 102 flowers  ReDO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='764  Kyriazi et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='541  Voynov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='540  SS-CPGAN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='791  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Conclusion  In this work, we proposed a novel Self-Supervised  Cut-and-Paste GAN method to learn object seg-  mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Specifically, we unify the cut-and-paste  adversarial training with the proposed segmenta-  tion based self-supervision learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content='  Unlike the  existing transformation based self-supervised meth-  ods, our method improves the discriminator’s rep-  resentation ability by enhancing structure learning  with global and local feedback from the synthesized  masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' Furthermore, SS-CPGAN overcomes the is-  sue of unwanted trivial solutions (generating con-  stant masks of only all-zeros or all-ones pixel values)  that plagues the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
+page_content=' The experimental results  show that our approach generates superior quality  images and achieves promising results on the bench-  mark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfgvjr/content/2301.00366v1.pdf'}
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+arXiv:2301.03822v1  [cs.IT]  10 Jan 2023
+1
+Transmission Design for Active RIS-Aided
+Simultaneous Wireless Information and Power
+Transfer
+Hong Ren, Member, IEEE, Zhiwei Chen, Guosheng Hu, Zhangjie Peng,
+Cunhua Pan, Member, IEEE, and Jiangzhou Wang, Fellow, IEEE
+Abstract—Reconfigurable intelligent surface (RIS) is a revolu-
+tionary technology to enhance both the spectral efficiency and
+energy efficiency of wireless communication systems. However,
+most of the existing contributions mainly focused on the study of
+passive RIS, which suffers from the “double fading” effect. On
+the other hand, active RIS, which is equipped with amplifiers, can
+effectively address this issue. In this paper, we propose an active
+RIS-aided simultaneous wireless information and power transfer
+(SWIPT) system. Specifically, we maximize the weighted sum
+rate of the information receivers, subject to the minimum power
+received at all energy receivers, amplification power constraint
+at the active RIS, and the maximum transmit power constraint
+at the base station (BS). By adopting alternating optimization
+framework, suboptimal solutions are obtained. Simulation results
+show that the active RIS-aided SWIPT system has higher
+performance gain with the same power budget.
+Index Terms—Reconfigurable intelligent surface (RIS), active
+RIS, wireless information and power transfer (SWIPT).
+I. INTRODUCTION
+Reconfigurable intelligent surface (RIS), composed of a
+large number of reflecting elements, has received extensive
+research attention in both academia and industry [1]–[4].
+Specifically, RIS can dynamically adjust the electromagnetic
+properties of the reflecting elements in a programmable way,
+and then reconfigure the wireless propagation environment in
+a desired way [5], [6].
+However, most of the existing contributions mainly focused
+on the passive RIS-aided communication systems, which suffer
+from the “double fading” effect [7]. To address this issue,
+active RIS has been proposed in [8], [9], which is equipped
+This work was supported in part by the National Natural Science Foun-
+dation of China (62201137), in part by the Natural Science Foundation
+of Shanghai under Grant 22ZR1445600, in part by the National Natural
+Science Foundation of China under Grant 61701307, in part by the open
+research fund of National Mobile Communications Research Laboratory,
+Southeast University under Grant 2018D14, and in part by the National
+Natural Science Foundation of China (62101128) and Basic Research Project
+of Jiangsu Provincial Department of Science and Technology (BK20210205).
+(Corresponding authors: Zhangjie Peng and Zhiwei Chen.)
+Hong Ren and Cunhua Pan are with the National Mobile Communications
+Research Laboratory, Southeast University, Nanjing 210096, China. (e-mail:
+[email protected]; [email protected]).
+Zhiwei Chen is with the College of Information, Mechanical and Electrical
+Engineering, Shanghai Normal University, Shanghai 200234, China (e-mail:
+[email protected]).
+Guosheng Hu is with the Shanghai Technical Institute of Electronics &
+Information, Shanghai 201411, China (e-mail: [email protected]).
+Zhangjie Peng is with the College of Information, Mechanical, and Electri-
+cal Engineering, Shanghai Normal University, Shanghai 200234, China, also
+with the National Mobile Communications Research Laboratory, Southeast
+University, Nanjing 210096, China, and also with the Shanghai Engineering
+Research Center of Intelligent Education and Bigdata, Shanghai Normal
+University, Shanghai 200234, China (e-mail: [email protected]).
+Jiangzhou Wang is with the School of Engineering, University of Kent,
+CT2 7NT Canterbury, U.K. (e-mail: [email protected]).
+with some amplifiers. Different from multiple input multiple
+output (MIMO) relay with high power cost and additional
+time/frequency resource, e.g., amplify and forward (AF) re-
+lay, the active RIS basically inherits the hardware structure
+of passive RIS, while equipping with a set of low-power
+reflection-type amplifiers. As a result, active RIS can not only
+tune the phase of the reflected signals, but also amplify the
+power of the reflected signals. Recently, the authors of [7] have
+rigorously demonstrated that the active RIS-aided single-input
+single-output (SISO) system is superior to the passive RIS in
+terms of the achievable date rate when both systems have the
+same power budget. On the other hand, simultaneous wireless
+information and power transfer (SWIPT) is envisioned as
+a promising technology in future internet of things (IoT)
+[10]–[12]. The authors of [10] investigated a passive RIS-
+aided SWIPT system and showed that the passive RIS can
+enhance the data rates at the information receivers (IRs), while
+ensuring the minimum power requirements of the received
+power at the energy receivers (ERs). In [11], the authors
+studied an optimization problem of maximizing the minimum
+rate of the IRs in the passive RIS-aided SWIPT system with
+imperfect channel state information (CSI). And the authors
+of [12] aimed to maximize the data rate by proposing a
+joint time-switching and phase-shifting solution for passive
+RIS-assisted SWIPT communications. However, the above
+literatures [10]–[12] about passive RIS-aided SWIPT systems
+suffer from the “double fading”, and for the SWIPT systems
+with requirements for both energy reception and information
+reception, there are no literature investigating whether the
+active RIS with low-power amplifiers performs better than the
+MIMO relay with complex hardware structure.
+Against the above background, we consider an active RIS-
+aided SWIPT downlink system, and aim to maximize the
+downlink weighted sum rate (WSR). Different from the pas-
+sive RIS [10]–[12], it is noted that the active RIS introduces
+a new optimization variable due to its ability to amplify the
+reflected signal, generates the non-ignorable thermal noise, and
+adds an output signal power constraint of the active RIS, which
+makes the optimization problem more challenging. To solve
+the non-convex problem, our contributions of this work are
+summarized as follows
+1) By considering the active RIS-aided SWIPT system, we
+aim to maximize the downlink weighted sum rate (WSR) of
+the IRs, by jointly optimizing the transmit beamforming at
+the base station (BS) and the reflecting coefficients at the
+active RIS, subject to the minimum power harvested at all
+ERs, amplification power constraint at the active RIS, and the
+
+2
+Active RIS
+�
+hr,l
+BS
+IRs
+gd,k
+hd,l
+gr,k
+ERs
+Q
+IR
+r,r l
+Fig. 1. System model.
+maximum transmit power constraint at the BS.
+2) By adopting alternating optimization (AO) framework,
+we transform the objective function by fractional programming
+(FP) method, and utilize the first-order Taylor approximation
+to linearize the non-convex constraint of the active RIS am-
+plification power. Then, we effectively solve the subproblems
+and obtain the suboptimal solutions.
+3) Simulation results show that the active RIS can achieve
+higher downlink WSR than the passive RIS/AF relay in an
+SWIPT system [10]–[13] with the same power budget. And
+as the location of the RIS is closer to the locations of the
+ERs, it can achieve higher downlink WSR. In addition, the
+appropriate number of RIS reflecting elements is enough to
+enable the SWIPT system to achieve good performance.
+II. SYSTEM MODEL
+As shown in Fig. 1, we consider an active RIS-aided mul-
+tiuser multiple input single output (MISO) downlink system,
+where an RIS with L reflecting elements is deployed to assist
+SWIPT. The system is composed of a BS with N antennas,
+KI single-antenna IRs, and KE single-antenna ERs.
+The signal transmitted from the BS is expressed as
+t =
+KI
+�
+k=1
+wksIR
+k
++ v,
+(1)
+where wk ∈ CN×1 is the beamforming vector for the k-th IR,
+sIR
+k
+∼ CN(0, 1), k ∈ {1, · · ·, KI} is the transmit information
+symbol for the k-th IR, and v ∈ CN×1 ∼ CN(0, V) is the
+energy signal vector, where V is the covariance matrix of the
+energy signal vector.
+The channels spanning from the BS to the active RIS, from
+the BS to the k-th IR, from the BS to the i-th ER, from the RIS
+to the k-th IR, and from the RIS to the i-th ER are denoted as
+Q ∈ CL×N, gd,k ∈ CN×1, hd,i ∈ CN×1, gr,k ∈ CL×1, and
+hr,i ∈ CL×1, respectively.
+The reflected and amplified signal at the active RIS can be
+modeled as follows
+tr = Φ (Qt + nRIS) ,
+(2)
+where Φ = diag
+�
+a1ejφ1, · · · , alejφl, · · · , aLejφL�
+denotes the
+reflection matrix of the active RIS, φl and al are the phase
+shift and the amplitude of the l-th element, respectively. The
+thermal noise generated by the active RIS cannot be neglected,
+and nRIS ∼ CN(0, δ2
+rI) denotes the thermal noise of the active
+RIS.
+The received signal at the k-th IR can be written as
+yIR,k = gH
+d,kt + gH
+r,ktr + nIR
+= gH
+k t + gH
+r,kΦnRIS + nIR
+= gH
+k wksIR
+k +
+KI
+�
+i=1,i̸=k
+gH
+k wisIR
+i
++gH
+k v+ gH
+r,kΦnRIS + nIR,
+(3)
+where gH
+k ≜ gH
+d,k +gH
+r,kΦQ, and nIR ∼ CN(0, δ2
+IR) is the ad-
+ditive white Gaussian noise (AWGN). Unlike the passive RIS
+model [10]–[12], the active RIS consisting of amplifiers has
+the ability to amplify the power of the reflected signal. Thus,
+the reflection matrix Φ has non-unity amplitude components.
+Then, the signal-to-interference-plus-noise ratio (SINR) of
+the k-th IR is expressed as
+γk =
+|gH
+k wk|2
+KI
+�
+i=1,i̸=k
+|gH
+k wi|2+gH
+k Vgk + δ2r∥gH
+r,kΦ∥2+δ2
+IR
+.
+(4)
+Thus, the rate at the k-th IR is expressed as
+Rk = log2 (1 + γk) .
+(5)
+The received signal at the i-th ER can be written as
+yER,i = hH
+d,it + hH
+r,itr + nER
+= hH
+i t + hH
+r,iΦnRIS + nER
+=
+KI
+�
+k=1
+hH
+i wksIR
+k
++ hH
+i v + hH
+r,iΦnRIS + nER,
+(6)
+where hH
+i ≜ hH
+d,i + hH
+r,iΦQ, and nER ∼ CN(0, δ2
+ER) is the
+AWGN at the i-th ER. Considering the fact that both the
+data and energy signals transmitted by the BS are carried by
+the beamforming, the harvested power at the i-th ER while
+ignoring the AWGN power is given by
+Ei = ηi
+� KI
+�
+k=1
+|hH
+i wk|2+hH
+i Vhi + δ2
+r∥hH
+r,iΦ∥2
+�
+,
+(7)
+where ηi is the energy harvesting efficiency of the i-th ER.
+III. PROBLEM FORMULATION
+To satisfy the requirements of both IRs and ERs, we
+consider an optimization problem of maximizing the WSR of
+all IRs, while satisfying the harvested power requirements of
+all ERs, subject to the power constraints at the BS and active
+RIS. Thus, the optimization problem is formulated as
+max
+{wk},V,Φ
+KI
+�
+k=1
+αkRk
+(8a)
+s.t.
+∥ΦQt∥2+δ2
+r∥Φ∥2⩽ P act
+RIS,
+(8b)
+∥t∥2⩽ P act
+BS ,
+(8c)
+Ei ⩾ Pi, i ∈ {1, · · · , KE},
+(8d)
+where αk is the weighting factor of the k-th IR, P act
+RIS is the
+output signal power of the active RIS, P act
+BS is the transmit
+power limit at the BS and Pi is the minimum harvested power
+threshold for the i-th ER.
+Due to the fact that variables {{wk}, V, Φ} are coupled
+together in the objective function of Problem (8), it is chal-
+lenging to solve Problem (8). We then exploit the fractional
+programming (FP) [14] method to decouple the objective
+function of Problem (8), and adopt the Alternate Optimization
+(AO) algorithm to obtain the solutions in the next subsection.
+A. FP method
+Firstly, we use the FP method to transform the ob-
+jective function. By introducing auxiliary variables ˜γ
+=
+[˜γ1, · · · , ˜γKI]T ∈ CKI×1, the objective function of Problem
+(8) is equivalent to
+
+3
+fa(˜γ, wk, V, Φ) =
+KI
+�
+k=1
+αklog (1 + ˜γk) −
+KI
+�
+k=1
+αk˜γk
++
+KI
+�
+k=1
+αk(1 + ˜γk)|gH
+k wk|2
+KI
+�
+i=1
+|gH
+k wi|2+gH
+k Vgk + δ2r∥gH
+r,kΦ∥2+δ2
+IR
+.
+(9)
+We adopt the AO framework to obtain the optimal solutions.
+For fixed variables {{wk}, V, Φ}, by setting ∂fa/∂˜γk to zero,
+the optimal ˜γopt
+k
+is obtained as
+˜γopt
+k
+= γk, k ∈ {1, · · ·, KI}.
+(10)
+Then, we fix ˜γk and define a new function as
+fb(˜γ, wk, V, Φ)
+=
+KI
+�
+k=1
+αk(1 + ˜γk)|gH
+k wk|2
+KI
+�
+i=1
+|gH
+k wi|2+gH
+k Vgk + δ2r∥gH
+r,kΦ∥2+δ2
+IR
+.
+(11)
+By introducing auxiliary variables ρ = [ρ1, · · · , ρKI]T ∈
+CKI×1 and adopting the quadratic transform [14], we further
+recast fb as
+fc(ρ, ˜γ, wk, V, Φ) = 2
+KI
+�
+k=1
+�
+αk(1 + ˜γk)R{ρ∗
+kgH
+k wk}
+−
+KI
+�
+k=1
+|ρk|2
+� KI
+�
+i=1
+|gH
+k wi|2+gH
+k Vgk + δ2
+r∥gH
+r,kΦ∥2+δ2
+IR
+�
+.
+(12)
+Similarly, by setting ∂fc/∂ρk to zero, we obtain the optimal
+ρopt
+k
+as
+ρopt
+k
+=
+�
+αk(1 + ˜γk)gH
+k wk
+KI
+�
+i=1
+|gH
+k wi|2+gH
+k Vgk + δ2r∥gH
+r,kΦ∥2+δ2
+IR
+,
+k ∈ {1, · · · , KI}.
+(13)
+After obtaining the above optimal auxiliary variables, in the
+next subsection, we then focus on optimizing {{wk}, V, Φ},
+given {ρ, ˜γ}.
+B. Optimizing wk and V
+By defining W ≜ [wT
+1 , · · · , wT
+KI]T, for fixed variables
+{ρ, ˜γ, Φ}, Problem (8) is expressed as
+max
+W,V
+R{bHW}−WHA1W−
+KI
+�
+k=1
+|ρk|2Tr{gkgH
+kV} (14a)
+s.t.
+WHBW + Tr{QHΦHΦQV} ⩽ ˆP act
+RIS,
+(14b)
+∥W∥2+Tr{V} ⩽ P act
+BS ,
+(14c)
+WHDiW+Tr{hihH
+i V}⩾P
+′
+i , i∈{1,· · ·, KE}, (14d)
+V ⪰ 0,
+(14e)
+where
+b = [bT
+1 , bT
+2 , · · · , bT
+KI]T, bH
+k = 2
+�
+αk(1 + ˜γk)ρ∗
+kgH
+k ,
+(15)
+A1 = IKI ⊗
+KI
+�
+i=1
+|ρi|2gigH
+i ,
+(16)
+B = IKI ⊗ QHΦHΦQ,
+(17)
+ˆP act
+RIS = P act
+RIS − δ2
+r∥Φ∥2,
+(18)
+Di = IKI ⊗ hihH
+i ,
+(19)
+P
+′
+i = Pi
+ηi
+− δ2
+r∥hH
+r,iΦ∥2.
+(20)
+However, it is noted that the constraint in (14d) is non-
+convex, which makes Problem (14) still intractable. We then
+approximate the constraint (14d) by its first-order Taylor
+expansion as
+WHDiW ⩾ 2R{WH(t)DiW} − WH(t)DiW(t),
+(21)
+where WH(t) is the beamforming matrix at the t-th iteration.
+Then, Problem (14) is written as
+max
+W,V
+R{bHW}−WHA1W−
+KI
+�
+k=1
+|ρk|2Tr{gkgH
+k V} (22a)
+s.t.
+WHBW + Tr{QHΦHΦQV} ⩽ ˆP act
+RIS,
+(22b)
+∥W∥2+Tr{V} ⩽ P act
+BS ,
+(22c)
+2R{WH(t)DiW}+Tr{hihH
+i V}⩾P
+′′
+i,
+i∈{1,· · ·, KE},
+(22d)
+V ⪰ 0,
+(22e)
+where P
+′′
+i = P
+′
+i + WH(t)DiW(t). Problem (22) is a convex
+problem which can be solved by CVX tools [15].
+C. Optimizing the Reflection Matrix Φ of the Active RIS
+Given {ρ, ˜γ, {wk}, V}, we consider to optimize Φ in this
+subsection. First, by assuming rank(V) = rE, we can express
+V as V =
+rE
+�
+k=1
+vkvH
+k based on the eigenvalue decomposition
+(EVD). Then, we define ˜Φ = [a1ejφ1, a2ejφ2, · · · , aLejφL]H.
+By substituting the expressions of gk and hk into Problem (8)
+and removing the constant terms, Problem (8) is rewritten as
+max
+˜Φ
+R{˜ΦHe} − ˜ΦHF˜Φ
+(23a)
+s.t.
+˜ΦHJ˜Φ ⩽ P act
+RIS,
+(23b)
+˜ΦHRi ˜Φ+2R{˜ΦHri}⩾ ˜Pi, i∈{1, · · · , KE}, (23c)
+where
+e = 2
+KI
+�
+k=1
+�
+αk(1 + ˜γk)diag(ρ∗
+kgH
+r,k)Qwk
+−
+KI
+�
+k=1
+|ρk|2�
+diag(gH
+r,k)Q
+KI
+�
+i=1
+wiwH
+i gd,k
++ diag(gH
+r,k)QVgd,k
+�
+,
+(24)
+F =
+KI
+�
+k=1
+|ρk|2�
+diag(gH
+r,k)Q
+KI
+�
+i=1
+wiwH
+i QHdiag(gr,k)
++ diag(gH
+r,k)QVQHdiag(gr,k)
++ δ2
+rdiag(gH
+r,k)diag(gr,k)
+�
+,
+(25)
+J =
+KI
+�
+k=1
+diag(Qwk)diag(wH
+k QH)
++
+rE
+�
+k=1
+diag(Qvk)diag(vH
+k QH) + δ2
+rIL,
+(26)
+Ri = diag(hH
+r,i)Q
+KI
+�
+k=1
+wkwH
+k QHdiag(hr,i)
++ diag(hH
+r,i)QVQHdiag(hr,i)
++ δ2
+rdiag(hH
+r,i)diag(hr,i),
+(27)
+
+4
+Algorithm 1 AO framework of solving Problem (8)
+1: Initial iteration number t = 1, maximum number of
+iterations tmax, feasible w(1),V(1),Φ(1), error tolerance ε
+and calculate the value of
+KI
+�
+k=1
+αkR(1)
+k ;
+2: Update ˜γ(t) by (10);
+3: Update ρ(t) by (13);
+4: Update W(t), V(t) by solving (22);
+5: Update Φ(t) by solving (34);
+6: If |
+KI
+�
+k=1
+αkR(t+1)
+k
+−
+KI
+�
+k=1
+αkR(t)
+k |/
+KI
+�
+k=1
+αkR(t+1)
+k
+< ε or t ≥
+tmax, terminate. Otherwise, set t ← t + 1 and go to step
+2.
+ri = diag(hH
+r,i)Q
+KI
+�
+i=1
+wiwH
+i hd,i+diag(hH
+r,i)QVhd,i,
+(28)
+˜Pi = Pi
+ηi
+− WHD
+′
+iW − VHE
+′
+iV,
+(29)
+D
+′
+i = IKI ⊗ hd,ihH
+d,i,
+(30)
+E
+′
+i = IKE ⊗ hd,ihH
+d,i.
+(31)
+Problem (23) is still non-convex due to the non-convex
+constraint (23c). Thus, we transform the non-convex constraint
+(23c) by its first-order Taylor expansion, and constraint (23c)
+is transformed as
+˜ΦHRi ˜Φ ⩾ 2R{˜ΦHRi ˜Φ(t)} − ˜ΦH(t)Ri ˜Φ(t),
+(32)
+where ˜Φ(t) is the phase shift vector at the t-th iteration. Thus,
+the constraint in (23c) is rewritten as
+2R
+�
+˜ΦH �
+ri + Ri ˜Φ(t)
+��
+⩾ ˜
+P
+′
+i , i ∈ {1, · · ·, KE},
+(33)
+where ˜
+P
+′
+i = ˜Pi + ˜ΦH(t)Ri ˜Φ(t). Problem (8) is reformulated
+asmax
+˜Φ
+R{˜ΦHe} − ˜ΦHF˜Φ
+(34a)
+s.t.
+˜ΦHJ˜Φ ⩽ P act
+RIS,
+(34b)
+2R
+�
+˜ΦH�
+ri+Ri ˜Φ(t)
+��
+⩾ ˜
+P
+′
+i , i∈{1, · · · , KE}, (34c)
+which is a quadratically constrained quadratic program
+(QCQP) problem and can be solved by CVX tools.
+D. Algorithm Complexity
+Finally, Problem (8) is solved by alternately solving Prob-
+lem (22) and Problem (34) until convergence. We summarize
+the proposed AO framework of solving Problem (8) in Al-
+gorithm 1. It is noted that the main computation to solve
+Problem (8) lies in alternately solving Problem (22) and
+Problem (34). We use Ia, Ib and I to denote the numbers of
+iterations for the convergence of Problem (22), Problem (34)
+and Problem (8), respectively. Then, the overall computational
+complexity of solving Problem (8) can be approximated by
+O
+�
+I
+�
+IaK2
+I N 2 + IbL2��
+.
+IV. SIMULATION RESULTS
+In this section, we provide numerical results to evaluate
+the performance of the active RIS-aided SWIPT system. We
+assume that the BS and the active RIS are respectively located
+at (0 m, 0 m), (10 m, 10 m) in a two-dimensional plane.
+KI = 4 IRs are randomly distributed in a circle centered at
+(30 m, 0 m) with a radius of 5 m, and KE = 4 ERs are
+10
+20
+30
+40
+50
+Total power(W)
+10
+20
+30
+40
+50
+60
+70
+Weighted sum rate(bps/Hz)
+Active RIS
+Passive RIS
+No RIS
+AF relay
+Fig. 2. The WSR versus the total power when N = 4 and
+L = 20.
+randomly distributed in a circle centered at (20 m, 0 m) with
+a radius of 5 m. The large-scale fading of the channels are
+modeled as PL=−30−10αlog10d (dB), where α is the path
+loss exponent and d is the link distance in meter. In this work,
+we set α = 2.3 for Q, α = 2.3 for hr,i, α = 2.5 for gr,k,
+α = 3.2 for gd,k, and α = 2.8 for hd,i. The small-scale fading
+is assumed to be Rician distributed. For simplicity, the Rician
+factor is assumed to be 5. The other parameters are set as
+follows: noise power of δ2
+r = δ2
+IR = δ2
+ER = −80 dBm, error
+tolerance of ε = 10−3, minimum harvested power threshold
+of Pi = 10−6 W.
+In order to illustrate the impact of the active RIS, we
+compare the active RIS-aided multiuser SWIPT system with
+the following schemes:
+• Passive RIS: It displays a passive RIS in the SWIPT sys-
+tem, which means that only the phase shifts of the transmission
+signals are adjusted and there is no power amplifier at the RIS.
+• No RIS: No RIS is to assist the SWIPT system, which
+means that the BS only transmits signals to IRs and ERs
+through the direct links.
+• AF relay: It displays an AF relay in the SWIPT system
+at the same location as the RIS in the SWIPT system.
+We adopt the power model in [7], [9]. Thus, the power
+consumption models corresponding to the above schemes are
+given by
+Ptotal = P act
+BS + P act
+RIS + L(PC + PDC),
+(35)
+Ptotal = P pas
+BS + LPC,
+(36)
+Ptotal = P no
+BS,
+(37)
+Ptotal = P af
+BS + Prelay + LPT,
+(38)
+where PC is the power consumption of the switch and control
+circuit at each reflecting element, PDC is the direct current
+biasing power used by each active reflecting element, PT is the
+dissipated power at each antenna of the AF relay, and Prelay
+is the transmit power limit at the AF relay. P act
+BS , P pas
+BS , P no
+BS,
+and P af
+BS are the maximum transmit power of the BS in the
+corresponding schemes.
+Power consumption parameters of
+hardware devices are set as follows: PC = −10 dBm, PDC =
+−5 dBm, and PT = 10 dBm. We assume that all schemes
+have the same total power Ptotal, and set P act
+BS = P act
+RIS, P af
+BS =
+Prelay.
+
+5
+0
+10
+20
+30
+40
+The location of the RIS (m)
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+Weighted sum rate(bps/Hz)
+Active RIS
+Passive RIS
+No RIS
+AF relay
+Fig. 3. The WSR versus the location of the RIS when N = 4
+and L = 20.
+Fig. 2 investigates the impact of the total system power
+on the WSR. As seen from Fig. 2, the WSRs of the scheme
+“Active RIS”, the scheme “Passive RIS”, the scheme “No RIS”
+and the scheme “AF relay” gradually increase with the transmit
+power. However, within the same total power consumption, it
+is observed that the scheme “Active RIS” achieves higher WSR
+than the scheme “Passive RIS”, the scheme “No RIS” and the
+scheme “AF relay”, which demonstrates that displaying an RIS
+can enhance the WSR of SWIPT systems and active RIS can
+achieve best performance in an SWIPT system.
+Fig. 3 depicts the WSR versus the location of the RIS. We
+assume that the location of the RIS changes in the horizontal
+direction. Comparing “Active RIS” with “Passive RIS”, “No
+RIS” and “AF relay”, it is observed that the WSR of the
+“Active RIS” is always higher than the others. In addition,
+we can find that the WSR increases as the RIS is close to the
+location of the ERs, and the WSR decreases as the location
+of the RIS is far away from the ERs, which is due to the
+fact that the channel gain between the RIS and the ERs will
+become greater as the RIS’s position approaches the ERs’
+positions, and the ERs are easier to reach the thresholds of
+energy reception. Thus, it leaves more space to jointly optimize
+the beamforming at the BS and reflection matrix at the RIS
+to further improve the WSR.
+Fig. 4 depicts the WSR versus the number of RIS reflecting
+elements. It is observed that the WSR of the scheme “Active
+RIS” increases slowly as the number of RIS reflecting ele-
+ments increases, when the number of RIS reflecting elements
+is small. However, both the WSRs of the scheme “Active RIS”
+and the scheme “Passive RIS” decrease as the number of RIS
+reflecting elements L exceeds about 50, which illustrates that
+the appropriate number of RIS reflecting elements is able to
+make the system achieve the good performance, and large
+number of RIS reflecting elements can cause performance loss,
+when the total power consumption is fixed.
+V. CONCLUSIONS
+This work studied an active RIS-aided SWIPT system. We
+focused on maximizing the WSR of the IRs, subject to the
+power requirements of all ERs, transmit power limit at the BS
+and the amplification power budget at the RIS. By adopting
+the FP method and quadratic transform, we transformed the
+original problem into a tractable form. Then, the beamforming
+vector at the BS and the optimal reflection matrix at the
+RIS were obtained via the AO framework. Under the same
+20
+30
+40
+50
+60
+70
+The number of RIS reflecting elements 
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+65
+Weighted sum rate(bps/Hz)
+Active RIS
+Passive RIS
+No RIS
+AF relay
+Fig. 4. The WSR versus the number of RIS reflecting
+elements L when N = 4.
+power consumption, we compared the system gains for “Active
+RIS”, “Passive RIS”, “No RIS” and “AF relay”. Simulation
+results demonstrated that the active RIS-aided SWIPT system
+can achieve better performance than the passive RIS/AF relay
+aided SWIPT system.
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+

PNE2T4oBgHgl3EQfVQfK/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf,len=414
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='03822v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='IT]  10 Jan 2023  1  Transmission Design for Active RIS-Aided  Simultaneous Wireless Information and Power  Transfer  Hong Ren, Member, IEEE, Zhiwei Chen, Guosheng Hu, Zhangjie Peng,  Cunhua Pan, Member, IEEE, and Jiangzhou Wang, Fellow, IEEE  Abstract—Reconfigurable intelligent surface (RIS) is a revolu-  tionary technology to enhance both the spectral efficiency and  energy efficiency of wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' However,  most of the existing contributions mainly focused on the study of  passive RIS, which suffers from the “double fading” effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' On  the other hand, active RIS, which is equipped with amplifiers, can  effectively address this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' In this paper, we propose an active  RIS-aided simultaneous wireless information and power transfer  (SWIPT) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Specifically, we maximize the weighted sum  rate of the information receivers, subject to the minimum power  received at all energy receivers, amplification power constraint  at the active RIS, and the maximum transmit power constraint  at the base station (BS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' By adopting alternating optimization  framework, suboptimal solutions are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Simulation results  show that the active RIS-aided SWIPT system has higher  performance gain with the same power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Index Terms—Reconfigurable intelligent surface (RIS), active  RIS, wireless information and power transfer (SWIPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' INTRODUCTION  Reconfigurable intelligent surface (RIS), composed of a  large number of reflecting elements, has received extensive  research attention in both academia and industry [1]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Specifically, RIS can dynamically adjust the electromagnetic  properties of the reflecting elements in a programmable way,  and then reconfigure the wireless propagation environment in  a desired way [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  However, most of the existing contributions mainly focused  on the passive RIS-aided communication systems, which suffer  from the “double fading” effect [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' To address this issue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  active RIS has been proposed in [8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' [9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' which is equipped  This work was supported in part by the National Natural Science Foun-  dation of China (62201137),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' in part by the Natural Science Foundation  of Shanghai under Grant 22ZR1445600,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' in part by the National Natural  Science Foundation of China under Grant 61701307,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' in part by the open  research fund of National Mobile Communications Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Southeast University under Grant 2018D14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' and in part by the National  Natural Science Foundation of China (62101128) and Basic Research Project  of Jiangsu Provincial Department of Science and Technology (BK20210205).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (Corresponding authors: Zhangjie Peng and Zhiwei Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=')  Hong Ren and Cunhua Pan are with the National Mobile Communications  Research Laboratory, Southeast University, Nanjing 210096, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' (e-mail:  hren@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' cpan@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Zhiwei Chen is with the College of Information, Mechanical and Electrical  Engineering, Shanghai Normal University, Shanghai 200234, China (e-mail:  1000497437@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='shnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Guosheng Hu is with the Shanghai Technical Institute of Electronics &  Information, Shanghai 201411, China (e-mail: huguosheng@stiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Zhangjie Peng is with the College of Information, Mechanical, and Electri-  cal Engineering, Shanghai Normal University, Shanghai 200234, China, also  with the National Mobile Communications Research Laboratory, Southeast  University, Nanjing 210096, China, and also with the Shanghai Engineering  Research Center of Intelligent Education and Bigdata, Shanghai Normal  University, Shanghai 200234, China (e-mail: pengzhangjie@shnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Jiangzhou Wang is with the School of Engineering, University of Kent,  CT2 7NT Canterbury, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' (e-mail: j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='wang@kent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  with some amplifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Different from multiple input multiple  output (MIMO) relay with high power cost and additional  time/frequency resource, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=', amplify and forward (AF) re-  lay, the active RIS basically inherits the hardware structure  of passive RIS, while equipping with a set of low-power  reflection-type amplifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' As a result, active RIS can not only  tune the phase of the reflected signals, but also amplify the  power of the reflected signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Recently, the authors of [7] have  rigorously demonstrated that the active RIS-aided single-input  single-output (SISO) system is superior to the passive RIS in  terms of the achievable date rate when both systems have the  same power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' On the other hand, simultaneous wireless  information and power transfer (SWIPT) is envisioned as  a promising technology in future internet of things (IoT)  [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The authors of [10] investigated a passive RIS-  aided SWIPT system and showed that the passive RIS can  enhance the data rates at the information receivers (IRs), while  ensuring the minimum power requirements of the received  power at the energy receivers (ERs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' In [11], the authors  studied an optimization problem of maximizing the minimum  rate of the IRs in the passive RIS-aided SWIPT system with  imperfect channel state information (CSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' And the authors  of [12] aimed to maximize the data rate by proposing a  joint time-switching and phase-shifting solution for passive  RIS-assisted SWIPT communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' However, the above  literatures [10]–[12] about passive RIS-aided SWIPT systems  suffer from the “double fading”, and for the SWIPT systems  with requirements for both energy reception and information  reception, there are no literature investigating whether the  active RIS with low-power amplifiers performs better than the  MIMO relay with complex hardware structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Against the above background, we consider an active RIS-  aided SWIPT downlink system, and aim to maximize the  downlink weighted sum rate (WSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Different from the pas-  sive RIS [10]–[12], it is noted that the active RIS introduces  a new optimization variable due to its ability to amplify the  reflected signal, generates the non-ignorable thermal noise, and  adds an output signal power constraint of the active RIS, which  makes the optimization problem more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' To solve  the non-convex problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' our contributions of this work are  summarized as follows  1) By considering the active RIS-aided SWIPT system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' we  aim to maximize the downlink weighted sum rate (WSR) of  the IRs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' by jointly optimizing the transmit beamforming at  the base station (BS) and the reflecting coefficients at the  active RIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' subject to the minimum power harvested at all  ERs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' amplification power constraint at the active RIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' and the  2  Active RIS  �  hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='l  BS  IRs  gd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k  hd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='l  gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k  ERs  Q  IR  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='r l  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' System model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  maximum transmit power constraint at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  2) By adopting alternating optimization (AO) framework,  we transform the objective function by fractional programming  (FP) method, and utilize the first-order Taylor approximation  to linearize the non-convex constraint of the active RIS am-  plification power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Then, we effectively solve the subproblems  and obtain the suboptimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  3) Simulation results show that the active RIS can achieve  higher downlink WSR than the passive RIS/AF relay in an  SWIPT system [10]–[13] with the same power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' And  as the location of the RIS is closer to the locations of the  ERs, it can achieve higher downlink WSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' In addition, the  appropriate number of RIS reflecting elements is enough to  enable the SWIPT system to achieve good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' SYSTEM MODEL  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 1, we consider an active RIS-aided mul-  tiuser multiple input single output (MISO) downlink system,  where an RIS with L reflecting elements is deployed to assist  SWIPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The system is composed of a BS with N antennas,  KI single-antenna IRs, and KE single-antenna ERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  The signal transmitted from the BS is expressed as  t =  KI  �  k=1  wksIR  k  + v,  (1)  where wk ∈ CN×1 is the beamforming vector for the k-th IR,  sIR  k  ∼ CN(0, 1), k ∈ {1, · · ·, KI} is the transmit information  symbol for the k-th IR, and v ∈ CN×1 ∼ CN(0, V) is the  energy signal vector, where V is the covariance matrix of the  energy signal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  The channels spanning from the BS to the active RIS, from  the BS to the k-th IR, from the BS to the i-th ER, from the RIS  to the k-th IR, and from the RIS to the i-th ER are denoted as  Q ∈ CL×N, gd,k ∈ CN×1, hd,i ∈ CN×1, gr,k ∈ CL×1, and  hr,i ∈ CL×1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  The reflected and amplified signal at the active RIS can be  modeled as follows  tr = Φ (Qt + nRIS) ,  (2)  where Φ = diag  �  a1ejφ1, · · · , alejφl, · · · , aLejφL�  denotes the  reflection matrix of the active RIS, φl and al are the phase  shift and the amplitude of the l-th element, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The  thermal noise generated by the active RIS cannot be neglected,  and nRIS ∼ CN(0, δ2  rI) denotes the thermal noise of the active  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  The received signal at the k-th IR can be written as  yIR,k = gH  d,kt + gH  r,ktr + nIR  = gH  k t + gH  r,kΦnRIS + nIR  = gH  k wksIR  k +  KI  �  i=1,i̸=k  gH  k wisIR  i  +gH  k v+ gH  r,kΦnRIS + nIR,  (3)  where gH  k ≜ gH  d,k +gH  r,kΦQ, and nIR ∼ CN(0, δ2  IR) is the ad-  ditive white Gaussian noise (AWGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Unlike the passive RIS  model [10]–[12], the active RIS consisting of amplifiers has  the ability to amplify the power of the reflected signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Thus,  the reflection matrix Φ has non-unity amplitude components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Then, the signal-to-interference-plus-noise ratio (SINR) of  the k-th IR is expressed as  γk =  |gH  k wk|2  KI  �  i=1,i̸=k  |gH  k wi|2+gH  k Vgk + δ2r∥gH  r,kΦ∥2+δ2  IR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (4)  Thus, the rate at the k-th IR is expressed as  Rk = log2 (1 + γk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (5)  The received signal at the i-th ER can be written as  yER,i = hH  d,it + hH  r,itr + nER  = hH  i t + hH  r,iΦnRIS + nER  =  KI  �  k=1  hH  i wksIR  k  + hH  i v + hH  r,iΦnRIS + nER,  (6)  where hH  i ≜ hH  d,i + hH  r,iΦQ, and nER ∼ CN(0, δ2  ER) is the  AWGN at the i-th ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Considering the fact that both the  data and energy signals transmitted by the BS are carried by  the beamforming, the harvested power at the i-th ER while  ignoring the AWGN power is given by  Ei = ηi  � KI  �  k=1  |hH  i wk|2+hH  i Vhi + δ2  r∥hH  r,iΦ∥2  �  ,  (7)  where ηi is the energy harvesting efficiency of the i-th ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' PROBLEM FORMULATION  To satisfy the requirements of both IRs and ERs, we  consider an optimization problem of maximizing the WSR of  all IRs, while satisfying the harvested power requirements of  all ERs, subject to the power constraints at the BS and active  RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Thus, the optimization problem is formulated as  max  {wk},V,Φ  KI  �  k=1  αkRk  (8a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  ∥ΦQt∥2+δ2  r∥Φ∥2⩽ P act  RIS,  (8b)  ∥t∥2⩽ P act  BS ,  (8c)  Ei ⩾ Pi, i ∈ {1, · · · , KE},  (8d)  where αk is the weighting factor of the k-th IR, P act  RIS is the  output signal power of the active RIS, P act  BS is the transmit  power limit at the BS and Pi is the minimum harvested power  threshold for the i-th ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Due to the fact that variables {{wk}, V, Φ} are coupled  together in the objective function of Problem (8), it is chal-  lenging to solve Problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' We then exploit the fractional  programming (FP) [14] method to decouple the objective  function of Problem (8), and adopt the Alternate Optimization  (AO) algorithm to obtain the solutions in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' FP method  Firstly, we use the FP method to transform the ob-  jective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' By introducing auxiliary variables ˜γ  =  [˜γ1, · · · , ˜γKI]T ∈ CKI×1, the objective function of Problem  (8) is equivalent to  3  fa(˜γ, wk, V, Φ) =  KI  �  k=1  αklog (1 + ˜γk) −  KI  �  k=1  αk˜γk  +  KI  �  k=1  αk(1 + ˜γk)|gH  k wk|2  KI  �  i=1  |gH  k wi|2+gH  k Vgk + δ2r∥gH  r,kΦ∥2+δ2  IR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (9)  We adopt the AO framework to obtain the optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  For fixed variables {{wk}, V, Φ}, by setting ∂fa/∂˜γk to zero,  the optimal ˜γopt  k  is obtained as  ˜γopt  k  = γk, k ∈ {1, · · ·, KI}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (10)  Then, we fix ˜γk and define a new function as  fb(˜γ, wk, V, Φ)  =  KI  �  k=1  αk(1 + ˜γk)|gH  k wk|2  KI  �  i=1  |gH  k wi|2+gH  k Vgk + δ2r∥gH  r,kΦ∥2+δ2  IR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (11)  By introducing auxiliary variables ρ = [ρ1, · · · , ρKI]T ∈  CKI×1 and adopting the quadratic transform [14], we further  recast fb as  fc(ρ, ˜γ, wk, V, Φ) = 2  KI  �  k=1  �  αk(1 + ˜γk)R{ρ∗  kgH  k wk}  −  KI  �  k=1  |ρk|2  � KI  �  i=1  |gH  k wi|2+gH  k Vgk + δ2  r∥gH  r,kΦ∥2+δ2  IR  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (12)  Similarly, by setting ∂fc/∂ρk to zero, we obtain the optimal  ρopt  k  as  ρopt  k  =  �  αk(1 + ˜γk)gH  k wk  KI  �  i=1  |gH  k wi|2+gH  k Vgk + δ2r∥gH  r,kΦ∥2+δ2  IR  ,  k ∈ {1, · · · , KI}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (13)  After obtaining the above optimal auxiliary variables, in the  next subsection, we then focus on optimizing {{wk}, V, Φ},  given {ρ, ˜γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Optimizing wk and V  By defining W ≜ [wT  1 , · · · , wT  KI]T, for fixed variables  {ρ, ˜γ, Φ}, Problem (8) is expressed as  max  W,V  R{bHW}−WHA1W−  KI  �  k=1  |ρk|2Tr{gkgH  kV} (14a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  WHBW + Tr{QHΦHΦQV} ⩽ ˆP act  RIS,  (14b)  ∥W∥2+Tr{V} ⩽ P act  BS ,  (14c)  WHDiW+Tr{hihH  i V}⩾P  ′  i , i∈{1,· · ·, KE}, (14d)  V ⪰ 0,  (14e)  where  b = [bT  1 , bT  2 , · · · , bT  KI]T, bH  k = 2  �  αk(1 + ˜γk)ρ∗  kgH  k ,  (15)  A1 = IKI ⊗  KI  �  i=1  |ρi|2gigH  i ,  (16)  B = IKI ⊗ QHΦHΦQ,  (17)  ˆP act  RIS = P act  RIS − δ2  r∥Φ∥2,  (18)  Di = IKI ⊗ hihH  i ,  (19)  P  ′  i = Pi  ηi  − δ2  r∥hH  r,iΦ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (20)  However, it is noted that the constraint in (14d) is non-  convex, which makes Problem (14) still intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' We then  approximate the constraint (14d) by its first-order Taylor  expansion as  WHDiW ⩾ 2R{WH(t)DiW} − WH(t)DiW(t),  (21)  where WH(t) is the beamforming matrix at the t-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Then, Problem (14) is written as  max  W,V  R{bHW}−WHA1W−  KI  �  k=1  |ρk|2Tr{gkgH  k V} (22a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  WHBW + Tr{QHΦHΦQV} ⩽ ˆP act  RIS,  (22b)  ∥W∥2+Tr{V} ⩽ P act  BS ,  (22c)  2R{WH(t)DiW}+Tr{hihH  i V}⩾P  ′′  i,  i∈{1,· · ·, KE},  (22d)  V ⪰ 0,  (22e)  where P  ′′  i = P  ′  i + WH(t)DiW(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Problem (22) is a convex  problem which can be solved by CVX tools [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Optimizing the Reflection Matrix Φ of the Active RIS  Given {ρ, ˜γ, {wk}, V}, we consider to optimize Φ in this  subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' First, by assuming rank(V) = rE, we can express  V as V =  rE  �  k=1  vkvH  k based on the eigenvalue decomposition  (EVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Then, we define ˜Φ = [a1ejφ1, a2ejφ2, · · · , aLejφL]H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  By substituting the expressions of gk and hk into Problem (8)  and removing the constant terms, Problem (8) is rewritten as  max  ˜Φ  R{˜ΦHe} − ˜ΦHF˜Φ  (23a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  ˜ΦHJ˜Φ ⩽ P act  RIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (23b)  ˜ΦHRi ˜Φ+2R{˜ΦHri}⩾ ˜Pi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' i∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' KE},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' (23c)  where  e = 2  KI  �  k=1  �  αk(1 + ˜γk)diag(ρ∗  kgH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)Qwk  −  KI  �  k=1  |ρk|2�  diag(gH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)Q  KI  �  i=1  wiwH  i gd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k  + diag(gH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)QVgd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (24)  F =  KI  �  k=1  |ρk|2�  diag(gH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)Q  KI  �  i=1  wiwH  i QHdiag(gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)  + diag(gH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)QVQHdiag(gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)  + δ2  rdiag(gH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)diag(gr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='k)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (25)  J =  KI  �  k=1  diag(Qwk)diag(wH  k QH)  +  rE  �  k=1  diag(Qvk)diag(vH  k QH) + δ2  rIL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (26)  Ri = diag(hH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='i)Q  KI  �  k=1  wkwH  k QHdiag(hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='i)  + diag(hH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='i)QVQHdiag(hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='i)  + δ2  rdiag(hH  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='i)diag(hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (27)  4  Algorithm 1 AO framework of solving Problem (8)  1: Initial iteration number t = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' maximum number of  iterations tmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' feasible w(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='V(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='Φ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' error tolerance ε  and calculate the value of  KI  �  k=1  αkR(1)  k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  2: Update ˜γ(t) by (10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  3: Update ρ(t) by (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  4: Update W(t), V(t) by solving (22);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  5: Update Φ(t) by solving (34);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  6: If |  KI  �  k=1  αkR(t+1)  k  −  KI  �  k=1  αkR(t)  k |/  KI  �  k=1  αkR(t+1)  k  < ε or t ≥  tmax, terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Otherwise, set t ← t + 1 and go to step  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  ri = diag(hH  r,i)Q  KI  �  i=1  wiwH  i hd,i+diag(hH  r,i)QVhd,i,  (28)  ˜Pi = Pi  ηi  − WHD  ′  iW − VHE  ′  iV,  (29)  D  ′  i = IKI ⊗ hd,ihH  d,i,  (30)  E  ′  i = IKE ⊗ hd,ihH  d,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (31)  Problem (23) is still non-convex due to the non-convex  constraint (23c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Thus, we transform the non-convex constraint  (23c) by its first-order Taylor expansion, and constraint (23c)  is transformed as  ˜ΦHRi ˜Φ ⩾ 2R{˜ΦHRi ˜Φ(t)} − ˜ΦH(t)Ri ˜Φ(t),  (32)  where ˜Φ(t) is the phase shift vector at the t-th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Thus,  the constraint in (23c) is rewritten as  2R  �  ˜ΦH �  ri + Ri ˜Φ(t)  ��  ⩾ ˜  P  ′  i , i ∈ {1, · · ·, KE},  (33)  where ˜  P  ′  i = ˜Pi + ˜ΦH(t)Ri ˜Φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Problem (8) is reformulated  asmax  ˜Φ  R{˜ΦHe} − ˜ΦHF˜Φ  (34a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  ˜ΦHJ˜Φ ⩽ P act  RIS,  (34b)  2R  �  ˜ΦH�  ri+Ri ˜Φ(t)  ��  ⩾ ˜  P  ′  i , i∈{1, · · · , KE}, (34c)  which is a quadratically constrained quadratic program  (QCQP) problem and can be solved by CVX tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Algorithm Complexity  Finally, Problem (8) is solved by alternately solving Prob-  lem (22) and Problem (34) until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' We summarize  the proposed AO framework of solving Problem (8) in Al-  gorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' It is noted that the main computation to solve  Problem (8) lies in alternately solving Problem (22) and  Problem (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' We use Ia, Ib and I to denote the numbers of  iterations for the convergence of Problem (22), Problem (34)  and Problem (8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Then, the overall computational  complexity of solving Problem (8) can be approximated by  O  �  I  �  IaK2  I N 2 + IbL2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, we provide numerical results to evaluate  the performance of the active RIS-aided SWIPT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' We  assume that the BS and the active RIS are respectively located  at (0 m, 0 m), (10 m, 10 m) in a two-dimensional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  KI = 4 IRs are randomly distributed in a circle centered at  (30 m, 0 m) with a radius of 5 m, and KE = 4 ERs are  10  20  30  40  50  Total power(W)  10  20  30  40  50  60  70  Weighted sum rate(bps/Hz)  Active RIS  Passive RIS  No RIS  AF relay  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The WSR versus the total power when N = 4 and  L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  randomly distributed in a circle centered at (20 m, 0 m) with  a radius of 5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The large-scale fading of the channels are  modeled as PL=−30−10αlog10d (dB), where α is the path  loss exponent and d is the link distance in meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' In this work,  we set α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='3 for Q, α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='3 for hr,i, α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='5 for gr,k,  α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='2 for gd,k, and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='8 for hd,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The small-scale fading  is assumed to be Rician distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' For simplicity, the Rician  factor is assumed to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The other parameters are set as  follows: noise power of δ2  r = δ2  IR = δ2  ER = −80 dBm, error  tolerance of ε = 10−3, minimum harvested power threshold  of Pi = 10−6 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  In order to illustrate the impact of the active RIS, we  compare the active RIS-aided multiuser SWIPT system with  the following schemes:  Passive RIS: It displays a passive RIS in the SWIPT sys-  tem, which means that only the phase shifts of the transmission  signals are adjusted and there is no power amplifier at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  No RIS: No RIS is to assist the SWIPT system, which  means that the BS only transmits signals to IRs and ERs  through the direct links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  AF relay: It displays an AF relay in the SWIPT system  at the same location as the RIS in the SWIPT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  We adopt the power model in [7], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' the power  consumption models corresponding to the above schemes are  given by  Ptotal = P act  BS + P act  RIS + L(PC + PDC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (35)  Ptotal = P pas  BS + LPC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (36)  Ptotal = P no  BS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (37)  Ptotal = P af  BS + Prelay + LPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  (38)  where PC is the power consumption of the switch and control  circuit at each reflecting element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' PDC is the direct current  biasing power used by each active reflecting element,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' PT is the  dissipated power at each antenna of the AF relay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' and Prelay  is the transmit power limit at the AF relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' P act  BS , P pas  BS , P no  BS,  and P af  BS are the maximum transmit power of the BS in the  corresponding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Power consumption parameters of  hardware devices are set as follows: PC = −10 dBm, PDC =  −5 dBm, and PT = 10 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' We assume that all schemes  have the same total power Ptotal, and set P act  BS = P act  RIS, P af  BS =  Prelay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  5  0  10  20  30  40  The location of the RIS (m)  10  15  20  25  30  35  40  45  50  55  60  Weighted sum rate(bps/Hz)  Active RIS  Passive RIS  No RIS  AF relay  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The WSR versus the location of the RIS when N = 4  and L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 2 investigates the impact of the total system power  on the WSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' As seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 2, the WSRs of the scheme  “Active RIS”, the scheme “Passive RIS”, the scheme “No RIS”  and the scheme “AF relay” gradually increase with the transmit  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' However, within the same total power consumption, it  is observed that the scheme “Active RIS” achieves higher WSR  than the scheme “Passive RIS”, the scheme “No RIS” and the  scheme “AF relay”, which demonstrates that displaying an RIS  can enhance the WSR of SWIPT systems and active RIS can  achieve best performance in an SWIPT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 3 depicts the WSR versus the location of the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' We  assume that the location of the RIS changes in the horizontal  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Comparing “Active RIS” with “Passive RIS”, “No  RIS” and “AF relay”, it is observed that the WSR of the  “Active RIS” is always higher than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' In addition,  we can find that the WSR increases as the RIS is close to the  location of the ERs, and the WSR decreases as the location  of the RIS is far away from the ERs, which is due to the  fact that the channel gain between the RIS and the ERs will  become greater as the RIS’s position approaches the ERs’  positions, and the ERs are easier to reach the thresholds of  energy reception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Thus, it leaves more space to jointly optimize  the beamforming at the BS and reflection matrix at the RIS  to further improve the WSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 4 depicts the WSR versus the number of RIS reflecting  elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' It is observed that the WSR of the scheme “Active  RIS” increases slowly as the number of RIS reflecting ele-  ments increases, when the number of RIS reflecting elements  is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' However, both the WSRs of the scheme “Active RIS”  and the scheme “Passive RIS” decrease as the number of RIS  reflecting elements L exceeds about 50, which illustrates that  the appropriate number of RIS reflecting elements is able to  make the system achieve the good performance, and large  number of RIS reflecting elements can cause performance loss,  when the total power consumption is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' CONCLUSIONS  This work studied an active RIS-aided SWIPT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' We  focused on maximizing the WSR of the IRs, subject to the  power requirements of all ERs, transmit power limit at the BS  and the amplification power budget at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' By adopting  the FP method and quadratic transform, we transformed the  original problem into a tractable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Then, the beamforming  vector at the BS and the optimal reflection matrix at the  RIS were obtained via the AO framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Under the same  20  30  40  50  60  70  The number of RIS reflecting elements  15  20  25  30  35  40  45  50  55  60  65  Weighted sum rate(bps/Hz)  Active RIS  Passive RIS  No RIS  AF relay  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' The WSR versus the number of RIS reflecting  elements L when N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  power consumption, we compared the system gains for “Active  RIS”, “Passive RIS”, “No RIS” and “AF relay”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Simulation  results demonstrated that the active RIS-aided SWIPT system  can achieve better performance than the passive RIS/AF relay  aided SWIPT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content='  REFERENCES  [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Peng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Pan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Li, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
+page_content=' Swindlehurst, “Multiuser  full-duplex two-way communications via intelligent reflecting surface,”  IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
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+page_content=' 69, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE2T4oBgHgl3EQfVQfK/content/2301.03822v1.pdf'}
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SdE3T4oBgHgl3EQfZwoa/content/tmp_files/2301.04499v1.pdf.txt

@@ -0,0 +1,1599 @@
+Precision measurement of an electron pump at 2 GHz
+Stephen P. Giblin,1 Gento Yamahata,2 Akira Fujiwara,2 and Masaya Kataoka1
+1)National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW,
+United Kingdom
+2)NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198,
+Japan
+(Dated: 12 January 2023)
+A well-characterised sample of silicon tunable-barrier electron pump has been operated at a frequency of
+2 GHz using a custom drive waveform, generating a pump current of 320 pA. Precision measurements of the
+current were made as a function of pump control parameters, using a blind protocol, over a 7-week campaign.
+The combined standard uncertainty for each ∼ 10 hour measurement was ∼ 0.1 parts per million.
+The
+pump current exhibits a plateau along the exit gate voltage flat to approximately 0.1 parts per million, but
+offset from ef by 0.2 parts per million. This offset may be a sign of errors in the current traceability chain,
+indicating a limit to the accuracy of small current scaling using existing methods based on cryogenic current
+comparators.
+PACS numbers: 1234
+I.
+INTRODUCTION
+Electron pumps are devices that aim to generate a ref-
+erence DC electric current by moving electrons one at a
+time in response to a periodic control signal at frequency
+f. They potentially offer a simple and elegant traceabil-
+ity route for small currents, direct to the SI definition of
+the ampere1,2. A class of pumps fabricated from semicon-
+ductor materials3 has demonstrated accurate and robust
+current generation at roughly the part-per-million (ppm)
+accuracy level, for currents IP up to 160 pA4. However,
+important questions must be answered before electron
+pumps can confidently be adopted as reference current
+standards at the uncertainty levels of primary electrical
+metrology. Most significantly, the robustness and device
+independence of the current needs to be demonstrated at
+at least the 0.1 ppm level, over a range of device designs
+and operating parameters. To date, two studies have fo-
+cused on the robustness of the current from GaAs pumps,
+at current levels of ∼ 100 pA, at uncertainty levels for
+each data point of ∼ 2 ppm5 and ∼ 0.5 ppm6.
+However, blind measurement techniques which have
+been implemented in other metrology areas to remove
+bias7 have not yet been applied to the study of electron
+pumps where the pump current is treated as an unknown
+and compared to a known reference current. Address-
+ing unconscious experimenter bias is particularly impor-
+tant in experiments where the expectation of the result
+is strongly constrained; in this case, we expect IP = ef,
+and there is a possibility that in a non-blind measure-
+ment, the experimenter may unconsciously favour pump
+control parameters that yield this result.
+Particularly
+important in the electron pump context is the lack of
+reproducibility in attempts to realise a capacitance stan-
+dard based on pumping a known number of electrons onto
+a cryogenic capacitor8,9. The authors of Ref. 9 were un-
+able to reproduce the results of Ref. 8, and identified
+components in the capacitance measurement uncertainty
+which had previously been under-estimated.
+Evaluating
+the
+robustness
+of
+the
+pump
+current
+presents a challenge due to the time-scales involved: the
+small currents require many hours of averaging to resolve
+0.1 ppm for a single data point, and the time-scale of the
+whole measurement campaign challenges the stability of
+the measurement system and the electron pump itself4.
+To reduce the measurement time, or equivalently, to al-
+low more data points to be measured within the timescale
+of a measurement campaign, the pump current should
+be increased as much as possible.
+Custom gate drive
+waveforms which slow down the electron capture pro-
+cess have been used to operate GaAs pumps accurately
+at much higher frequencies than were possible with sine
+wave drive6,10,11. With these pumps the upper frequency
+limit for accurate pumping was f ∼ 1 GHz even with the
+custom waveforms. Silicon pumps, on the other hand,
+have demonstrated accurate pumping at f = 1 GHz with
+sine wave drive12,13, and the possibility of increasing the
+frequency further while maintaining sub-ppm pumping
+accuracy using custom waveforms has not yet been ex-
+plored.
+II.
+EXPERIMENTAL METHOD AND BLIND
+PROTOCOL
+We investigate a single well-characterised sample of
+silicon pump which has previously been the subject of
+two precision measurement campaigns13,14. The pump
+is a silicon nanowire-MOSFET, in which charge carri-
+ers are induced by a positive voltage applied to a global
+top gate14,15 which was set to 4 V for all the measure-
+ments. Two finger gates, denoted the entrance gate and
+exit gate, define the region of the nanowire where a sin-
+gle electron can be trapped. Negative DC voltages VENT
+and VEXIT applied to these gates define the pump operat-
+ing point, and the periodic pump drive signal is added to
+VENT using a room-temperature bias-tee. A 50 Giga sam-
+ples/s arbitrary waveform generator (AWG, Tektronix
+arXiv:2301.04499v1  [cond-mat.mes-hall]  11 Jan 2023
+
+2
+FIG. 1. (a): Grey-scale derivative pump map using sine wave
+drive at 2.062 GHz, PRF = 13.2 dBm. (b): Pump map using
+a waveform from an AWG at repetition rate f = 2 GHz. One
+cycle of the AWG waveform is shown in the inset. Note that this
+waveform is subsequently amplified by an inverting amplifier to
+yield the correct polarity of gate voltage, whereby the negative
+voltage pulse on the entrance gate raises the entrance barrier
+to pump an electron. (c): Log-scale plots of the pump current
+along the horizontal dashed lines in plots (a) and (b).
+70001A) was used to generate a custom waveform for
+the pump drive. Because the AWG output had a max-
+imum peak-peak amplitude of VAC = 0.5 V, the output
+was amplified by a wide-band inverting RF amplifier with
++15 dB gain before the bias-tee. The AWG is referenced
+to a 10 MHz frequency reference derived from a hydrogen
+maser.
+Figure 1 shows characterisation data using both sine
+wave drive and the custom AWG waveform at a repeti-
+tion frequency of 2 GHz. It is clear from the log-scale
+plots of figure 1 (c) that there is a substantial plateau
+along the exit gate axis when using the AWG drive wave-
+form, but not when using sine wave drive. The inset to
+figure 1(b) shows the AWG waveform used for all the
+measurements reported in this paper. Characterisation
+data at other frequencies is inlcuded in supplementary
+sections A and B.
+The experimental apparatus and methods used for this
+study are in many respects identical to that used in Ref.
+13. As in those experiments, the pump is cooled to a
+temperature close to 4 K by suspending it above a liquid
+helium surface. The pump current IP is measured us-
+ing a noise-optimised ultrastable low-noise current ampli-
+fier (ULCA)16, with a precision digital voltmeter (DVM)
+recording the ULCA output. As in Ref. 13, the DVM was
+calibrated roughly once every hour by switching its input
+to a Josephson voltage standard (JVS). A single precision
+measurement typically lasted between 8 and 10 hours and
+included between 7 and 11 voltmeter calibrations. To re-
+move offset drifts in the measurement system during pre-
+cision measurements, the pump drive signal was toggled
+on and off with a cycle time of 228 s. Roughly the first
+34 seconds of each data segment (300 out of 1000 data
+points) following each on or off switch was rejected from
+the analysis to remove transient effects.
+More details
+of the measurement protocol are given in supplementary
+section C.
+The pump current IP is calculated from the on-off
+difference in the DVM voltages ∆V using the equation
+IP = ∆V/ATR. Here, ATR is the trans-resistance gain
+of the ULCA, nominally equal to 109 V/A. This gain is
+calibrated against the quantum Hall resistance (QHR) in
+2 stages17 and via some intermediate transfer standards,
+using a cryogenic current comparator (CCC) with rel-
+ative uncertainty less than 0.1 ppm18. Detailed calibra-
+tion results are reported in supplementary section E. The
+measurement of the pump current was therefore traceable
+to the SI unit ampere via the JVS, the QHR, and the re-
+lationship I = V/R. For characterisation measurements
+such as those reported in figures 1, 2a, and small filled
+points in figures 2b and 2c, no offset subtraction was per-
+formed: the pump drive signal was left on, and each data
+point is a single 20 power line cycle DVM measurement.
+A blind protocol was implemented so that the lead ex-
+perimenter could not see the true value of IP while the
+measurements and data analysis were in progress. This
+is achieved by multiplying all the DVM readings by a
+hidden scaling factor β = 1.00000387, at a low level in
+the measurement software.
+An exception occurs when
+when the DVM is connected to the JVS for calibration,
+in which case β = 1. While tuning the pump and per-
+forming measurements, the experimenter does not know
+the scaling factor and can only access the scaled pump
+current IP,B = β∆V/ATR. Therefore, the tuning of the
+pump operating parameters and the choice of parame-
+ters for the precision measurements can only be made
+with reference to the flatness of the current plateau, not
+the deviation of the current from ef. The scaling factor
+was programmed and password-protected by a member
+of the team who was not otherwise involved in the exper-
+iments. The experimenter knew that it was constrained
+such that |1 − β| < 5 × 10−6 so that gross failures of the
+pump or apparatus would be apparent during character-
+ization measurements. The scaling factor was revealed
+after the experiments were finished and data analysis,
+including analysis of the ULCA calibrations, completed.
+III.
+PRECISION MEASUREMENT CAMPAIGN
+The aim of the measurements was to study the pump
+current as a function of control parameters VENT, VEXIT
+and VAC. To this end, a total of 67 precision measure-
+ments were made during a 7-week campaign, employing
+the apparatus and blind protocol described in section II.
+The measurements were divided into 17 ‘runs’. For most
+of the runs, several measurements were made while vary-
+ing one control parameter. Runs 11-13 consisted of sin-
+
+(a) Sine 2.062 GHz
+Sine 2.062 GHz
+-0.8
+VENT
+AWG 2.000 GHz
+0
+/ V
+lp/ef
+(c)
+-1.6
+-2
+Log I 1 - /
+-1.6
+-1.2
+VEXIT / V
+4
+-6
+(b) AWG 2 GHz
+-1.6
+-1.4
+-1.2
+-0.6
+VEXIT / V
+VENT
+10
+/ V
+plots a, b:
+-1.2
+d/p / dVExIT / nAV-1
+-1.6
+-1.4
+-1.2
+-1.0
+0
+VEXIT / V3
+FIG. 2. (a): Derivative pump map measured after precision run
+4 and before precision run 5.
+(b) and (c): Line-scans of the
+pump current measured along the gate voltage axes indicated
+by solid colored lines in (a), and plotted on a logarithmic scale.
+The vertical dashed lines indicate the fixed value of entrance
+(exit) gate voltage used for the exit (entrance) gate scan. The
+diagonal dashed lines are guides to the eye extrapolating the
+exponential edges of the plateau.
+Larger filled points are the
+precision measurement data for runs 1-5, with the run number
+indicated in the plot legend. Data points indicated with a star
+(∗) failed the stationary-mean test.
+gle measurements without varying a parameter. Further
+detail of the measurement chronology is given in supple-
+mentary section D. To monitor the stability of the pump,
+a ‘fingerprint’ pump map was obtained before and after
+each run, apart from a few occasions when it was pre-
+vented by an experimental difficulty. For completeness,
+all of these pump maps are shown in supplementary sec-
+tion H. Additional line scans of current as a function of
+one or more control parameters were also measured to
+assess the optimal values of fixed control parameters for
+the next precision run. Typically, these scans were used
+to find the value of the control parameter that maximised
+the plateau width. They used a single 20 PLC measure-
+ment for each data point, with a relative uncertainty of
+approximately 10 ppm per data point. A pass / fail sta-
+tionary mean statistical test, described in supplementary
+section G, was applied at the data analysis stage to each
+precision measurement to evaluate whether the current
+was stable during the measurement time.
+IV.
+RESULTS OF PRECISION MEASUREMENTS
+A.
+Precision results
+After run 5, the pump became less stable, (supple-
+mentary section H), making it difficult to establish the
+flatness of plateaus along VENT and VEXIT axes. For this
+reason, we concentrate here on the data from runs 1-5,
+although the full precision data set is presented in sup-
+plementary figure S9. In figure 2, we present the data
+from the first 5 precision runs. Panel (a) shows a pump
+map recorded between runs 4 and 5, and panels (b) and
+(c) show line-scans on a log scale which highlight the de-
+viation of the current from the ideal value on the 1ef
+plateau. The fixed value of VENT (VEXIT) for the VEXIT
+(VENT) line-scan was adjusted in order to maximise the
+width of the plateau in the log-scale plot. The results of
+precision runs 1-5 are plotted as solid points in figures 2
+(b) and (c). Runs 1-4 were VEXIT scans, plotted in figure
+2 (c), and run 5 was a VENT scan, plotted in figure 2 (b).
+The 18 data points along the VEXIT axis (figure 2 (c))
+define a plateau in agreement with an extrapolation of
+the standard-accuracy measurement. The precision data
+point marked with a star (∗), failed the stationary-mean
+test, presumably because it was close to the edge of the
+plateau, and small fluctuations in offset charge, equiva-
+lent to shifts in VEXIT, caused fluctuations in the pumped
+current to be resolved on the time-scale of the precision
+measurement.
+The precision IP(VEXIT) data for runs 1-4 (apart from
+the point that failed the stationary mean test) are re-
+plotted on a linear y-axis in figure 3a as ∆IP = (IP −
+ef)/ef. The mean of these 17 points is ∆IP = 0.22 ppm,
+with a standard deviation σ of 0.14 ppm. The individ-
+ual data points have a mean combined uncertainty ⟨U T ⟩
+of 0.102 ppm, although the uncorrelated random uncer-
+tainty, UA, for each data point is smaller, in the range
+0.08 − 0.09 ppm. The scatter of the points is therefore
+slightly larger than what would be expected from the
+type A uncertainty of each point (σ > ⟨U A⟩), although
+not statistically incompatible with the assumption that
+the data is sampling a stationary mean along the plateau.
+We can therefore conclude that this data is consistent
+with a plateau along the VEXIT axis, flat at the 0.1 ppm
+level, but significantly offset from ef by 0.22 ppm. Fig-
+ure 4 (b) shows the same data re-analysed with the first
+700 data points rejected from the beginning of each 1000-
+point data segment instead of the standard 300. This was
+to test for the presence of a time constant in the current,
+as discussed in section V.
+Only one precision run was performed along the VENT
+axis before the interruption, illustrated by the heavy
+filled points in figure 2 (b).
+One data point, marked
+with a ∗, failed the stationary-mean test. It is not clear
+from this single run whether this data point indicates
+real structure to the plateau at level of ∼ 5 ppm, or if it
+is the result of a drift in the device state. The remain-
+ing 4 data points mark a plateau region which, combined
+
+(a)
+-0.4-
+8
+^/
+-0.8
+ENT
+EXIT
+/ nA V-1
+-1.2
+-1.6
+-1.2
+V
+V
+-0.8
+EXIT
+1
+(q)
+2
+(c)
+3
+0
+5
+/ef
+4
+-2
+4
+Log 
+-8
+-1.0
+-0.5
+-1.6
+-1.4
+-1.2
+V
+ENT
+EXIT4
+FIG. 3.
+Results of precision measurements for runs 1-4 as a
+function of VEXIT, expressed as ∆IP = (IP − ef)/ef. The data
+have been analysed with (a): 300 and (b): 700 data points
+rejected from the start of each 1000-point data segment. Error
+bars show the combined standard uncertainty UT The horizontal
+dashed lines show the weighted means of each data set. Arrows
+highlight run 1, measurement 4 and run 4, measurement 4. A
+breakdown of the uncertainty for these measurements is given in
+table I.
+with the stability of the pump map from runs 1-5, gives
+confidence that the fixed value of of VENT selected for
+runs 1-4 is in the middle of an experimentally-determined
+plateau. The mean of the 4 measurements from run 5 is
+∆IP = 0.15 ppm, with a standard deviation of 0.09 ppm.
+This is consistent with the deviation measured in runs
+1-4 given the much smaller sample size.
+B.
+Uncertainty
+In table I, the breakdown of the uncertainty is given
+for two measurements indicated by arrows in figure 3a.
+The uncertainties due to the two stages of the ULCA cal-
+ibration are presented as separate components, with the
+uncertainty due to the drift of the ULCA gains in between
+calibrations included in these two terms. This was sig-
+nificantly reduced by performing frequent ULCA calibra-
+tions, with more detail given in supplementary sections
+D and E. Run 1, measurement 4 is a typical represen-
+tative measurement, and run 4, measurement 4 had the
+lowest combined uncertainty of the campaign. As in pre-
+vious measurement campaigns, the type A uncertainty
+of the pump measurement is the largest single contribu-
+tion, but the larger pump current achieved in this study
+has reduced UA to below 0.1 ppm and the uncertainty in
+the ULCA calibration is now a significant contribution.
+Specifically, the uncertainty in the output stage gain RIV
+(nominal value 1 MΩ) is limited by the 0.04 ppm uncer-
+tainty in the 100 kΩ reference resistor traceable to the
+TABLE I. Uncertainty breakdown for run 1, measurement 4,
+and run 4, measurement 4. All entries in the table are dimen-
+sionless relative uncertainties (k = 1) in parts per million.
+Contribution
+Meas. 1.4
+Meas. 4.4
+ULCA GI Cal.
+0.024
+0.024
+ULCA RIV Cal.
+0.062
+0.043
+ULCA Temp. corr.
+0.023
+0.023
+DVM Cal.
+0.014
+0.016
+Pump UA
+0.088
+0.061
+Total UT
+0.111
+0.084
+QHR via a chain of 3 intermediate measurements18,19.
+C.
+stability of the pump
+The measurement campaign was divided into two parts
+by an instrument issue which forced a period of 8 days’
+down-time between runs 5 and 6. During this time, the
+pump was thermally cycled to room temperature and
+back to 4 K twice. From examination of the pump maps
+in supplementary section H, it is clear that the pump
+became less stable after this interruption, although even
+before the interruption, small changes in the ‘nose’ (the
+onset of pumped current as VEXIT is made less negative)
+of the pump map are visible. This contrasts with the data
+of Ref. 13 showing this sample of pump to be extremely
+stable over multiple cool-downs in different laboratories,
+when driven with a sine wave at ∼ 1 GHz. We conjecture
+that at least some of the changes visible in the pump
+maps during the present campaign may be due to changes
+in the transmission of the cryogenic microwave line at
+frequencies ≫ 1 GHz.
+This could plausibly arise due
+to changes in the temperature gradient along the line,
+and would affect the high frequency components of the
+AWG waveform, causing distortion of the waveform at
+the pump entrance gate.
+V.
+DISCUSSION
+The study was complicated by instability in the pump
+map, which made it difficult in the later parts of the mea-
+surement campaign to interpret the results as sampling a
+stable state of the device. However, enough results were
+obtained from runs 1-5 to establish that the pump cur-
+rent is invariant in the exit gate voltage at the level of 1
+part in 107. Averages over these data points presented
+in the previous section give ∆IP ∼ 2×10−7, a significant
+offset from the ideal current IP = ef. The flatness of
+the plateau suggests that the offset is due to an error in
+the measurement system which applies a constant offset
+to all the measurements, rather than an error due to the
+physics of the pump itself.
+One possible cause of error is a time constant in the
+pump current. This was discussed in Ref. 13, and could
+
+4.4
+1.4
+(a)
+0.5
+(udd) d/v
+---在----0.22 ppm
+0.0
+王
+-0.5
+-1.40
+-1.35
+-1.30
+Run number
+1
+3
+2
+4
+(b)
+0.5
+(wdd)
+0.13 ppm
+△/p (
+0.0
+工
+-0.5
+-1.40
+-1.35
+-1.30
+VEXIT (V)5
+plausibly arise from heating due to the relatively large
+RF powers applied to the device gate.
+Repeating the
+data analysis of runs 1-4 with 700 data points rejected
+from the start of each segment instead of 300 did indeed
+yield an average pump current closer to ef, as shown
+in figure 3b. However, the larger type A uncertainties
+in this analysis make it difficult to draw a firm conclu-
+sion regarding possible time constants.
+Measurements
+with much longer on-off cycle times could potentially
+resolve this question, but require the 1/f noise corner
+of the ULCA current measurement to be at frequencies
+well below 1 mHz. ULCA units have demonstrated this
+performance in bench tests16, but the cryogenic wiring
+involved in a pump measurement introduces additional
+sources of noise and drift. Another possible cause of er-
+ror is a non-linearity in the gain of the ULCA. The ULCA
+input stage gain GI is calibrated at a current of 6 nA, and
+the pump current is 320 pA. Comparisons of the gains of
+two ULCA units with different input stage designs, de-
+tailed in supplementary section F, set an upper limit to
+possible non-linearity of a few parts in 108, so this is
+unlikely to cause errors of a part in 107.
+Possibly the most important cause of error could arise
+from the CCC calibration of the ULCA GI. This could
+result from rectification of noise by the CCC’s SQUID
+detector leading to different SQUID offsets for the two
+polarities of current used in the ULCA calibration20.
+One study on CCC performance in the low-flux regime21
+concluded that noise pickup might lead to this type of
+error at SQUID flux levels below 1 µΦ0, although this
+number was based on a limited number of measurements
+and is specific to a particular CCC design22, different
+in detail to the CCC used to calibrate the ULCA in
+our experiments. We calibrated the ULCA GI using a
+CCC18 with a 10000 : 10 turns ratio, and a sensitiv-
+ity of 6 µA turns/Φ0. The current in the large winding
+was approximately ±5 nA, giving a full-signal ampere-
+turns product of 100 µA turns, corresponding to a flux
+of 16.7 Φ0.
+A flux of 1 µΦ0 therefore corresponds to
+0.06 ppm of the full signal in the ULCA GI calibra-
+tion, three times smaller than the observed discrepancy
+in the electron pump current.
+However, no investiga-
+tions have yet been carried out on the performance of
+our CCC in the low-flux regime, so the size of possible
+noise-rectification errors is not known. Low flux ratio ac-
+curacy tests such as those presented in Ref. 21 should
+provide useful information on the scale of possible errors.
+We note that if these errors are affecting the ULCA cali-
+brations in our experiment, they are remarkably constant
+in time, as shown by the ∼ 5 × 10−8 relative stability of
+the ULCA input gain over the duration of the measure-
+ment campaign illustrated in supplementary section E.
+This indicates that if noise is affecting the SQUID, its
+most likely source is the CCC bridge electronics, rather
+than external sources.
+The upper frequency limit for accurate pumping with
+tunable-barrier pumps has previously been empirically
+established at around 1 GHz4. We have shown that this
+can be increased, albeit in a rather exceptional sample
+of pump. In this study, the practical upper frequency
+limit was determined by a combination of plateau round-
+ing, and increased incidence of switching events which
+shifted the pump operating point in the VENT − VEXIT
+plane.
+This hints at device-physics factors which may
+limit the practical upper operation frequency, possibly
+charge traps which are activated by high frequency com-
+ponents present in the drive signal. Further investigation
+of more samples of pump could shed fruitful light on this
+question.
+VI.
+CONCLUSIONS
+Precision measurements have been made of the cur-
+rent from a silicon electron pump driven at a frequency
+of 2 GHz using a custom drive waveform applied to the
+entrance gate. The pump current is invariant in exit gate
+voltage with a precision of 0.1 ppm (32 aA), but offset by
+roughly 0.2 ppm from the expected current corresponding
+to one electron for each pump cycle. The application of
+a blind measurement protocol provides added confidence
+that this result is not affected by experimenter bias. At
+this accuracy level, the measurement of the pump cur-
+rent challenges the state of the art in existing electrical
+metrology methods, with scaling of small currents using
+CCCs at low flux levels posing a particularly interesting
+problem. The recent demonstration of current plateaus
+due to the dual Josephson effect23 raises the possibility
+of a metrological investigation of the dual Josephson ef-
+fect in the near future, providing added motivation for a
+better understanding of low current scaling.
+ACKNOWLEDGMENTS
+The authors would like to thank Colin Porter and Scott
+Wilkins for making the NPL primary Josephson voltage
+standard available, and for assistance with setting up the
+voltmeter calibration. This research was supported by
+the UK department for Business, Energy and Industrial
+Strategy. A.F. and G.Y. are supported by JSPS KAK-
+ENHI Grant Number JP18H05258.
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+
+7
+FIG. S1. (a): Grey-scale derivative pump map using sine wave
+drive at 1.05 GHz, PRF = 11.6 dBm. (b): Pump map using a
+waveform from an AWG at repetition rate f = 1.04 GHz. One
+cycle of the AWG waveform is shown in the inset. Note that this
+waveform is subsequently amplified by an inverting amplifier to
+yield the correct polarity of gate voltage, whereby the negative
+voltage pulse on the entrance gate raises the entrance barrier
+to pump an electron. (c): Log-scale plots of the pump current
+along the horizontal dashed lines in plots (a) and (b).
+VII.
+SUPPLEMENTARY INFORMATION
+A.
+AWG waveform at 1 GHz
+The silicon pump in this study has already exhibited
+robust quantisation at 1.05 GHz with sine wave drive13,
+as illustrated in the pump map and log plot of figure
+S1 (a) and (c). As an initial part of the setup process,
+we tested the pump operation using an AWG waveform
+at a similar frequency of 1.04 GHz. This resulted in a
+substantially wider plateau, seen by comparing the log
+plots with sine wave and AWG drive in figure S1 (c). Note
+that the AWG waveform leads to substantial distortion
+of the pump map (figure S1 (b)), due to the electron
+capture occurring at different rates dVENT/dt as VENT is
+scanned. This data was an important motivator towards
+the main study because it showed for the first time that
+the type of waveform first used on GaAs pumps in Ref.
+10 could also yield a substantial improvement in plateau
+flatness with Si pumps.
+B.
+Exploration of higher frequencies
+During the setup of the experiments reported in the
+main text, frequencies above 2 GHz were explored using
+custom waveforms (figure S2). The data at 4 GHz shows
+a feature which may be attributable to non-adiabatic
+excitation24 resulting from the rapid deformation of the
+FIG. S2. Exit gate scans of pump current using AWG waveforms.
+The waveforms are illustrated as insets at the top of the plot on
+a common time axis. Dashed horizontal lines indicate the cur-
+rent ef at each frequency. Entrance gate voltages are −0.79 V,
+−1.08 V, and −1.35 V at frequencies of 2 GHz, 2.574 GHz and
+4 GHz respectively. The AWG output amplitude was 0.47 V pp
+for all measurements prior to amplification by a 15 dB wide-band
+inverting amplifier. The arrow indicates a feature possibly due
+to non-adiabatic excitation.
+confining potential formed by the entrance and exit gates.
+Although the 1ef plateau at 4 GHz looks superficially
+flat on this expanded current scale, its slope could easily
+be resolved by zooming the data and no precision mea-
+surements were attempted. The plateau at 2.574 GHz
+was sufficiently flat for metrological investigation, but
+the stability of the pump map was degraded compared
+to 2 GHz, with sudden shifts along the entrance and
+exit gate axes becoming common on time-scales of a few
+hours.
+Switches in the pump state generally occurred
+more frequently as f was increased, and we speculate that
+high frequency components in the drive signal may acti-
+vate charge traps in the device structure. Consequently,
+all the precision measurements reported in the main text
+used f = 2 GHz, with the waveform shown in the inset
+of figure S2, and also the inset of figure 1 (b) of the main
+text.
+C.
+Raw data
+The measurement apparatus and procedure, with two
+exceptions, are the same as described in Ref. 13 and its
+supplementary information. The exceptions are firstly,
+the use of a blind protocol as discussed in the main text,
+and secondly, the use of a noise-optimised ULCA16 in-
+stead of a standard ULCA17. All measurements are per-
+formed as on-off cycles.
+For pump measurements, the
+‘on’ and ‘off’ states correspond to the entrance gate drive
+waveform from the arbitrary waveform generator (AWG)
+being turned on and off respectively. For calibrations of
+the digital voltmeter (DVM) used to read out the ULCA,
+
+(a) Sine 1.05 GHz
+Sine 1.05 GHz
+-1.2-
+AWG 1.04 GHz
+VENT
+0
+lp/ef
+(c)
+/ V
+-1.8
+-2
+11-1
+-1.6
+-1.2
+4
+Log 1
+6
+(b) AWG 1.04 GHz
+-1.6
+-1.4
+-1.2
+-0.8
+VEXIT / V
+VENT
+/ V
+8
+plots a, b:
+-1.6
+-1.2
+-1.6
+VEXIT / V2 GHz
+1000
+2.574 GHz
+4 GHz
+lp / pA
+500
+e1
+0
+-1.6
+-1.4
+-1.2
+-1.0
+-0.8
+VEXIT / V8
+FIG. S3. Raw data, as viewed in a LabView program used to visualise the data during the measurement campaign. The top pair
+of plots show the raw voltmeter data from one measurement - run 16, measurement 3. The plots are zoomed to highlight the ‘on’
+(upper plot, yellow points) and ‘off’ (lower plot, blue points) pump data. The voltmeter calibration data are off the scale of these
+plots. The lower pair of plots show the beginning of the measurement on an expanded y-axis, and a zoomed x-axis. The first 800
+data points are voltmeter calibrations. The x-axis is simply the sequential data point number. This does not quite map linearly onto
+time, because the cal cycles used a voltmeter auto zero with every data point, whereas an auto zero was performed after every 25th
+data point for the measure cycles. All data points were integrated over 10 power line cycles. On both plots, vertical bars indicate
+the y-scale in raw voltmeter units (ULCA input current).
+the ‘on’ and ‘off’ states correspond to the Josephson volt-
+age standard programmed to output 0.32 V and 0 V re-
+spectively.
+In figure S3 we illustrate some raw data, and explain
+the nomenclature used to describe the data files. The
+illustrated data is measurement 3 from run 16. The data
+are the blind-scaled readings of the Agilent 3458A DVM,
+connected to either the Josephson voltage standard for
+the calibration cycles, or the ULCA for the measure cy-
+cles. For the calibration cycles, the data are the com-
+pletely raw readings from the voltmeter, and for the mea-
+sure cycles the raw readings have been multiplied by the
+blind scaling factor β = 1.00000387. The particular mea-
+surement illustrated here consisted of 7 ‘sequences’. Each
+sequence starts with 8 voltmeter calibration cycles. The
+calibration cycles were done with the DVM auto zero
+turned on, and 50 data points for each on or off segment.
+After the calibration cycles, the voltmeter was connected
+to the ULCA output, and a set of pump measurement
+cycles were done with 1000 data points for each segment,
+auto zero off, and an auto zero operation every 25 data
+points (optimisation of the DVM auto zero interval in the
+context of single-electron pump measurements was first
+discussed in Ref. 6). For the illustrated measurement,
+there were 8 measurement cycles in one sequence. Other
+measurements in the campaign used from 7 to 11 cycles
+per sequence. After the 7 cal-measure sequences, a final
+set of 8 calibration cycles was performed, so that each
+set of measure cycles had a calibration cycle before and
+after, for evaluating the calibration factor to apply to the
+measurement data as described in the supplementary in-
+formation to Ref. 13. The data analysis evaluated the
+pump current separately for each sequence, and the sta-
+tistical properties of this data was used as a pass / fail
+criteria for the measurement, as described in supplemen-
+tary section G. The current reported for the measurement
+was the weighted mean over the sequences.
+Two points are worth remarking in the data.
+The
+first is that the hysteretic Josephson voltage standard
+does not always yield the same step number (it was pro-
+grammed to switch between nominal values of 320 mV
+and 0 V). As discussed in the supplementary informa-
+tion to Ref. 13, this is not an issue as long as the DVM
+is linear over the narrow range of voltages sampled by
+the different calibration steps. The second is the remark-
+able stability of the ULCA offset. By eye, it does not
+appear to drift by more than about 1 fA over the course
+of the measurement. We will examine the stability of the
+ULCA gain and offset in more detail in supplementary
+section E.
+
+measurement
+0.32048-
+20UV(20fA)
+0.3204-
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000 
+45000
+50000
+55000
+60000
+65000
+70000
+75000
+8000
+85000
+90000
+95000
+100000
+105000
+110000
+115000
+12000
+Data point number
+20
+(20 fA
+I reading (M)
+sequence
+2E-5
+:
+-2E-5
+-4E-5-
+ 5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+45000
+ 50000
+55000
+60000
+65000
+7000
+75000
+80000
+85000
+90000
+95000
+10000010500011000
+115000120000
+cal cycle
+0.323
+pump cycle
+VM reading 
+0.32
+2 mV (2 pA)
+ 0.318
+0.317-
+100　200　300　400　500
+1400 1500 1600 1700 1800
+700
+900
+10001100
+1200
+1300
+Data point number
+0.003
+M 0.002-
+2mV(2pA
+-0.003-
+100 200 300 400 500 600 700 800
+1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 3900 40009
+FIG. S4. Plot (a): filled circles, left axis: Calibration factor kDVM
+of the DVM. line, right axis: laboratory temperature. Coloured
+blocks at the bottom of the plot, and events labeled ‘E1’ and
+‘E2’ are explained in the supplementary text. Inset: Log-scale
+histogram of the difference between adjacent measurements of
+kDVM. The black square in the main plot shows a range of DVM
+calibration data plotted on expanded axes in plot (b).
+D.
+Voltmeter calibrations and measurement time-line
+In figure S4 we have combined several pieces of infor-
+mation pertinent to the measurement campaign.
+The
+main graph of plot (a) shows, on the left axis, the cali-
+bration factors, kDVM, of the DVM recorded during the
+measurement campaign. We define the calibration factor
+as kDVM∆VIND = ∆VREF, where ∆VIND is the change
+in indicated voltage and ∆VREF is the change in applied
+reference voltage evaluated from an on-off cal cycle. Each
+plotted point is averaged from a set of 8 calibration cycles
+directly against the Josephson array at a nominal volt-
+age of 0.32 V. No data points have been omitted from
+this plot, and some outlying data points with large er-
+ror bars are the result of failure of the frequency lock to
+the Josephson array control electronics. The pink line
+plotted on the right axis shows the laboratory tempera-
+ture, as measured by a sensor integrated into the ceiling.
+Periods when the experiment was not running are vis-
+ible as gaps in the voltmeter calibration data, and to
+clarify the experimental time-line, shaded blocks at the
+bottom of the plot indicate what was happening. Four
+types of activity are indicated: The experimental runs,
+numbered 1-17; the weekend calibrations of the ULCA
+input stage gain GI; The short calibrations of the ULCA
+output stage RIV, and finally a period of down-time indi-
+cated by a cross-hatched block when the experiment was
+stopped due to a fault in the AWG used to generate the
+pump drive signal.
+Two events marked E1 and E2 are indicated. E1 marks
+when an un-used instrument in the experimental rack
+(a sine wave generator) was switched off.
+The reduc-
+tion of heat produced in the rack caused a noticeable
+change in the calibration factor of the voltmeter, which
+was mounted directly above the sine wave generator. The
+fact that this is visible in the kDVM data illustrates the
+sensitivity of the direct calibrations of the DVM against
+the Josephson array. The event E1 also lowered the tem-
+perature of the ULCA, mounted higher up in the rack,
+reducing ATR by roughly 0.15 ppm. Event E2 marks a
+dramatic excursion of the laboratory temperature caused
+by planned maintenance of the air conditioning. This re-
+sulted in a larger uncertainty assigned to some of the
+measurements of run 15 because of rapid changes in the
+ULCA temperature. The transition from stable to fluc-
+tuating temperature roughly half-way through the mea-
+surement campaign was co-incident with a transfer of
+liquid helium into the experimental dewar. It may also
+be related to increased activity in adjacent laboratories
+as activities were re-started and staff returned following
+relaxation of covid-19 control measures.
+One important contribution to the uncertainty of the
+current measurement is the stability of the DVM on the
+1-hour time taken for a cal-measure sequence. The in-
+set to figure S4 (a) shows a histogram of the difference
+in kDVM between adjacent calibrations during measure-
+ments, denoted ∆kDVM. Generally, the DVM is stable
+to better than 0.2 ppm on time-scales of an hour, but
+jumps in kDVM of up to 0.5 ppm sometimes occur. As in
+our previous study13, the uncertainty due to the drift in
+kDVM was evaluated using a rectangular distribution as
+∆kDVM/2
+√
+3, so a jump in kDVM of 0.2 ppm contributes
+0.057 ppm to the combined uncertainty in the pump cur-
+rent. The 1-hour DVM calibration interval is therefore
+consistent with achieving a combined uncertainty in the
+pump measurement of 0.1 ppm. To visualise the short-
+term stability of the DVM in the time domain, plot (b)
+shows a portion of the main plot on an expanded time
+axis. Over this 3-day period, the voltmeter calibration
+did not drift by more than 0.3 ppm. The voltmeter cali-
+bration data are of general interest for electrical metrol-
+ogy, where voltmeters such as the 3458A are commonly
+used as transfer standards. From the general perspec-
+tive of evaluating the DVM performance in metrological
+applications, this data set shows the DVM comfortably
+exceeding its manufacturer’s 24-hour accuracy specifica-
+tion of 1.5 ppm on the 1 V range. Calibrations over longer
+time-scales (not shown) show that the 90-day specifica-
+tion of 4.6 ppm is also exceeded by typically a factor 5.
+E.
+ULCA calibrations
+The noise-optimised ULCA was calibrated using a
+cryogenic current comparator (CCC) bridge, as described
+in Ref. 18. For the calibrations, the ULCA was hand-
+carried to an adjacent laboratory. It was specifically car-
+
+100
+Counts
+10
+(a)
+0.999983
+E1
+0.0 △kDVM0.5
+/ ppm
+(00)
+0.999982
+KDVM
+20
+Lab Temp.
+0.999981
+E2
+0.999980
+16,
+G
+6,7G.
+G.
+1-3
+G.
+4.5
+8-11
+12-15
+15
+17
+0.999979
+15 Jan
+29 Jan
+12 Feb
+26 Feb
+Date (2021)
+(b)
+0.9999814
+KDVM
+HH
+0.9999812
+0.9999810
+19 Feb
+21 Feb
+Date10
+FIG. S5. (a): Deviations from nominal of (upper plot): ULCA
+input current gain and (lower plot): ULCA output stage gain,
+corrected to a standard temperature.
+The plot shows all the
+calibrations performed on this ULCA since its delivery to NPL.
+The time period covered by the measurement campaign is shown
+as a purple shaded box, and the fixed value adopted for the
+input stage gain during the measurement campaign is shown as
+a horizontal dashed line. (b): Deviation from nominal of the
+ULCA transresistance gain calculated from the data in plot (a),
+for use during the measurement campaign.
+The boxes above
+each data point show the run numbers covered by the 6 values
+of trans-resistance gain.
+ried by hand rather than on a trolley to minimise the
+possibility of mechanical shocks.
+As illustrated in the
+time-line of figure S4 (a), a total of 4 calibrations of the
+input stage gain GI (nominal value 1000), and 6 calibra-
+tions of the output gain RIV (nominal value 1 MΩ) were
+preformed during the measurement campaign. The over-
+all trans-resistance gain of the ULCA is ATR = GIRIV
+(nominal value 1 GΩ)17. The results of all calibrations
+of this ULCA unit since its delivery to NPL are shown in
+figure S5 (a). The historical behaviour of the input and
+output gains is different, and resulted in different statis-
+tical treatments. The input stage gain does not show any
+significant drift over the measurement campaign, and fur-
+thermore, the limited number of additional calibrations
+before and after the campaign did not give any evidence
+for long-term drift. Consequently it was assumed to be
+constant during the measurement campaign.
+Its value
+was taken to be the weighted mean of the four calibra-
+tions during the campaign, shown as a horizontal dashed
+FIG. S6. Averaged DVM readings with the pump (a): on, and
+(b): off. Each data point is averaged from one measurement, so
+is the mean of ∼ 60, 000 DVM readings. A scale bar indicates 1
+ppm of the 320 pA pump current.
+line in figure S5 (a). On the other hand, the output stage
+gain shows some drift over time.
+Values of RIV were
+chosen half way between ‘before’ and ‘after’ calibration
+values, with uncertainties which included a drift term de-
+rived from a rectangular distribution. In this way, five
+values of ATR were calculated to cover runs 4-17. Runs
+1-3 were not preceded immediately by any ULCA calibra-
+tions, so the value of RIV was taken to be the first RIV
+calibration, in between runs 3 and 4, with an uncertainty
+derived from a rectangular distribution bounded by the
+highest and lowest RIV calibrations during the measure-
+ment campaign. In other words, we assumed that the
+drift behaviour of RIV for the few days covering runs 1-3
+was similar to the behaviour during the rest of the mea-
+surement runs. The 6 values of ATR with their combined
+standard uncertainties used to analyse the measurements
+are shown in figure S5 (b).
+The remarkable stability of the ULCA offset current
+is already visible in the raw data of figure S3 (a), and
+in figure S6 we go further and show the averaged values
+of the ‘ON’ and ‘OFF’ signals measured by the DVM.
+Each data point in this graph is the average of all the
+ON (plot (a)) or OFF (plot (b)) DVM readings after
+rejecting the first 300 readings in each segment.
+The
+offset current does not change by more than 2 fA over
+the 2-month period covered by the measurements. The
+drift in offset current may be partially attributable to
+changes in ULCA temperature, but there may also be
+contributions due to changes in leakage currents through
+
+(a)
+7.8-
+SG, / ppm
+7.7
+-- 1
+7.6
+7.5
+7.4
+7.3
+27.0
+TI
+27.2
+-27.4
+01/10/2019
+01/10/2020
+Date
+(b)
+8-11
+12-15
+1-3
+4,5
+6,7
+ppm
+-19.4
+16,17
+T
+-19.5
+-19.6
+15 Jan
+29 Jan
+12 Feb
+26 Feb
+Date in 2021(a)
+0.320433
+0.320432
+1 ppm = 0.32 fA
+0.320431
+15 Jan
+29 Jan
+12 Feb
+26 Feb
+(b)
+-0.000003
+V
+->
+1 ppm = 0.32 fA
+0.000004
+-0.000005
+15 Jan
+29 Jan
+12 Feb
+26 Feb11
+FIG. S7. (a): Difference in input current gains GI of two ULCA
+units for two series of measurements in self-test configuration, in
+which the test current was alternated between a ‘high’ current of
+4.8 nA and a ‘Low’ current, either 320 pA or 640 pA. (b): Differ-
+ence in trans-resistance gains ATR, alternating the test current
+between 4.8 nA and 320 pA. The data plot legend refers to both
+panels (a) and (b).
+the electron pump control gates. The possible leakage
+current paths through the device gates were discussed in
+the supplementary information to Ref. 13.
+F.
+ULCA linearity
+The linearity of the ULCA gain is a key assumption in
+this experiment, because the calibration of GI is done at
+an input current of ∼ 5 nA and the pump current during
+the measurement is 320 pA. One previous investigation
+set an upper bound on the non-linearity of the overall
+UCLA transresistance gain ATR at around the 0.1 ppm
+level16. We attempted to reduce this upper bound, using
+two test methods previously demonstrated for the ULCA.
+First, we compared the input stage current gains of two
+ULCA units, as was first demonstrated in Ref. 17. This
+is called the ‘self-test’ configuration. A standard ULCA
+unit, not otherwise used in our experiment, was used as a
+source to generate a test current for comparing its input
+stage gain GI,source with the input stage gain of the noise-
+optimised experimental ULCA GI,measure. This self-test
+configuration is quite straightforward to implement, be-
+cause the readout DVM measures a small signal derived
+from the difference in the input gains of the two ULCAs,
+denoted αGI = GI,source − GI,measure. We alternated sets
+of forward-reverse cycles with test currents of ±4.8 nA,
+±320 pA and ±640 pA to obtain the data of figure S7
+(a). The forward-reverse cycle time was 60 s, and the
+data points are averaged from 100 and 1000 cycles for the
+±4.8 nA and ±320 pA currents respectively. The back-
+ground drift of αGI visible in the high current data is due
+to temperature variation of the ULCAs, but by evaluat-
+ing the difference between each low-current data points
+(orange triangles) and the mean of the two adjacent high
+current data points (green circles), we can extract the
+current dependence as a mean over 6 cycles of high-low-
+high current. We obtain the current dependence in αGI
+between 4.8 nA and 320 pA as 0.002 ± 0.029 ppm. An
+additional run examined the current dependence between
+4.8 nA and 640 pA (blue diamonds). This data was not
+evaluated, but clearly the current dependence is around
+a part in 108 or less.
+For the second test, we measured the current depen-
+dence of the difference in the overall trans-resistance
+gains of the two ULCAs, again with the standard ULCA
+in ‘source’ mode, and the noise-optimised experimental
+ULCA in ‘measure’ mode. This test configuration is il-
+lustrated in figure 6 of Ref. 16. It is less straightforward
+to implement than the self-test configuration, because
+the voltage outputs of the source and measure ULCAs
+have opposite signs. We implemented a protocol equiv-
+alent to figure 7b of Ref. 16. A single DVM could be
+connected to either the source or measure ULCA us-
+ing an automated switch - the same switch that was
+used in the main experiment to connect the DVM ei-
+ther to the ULCA output or the JVS. One cycle con-
+sisted of four segments of data:
+the test current was
+applied with both polarities with the DVM connected
+to the source ULCA, recording a forward-reverse differ-
+ence voltage ∆Vsource and then the test current was ap-
+plied with both polarities with the DVM connected to the
+measure ULCA, recording a difference voltage ∆Vmeasure.
+Acquiring one cycle took 2 minutes. Assuming that the
+DVM calibration factor does not change on this time-
+scale, The ratio of ULCA transresistance gains is given by
+αATR = ATR,source/ATR,measure = ∆Vmeasure/∆Vsource.
+We are interested in whether the ratio of gains depends
+on current, so as in the tests of GI linearity, we alternated
+1000 cycles at ±320 pA test current, with 100 cycles at
+±4.8 nA test current to yield the averaged data points
+in figure S7 (b). Similarly to the data of figure S7 (a),
+we averaged the high-low-high differences, to obtain the
+current dependence of αATR as 0.006 ± 0.023 ppm.
+Of course, this data does not conclusively rule out non-
+linearity in the ULCA unit used for the measurements.
+It only gives information on the linearity of the differ-
+ence in the gains of the two ULCA units. It is a slightly
+stronger test than the one published in Ref. 16, how-
+ever. While that measurement used two nominally iden-
+tical noise-optimised ULCAs, our measurement used a
+standard ULCA in the ‘source’ role. The different values
+of resistors used in the current scaling networks make it
+less likely that both ULCA units would have the same
+
+(a)
+8.8×106
+αG, / ppm
+-8.9×10-6
+-9.0×10-6
+-9.1×106
+-9.2×10-6
+Measurement number
+(b)
+± 4.8 nA
+-35.6
+± 320 pA
+± 640 pA
+-35.7
+-35.8
+-35.9
+Measurement number12
+current-dependence to the gain.
+G.
+Statistical tests and data set rejection
+As mentioned in the main text, the pump state, as doc-
+umented by the ‘pump maps’, changed during the mea-
+surement campaign, with some obvious dramatic changes
+occurring during some measurements, and more subtle
+changes during other measurements. Even if the pump
+map was stable, some of the measurements close to the
+edges of the current plateaus could be affected by small
+fluctuations in offset charge, leading to relatively large
+changes in pump current as the operating point drifted
+on and off the plateau.
+It could not generally be as-
+sumed that the pump current sampled by a measure-
+ment lasting more than 10 hours represented a station-
+ary mean. Each measurement was therefore subjected to
+a statistical test. Recall from supplementary section S3,
+that the pump current from each sequence was evaluated
+separately.
+This yielded m values of IP, denoted IP,m
+with uncertainties U(IP,m), where m is the number of se-
+quences in the measurement. If all the IP,m are sampling
+the same value of pump current, on average the stan-
+dard deviation of the IP,m, σ(IP,m) will be equal to the
+mean of the uncertainties, ⟨U(IP,m)⟩. We propose the ra-
+tio σ(IP,m)/⟨U(IP,m)⟩ = R as a statistical measure of the
+stationarity of the data, and in figure S8, we plot a his-
+togram of this quantity (grey bars, right axis) for the 64
+measurements performed during our campaign. We also
+plot (red bars, left axis) a histogram of the same quantity
+obtained from 1000 simulated measurements, in which
+the simulated raw data, both for the measurement and
+calibration cycles, was generated from a stationary mean
+multiplied by Gaussian white noise with the same stan-
+dard deviation as the real data. As expected, the most
+probable value of R for this simulated stationary data is
+1, and the probability of obtaining a measurement with
+R > 2 from a set of 1000 measurements becomes neg-
+ligible. Since we only performed 64 measurements, we
+assigned a cutoff of R = 1.7, and rejected measurements
+with R > 1.7. Comparing the histogram of the measured
+data with the simulation, it is clear that a significant
+number of data sets have an R value which would be
+improbably high if the pump current was constant dur-
+ing the measurement. This is actually expected, for the
+reason that some of the precision measurements were se-
+lected with control parameter values close to the edges of
+the current plateau. For these measurements, small fluc-
+tuations in offset charge during the measurement (equiv-
+alent to a drift in the control parameters) would cause
+the pump current to drift away from ef.
+To see the accept / reject criteria in action, two exam-
+ple measurements from run 10 are plotted in figures S8
+(b) and (c), with the corresponding R values marked with
+red and green arrows on the x-axis of panel (a). The data
+of panel (b) clearly shows a decrease in the pump current,
+and it would be tempting to reject this data set based just
+FIG. S8. (a): Histograms of the quantity R, defined in the sup-
+plementary text as the ratio of the standard deviation of the m
+values of IP calculated for each measurement (m is sequence
+number) to the average uncertainty of IP,m. The grey bars re-
+ferred to the right axis are for the measured data, and the red
+cross-hatched bars referred to the left axis are for 1000 sim-
+ulated measurements assuming a stationary mean.
+A vertical
+dashed line shows the R = 1.7 rejection threshold derived from
+the probability of the simulated data having an R greater than
+this value. Panels (b) and (c) show IP,m for two example mea-
+surements from run 10, with (b): R > 1.7 and (c): R < 1.7.
+The R values for these data sets are indicated with red and green
+arrows respectively on panel (a).
+on this time-domain visualisation of IP,m. However, the
+definition of the R parameter makes this otherwise sub-
+jective process more quantitative. Altogether, 14 mea-
+surements during the entire measurement campaign had
+R > 1.7.
+H.
+full data set
+Due to instability of the pump after run 5, only the
+data from runs 1-5 are analysed in the main text. The
+increasing instability is visible in the pump maps of figure
+S10, and also in the increasing number of runs which
+failed the stationary mean test. In figure S9, we present
+all of the precision data on linear axes.
+Plots (a,b,c)
+show all of the measurements on expanded y-axes, and
+plots (d,e,f) show the sub-set of the measurements which
+passed the stationary-mean test. Figure S10 shows the
+full set of ‘fingerprint’ pump maps obtained before and
+after each precision measurement run. For data integrity
+
+(a)
+200
+15
+Simulated data
+measured data
+Count (Simulated)
+150.
+Count (Measured)
+REJECT
+10
+100
+5
+50.
+0
+0
+0
+2
+4
+5
+6
+7
+8
+R
+(b)
+(c)
+Run 10, V.
+= -0.835 V
+Run 10, V
+= -0.775 V
+ENT
+ENT
+wdd
+REJECT
+0
+ACCEPT
+p,m
+-2
+12 hours
+0
+2
+4
+6
+8
+10
+0
+2
+4
+6
+8
+10
+seguence m
+seguence m13
+FIG. S9. a-c: Deviation of the pump current from its nominal value as a function of (a): Exit gate voltage, (b): Entrance gate
+voltage and (c): AWG output amplitude. All of the measurements from the 17 runs are shown in these plots. One data point in panel
+(a) is off the y-axis scale, and is indicated by an arrow. (d), (e) and (f): the sub-set of data in plots (a), (b) and (c) respectively,
+which passed the stationary mean test, on expanded axes. In each plot, vertical dotted lines indicate fixed values of the scanned
+parameter for runs in the other plots. Error bars indicate combined standard uncertainties UT.
+purposes, this figure also includes the 4-digit hexadecimal
+file identifier for the precision raw data.
+
+(a)
+(d)
+1.0
+9,105,7,15
+8
+1
+4
+11
+2
+12
+14
+17
+6
+6
+run 6
+0.5
+3
+8
+16
++27
+udd
+0.0
+△/p 
+4
+11
+-0.5
+2
+6
+12
+0.
+3
+8
+16
+-2 
+-1.0
+-1.40
+-1.35
+-1.30
+-1.25
+-1.40
+-1.35
+-1.30
+-1.25
+VeXIT / V
+VEXIT / V
+5
+13
+(b)
+(c)
+(e)
+(f)
+6
+7
+14
+1.0
+16,17
+1,2,3,4,6,8,11,12
+1-16
+9
+15
+4
+0.5
+10
+17
+udd
+udd
+2
+0.0
+N/p 
+0
+△/p /
+0.5
+-2 
+5
+14
+17
+7
+10
+15
+-1.0
+4
+0.90 -0.85 -0.80 -0.75 -0.70
+0.460.470.48
+0.85-0.80-0.75-0.70
+0.46
+0.47
+0.48
+VENT / V
+VAc / V
+VENT / V
+VAc / V14
+FIG. S10. Thumbnail pump maps measured before and after each precision scan. Each pump map is an inverted grey-scale derivitive
+plot of the current similar to the one shown in figures 1(b) and 2(a) of the main text. The axis limits are the same for each thumbnail:
+The x-axis is VEXIT, from −1.7 V to −0.8 V, and the y-axis is VENT, from −1.4 V to −0.4 V. Each horizontal row of the table represents
+a precision run. The middle cell contains some text data describing the run, including the 4-digit hexadecimal file number identifying
+the raw data set for the precision run. The left-most cell shows the pump map recorded before the run, and the right-most cell shows
+the pump map recorded after the run. Missing pump maps for runs 9,10 and 11 were due to software crashes. Red arrows highlight
+runs in which the pump map changed dramatically during the run.
+
+8 Days down-time due to AWG issue

VNAyT4oBgHgl3EQfhfi9/content/tmp_files/2301.00379v1.pdf.txt

@@ -0,0 +1,1294 @@
+ 
+Review Article 
+A review of Implementation and Challenges of Unmanned Aerial Vehicles for Spraying 
+Applications and Crop Monitoring in Indonesia 
+ 
+Authors:  
+Muhamad Rausyan Fikri1,2, Taufiq Candra3, Kushendarsyah Saptaji4, Ajeng Nindi Noviarini5, Dilla Ayu Wardani3 
+1. 
+Automation Technology and Mechanical Engineering, Faculty of Engineering and Science, Tampere 
+University, Tampere, 33720, Finland 
+2. 
+Information Systems, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, 
+Indonesia 
+3. 
+Industrial Engineering, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, 
+Indonesia 
+4. 
+Mechanical Engineering, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, 
+Indonesia  
+5. 
+Computer Science, Faculty of Engineering and Technology, Sampoerna University, Jakarta, 12780, 
+Indonesia 
+ 
+Abstract: 
+The rapid development of technology has brought unmanned aerial vehicles (UAVs) to become 
+widely known in the current era. The market of UAVs is also predicted to continue growing with 
+related technologies in the future. UAVs have been used in various sectors, including livestock, 
+forestry, and agriculture. In agricultural applications, UAVs are highly capable of increasing the 
+productivity of the farm and reducing farmers' workload. This paper discusses the application of 
+UAVs in agriculture, particularly in spraying and crop monitoring. This study examines the 
+urgency of UAV implementation in the agriculture sector. A short history of UAVs is provided in 
+this paper to portray the development of UAVs from time to time. The classification of UAVs is 
+also discussed to differentiate various types of UAVs. The application of UAVs in spraying and 
+crop monitoring is based on the previous studies that have been done by many scientific groups 
+and researchers who are working closely to propose solutions for agriculture-related issues. 
+Furthermore, the limitations of UAV applications are also identified. The challenges in 
+implementing agricultural UAVs in Indonesia are also presented. 
+ 
+Keywords:  
+Unmanned aerial vehicle, agricultural UAV, spraying, crop monitoring.  
+
+ 
+ 
+1. Introduction 
+According to the United Nations (UN), the world population is projected to reach 9.7 billion people 
+in 2050 (UN, 2015). This vast population would potentially double the food demand in the future 
+(Hunter et al., 2017). Consequently, the ever-growing population that would emerge could cause 
+food shortages in the future. This issue has become a severe problem since the Food and 
+Agriculture Organization (FAO) announced similar speculation in which the current agricultural 
+production must be increased by 70 percent by 2050 to meet the increasing demand for high-
+quality food (Mundial, 2021). Many people suffering from hunger become a signal of how severe 
+the food shortage is, and it was reported that more than 820 million people in 2018 were considered 
+undernutrition (WHO, 2019). Surprisingly, the earlier data mentioned shows the increasing 
+tendency towards people suffering from hunger since only around 690 million people were 
+considered suffering from hunger in 2015. This kind of data indeed contradicts the second 
+Sustainable Development Goals (SDGs) approved by the United Nations (UN) in 2015 with the 
+aims to eradicate hunger and ensure access to food for all people (UN, 2015). On the other hand, 
+the labor shortages in the agricultural sector due to the aging population and the decreasing number 
+of workers have exacerbated the situation. The lack of laborers in the agricultural field would 
+expand the cultivation area per worker and increase the workload of workers (Seo & Umeda, 
+2021). 
+Alternative and innovative solutions to increase food production in dealing with those 
+issues are needed. One way to increase food production is to promote the internet of things (IoT), 
+robotics, and artificial intelligence (AI). By shifting the workforce to the technology's utilization, 
+it is expected to solve the labor shortages and improve farmers' skills. This transformation is 
+inseparable from revolutionary industries that constantly bring industrial innovations. Some 
+industrial innovations found in recent years, such as sensor technologies, big data, and artificial 
+intelligence (AI), have been considered as the beginning of the "Industry 5.0" era by the European 
+Commission (EC) (EC, 2021). The emergence of technologies characterized by advanced 
+digitalization is believed to play a significant role in increasing production flexibility and making 
+the value chain more robust so that technology could minimize the farmers' workload and improve 
+the speed and accuracy of the work. 
+
+ 
+Among the technologies mentioned earlier, unmanned aerial vehicles (UAVs) are one 
+viable way to increase food production. UAVs are less expensive and have contributed to many 
+areas in agriculture, including spraying, weed recognition, and crop monitoring (Mogili & Deepak, 
+2018). UAVs' timely and reliable information about the production, yield and crop management 
+would become beneficial to ensure food safety and security for stakeholders such as farmers and 
+sales units (Martos et al., 2021).  UAV technologies in agriculture could also enable the complete 
+monitoring of crop conditions from the beginning of the growing season until the end of harvest 
+(Silver et al., 2017).  
+Some leading technologies are possible by implementing UAVs in the agriculture sector. 
+Therefore, this paper focuses on reviewing UAVs applications for spraying and crop monitoring 
+in the agricultural field. Some research results on the use of UAVs in spraying and crop monitoring 
+are discussed thoroughly to highlight the use of UAVs and the characteristics of the farming sector. 
+Some limitations exist during UAVs implementation are also reviewed to reveal the gap of UAV 
+implementation in the agriculture field. The rest of this paper is organized as follows. Section 2 
+describes UAVs' history and the classification of UAVs. Section 3 describes the application of 
+UAVs focusing on spraying and crop monitoring. Section 4 provides some limitations in adopting 
+UAV technologies in the agriculture sector. In section 5, the challenge in implementing UAVs in 
+Indonesia is discussed. The last section provides the conclusion.  
+ 
+2. Agriculture in Indonesia and Its Challenge 
+2.1. Current Condition 
+Agriculture programs in Indonesia have been a big agenda at the national level, such as 
+National Agenda 21, National Development Programs, and Agricultural and Forestry 
+Revitalization Strategies, encouraging Indonesia to adopt sustainable agriculture. The Central 
+Planning Authority (BAPPENAS), the Ministry of Agriculture, and the Environment Ministry 
+have implemented these ideas. Most of these plans include components suitable for effective 
+environmental management of Indonesian agricultural exports. 
+The motivations for using these tactics have shifted over time, and they seem to be 
+responding to a variety of distinct trends. First and foremost, Indonesian national plans have 
+prioritized socio-economic objectives above ecologically sustainable ones. Nonetheless, 
+environmental concerns have become more critical, as evidenced by recent reforms and the 
+
+ 
+increasing frequency of ecological issues in strategic documents. Second, strategy papers also 
+show a change in direction as the combination of means changes, with less focus on laws and 
+regulations and more attention to the means for market creation and voluntary methods over time. 
+The tensions between diverse skills and conservation goals and local revenue-generating needs 
+have led to different patterns of success in different states across the country. Significant 
+advancements have been achieved in modernizing agro-environmental rules, made possible by 
+increased information and worldwide best practices. The extent to which environmental hazards 
+pose local or global dangers, the degree of environmental degradation of a particular product, and 
+the availability of legal, enforcement, budgetary, and regulatory capacities for sub-national 
+governments all influence the choice of the policy tool. 
+For practical reasons, Indonesian policymakers have used a range of mechanisms to 
+minimize agriculture's environmental footprint, including direct regulation, market creation or 
+market modification incentives, voluntary and beneficial solutions, and market modification 
+incentives. Policies are implemented via legislative and regulatory mechanisms, which are 
+probably targeted at plantation states and large farms. It is essential to note the existence of 
+obligatory ISPO standards (in the section on local regulatory instruments), since they have just 
+recently been adopted as a result of voluntary standards being adopted as mandatory. Additional 
+factors that impact policymakers' choices to implement a particular instrument include the 
+potential efficacy of the instrument in comparison to its costs and the capacity of the policymaker 
+to enforce the instrument in the face of likely political opposition. In this respect, implementing 
+regulatory and legislative tools seems to be the most effective method of monitoring prominent 
+investments, such as planting restrictions and the demand for environmental impact assessments. 
+According to the findings of the Indonesian research, foreign pressure had a role in the spread of 
+planting restrictions throughout the country. In addition, the implementation of regulatory 
+instruments may be most effective when their administrative and monitoring costs are already 
+integrated into a current administration, such as indirect product charges for import limitations, 
+which are already embedded into an existing administration. 
+ 
+2.2. The Challenge in Indonesia’s Agriculture 
+One of the factors is the limited availability of agricultural land in Indonesia due to land 
+reform, which is widespread in big provinces. As the population rose, so did the need for housing. 
+
+ 
+As a result, developers exploit a large portion of agricultural land to construct real estate. The 
+growth in the number of people also increased demand for trade and tourism, contributing to 
+increased demand for land. Farmers could not be faulted for selling their farms in this scenario. 
+Farmers were driven to sell their lands due to a lack of knowledge and technology, high agricultural 
+costs, and rising necessities. Farmers in Indonesia with low levels of education have little choice 
+except to work outside the agricultural industry; therefore, those who do not own land are tenant 
+farmers. Food price increases should be a dream come true for farmers, as their revenue would 
+almost certainly rise. Unfortunately, because most farmers in Indonesia are tenant farmers, it has 
+become a boomerang for their wellbeing. The rise in food prices has little effect on the well-being 
+of Indonesian farmers. Their income remains minimal, and they must continue to purchase their 
+basic necessities at market prices. Those who own land have benefited from the growing price. 
+Furthermore, the general public's perception of farmers is that they do not do a good job. The 
+younger generation is interested in non-agricultural jobs, such as parenting a farmer's child. 
+Farmers' regeneration is hampered as a result, and many opt to sell their land to be established as 
+capital or to work in the non-agricultural sector.  
+As the world's population rises at an alarming rate, agriculture must expand to supply the 
+growing demand for food against all odds. The agriculture sector is the most vulnerable to the 
+impact of integrating fresh, modern innovations in eradicating environmental-related challenges 
+and enhancing the current productivity rate. Now, the question is, how could we possibly do this? 
+Marking a third wave of the “Green Revolution”, the concept of precision agriculture with 
+technological help such as Unmanned Aerial Vehicle (UAV) has become popular nowadays in the 
+vast area of agriculture due to its tremendous benefits. Farmers and managers can boost operational 
+efficiency, cut expenses, minimize waste, and improve the quality of crops with the aid of accurate 
+data. Overall, technology has been a key component behind agricultural development and other 
+discoveries brought into the industry. 
+ 
+3. Unmanned Aerial Vehicle (UAV) 
+3.1 UAVs History 
+An unmanned Aerial Vehicle (UAV) is an aircraft with no pilot on board; in other words, 
+it refers to auto-piloted aircraft (Ahmad et al., 2021). The unmanned type of aircraft can be 
+operated in two ways, either by a human operator or autonomously operated under the control of 
+
+ 
+an onboard computer (Pablo et al., 2020). According to the US Department of Defense (DOD), 
+UAV can be described as either a single air vehicle (with equipped surveillance sensors) or a UAV 
+system (UAS) that consists of three to six air vehicles, a ground control station, and support 
+equipment (Gertler, 2012). Furthermore, UAVs are often associated with remote sensing in 
+carrying out their task. This remote sensing is commonly known as UAV remote sensing, which 
+combines UAV and remote sensing technology that can quickly capture information about land, 
+environment, and resources for further data processing (Shi & Liu, 2011). In the US and other 
+developed countries, UAV remote sensing has been applied in many fields such as forestry, 
+environmental protection, land, and military (Xiang & Tian, 2011). UAV remote sensing are used 
+because it could be deployed quickly in repeated times. In addition, they are less costly, safer than 
+piloted aircraft, flexible in terms of flying height, and able to obtain very high-resolution imagery 
+(Yang et al., 2011).  
+The term unmanned aerial vehicles are also known as remotely piloted aircraft (RPA). 
+Even though the terms UAV and RPA are interchangeable, the term UAV is commonly used by 
+aviation organizations (Santos et al., 2019), while the term RPA is widely used in Europe 
+(Gallardo- Saavedra et al., 2018). Back then, in 1930, UAVs were also known as “Queen Bees” 
+(Vroegindeweij et al., 2014) and were initially used for military purposes (Muchiri & Kimathi, 
+2016). In 1986, UAVs that work specifically in agricultural contexts were introduced by launching 
+UAVs for Montana’s forest fires monitoring and followed by the capture of enhanced image 
+resolution using UAVs in 1994 (Muchiri & Kimathi, 2016). Then, a more complex UAV model 
+was finally developed by Yamaha through “Yamaha RMAX,” with the primary function for pest 
+control and crop monitoring application (Mogili & Deepak, 2018).  This UAV model is used for 
+pesticide spraying in rice fields of Asia. As opposed to ground-based sprayers, the pesticides 
+deposition of this UAV model is quite similar, but this UAV model is used explicitly for a high-
+value crop environment (Giles & Billing, 2015).  
+ 
+3.2 Classification of UAVs 
+Generally, there are three types of UAV platforms: fixed-wing, rotary-wing UAVs, and 
+non-wing UAVs (Figure 1). A fixed-wing UAV resembles an airplane and requires a runway or 
+Modelsurface (meadow or road) for take-off and landing (Pederi & Cheporniuk, 2015). This kind 
+of UAV uses thrust and aerodynamic lifting forces to fly. It has a larger size than a rotary-wing 
+
+ 
+model and is mainly used for aerial mapping, spraying, and photography over a wide range of time 
+(Li & Yang, 2012). This UAV type typically lacks hovering while offering high-speed flights for 
+longer durations (Ahmad et al., 2021). The gliding capabilities possessed by fixed-wing aircraft 
+could enable greater flight endurance, allowing them to operate over longer distances (up to 15-20 
+km) (Paneque-Gálvez et al., 2014). 
+ 
+ 
+D 
+Figure 1. Illustration of Basic UAVs (A) Fixed-Wing UAV (B) Rotary-Wing UAV (C) Combinational Concepts 
+(source: Ahmad et al., 2021), (D) Blimps (source: Tao et al., 2018) 
+On the other hand, rotary-wing UAVs is primarily categorized into the helicopter and 
+multi-rotor types. The helicopter type of rotary-wing UAV has a unique feature with a large 
+propeller atop the aircraft. It is widely used for spraying and aerial photography (see Figure 2) 
+(Swain et al., 2010). They can hover, vertical takeoff, and land with nimble maneuverability while 
+exhibiting low-speed flight for a shorter duration (Ahmad et al., 2021). In comparison, the multi-
+rotor models are called according to the number of rotors (Kim et al., 2019). Quadcopter, 
+hexacopter, and octocopter are some multi-rotors UAVs that are widely known (Figure 3). These 
+UAVs are lifted and propelled according to the number of rotors (Mogili & Deepak, 2018).  
+ 
+
+ 
+ 
+Figure 2. Single Rotor/Helicopter UAV Type (source: Huang et al., 2009) 
+ 
+Figure 3. Multirotor UAVs, (A) Quadcopter (source: Spoorthi et al., 2017) (B) Hexacopter (source: Yallappa et al., 
+2017) (C) Octocopter (source: Wallace et al., 2016) 
+For example, the rotor movement of a quadcopter is responsible for generating the lift of a 
+quadcopter. In a quadcopter, each of two rotors moves in an opposite way of which two rotors turn 
+in the clockwise direction and the other two turn in the anticlockwise direction. The movement of 
+the quadcopter around the axis consists of yaw (clockwise and anticlockwise), pitch (backward 
+and forward), and roll angles (right and left). The quadcopter uses a control system to balance the 
+
+SR200
+SR20B
+c 
+thrust of each rotor in order to support the UAVs' lift and yaw, pitch, and roll angles (Mogili & 
+Deepak, 2018). This control system turned out to be practical to produce a stable flight of the 
+UAVs (Patel et al., 2013). Moreover, two quadcopter configuration types include the plus (+) and 
+cross (X) models, as shown in Figure 4. The cross model is more popular between the two models 
+due to its stability (Kedari et al., 2016). 
+ 
+ 
+Figure 4. Quadcopter Configuration Model (A) Plus configuration (B) Cross Configuration (source: Mogili & 
+Deepak, 2018) 
+Furthermore, these multirotor UAVs have extended their functionalities by equipping appropriate 
+sensors such as vision, infrared, multispectral, and hyperspectral cameras. The expansion of UAV 
+features brings great influences, especially in adding the capabilities of the UAVs. Those sensors 
+are used to obtain data such as vegetation, reflectance indexes, and leaf areas in order to provide 
+information about the current state of crops. With this information, farmers can make possible 
+remedies or policies (weed control, fertilization, irrigation) according to the condition of the crops 
+(Gonzalez-De-Santos, 2016).   
+Further, non-wing UAVs have been developed to cope with the long endurance of flying robots 
+and are lighter than air (LTA) such as Blimp. A blimp is identically has a larger size than fixed-
+wing and rotary-wing and cushioned with a helium-filled envelope, making the robot safe to fly 
+indoors, causing no threat to humans and the surroundings even with collisions (Tao et al., 2018). 
+With the lifting force provided by air buoyancy, the blimp has flight endurance for more than 2 
+hours (Cho et al., 2017). Blimp is the one type of UAV lighter than the air UAVs (Krishna, 2021a; 
+Thusoo, 2021; Tsouros et al., 2019). It has a balloon-like body created from tough fabric and filled 
+with helium gas (Prisacariu et al., 2019). Blimp is notorious as the dirigible and was firstly 
+
+PitchForward
+PitchForward
+Roll Left
+Roll Left
+Roll Right
+Roll Right
+A
+B
+PitchBackward
+PitchBackward 
+designed in 1852 by Henri Giffard (Krishna, 2021a). This type of UAV has high endurance and 
+can flow longer than other types of UAV, approximately 1-3 weeks travel (Krishna, 2021a). Due 
+to its characteristics, the blimp is advantageous in numerous aspects of life, including the military 
+and agriculture. It was purposeful as the cargo transit and the sentinels between the missile site 
+and the military camp (Krishna, 2021a). It is also used to monitor the long-distance aspect, 
+especially in urban traffic and buildings. In PA, it monitors crop production, identifies the plant’s 
+disease, erosion, and detects either flood or drought conditions (Krishna, 2021a). Unlike the other 
+UAV, the blimp is also considered a safe technology since it remains in the air and did not collide 
+even if it loses its power (Krishna, 2021a; Tsouros et al., 2019). Besides, the University of Leeds 
+research reveals that blimp is chosen as the cheapest UAV to conduct a terrain survey (Krishna, 
+2021a). Thus, it could provide a detailed but further explanation of crops, surface minerals, 
+vegetation, and water quality. 
+4. Application of Blimp in Agriculture 
+Compared to other UAVs, the blimp has a pivotal function, including load capacity, safety, 
+quality, and environmental safes, which make it useful for everyday life. Researchers stated that 
+blimp could carry up to 400 tons of load with 110-160 km speed travel (Krishna, 2021a). Besides 
+its ability to fly longer, blimps could land on every land’s surface. In the case of environmental 
+footprint, blimp could reduce carbon dioxide emission (Krishna, 2021a). 
+Nowadays, there are several features of blimps with specific purposes. Those five types 
+include tethered, untethered, remote-controlled blimp, Giga blimps, and hybrid blimps (Krishna, 
+2021a). Tethered blimp (aerostats) ables for free flight and a steady anchored flight using solid 
+tethers (Mahmood & Ismail, 2020). A tethered blimp enables accurately obtaining a stereoscopic 
+image that might cover 35 m2 to 20,000 m3 (Krishna, 2021a). Meanwhile, the untethered is 
+commonly used in cargo transit, travel, and aerial surveillance. Thirdly, the remote-controlled 
+blimp uses the robotic that use the program flight plan to fly. There are two different types of 
+remote control blimps; small and large. The small blimp is commonly used for advertisement, and 
+the giant blimp (Giga blimp) is significant for military purposes. Lastly, the hybrid blimp, a new 
+modification of blimp that is prone to extreme conditions, can transport goods and civilian travel. 
+Unlike the other popular UAVs, multirotor and fixed-wing, the number of blimps applied 
+in PA is insignificant (Krishna, 2021b; Tsouros et al., 2019). However, considering its strength 
+characteristic and function, blimp could start considering the blimp as the priority to improve the 
+
+ 
+PA. (Mogili, Rao Deepak, 2021) stated that the integration of blimp with quadcopter aerial 
+automated pesticide sprayer (AAPS) is pivotal for pesticide spraying in lower altitudes by 
+following the GPS altitude. This technology is controlled with an android app to create an effective 
+cost-saving (Mogili, Rao Deepak, 2021). Besides, other researchers use the blimp with a Charge-
+Couple Device to identify the Leaf Arena Index and biomass in soybean and paddy fields 
+(Chilonga & Kiswisch, 2016). The results showed that the technology is stable and provides high-
+resolution images (Chilonga & Kiswisch, 2016). (Ponti et al., 2016) also stated that the blimp could 
+be practically used to monitor the bean crop dataset using the combination of 1/2.3 inch of CCD 
+sensor, 6.3 to 18.99 lens focal, and 10 Mega Pixel digital camera. The research found 29,556 
+examples of the positive dataset and 11404 negative datasets in Brazil (Ponti et al., 2016). 
+         The blimp could be significantly used in monitoring agriculture (Mahmood & 
+Ismail, 2020). For instance, the research conducted by (Bajoria et al., 2017) proposed a tethered 
+aerostat system that could be used to mitigate the vertebrate mammal and bird hazard, which is 
+positively contributed to 18%-43% of crop loss in India. The proposed design has been proven to 
+carry about a 50 kg payload and 25 m/s ambient wind speed (Bajoria et al., 2017). Other research 
+revealed that tethered aerostat combined with the electro-optical, acoustic, and laser-based sensors 
+could scare the bird and other pests (Krishna, 2021a). To mitigate the occurrence of pests, the other 
+researchers also create a Hawk Kite and Helikite aerostat hybrid that is purposeful to scare some 
+of the bird’s species, including the pigeons, seagulls, parrots, rooks, blackbird, etc. (Perigrine Ltd. 
+2018). 
+         Besides, a tethered blimp (aerostat) could provide aerial images surrounding the 
+natural disaster zone. This image helps identify the cropped field due to the flood, large soil 
+erosion, drought, and crop loss due to the pest attack (Krishna, 2021a). It is also used to maintain 
+the field quality because the aerostat could lofty 24 hours surveillance above. Thus, it could help 
+the farmer control and watch the field without going directly to the farm. Besides the 
+aforementioned reasons, the aerostat could reduce the enormous cost of capturing the crop’s data. 
+The farmer could use the aerostat to pertain the crop data and send its digital data in a computer 
+program (Krishna, 2021a). Furthermore, the integration of aerostat with the sensor could help the 
+farmer obtain continuous data of Nutrients crop status. Thus, it could help the farmer evaluate the 
+number of nutrients placed for the crop (Krishna, 2021a).  
+5. Agricultural Unmanned Aerial Vehicle 
+
+ 
+In recent years, the application of more advanced technology in agriculture has gained 
+more attention. Several technologies such as satellites, UAVs, Geographic Information System 
+(GIS), Global Positioning System (GPS), and many other applications of technologies have been 
+able to pave their way into the agricultural field.  The process modernization and industrial 
+revolution that brought many innovations in technology applications have opened the gate of 
+precision agriculture (Ahmad et al., 2021). Precision agriculture is defined as the utilization of 
+technology in the agricultural production system in order to determine, analyze, and manage the 
+farming factors to increase crop productivity, ensure environmental sustainability, and improve 
+business profitability (Unal & Topakci, 2013). This precision agriculture is seemingly possible to 
+increase food production due to its effective functionalities under pressure conditions such as the 
+ongoing reduction of arable land, the increase in global population, and the high cost of farming 
+due to wastage in the use of water and chemicals (Abdullahi et al., 2015).   
+UAVs have gained popularity as a pivotal part of precision agriculture to ensure 
+agricultural sustainability (Rani et al., 2019). The use of UAV, which plays a key role in reducing 
+the data acquisition time and processing cost, is considered as the main reason for its popularity 
+(Berni et al., 2009). The rapid development of UAVs that extend its functions to aerial photography 
+and video and weather forecasting, with the support of spatial data collection to help stakeholders 
+create policies and decisions, has attracted many parties to UAVs (Sylvester, 2018). 
+Moreover, the market of UAVs that is estimated to reach up to US$200 billion by the end 
+of 2020 has successfully described the popularity of UAVs as well (Puri et al., 2017).  The huge 
+estimation of the total market of UAVs has shown that the market value of UAVs has doubled 
+within three years. PwC’s Drone Powered Solutions team quantified that the total market value of 
+UAVs is about US$127.3 billion in 2017 (Silver et al., 2017).  Although the estimation of the 
+market value of UAVs in 2020 and the total market value of UAVs in 2017 are not exclusively 
+focused on agriculture sectors, this number was sufficient to portray the UAV market development.  
+In addition, the affordable cost of UAVs is another factor that influences its popularity 
+nowadays. This low-cost factor motivates many small companies to switch to using UAVs with 
+its simple and easy-to-understand operating systems in serving some activities in the agriculture 
+sector, including area measurement and crop monitoring (Hatfield & Prueger, 2010).  
+ 
+6.  Applications of Unmanned Aerial Vehicles in Precision Agriculture 
+
+ 
+Currently, there are numerous applications of UAVs in precision agriculture. They are used 
+in many areas of crops. This section introduces two applications of agricultural UAVs i.e., spraying 
+and crop monitoring. The summary is shown in Table 1. 
+ 
+6.1 Spraying 
+Prior to the implementation of UAVs for spraying, the farmers used spraying bags to spray 
+pesticides all over the farm (Spoorthi et al., 2017). Manual spraying is very dangerous for the 
+workers because the measure of pesticides per hectare of agricultural land correlates to the risk of 
+worker ailments. The heavy bag carried by the farmers could also make them get strained. 
+Fortunately, the use of UAVs can reduce the usage of pesticides, maximize efficiency, and improve 
+the well-being of the workers (Luck et al., 2010; Pyo, 2006).  
+ 
+Manual spraying is also considered ineffective for spraying the farmland because the 
+pesticides may not spread evenly in every area. The excessive use of chemicals or pesticides in 
+certain agricultural land is responsible for loss of soil fertility, soil degradation, and subsequent 
+degradation of water-related ecosystems. In addition, the chemicals or pesticides absorbed by the 
+crops and natural resources such as water and soil might cause pollution risk and severe health 
+impacts for the environment. Therefore, UAVs are required to minimize such dangers by helping 
+the spraying process specifically in the targeted area (Daponte et al., 2019). 
+ 
+In addition, to pave the way towards sustainable agriculture, the employment of UAVs in 
+the agriculture sector also offers other benefits in terms of their operation. The implementation of 
+UAVs can make the process relatively faster and cheaper than other methods (Rani et al., 2019). 
+The efficient usage of UAVs was also widely reported in the literature. The use of 3WWDZ-10A, 
+XAG is successfully effective in controlling Spodoptera frugiperda, an invasive sugarcane crop 
+pest, by spraying pesticides (Song et al., 2020). In addition, the use of UAV (DJI Phantom 3) is 
+found to be effective in spraying pesticides in the nominated areas using electronic traps (E-traps), 
+which can count the insect and transmit the data to the server (Psirofonia et al., 2017). Studies have 
+also found that UAVs might improve the accuracy of control over crops by equipping the UAVs 
+with precision control algorithms (Faiçal et al., 2016). In summary, it is proved that the UAV 
+application offers several advantages in reducing the workload of the farmers and providing 
+efficient and low-cost service in the agriculture field.  
+
+ 
+ 
+Some issues have been identified regarding the use of UAVs in crop areas overlapping and 
+outer edges, despite the advantages during UAVs implementation. These issues arise because some 
+crop fields are not fully covered properly during spraying, leading to reduced crop quality in 
+particular areas. To overcome this problem, the swarm of UAVs was introduced in a control loop 
+algorithm during UAV operation (Yao et al., 2016). Swarm control is considered a practical 
+technology since it could control multiple UAVs via one operator or program. In the swarm control 
+method, the operator can select an efficient shape based on the application so that the swarm can 
+be centralized, decentralized, and distributed according to the desired shape (Kim et al., 2019). In 
+addition, the spraying pesticides process on the crop is then organized by considering the feedback 
+from the Wireless Sensor Networks (WSNs) deployed in the field (Costa et al., 2012). The control 
+loop is responsible for the communication of each UAV in adjusting the UAVs’ route according 
+to the changes in wind speed and the number of messages exchanged in between (Faiçal et al., 
+2014). During this communication process, a short delay might exist in the control loop since the 
+UAVs need time to analyze the data from WSN to route further (Kale et al., 2015). An automatic 
+navigation spraying system of UAV was developed to direct the UAV in a particular area (Xue et 
+al., 2016). 
+ 
+Another way in using swarm control is through task allocation technology. This technology 
+is currently used in mapping agricultural lands (Barrientos et al., 2011). In order to use swarm 
+technology, a route is assigned to each UAV. A route is built by dividing each region or area 
+among several UAVs. A map of the area is obtained by capturing a single picture through a camera 
+sensor attached to a UAV (Ju & Son, 2018). This kind of technique requires K-mean algorithms 
+in order to reduce complexity and prevent collision among UAVs. The most significant aspect of 
+this swarm technique is the combination of algorithms that come in handy in maintaining the 
+consistent distances between UAVs. These consistent distances allow linear and nonlinear control 
+that resist strong external influences (Kim et al., 2019). The implementation of swarm techniques 
+and task allocation in agriculture can be seen in Figure 5. This application most likely improves 
+the accuracy of agricultural operations, reduces operator control efforts, reduces work time, and 
+induces battery and payload shortages.  
+
+ 
+ 
+Figure 5. (A) Swarm Control (source: Ju & Son, 2018) (B) Task allocation (source: Barrientos et al., 2017) 
+In the spraying using UAVs, the sprinkling system is mounted at the lower region of the 
+UAV, which has a nozzle under the pesticide tank in order to sprinkle the pesticide downstream in 
+the field. An appropriately selected nozzle is a significant part of pesticide application since it is a 
+significant factor in determining the amount of spray applied to an area, the coverage obtained on 
+the target surface, the amount of potential drift, and the uniformity of application (Ru et al., 2014).  
+Furthermore, the sprinkling system generally has two modules: the controller and the sprinkling 
+system. The sprinkling system consists of the spraying content, either pesticides or fertilizers. 
+Meanwhile, the controller is used to trigger the nozzle of the sprayer. The controller efficiency 
+could be increased by using a PWM controller in pesticide applications (Zhu et al., 2010; Huang 
+et al., 2009).  
+Another important component of the sprinkling system is a pressure pump used to put 
+pressure into the pesticide in the tank to flow through the nozzle (Tang et al., 2018). This pressure 
+pump works closely with the motor driver integrated circuit in completing their task in putting the 
+pressure to sprinkle the pesticide (Mogili & Deepak, 2018). The full spraying system can be seen 
+in Figure 6. The integration between UAV and the spraying system is expected to provide a 
+potential platform for pest management and vector control, an accurate site-specific application 
+for a large crop field. For this objective, a heavy lift of UAVs is required to cover many areas 
+(Sarghini & De Vivo, 2017). 
+
+RemoteSensingTaskusingcameramountedonUAV
+(MultipleUAVs)
+contro
+UAV
+ensing
+Agriculturalfield
+Teleoperationcontrolwith
+device
+Hapticdevice(NovintFalcon)
+A
+B 
+ 
+Figure 6. Spraying System Structure Diagram (source: Tang et al., 2018) 
+ 
+ 
+ 
+6.2 Crop Monitoring 
+A crop monitoring is defined as predicting the yield or crop quality by analyzing the available crop 
+data (Kim et al., 2019). It is essential for optimizing crop production because it can assess crop 
+health and indicate bacterial or fungal infections. Furthermore, the crop scanning produced by 
+visible and near-infrared (NIR) light could reflect the different amounts of green light and NIR 
+light that are extremely essential in producing multispectral images that can track changes in crops 
+assess their health (Costa et al., 2012). The farmers can plan and apply remedies more precisely 
+according to the identified issues with such information. It makes the fast response to bacterial or 
+fungal infection, and infestation comes in handy and increases crop endurance into future issues.  
+ 
+The use of UAVs for crop monitoring is also highlighted due to their ability to monitor a 
+large farm. By utilizing the UAVs, a large area of farmland can be fully monitored. It reduces the 
+significant time and labor required for monitoring large farm areas manually (Kim et al., 2019). 
+Aasen et al. (2015) reported that the UAVs application offers low crop monitoring costs. This is 
+due to the use of lightweight sensors and the implementation of low-flying UAVs (see Figure 7).  
+
+Electronic speed control
+Tank
+Sprayboom
+Nozzle1
+Nozzle2
+Waterpump 
+ 
+Figure 7. UAV Platform (source: Aasen et al., 2015) 
+A camera-equipped UAV can also observe the crop with different indices (Simelli & 
+Tsagaris, 2015). Turner et al. (2011) used multispectral cameras mounted in UAVs to analyze the 
+vegetation index of grapes obtained from vineyards. These vegetation index data are considered 
+very important to emphasize the significant indicators to increase productivity and improve the 
+shortcoming from farming activity. Furthermore, the application of UAVs in crop monitoring 
+could also be seen through UAVs' capability to fly up to hectares of a field in one single flight. For 
+this purpose, multispectral and thermal cameras are mounted at the UAVs' downside to recording 
+the vegetation canopy's reflection (Bendig et al., 2012; Colomina & Molina, 2014). These cameras 
+can take one capture per second and store it in the memory. The images are captured in the visible 
+five bands with five different wavelengths (i) blues wavelength 440-510 nm (ii) green wavelength 
+520-590 nm, (iii) red wavelength 630-685 nm, (iv) red edge wavelength 690-730 nm, (v) near-
+infrared wavelength 760-850 nm. Then, those images were sent to the ground station through 
+telemetry. The process of communication used the MAVLINK protocol. The data collected from 
+the multispectral camera was analyzed by the geographic indicator Normalized Difference 
+Vegetation Index (NDVI) (Reinecke & Prinsloo, 2017; Bhandari et al., 2012).  
+Moreover, the application of UAVs for crop monitoring has been implemented for 
+conducting several tasks including monitoring crop growth, chlorophyll, and phenology 
+measurement, and counting plants (Pino, 2019). These tasks are performed using SenseFly's e Bee 
+Ag that has NIR and NDVI sensors. These sensors can replace traditional farm scouting by 
+minimizing human error (Natu & Kulkarni, 2016). In addition, UAVs are involved in monitoring 
+crops in hilly areas that are considered to be difficult for traditional scouting (Rani et al., 2019).
+
+OktoXL
+UHD185
+Gimbal
+CP
+ICS
+SBC 
+Table 1. Applications of Agricultural UAVs 
+Task  
+UAV 
+Model 
+Indices  
+Crop 
+Flight 
+Altitude 
+(m) 
+Sensors 
+Task Period 
+Reference 
+Type 
+Model 
+Spraying 
+Fixed-wing 
+UAV 
+Normalized Difference Vegetation 
+Index (NDVI) 
+Maize Silage 
+150 
+- 
+Canon s110 Throughout 
+the year 
+(Castaldi et 
+al., 2017) 
+Helicopter 
+Spray Work Rate 
+Vineyard 
+3-4 
+Digital Camera 
+- 
+May 
+(Giles & 
+Billing, 
+2015) 
+Route Precision, Spraying 
+Uniformity 
+Wheat 
+5, 7, 9 
+Image Transmitter 
+- 
+Summer 
+(Xue et al., 
+2016) 
+Droplet size, Flow rate 
+Field 
+6 
+Proprietary Radio 
+Receiver 
+- 
+Throughout 
+the year 
+(Huang et 
+al., 2009) 
+Leaf Area Index (LAI), 
+Normalized Difference Vegetation 
+Index (NDVI) 
+Maize Silage 
+35 
+Multi-Spectral Camera 
+Agrosenso 
+Throughout 
+the year 
+(Castaldi et 
+al., 2017) 
+- 
+Field 
+20 
+Wireless Sensor Networks 
+-  
+- 
+(Faiçal et al., 
+2017) 
+Quadcopter 
+ 
+Time of Communication between a 
+Sensor 
+Soy, Rice, Corn 
+Gapes, 
+Sugarcane 
+5, 10, 20 
+RF Module 
+XBee-PRO 
+series 2 
+Summer 
+(Faiçal et al., 
+2014) 
+Droplet coverage rate, Density, 
+Droplet size 
+Cocktail, 
+Grapefruit, 
+Citrus 
+3.5, 4, 4.5 
+Digital Plant Canopy 
+Imager 
+Camas CI-
+110 
+Spring-
+Summer 
+(Pan et al., 
+2016) 
+Observed Deposition Rate, Field 
+Work Rate 
+Field 
+Few 
+meters 
+Multi-Spectral camera, 
+Hyper-Spectral camera, 
+Near-Infrared, Color-
+Infrared 
+- 
+Throughout 
+the year 
+(Meivel et 
+al., 2016) 
+Droplet Coverage Rate, Density, 
+Droplet size 
+Citrus 
+0.6, 1.2, 
+1.8 
+Water-Sensitive Paper 
+Cards (WSPs) 
+- 
+One day 
+(Tang et al., 
+2018) 
+Hexacopter 
+Discharge and Pressure of Spray 
+Liquid, Spray Uniformity, Spray 
+Liquid Loss, Droplet Size and 
+Density.  
+Paddy and 
+groundnut  
+1 
+HD FPV camera 
+- 
+Throughout 
+the year 
+(Yallappa et 
+al., 2017) 
+ 
+
+ 
+Table 1. Cont. 
+Task  
+UAV Model 
+Indices  
+Crop 
+Flight 
+Altitude 
+(m) 
+Sensors 
+Task Period 
+Reference 
+Type 
+Model 
+Crop 
+Monitoring 
+Fixed-wing 
+UAV  
+Normalized Difference Vegetation 
+Index (NDVI) 
+Arable crops (corn, cotton, 
+sunflower) 
+120 
+Multi-
+Spectral 
+Camera 
+Parrot 
+Sequoia Plus June-October (Bollas et al., 
+2021) 
+Normalized Difference Vegetation 
+Index (NDVI) 
+Rice 
+20 
+Multi-
+Spectral 
+Camera 
+Tetracam 
+ADC  
+camera 
+95 days 
+(Swain et al., 
+2010) 
+Quadcopter 
+NDVI, Ontario Soil and Crop 
+Improvement Association 
+Soybean, Wheat, Barley, 
+Oat, Canola 
+120 
+Digital 
+Camera 
+Aeryon 
+Photo3S 
+Spring-
+Autmn 
+(Zhang et al., 
+2014) 
+Visible-Band Difference 
+Vegetation Index, Normalized 
+Green-Blue Difference Wheat 
+Index, Green-Red Ratio Index 
+Wheat 
+100 
+Digital 
+Camera 
+SONY 
+ILCE-6000 
+September-
+July 
+(Du & 
+Noguchi, 
+2017) 
+Leaf Area Index (LAI), Total Dry 
+Weight (TDW), Plant Lenght (PL) 
+Three Rice Cultivars: 
+Nipponbae (Japonica), 
+IR64 (Indica), Basmati370 
+(Indica) 
+30 
+RGB 
+Camera 
+Zenmuse 
+X4s 
+Summer 
+(Peprah et al., 
+2021) 
+Vegetation Index (VI), Leaf Area 
+Index (LAI) 
+Coffee 
+30 
+Digital 
+RGB 
+Camera 
+Sony 
+EXMOR 
+1/2.3" 
+Throughout 
+the year 
+(Barbosa et 
+al., 2021) 
+Soil-Adjusted Vegetation Index 
+(SAVI), Leaf Area Index (LAI), 
+Normalized Difference Vegetation 
+Index (NDVI) 
+Sunflower 
+75 
+Digital 
+Camera 
+Tetracam 
+ADC Lite 
+four days 
+(Vega et al., 
+2015) 
+- 
+Field 
+- 
+RGB 
+Camera 
+- 
+- 
+(Doering et 
+a., 2014) 
+Hexacopter Normalized Difference Vegetation 
+Index (NDVI) 
+Vineyard 
+150 
+ADC-Lite 
+Camera 
+Tetracam 
+ADC-lite 
+camera 
+One day 
+(Primicerio et 
+al., 2012) 
+ 
+
+ 
+Table 1. Cont. 
+Task  
+UAV 
+Model 
+Indices  
+Crop 
+Flight 
+Altitude 
+(m) 
+Sensors 
+Task 
+Period 
+Reference 
+Type 
+Model 
+Crop 
+Monitoring 
+Hexacopter 
+Normalized Green-Red Difference 
+Index (NGRDI) 
+Pea, Oat 
+30 
+RGB Camera 
+Panasonic Lumix 
+DMC-GF1 
+April-
+August 
+(Jannoura et al., 
+2015) 
+Blue Green Pigment Index 2 
+(BGI2), Reformed Difference 
+Vegetation Index (RDVI) 
+Barley 
+30 
+Hyper-Spectral 
+Camera 
+Firefly ultra-high 
+definition 185 
+Summer 
+(Aasen et al., 
+2015) 
+Octocopter 
+NDVI, Soil Adjusted Vegetation 
+Index (SAVI), Optimized SAVI 
+(OSAVI) and Li 
+Barley 
+50 
+RGB-Sensor 
+Panasonic Lumix 
+GXI 
+April-
+July  
+(Bending et al., 
+2015) 
+Structure-from-Motion (SfM), 
+airborne laser scanning (ALS) 
+Eucalyptus 
+Pulchella 
+30 
+RGB Camera 
+Canon55D 
+One day (Wallace et al., 
+2016 
+Normalized Difference Vegetation 
+Index (NDVI), thermal temperature 
+Sugarbeet 
+55 
+Multiple Camera 
+Array (MCA) 
+Camera 
+Tetracam mini 
+MCA 
+One day 
+(Bendig et al., 
+2012) 
+- 
+Sunflower 
+122 
+Multi-Spectral 
+Camera 
+ADC Snap 
+- 
+(Noriega & 
+Anderson, 
+2016) 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+7. Limitations in Adopting UAVs Technologies in Agriculture Sector 
+7.1 Technical Decisions 
+
+ 
+Various types of UAVs have been produced in the commercial market by many manufacturers and companies, starting from hobby-type 
+products up to industrial model aircraft. Since there is no specific standard about the UAV development for agricultural purposes, it is 
+hard to find a UAV built specifically for the agricultural context (Huang et al., 2013). Moreover, suppose the available commercial 
+software packages, which support the photogrammetric data processing, are not standardized for agricultural purposes. In that case, the 
+desired UAV images may not be appropriately captured by the sensor. Therefore, it can prevent the users from taking the right actions 
+if unexpected situations such as a collision with another flying object occur (Abdullahi, 2015).  
+Another major problem associated with technical decisions is the battery usage and flight time limitations. The lithium-ion 
+batteries currently used in UAVs have an advantage over conventional batteries, especially in their larger capacity. However, the larger 
+capacity affects the weight of the batteries that become heavier in return (Saha et al., 2011). Unfortunately, this issue is challenging to 
+be solved within this day. Another problem related to battery usages is battery management. Even though it is known that the batteries 
+of UAVs need to have constant maintenance, most UAV operators often forget and do not carefully pay attention to this issue. As a 
+result, it caused periodic replacement that required additional cost (Lee et al., 2012). Lastly, the possible time for UAVs to fly, which is 
+around 20-30 minutes with a fresh battery, can still provide enough time for complete crop monitoring (Baha et al., 2012). Researchers 
+try to develop optimized hybrid batteries as solutions in dealing with this issue.  
+7.2 Cost 
+The lack of awareness of the UAVs' cost, is one of the reasons for the slower adoption of this technology in the agriculture sector. For 
+a starter system, agricultural UAVs can range from US$1,000 that might go up to US$10,000 or US$20,000, depending upon the cameras 
+and the features (Stehr, 2015). This cost is not quite affordable and surely will be an impending stop to adopt UAVs technology for 
+smallholder farmers (Ahmad et al., 2021). The interested farmers who could not afford the cost of UAVs may need to contract as a 
+group to get UAV services to reduce the individual expenses.  
+Another possible solution to minimize the cost of UAVs by purchasing inexpensive airframes and low-cost cameras. However, 
+this solution could build up a short endurance of the UAV platform. Moreover, the low-cost UAVs are usually equipped with lightweight 
+
+ 
+engines that might limit the reachable altitude of the UAVs. The low cost of cameras also limited the sensor payload both in dimension 
+and weight, and reduced image quality (Abdullahi, 2015). In addition, the separate purchases of UAV components require highly skilled 
+engineers or technicians to integrate and assembly, which may increase the total expenses (Huang et al., 2013). 
+ 
+Apart from the cost of vehicles equipped with cameras and software for aerial imagery processing, the farmers need to consider 
+the expenditures for the operator's license. The presence of this operator implies extra time and cost that need to be spent since not 
+everyone is allowed to operate the UAVs. Nevertheless, all these costs will constantly decrease over the years (Bollas et al., 2021). 
+7.3 Payload 
+Payload weight and size are critical factors for UAVs because they need to be carefully configured based on the specific 
+application of the UAV. When the UAV is ready to use, it needs to be configured by paying attention to payload design, mechanical 
+and electrical accommodation even though there is no specific engineering guideline to be followed (Huang et al., 2013). 
+7.4 Operation 
+In the UAV operation, most UAV types do not have the capability of automated take-off and landing (Huang et al., 2013). Furthermore, 
+the frequency of flying UAVs should be carefully selected because there are insufficient regulations about flying UAVs. Even certain 
+regions restrict the usage of UAVs as a security precaution (Eisenbeiss, 2009). Another challenge is the UAVs' inability to take readings 
+during extreme weather conditions like rain or storm (Abdullahi et al., 2015). Therefore, highly skilled operators for remote control are 
+required. However, the demand for skilled users to operate the UAVs is a problem for small and medium producers to adopt UAV 
+technology. Training issues and lack of demonstrated financial returns in the short and medium term are considered the reason for this 
+issue (Abdullahi et al., 2015). Thus, autonomous flight according to georeferenced coordinates has then become a highly desirable 
+component for practical use of UAVs in agriculture (Huang et al., 2013).  
+The swarm-control techniques can be applied to efficiently control multiple UAVs in performing a wide range of tasks. Although 
+swarm-control can provide practical techniques to lower the battery cost and operate more efficiently with shorter flight times, there is 
+a need for user interface improvement so that people who are older or unfamiliar with UAVs can easily control the UAVs. The user 
+
+ 
+interface improvement is made by considering multimodal feedback, including visual, auditory, and haptic feedback. Therefore, an 
+improvement that mainly focused on human-centered user interface and feedback are two ways that seem to be effective to deal with 
+multiple UAVs (Hong et al., 2017).  
+ 
+8. Challenges in Implementing UAVs in Indonesia 
+Kavianand et al. (2016) have reported that agricultural development in Indonesia is critical since it has primarily contributed to 
+Indonesia's GDP.  Roughly about 14.4 percent of Indonesia's total GDP comes from the agriculture sector and has reduced the 
+unemployment rate by absorbing 38.6 percent of the workforce (David & Ardiansyah, 2017). Despite its considerable contribution to 
+Indonesia's GDP, the contribution of agriculture to Indonesia's GDP has been remarkably decreasing for the last five decades due to low 
+productivity. Some natural phenomena, such as extreme weather changes, have also influenced Indonesia's agriculture (Syuaib, 2016).   
+ 
+Many researchers have suggested implementing precision agriculture via UAVs to improve the productivity of the work in 
+agriculture. The application of UAVs offered many benefits that could grow the economic profit and provide a proper solution in 
+solving current issues in agriculture. However, as much as Indonesia depends upon agriculture, the application of UAVs in the 
+agriculture field is relatively far from adopting the latest technology into farms. Even though some developed countries have started to 
+use UAVs in their precision agriculture and proved that this technology is essential in reducing farmers' workload, Indonesia seems to 
+fall behind and keep using manual operating for farm activity.   
+One of the viable reasons for preventing the adoption of UAVs is the education level of farmers. The majority of farmers in 
+Indonesia do not complete their high school education in which 38 percent of local farmers have graduated from primary school (Haq 
+et al., 2016). Furthermore, only 6 percent of the local farmers can complete high school or university. These numbers significantly 
+describe the current state of Indonesia's farmers. The lack of education could cause a low understanding of the technology application. 
+Moreover, this low level of understanding can lead to the anxiety of relearning integrating agriculture and technology (Suryanegara et 
+al., 2019).  
+
+ 
+Despite the reasons mentioned above, researchers have tested the application of UAV in several agricultural sectors. For instance, 
+the UAV has been practically implemented or tested in Indonesia’s agriculture, including the sugar cane plantation in PTPN 
+(Perkebunan Nusantara Maospati East Java) and a paddy field near Menara Cigarette Factory, and the Teak Wood Forest in Madiun 
+(Rokhmana, 2015). Besides, the application of UAV has been significantly tested in one of the paddy fields in Parankasalak, Sukabumi, 
+West Java, to monitor the crop by mapping the paddy field through differentiating them based on their spectral characteristic 
+(Rokhmatuloh et al., 2019). However, the research found that the implementation of UAV poses several limitations and challenges that 
+become a prohibitive factor for broader use in Indonesia’s agriculture. 
+The application of UAV in PA requires a high investment cost to purchase the technology and the maintenance cost (Tsouros et 
+al., 2019). Furthermore, due to the limited space of agricultural land and the unstable market price for the crop yield, the implementation 
+of UAV might pose another operational cost to the farmers (Tsouros et al., 2019; Vera et al., 2021). Even though the market has 
+commercially offered an amateur and cheaper UAV, the product has several limitations related to stability, accuracy, and quality 
+(Norasma et al., 2019). The cheaper UAV has a low ability to reach a certain altitude due to its low power engine (Norasma et al., 
+2019). This notion is reinforced by (Rokhmana, 2015) who notes that amateur UAVs generally have an error in their camera lens. This 
+case happens because both the stability and accuracy of the non-metric lens are low. Besides, the UAV is relatively light, possesses 
+only 3 kg weight, which causes them to be easily disturbed by the wind and air turbulence when the weather is windy (> 40km/h) and 
+rainy days (Norasma et al., 2019; Rokhmana, 2015; Tsouros et al., 2019). This case requires huge attention because Indonesia is located 
+along the equatorial belt region to have periodic heavy rain. 
+ Furthermore, the UAV requires data-intensive procedures and skilled personal for exploiting the acquainted imagery data. 
+(Tsouros et al., 2019). Hence, the farmers need to hire the expertise of UAV technology or do intensive training that may be costly. 
+This case requires intensive consideration because the average farmer in Indonesia is not in productive ages with low educational 
+background (Haq et al., 2016). It reported that 88% of the average farmer in Bantarkawung is on the 15-60 years and the remaining 
+farmer is on non-productive ages (Haq et al., 2016). Moreover, it discovered that only 6% of the majority graduated from high school 
+
+ 
+and university; the remaining only attended primary school, and 38% were in junior high school (Haq et al., 2016). As a result, most 
+farmers have a common understanding of technology, IoT and little comprehension of imagery data. Another viable reason for 
+preventing people from using the UAV technology is its limited flight time. (Tsouros et al., 2019) revealed that most commercial UAVs 
+only have 20 min to 1 hour flight time. Moreover, it only covers a small restricted area for each flight. Thus, the total cost expenses to 
+purchase the UAV technology for PA might not be advantageous.  
+9. Conclusion 
+The application of UAVs in current days has opened unlimited potential, especially in the agriculture sector. Two main UAVs 
+applications in agriculture sectors, such as spraying, and crop monitoring have been discussed. The urgency of UAVs and the 
+implementation of UAVs were necessary to be implemented in order to establish precision agriculture. Numerous issues and problems 
+that might occur in the future have also been highlighted to build awareness about the issues by providing various data and sources. The 
+application of UAVs in spraying and crop monitoring are the main parts of this paper since we were thoroughly investigating the 
+application of UAVs that includes the benefits obtained, various application forms of UAVs from several types of research, and the flow 
+of operating the UAVs. Moreover, the limitation found in the application of UAVs was also identified to reveal the gap of UAVs 
+implementation in the agriculture field. Lastly, the challenges in implementing UAVs are also being discussed, especially in Indonesia. 
+ 
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+arXiv:2301.01330v1  [math.GR]  3 Jan 2023
+ON REPRESENTATIONS OF DIRECT PRODUCTS AND THE
+BOUNDED GENERATION PROPERTY OF BRANCH GROUPS
+STEFFEN KIONKE AND EDUARD SCHESLER
+Abstract. We prove that the minimal representation dimension of a direct
+product G of non-abelian groups G1, . . . , Gn is bounded below by n + 1 and
+thereby answer a question of Ab´ert. If each Gi is moreover non-solvable, then
+this lower bound can be improved to be 2n. By combining this with results of
+Pyber, Segal, and Shusterman on the structure of boundedly generated groups
+we show that branch groups cannot be boundedly generated.
+Introduction
+An infinite group G is called just-infinite if all of its proper quotients are finite.
+Obvious examples of just-infinite groups are virtually simple groups. Other exam-
+ples arise from irreducible lattices in higher rank semisimple Lie groups, such as
+SLn(Z) for n ≥ 3, after dividing out their centers, see [12, Chapter IV]. Such groups
+are in fact hereditarily just-infinite, which means that they are residually finite and
+all of their finite index subgroups are just-infinite. Grigorchuk’s group [10] provided
+the first example of a just-infinite group that is not virtually a finite direct power of
+a simple or a hereditarily just infinite group. Grigorchuk’s group is a just-infinite
+branch group, which means that its commensurability classes of subnormal sub-
+groups form a lattice that is isomorphic to the lattice of open and closed subsets
+of a Cantor set. By Wilson’s classification [18] just-infinite groups fall into three
+classes. Every just-infinite group G is either a branch group or virtually a direct
+power of a simple or a hereditarily just-infinite group.
+Since its introduction by McCarthy [13] in the late 1960’s, the class of just-infinite
+groups remained an active field of research. One reason might be that every finitely
+generated infinite group admits a just-infinite quotient. Thus whenever there is
+some finitely generated, infinite group G that admits a property P that is preserved
+under homomorphic images, then there is also a finitely generated just-infinite group
+with P. Following [3], we call a property P that is preserved under homomorphic
+images an H-property. Well-known examples of H-properties include amenability,
+property (T), bounded generation, being a torsion group, having subexponential
+growth etc. In view of Wilson’s classification, it is natural to investigate which
+of the three classes of just-infinite groups contain groups that satisfy a given H-
+property P. For the H-property “being a torsion group” this question is settled. In
+this case it is know that there are finitely generated simple groups [15], just-infinite
+branch groups [10], and hereditarily just-infinite groups [9] that are torsion. On
+the other hand, there are torsion-free, finitely generated, just-infinite groups that
+are simple [4], branch [2], and hereditarily just-infinite (e.g. Z).
+The purpose of this note is to study this question for the bounded generation
+property. Recall that a group G is boundedly generated if it contains a finite subset
+2010 Mathematics Subject Classification. Primary 20E08; Secondary 20E26.
+Key words and phrases. bounded generation, branch group, faithful representations of direct
+products.
+Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -
+441848266.
+1
+
+2
+S. KIONKE AND E. SCHESLER
+{g1, . . . , gn} such that every g ∈ G can be written as g = gk1
+1 · · · gkn
+n for appropriate
+ki ∈ Z. Since infinite torsion groups are not boundedly generated, it follows that
+each of the three classes of just-infinite groups contains a finitely generated group
+that does not have the bounded generation property. On the other hand, it was
+proven by Carter and Keller [7] that PSLn(Z) is boundedly generated for n ≥ 3,
+which provides an interesting boundedly generated hereditarily just-infinite group.
+The existence of boundedly generated, infinite, simple groups was established by
+Muranov [14], whose construction seems to be the only one available at present. It
+remains to study just-infinite branch groups. The question of existence of boundedly
+generated just-infinite branch groups was raised by Bartholdi, Grigorchuk, and
+ˇSuni´k [3, Question 12] and remained open to the best of our knowledge.
+The
+purpose of the paper is to show that the answer is negative for arbitrary branch
+groups (even without the assumption of being just-infinite).
+Theorem 1. There is no boundedly generated branch group.
+As a consequence, it follows from Wilson’s classification of just-infinite groups
+that every boundedly generated infinite group has a quotient that is virtually a
+product of finitely many copies of a boundedly generated simple or hereditarily
+just infinite group. The proof the Theorem 1 is a rather direct combination of
+results of Pyber and Segal [16], Shusterman [17], and Ab´ert [1].
+Ab´ert proved that weakly branch groups are not linear over any field (for branch
+groups this result is due to Grigorchuk and Delzant). More precisely, he defined
+for every field k the natural number matk(n) to be the minimal r such that every
+graph on n vertices can be represented in the matrix algebra Mr,r(k) where the
+graph’s edges encode non-commutation. Ab´ert showed that
+�
+⌊n/2⌋ ≤ matk(n) ≤
+2(n−⌊log2(n)⌋+1) and asked for a linear lower bound [1, Question 4]. The following
+result answers this question in the affirmative.
+Theorem 2. Let k be a field and let r ≥ 1. Suppose that there are (r × r)-matrices
+a1, . . . , an, b1, . . . , bn ∈ Mr,r(k) such that all pairwise commutators are trivial except
+for [ai, bi] = aibi − biai for all i ∈ {1, . . . , n}. Then r ≥ n + 1.
+The lower bound in Theorem 2 is sharp. Let λ ∈ k×. Consider the matrices
+ai = I + E1,i+1, bi = I − λEi+1,i+1 ∈ Mn+1,n+1(k) for i = 1, . . . , n, where I is the
+identity matrix and Ei,j denotes the elementary matrix whose (i, j)-entry is 1 and
+all other entries are 0. Then ai, bi satisfy the assumptions of Theorem 2. If λ ̸= 1,
+then ai, bi are invertible. If |k| > 2, this shows with [1, Prop. 5] that
+�n
+2
+�
++ 1 ≤ matk(n) ≤ n + 1.
+The non-linearity of weakly branch groups follows since these groups contain
+infinite products of non-abelian groups. Let µk(G) denote the minimal dimension
+of a faithful, finite dimensional representation of a group G over a field k (we write
+µk(G) = ∞ if G is not linear over k). Theorem 2 directly implies a lower bound
+µk(G1 × . . . × Gn) ≥ n + 1 for direct products of non-abelian groups. Similarly
+Theorem 2 provides lower bounds for representations of products non-commutative
+(Lie) algebras. If the factors Gi are assumed to be non-solvable, the lower bound
+can be improved further.
+Theorem 3. Let k be a field, let G1, . . . , Gn be groups and let G = G1 × · · · × Gn
+denote their direct product.
+(1) If the groups G1, . . . , Gn are non-abelian, then µk(G) ≥ n + 1.
+(2) If the groups G1, . . . , Gn are non-solvable, then µk(G) ≥ 2n.
+Both lower bounds in Theorem 3 are sharp. Let ai = I + E1,i+1, bi = I −
+2Ei+1,i+1 ∈ GLn+1(Q) be as above. Setting Gi = ⟨ai, bi⟩ we can therefore deduce
+
+BRANCH GROUPS ARE NOT BOUNDEDLY GENERATED
+3
+that µQ(G1 ×. . .×Gn) = n+1. Suppose that the groups Gi in Theorem 3 are non-
+solvable subgroups of GL2(k) for some field k. Then each Gi can be embedded in a
+2×2-diagonal block in GL2n(k), which gives us an embedding of G = G1 ×· · ·×Gn
+in GL2n(k). Together with Theorem 3 this implies µk(G) = 2n. In particular,
+this applies to the case where each Gi is a non-abelian free group and thereby
+recovers [6, Theorem 3] in the SLn-case.
+1. Branch groups are not boundedly generated
+There are several characterizations of branch groups. The following one, which
+is a slight reformulation of [3, Definition 1.1], does not involve a rooted tree which
+makes it rather abstract. However it suits well for our purposes. A more geometric
+definition can be found in [3, Definition 1.13].
+Definition. A group G is called a branch group if it admits a decreasing sequence
+of subgroups (Hi)i∈N0 with H0 = G and
+�
+i∈N0
+Hi = 1, and a sequence of integers
+(ki)i∈N0 with k0 = 1 such that for each i the following hold:
+(1) Hi is a normal subgroup of finite index in G.
+(2) Hi splits as a direct product Hi = H(1)
+i
+× . . . × H(ki)
+i
+, where the factors are
+pairwise isomorphic.
+(3) the quotient mi+1 := ki+1/ki is an integer with mi+1 ≥ 2, and the product
+decomposition of Hi+1 refines the product decomposition of Hi in the sense
+that each factor H(j)
+i
+of Hi contains the factors H(ℓ)
+i+1 of Hi+1, where ℓ
+satisfies (j − 1) · mi+1 + 1 ≤ ℓ ≤ j · mi+1.
+(4) G acts transitively by conjugation on the set of factors H(j)
+i
+of Hi.
+As indicated in the introduction, not every branch group is just-infinite. In fact
+there is no need for finitely generated branch groups to admit a just-infinite quotient
+that is a branch group. See [8, Theorem 2] for an example of finitely generated
+branch group that maps homomorphically onto Z, which is of course just-infinite
+and bounded generated. As a consequence, to prove that branch groups cannot be
+boundedly generated, it is not sufficient to consider the just-infinite case, in which
+the claim turns out to be a direct consequence of results of Ab´ert [1], Pyber and
+Segal [16].
+Proof of Theorem 1. Suppose there is a branch group G that is boundedly gener-
+ated. Then [16, Corollary 1.6] tells us that G admits an epimorphism π: G → Q,
+where Q is an infinite linear group. However, by [1, Corollary 7] branch groups are
+not linear over any field. Thus Q is a proper quotient of G. As such Q is virtu-
+ally abelian by [8, Proposition 6]. Since G, being a boundedly generated group,
+is finitely generated, it follows that Q has a (non-trivial) free abelian finite index
+subgroup Q0. We can therefore consider the finite index subgroup G0 := π−1(Q0)
+of G, which by construction maps onto Z. Let us now fix an arbitrary number
+n ∈ N. From the definition of a branch group it follows that G contains a finite
+index subgroup of the form Hi = H(1)
+i
+× . . . × H(ki)
+i
+, where the factors are pairwise
+isomorphic and ki ≥ n. Then Hi ∩ G0 is a finite index subgroup of Hi. In this
+case it can be easily seen that there are pairwise isomorphic finite index subgroups
+K(j)
+i
+≤ H(j)
+i
+such that Ki := K(1)
+i
+× . . . × K(ki)
+i
+≤ Hi ∩ G0. In particular we see
+that Ki has finite index in G0, which implies that it maps onto Z. Thus some, and
+hence every, factor K(j)
+i
+maps onto Z. We can therefore deduce that the torsion-free
+part of the abelianization of Ki has rank at least ki ≥ n. As a consequence, this
+holds for every finite index subgroup of Ki. In particular this tells us that there
+is no finite index subgroup of Ki that can be generated with less then n elements.
+
+4
+S. KIONKE AND E. SCHESLER
+Since n ∈ N was arbitrary this contradicts a result of Shusterman [17, Theorem
+1.1], which tells us that for every boundedly generated group H there is a constant
+C > 0 such that every finite index subgroup of H contains a finite index subgroup
+that can be generated by at most C elements.
+□
+2. Lower bounds for the minimal representation dimension of directs
+products
+Let us now prove the results concerning the minimal representation dimensions.
+Proof of Theorem 2. For the proof we combine ideas from [1] and [5]. Extending
+scalars, we may assume that k is an infinite field. Recall that I ∈ Mr,r(k) denotes
+the identity matrix. We claim that I, a1, . . . , an, b1, . . . , bn are linearly independent.
+This follows along the lines of [1, Proof of Thm. 3]. Suppose that cI + �
+j λjaj +
+�
+j λ′
+jbj = 0 for c, λ1, . . . , λn, λ′
+1, . . . , λ′
+n ∈ k. Taking commutators with ai (resp.
+bi) shows λi = 0 (resp. λ′
+i = 0); since I ̸= 0 the last remaining coefficient c vanishes
+as well.
+Let V = kr and let C denote the linear span of {I, a1, . . . , an, b1, . . . , bn} in
+Mr,r(k). Consider the linear map Ψ: C → V defined by Ψ(X) = Xv for some
+v ∈ V . We will see that the image of Ψ has dimension at least n + 1 if v is chosen
+appropriately. As the commutators zi = [ai, bi] are non-trivial, the kernel of each
+zi is a proper subspace of V . However, V cannot be covered by a finite union of
+proper subspaces (as k is infinite). Thus there is a vector v ∈ V such that ziv ̸= 0
+for all i ∈ {1, . . . , n}. Let α: V → k be a linear form such that α(v) ̸= 0 and
+α(ziv) ̸= 0 for all i ∈ {1, 2, . . ., n} (such a linear form α exists, as the dual space
+V ∗ cannot be covered by finitely many proper subspaces). Now β : C × C → k
+defined by β(x, y) = α([x, y](v)) is an alternating form on C. It is not difficult to
+see that β is non-degenerate on the subspace ⟨a1, . . . , an, b1, . . . , bn⟩ ⊆ C (e.g. the
+matrix representation has full rank). Let us observe that kI +ker(Ψ) is an isotropic
+subspace, since for x, y ∈ kI + ker(Ψ) we have [x, y](v) = xyv − yxv = 0. As v ̸= 0
+we have I ̸∈ ker(Ψ) and thus dimk ker(Ψ) + 1 ≤ n + 1. This allows us to conclude
+that
+r ≥ dimk(im(Ψ)) = 2n + 1 − dimk ker(Ψ) ≥ n + 1.
+□
+Proof of Theorem 3. The first assertion follows immediately from Theorem 2. As-
+sume now that each Gi is non-solvable. If G is not linear, there is nothing to show.
+Assume that (ρ, V ) is a finite dimensional faithful representation over k. By exten-
+sion of scalars, we may assume that k is algebraically closed. Let V 1, . . . , V t denote
+the composition factors of V considered as G-module.
+Since k is algebraically
+closed, the composition factor V j is isomorphic to a tensor product
+V j = V j
+1 ⊗k V j
+2 ⊗k · · · ⊗k V j
+n
+where V j
+i is an irreducible Gi-representation; see e.g. [11, Prop. 2.3.23]. The compo-
+sition factors of V |Gi are the irreducible representations V 1
+i , . . . , V t
+i each one possi-
+bly occurring several times. Suppose for a contradiction that V j
+i is one-dimensional
+for all j.
+Then there is a basis of V such that ρ(Gi) is represented by upper
+triangular matrices. This gives a contradiction, since Gi is not solvable.
+For each j let Sj ⊆ {1, . . ., n} be the set of i such that dimk V j
+i ≥ 2. By the
+observation above, each i ≤ n belongs to at least one of the sets Sj. This implies
+dimk V =
+t
+�
+j=1
+n
+�
+i=1
+dimk V j
+i ≥
+t
+�
+j=1
+2|Sj| ≥
+t
+�
+j=1
+2|Sj| ≥ 2n.
+□
+
+BRANCH GROUPS ARE NOT BOUNDEDLY GENERATED
+5
+References
+1. Mikl´os Ab´ert, Representing graphs by the non-commuting relation, Publ. Math. Debrecen 69
+(2006), no. 3, 261–269. MR 2273978
+2. Laurent Bartholdi and Rostislav I. Grigorchuk, On parabolic subgroups and Hecke algebras of
+some fractal groups, Serdica Math. J. 28 (2002), no. 1, 47–90. MR 1899368
+3. Laurent Bartholdi, Rostislav I. Grigorchuk, and Zoran ˇSuni´k, Branch groups, Handbook of
+algebra, Vol. 3, Handb. Algebr., vol. 3, Elsevier/North-Holland, Amsterdam, 2003, pp. 989–
+1112. MR 2035113
+4. Marc Burger and Shahar Mozes, Lattices in product of trees, Inst. Hautes ´Etudes Sci. Publ.
+Math. (2000), no. 92, 151–194 (2001). MR 1839489
+5. Leandro Cagliero and Nadina Rojas, Faithful representations of minimal dimension of current
+Heisenberg Lie algebras, Internat. J. Math. 20 (2009), no. 11, 1347–1362. MR 2584190
+6. Caterina Campagnolo and Holger Kammeyer, Products of free groups in lie groups, Journal
+of Algebra 579 (2021), 237–255.
+7. David Carter and Gordon Keller, Bounded elementary generation of SLn(O), Amer. J. Math.
+105 (1983), no. 3, 673–687. MR 704220
+8. Thomas Delzant and Rostislav Grigorchuk, Homomorphic images of branch groups, and
+Serre’s property (FA), Geometry and dynamics of groups and spaces, Progr. Math., vol.
+265, Birkh¨auser, Basel, 2008, pp. 353–375. MR 2402409
+9. Mikhail Ershov and Andrei Jaikin-Zapirain, Property (T) for noncommutative universal lat-
+tices, Invent. Math. 179 (2010), no. 2, 303–347. MR 2570119
+10. R. I. Grigorˇcuk, On Burnside’s problem on periodic groups, Funktsional. Anal. i Prilozhen.
+14 (1980), no. 1, 53–54. MR 565099
+11. Emmanuel Kowalski, An introduction to the representation theory of groups, Graduate Studies
+in Mathematics, vol. 155, American Mathematical Society, Providence, RI, 2014. MR 3236265
+12. G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik
+und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-
+Verlag, Berlin, 1991. MR 1090825
+13. Donald McCarthy, Infinite groups whose proper quotient groups are finite. I, Comm. Pure
+Appl. Math. 21 (1968), 545–562. MR 237637
+14. Alexey Muranov, Diagrams with selection and method for constructing boundedly generated
+and boundedly simple groups, Comm. Algebra 33 (2005), no. 4, 1217–1258. MR 2136699
+15. Alexander Yu. Ol’shanskii, Infinite groups with cyclic subgroups, Dokl. Akad. Nauk SSSR 245
+(1979), no. 4, 785–787. MR 527709
+16. L´aszl´o Pyber and Dan Segal, Finitely generated groups with polynomial index growth, J. Reine
+Angew. Math. 612 (2007), 173–211. MR 2364077
+17. Mark Shusterman, Ranks of subgroups in boundedly generated groups, Bull. Lond. Math. Soc.
+48 (2016), no. 3, 539–547. MR 3509913
+18. J. S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc. 69 (1971),
+373–391. MR 274575
+FernUniversit¨at in Hagen, Fakult¨at f¨ur Mathematik und Informatik, 58084 Hagen
+Email address: [email protected]
+Email address: [email protected]
+

VdAzT4oBgHgl3EQfYPwk/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf,len=351
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='01330v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='GR]  3 Jan 2023  ON REPRESENTATIONS OF DIRECT PRODUCTS AND THE  BOUNDED GENERATION PROPERTY OF BRANCH GROUPS  STEFFEN KIONKE AND EDUARD SCHESLER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' We prove that the minimal representation dimension of a direct  product G of non-abelian groups G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , Gn is bounded below by n + 1 and  thereby answer a question of Ab´ert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' If each Gi is moreover non-solvable, then  this lower bound can be improved to be 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' By combining this with results of  Pyber, Segal, and Shusterman on the structure of boundedly generated groups  we show that branch groups cannot be boundedly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Introduction  An infinite group G is called just-infinite if all of its proper quotients are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Obvious examples of just-infinite groups are virtually simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Other exam-  ples arise from irreducible lattices in higher rank semisimple Lie groups, such as  SLn(Z) for n ≥ 3, after dividing out their centers, see [12, Chapter IV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Such groups  are in fact hereditarily just-infinite, which means that they are residually finite and  all of their finite index subgroups are just-infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Grigorchuk’s group [10] provided  the first example of a just-infinite group that is not virtually a finite direct power of  a simple or a hereditarily just infinite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Grigorchuk’s group is a just-infinite  branch group, which means that its commensurability classes of subnormal sub-  groups form a lattice that is isomorphic to the lattice of open and closed subsets  of a Cantor set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' By Wilson’s classification [18] just-infinite groups fall into three  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Every just-infinite group G is either a branch group or virtually a direct  power of a simple or a hereditarily just-infinite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Since its introduction by McCarthy [13] in the late 1960’s, the class of just-infinite  groups remained an active field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' One reason might be that every finitely  generated infinite group admits a just-infinite quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Thus whenever there is  some finitely generated, infinite group G that admits a property P that is preserved  under homomorphic images, then there is also a finitely generated just-infinite group  with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Following [3], we call a property P that is preserved under homomorphic  images an H-property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Well-known examples of H-properties include amenability,  property (T), bounded generation, being a torsion group, having subexponential  growth etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' In view of Wilson’s classification, it is natural to investigate which  of the three classes of just-infinite groups contain groups that satisfy a given H-  property P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' For the H-property “being a torsion group” this question is settled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' In  this case it is know that there are finitely generated simple groups [15], just-infinite  branch groups [10], and hereditarily just-infinite groups [9] that are torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' On  the other hand, there are torsion-free, finitely generated, just-infinite groups that  are simple [4], branch [2], and hereditarily just-infinite (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  The purpose of this note is to study this question for the bounded generation  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Recall that a group G is boundedly generated if it contains a finite subset  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Primary 20E08;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Secondary 20E26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' bounded generation, branch group, faithful representations of direct  products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -  441848266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  1  2  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' KIONKE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' SCHESLER  {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , gn} such that every g ∈ G can be written as g = gk1  1 · · · gkn  n for appropriate  ki ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Since infinite torsion groups are not boundedly generated, it follows that  each of the three classes of just-infinite groups contains a finitely generated group  that does not have the bounded generation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' On the other hand, it was  proven by Carter and Keller [7] that PSLn(Z) is boundedly generated for n ≥ 3,  which provides an interesting boundedly generated hereditarily just-infinite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  The existence of boundedly generated, infinite, simple groups was established by  Muranov [14], whose construction seems to be the only one available at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' It  remains to study just-infinite branch groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' The question of existence of boundedly  generated just-infinite branch groups was raised by Bartholdi, Grigorchuk, and  ˇSuni´k [3, Question 12] and remained open to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  The  purpose of the paper is to show that the answer is negative for arbitrary branch  groups (even without the assumption of being just-infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' There is no boundedly generated branch group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  As a consequence, it follows from Wilson’s classification of just-infinite groups  that every boundedly generated infinite group has a quotient that is virtually a  product of finitely many copies of a boundedly generated simple or hereditarily  just infinite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' The proof the Theorem 1 is a rather direct combination of  results of Pyber and Segal [16], Shusterman [17], and Ab´ert [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Ab´ert proved that weakly branch groups are not linear over any field (for branch  groups this result is due to Grigorchuk and Delzant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' More precisely, he defined  for every field k the natural number matk(n) to be the minimal r such that every  graph on n vertices can be represented in the matrix algebra Mr,r(k) where the  graph’s edges encode non-commutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Ab´ert showed that  �  ⌊n/2⌋ ≤ matk(n) ≤  2(n−⌊log2(n)⌋+1) and asked for a linear lower bound [1, Question 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' The following  result answers this question in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let k be a field and let r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Suppose that there are (r × r)-matrices  a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , bn ∈ Mr,r(k) such that all pairwise commutators are trivial except  for [ai, bi] = aibi − biai for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Then r ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  The lower bound in Theorem 2 is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let λ ∈ k×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Consider the matrices  ai = I + E1,i+1, bi = I − λEi+1,i+1 ∈ Mn+1,n+1(k) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , n, where I is the  identity matrix and Ei,j denotes the elementary matrix whose (i, j)-entry is 1 and  all other entries are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Then ai, bi satisfy the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' If λ ̸= 1,  then ai, bi are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' If |k| > 2, this shows with [1, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 5] that  �n  2  �  + 1 ≤ matk(n) ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  The non-linearity of weakly branch groups follows since these groups contain  infinite products of non-abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let µk(G) denote the minimal dimension  of a faithful, finite dimensional representation of a group G over a field k (we write  µk(G) = ∞ if G is not linear over k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Theorem 2 directly implies a lower bound  µk(G1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' × Gn) ≥ n + 1 for direct products of non-abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Similarly  Theorem 2 provides lower bounds for representations of products non-commutative  (Lie) algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' If the factors Gi are assumed to be non-solvable, the lower bound  can be improved further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let k be a field, let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , Gn be groups and let G = G1 × · · · × Gn  denote their direct product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  (1) If the groups G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , Gn are non-abelian, then µk(G) ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  (2) If the groups G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , Gn are non-solvable, then ��k(G) ≥ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Both lower bounds in Theorem 3 are sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let ai = I + E1,i+1, bi = I −  2Ei+1,i+1 ∈ GLn+1(Q) be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Setting Gi = ⟨ai, bi⟩ we can therefore deduce  BRANCH GROUPS ARE NOT BOUNDEDLY GENERATED  3  that µQ(G1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='×Gn) = n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Suppose that the groups Gi in Theorem 3 are non-  solvable subgroups of GL2(k) for some field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Then each Gi can be embedded in a  2×2-diagonal block in GL2n(k), which gives us an embedding of G = G1 ×· · ·×Gn  in GL2n(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Together with Theorem 3 this implies µk(G) = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' In particular,  this applies to the case where each Gi is a non-abelian free group and thereby  recovers [6, Theorem 3] in the SLn-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Branch groups are not boundedly generated  There are several characterizations of branch groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' The following one, which  is a slight reformulation of [3, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='1], does not involve a rooted tree which  makes it rather abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' However it suits well for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' A more geometric  definition can be found in [3, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' A group G is called a branch group if it admits a decreasing sequence  of subgroups (Hi)i∈N0 with H0 = G and  �  i∈N0  Hi = 1, and a sequence of integers  (ki)i∈N0 with k0 = 1 such that for each i the following hold:  (1) Hi is a normal subgroup of finite index in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  (2) Hi splits as a direct product Hi = H(1)  i  × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' × H(ki)  i  , where the factors are  pairwise isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  (3) the quotient mi+1 := ki+1/ki is an integer with mi+1 ≥ 2, and the product  decomposition of Hi+1 refines the product decomposition of Hi in the sense  that each factor H(j)  i  of Hi contains the factors H(ℓ)  i+1 of Hi+1, where ℓ  satisfies (j − 1) · mi+1 + 1 ≤ ℓ ≤ j · mi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  (4) G acts transitively by conjugation on the set of factors H(j)  i  of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  As indicated in the introduction, not every branch group is just-infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' In fact  there is no need for finitely generated branch groups to admit a just-infinite quotient  that is a branch group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' See [8, Theorem 2] for an example of finitely generated  branch group that maps homomorphically onto Z, which is of course just-infinite  and bounded generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' As a consequence, to prove that branch groups cannot be  boundedly generated, it is not sufficient to consider the just-infinite case, in which  the claim turns out to be a direct consequence of results of Ab´ert [1], Pyber and  Segal [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Suppose there is a branch group G that is boundedly gener-  ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Then [16, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='6] tells us that G admits an epimorphism π: G → Q,  where Q is an infinite linear group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' However, by [1, Corollary 7] branch groups are  not linear over any field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Thus Q is a proper quotient of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' As such Q is virtu-  ally abelian by [8, Proposition 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Since G, being a boundedly generated group,  is finitely generated, it follows that Q has a (non-trivial) free abelian finite index  subgroup Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' We can therefore consider the finite index subgroup G0 := π−1(Q0)  of G, which by construction maps onto Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let us now fix an arbitrary number  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' From the definition of a branch group it follows that G contains a finite  index subgroup of the form Hi = H(1)  i  × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' × H(ki)  i  , where the factors are pairwise  isomorphic and ki ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Then Hi ∩ G0 is a finite index subgroup of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' In this  case it can be easily seen that there are pairwise isomorphic finite index subgroups  K(j)  i  ≤ H(j)  i  such that Ki := K(1)  i  × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' × K(ki)  i  ≤ Hi ∩ G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' In particular we see  that Ki has finite index in G0, which implies that it maps onto Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Thus some, and  hence every, factor K(j)  i  maps onto Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' We can therefore deduce that the torsion-free  part of the abelianization of Ki has rank at least ki ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' As a consequence, this  holds for every finite index subgroup of Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' In particular this tells us that there  is no finite index subgroup of Ki that can be generated with less then n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  4  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' KIONKE AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' SCHESLER  Since n ∈ N was arbitrary this contradicts a result of Shusterman [17, Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='1], which tells us that for every boundedly generated group H there is a constant  C > 0 such that every finite index subgroup of H contains a finite index subgroup  that can be generated by at most C elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Lower bounds for the minimal representation dimension of directs  products  Let us now prove the results concerning the minimal representation dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' For the proof we combine ideas from [1] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Extending  scalars, we may assume that k is an infinite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Recall that I ∈ Mr,r(k) denotes  the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' We claim that I, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , bn are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  This follows along the lines of [1, Proof of Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Suppose that cI + �  j λjaj +  �  j λ′  jbj = 0 for c, λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , λn, λ′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , λ′  n ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Taking commutators with ai (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  bi) shows λi = 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' λ′  i = 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' since I ̸= 0 the last remaining coefficient c vanishes  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Let V = kr and let C denote the linear span of {I, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , bn} in  Mr,r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Consider the linear map Ψ: C → V defined by Ψ(X) = Xv for some  v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' We will see that the image of Ψ has dimension at least n + 1 if v is chosen  appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' As the commutators zi = [ai, bi] are non-trivial, the kernel of each  zi is a proper subspace of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' However, V cannot be covered by a finite union of  proper subspaces (as k is infinite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Thus there is a vector v ∈ V such that ziv ̸= 0  for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let α: V → k be a linear form such that α(v) ̸= 0 and  α(ziv) ̸= 0 for all i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=', n} (such a linear form α exists, as the dual space  V ∗ cannot be covered by finitely many proper subspaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Now β : C × C → k  defined by β(x, y) = α([x, y](v)) is an alternating form on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' It is not difficult to  see that β is non-degenerate on the subspace ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , an, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , bn⟩ ⊆ C (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' the  matrix representation has full rank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let us observe that kI +ker(Ψ) is an isotropic  subspace, since for x, y ∈ kI + ker(Ψ) we have [x, y](v) = xyv − yxv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' As v ̸= 0  we have I ̸∈ ker(Ψ) and thus dimk ker(Ψ) + 1 ≤ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' This allows us to conclude  that  r ≥ dimk(im(Ψ)) = 2n + 1 − dimk ker(Ψ) ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  □  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' The first assertion follows immediately from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' As-  sume now that each Gi is non-solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' If G is not linear, there is nothing to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Assume that (ρ, V ) is a finite dimensional faithful representation over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' By exten-  sion of scalars, we may assume that k is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Let V 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , V t denote  the composition factors of V considered as G-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Since k is algebraically  closed, the composition factor V j is isomorphic to a tensor product  V j = V j  1 ⊗k V j  2 ⊗k · · · ⊗k V j  n  where V j  i is an irreducible Gi-representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' [11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' The compo-  sition factors of V |Gi are the irreducible representations V 1  i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' , V t  i each one possi-  bly occurring several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Suppose for a contradiction that V j  i is one-dimensional  for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Then there is a basis of V such that ρ(Gi) is represented by upper  triangular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' This gives a contradiction, since Gi is not solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  For each j let Sj ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=', n} be the set of i such that dimk V j  i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' By the  observation above, each i ≤ n belongs to at least one of the sets Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' This implies  dimk V =  t  �  j=1  n  �  i=1  dimk V j  i ≥  t  �  j=1  2|Sj| ≥  t  �  j=1  2|Sj| ≥ 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  □  BRANCH GROUPS ARE NOT BOUNDEDLY GENERATED  5  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Mikl´os Ab´ert, Representing graphs by the non-commuting relation, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Debrecen 69  (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 3, 261–269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' MR 2273978  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Laurent Bartholdi and Rostislav I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Grigorchuk, On parabolic subgroups and Hecke algebras of  some fractal groups, Serdica Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 28 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 1, 47–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' MR 1899368  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Laurent Bartholdi, Rostislav I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Grigorchuk, and Zoran ˇSuni´k, Branch groups, Handbook of  algebra, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 3, Handb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 3, Elsevier/North-Holland, Amsterdam, 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 989–  1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' MR 2035113  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Marc Burger and Shahar Mozes, Lattices in product of trees, Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Hautes ´Etudes Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' (2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 92, 151–194 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' MR 1839489  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Leandro Cagliero and Nadina Rojas, Faithful representations of minimal dimension of current  Heisenberg Lie algebras, Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 20 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 11, 1347–1362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' MR 2584190  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Caterina Campagnolo and Holger Kammeyer, Products of free groups in lie groups, Journal  of Algebra 579 (2021), 237–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' David Carter and Gordon Keller, Bounded elementary generation of SLn(O), Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  105 (1983), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 3, 673–687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' MR 704220  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Thomas Delzant and Rostislav Grigorchuk, Homomorphic images of branch groups, and  Serre’s property (FA), Geometry and dynamics of groups and spaces, Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content='  265, Birkh¨auser, Basel, 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' 353–375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' MR 2402409  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Mikhail Ershov and Andrei Jaikin-Zapirain, Property (T) for noncommutative universal lat-  tices, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdAzT4oBgHgl3EQfYPwk/content/2301.01330v1.pdf'}
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+We are Going to the Space - Part 1:
+Which device to deploy in a satellite?
+Robert Bayer
+IT University of Copenhagen
+[email protected]
+Julian Priest
+IT University of Copenhagen
+[email protected]
+Pınar Tözün
+IT University of Copenhagen
+[email protected]
+ABSTRACT
+The shrinkage in sizes of components that make up satellites led to
+wider and low cost availability of satellites. As a result, there has
+been an advent of smaller organizations having the ability to deploy
+satellites with a variety of data-intensive applications to run on
+them. One popular application is image analysis to detect, for exam-
+ple, land, ice, clouds, etc. However, the resource-constrained nature
+of the devices deployed in satellites creates additional challenges
+for this resource-intensive application.
+In this paper, we investigate the performance of a variety of edge
+devices for deep-learning-based image processing in space. Our
+goal is to determine the devices that satisfy the latency and power
+constraints of satellites while achieving reasonably accurate results.
+Our results demonstrate that hardware accelerators (TPUs, GPUs)
+are necessary to reach the latency requirements. On the other hand,
+state-of-the-art edge devices with GPUs could have a high power
+draw, making them unsuitable for deployment on a satellite.
+1
+INTRODUCTION
+In the last century, most innovation in real-world satellite appli-
+cations were only available to the largest countries such as USA
+and Russia. These innovations led to significant reductions in the
+size of the components that make up a satellite and in the cost of
+the manufacturing and deployment process of a satellite. This, in
+turn, introduced a new CubeSat class of miniature satellites. Their
+format is based on a 10cm cube, with the possibility of combining
+multiple modules to create a larger satellite. This standardization
+makes a batch deployment of satellites easier as the format affords
+tight configuration. The reduction in costs that came as a result of
+this new deployment method led to the advent of satellites owned
+by small or private organizations.
+However, the size of the CubeSat class satellites poses new com-
+plex challenges such as the power and thermal constraints as well
+as the physical dimensions of components. These CubeSats often
+perform resource-intensive tasks, which clashes with the resource-
+constrained nature of their format.
+Image processing and analysis is one class of possible satellite
+workloads. Satellites with this type of workload take large-scale
+images, which they have to store and send back to a ground station.
+The link between the satellite and a ground station is of limited
+bandwidth and short-lived. The images must be highly compressed
+to send all of them or filtered by quality and areas of interest. The
+images already have a resolution of tens or hundreds of meters
+per pixel, and lossy compression would lead to even more loss of
+detail. Filtering preserves the details of the images, but is more
+resource-intensive.
+This paper is a step toward understanding the requirements
+and limitations of deep-learning-based image filtering systems on
+small satellites. More specifically, we characterize the requirements
+and constraints of an image processing unit (IPU) deployed on a
+satellite and outline possible use case scenarios of such a system,
+each introducing a different degree of resource constraints and
+intensity. We then further analyze the performance of multiple
+edge devices on these different scenarios and their suitability for
+possible deployment on one of the Danish Student CubeSat Project’s
+(DISCO) [7] satellites.
+These devices are based on different hardware architectures,
+ranging from a microcontroller to a tensor processing unit (TPU).
+While microcontrollers or more complex CPUs have extensive flight
+history (have already been deployed in satellites before) and low
+power footprint, they were not built with running deep learning
+workloads in mind. On the other hand, GPUs have been utilized
+more in the past decade in satellites, with one of the leading work-
+loads being machine learning, based on the success of GPUs in
+terrestrial use cases of machine learning. The latest of these archi-
+tectures, TPU, has not been extensively researched for deployment
+in satellites, even though TPUs were designed to perform fast neural
+network inference, promising low latency with high efficiency.
+The contributions of this study are as follows:
+• We illustrate the differences between the terrestrial and space
+edge IoT. The edge devices deployed in space have to perform
+very resource-intensive workloads with high reliability and do
+so in a highly resource-constrained environment.
+• We characterize a set of requirements specific to IPU on a
+CubeSat. We do this by outlining multiple use case scenarios
+and showing how they affect the required latency and possible
+power draw of such a system.
+• We compare the performance of multiple edge devices based
+on different architectures and show the need and the bene-
+fit of using highly specialized hardware for machine learning
+workloads on satellites.
+2
+BACKGROUND
+We first introduce DISCO project and survey related work that
+specifically targets data processing in satellites.
+2.1
+DISCO
+The Danish Student Cubesat Program (DISCO) [7] offers Danish
+university students the opportunity to design and operate a small
+satellite and to gain space flight experience. DISCO is a collaboration
+between four universities in Denmark, which will initially launch
+three student Cubesats into Low Earth Orbit with the first launch
+scheduled for 2023.
+DISCO2 has been designed by students to carry an Earth imag-
+ing payload, infrastructure to capture images in space, into a solar
+synchronous polar orbit. The instrument is a collaboration with
+arXiv:2301.04954v1  [cs.LG]  12 Jan 2023
+
+Bayer, et al.
+Image process-
+ing Unit
+Nominal
+Power
+Peak Power
+Design Margin
+Nominal Cycle
+Peak Cycle
+Power
+Mass budget
+Dimensions
+Constraints
+2.00 W
+5.00 W
+5%
+20%
+20%
+1.48 W
+0.15 Kg
+10x70x80 mm
+Table 1: List of constraints of the IPU on board of the DISCO2 satellite.
+the Arctic Research Centre at Aarhus University and will support a
+range of field research in Greenland. Initially focused on a long term
+marine systems and cryosphere monitoring projects, the satellite
+will supplement ground based observations with remote sensing
+data, providing both good polar coverage and on demand availabil-
+ity for the projects.
+The payload will include one or more high resolution cameras
+as well as a dedicated IPU, which will be capable of simple machine
+learning applications for image classification. Students will be able
+to carry out conventional on-satellite image processing in addi-
+tion to running machine learning applications for classification,
+discrimination, and feature identification.
+2.2
+Related Work
+The challenge of low bandwidth on small satellites creates the need
+for data post-processing to filter the data on the satellite before
+sending elsewhere. Neural-network-based filtering has been stud-
+ied for this purpose in recent years. Specifically, several works have
+tested the suitability of small system-on-chip (SoC) devices leverag-
+ing the power of GPUs to accelerate machine learning workloads
+[3, 13]. Moreover, some works have also explored the use of ASICs,
+such as Intel’s Movidius Myriad vision processing units (VPUs)
+[8, 9]. This VPU has even seen real-life deployment [8].
+Evaluation of TPUs for deployment on satellites is limited, how-
+ever. To the best of our knowledge, only one work has explored this
+option [10], with no real-life deployments. DISCO project would
+therefore be the first satellite to leverage TPU for onboard deep
+learning applications, based on the findings of our study.
+In this work, we aim to characterize the performance of the dif-
+ferent flavors of off-the-shelf edge devices to check their suitability
+for being deployed on a satellite in the context of DISCO project.
+Data management and processing for internet-of-things and edge
+computing has been important in the data management community
+as well [4, 14, 16, 18]. We are complementing these works with a
+very specific application focus, which is data-intensive applications
+deployed on satellites.
+3
+REQUIREMENTS
+The requirements and constraints used in this study are based on
+the DISCO2 Arctic imaging mission use case. The power and mass
+constraints of the edge device to be deployed on the satellite are
+based on the power and mass budgets developed in the DISCO2 en-
+gineering design. The satellite design is a modular 3U Cubesat with
+off-the-shelf modules for attitude control, power, communications,
+and flight control. The power budget values are typical for a 3U
+earth imaging Cubesat of this type. The resulting payload capacity
+is 1U (10x10x10cm) and 1.3kg and this must include the cameras,
+module enclosure, optics. and image processing unit (IPU). The full
+list of constraints of the IPU is shown in Table 1.
+The planned orbit of DISCO2 is a solar synchronous polar or-
+bit at 550 km altitude. The camera sensor is based on an Alvium
+1800 C-2040 and the lens focal length is the maximum that could
+be considered for the mission. Assuming a 50% overlap between
+images, i.e., capturing the same land area, this provides a minimum
+time of 4.42s between consecutive images (𝑇𝑖), which was derived
+as follows:
+𝑇𝑖 = 𝐺𝑆𝐷 ∗𝑊𝑖 ∗ ∩𝑖
+𝑣
+= 14.8495𝑚/𝑝𝑥 ∗ 4512𝑝𝑥 ∗ 0.5
+7585.16𝑚/𝑠
+= 4.42𝑠
+(1)
+, where 𝐺𝑆𝐷 (ground sample distance) is the spatial resolution of
+the image, 𝑊𝑖 is the width of image in pixels, ∩𝑖 is the overlap
+between images and 𝑣 is the orbital velocity of the satellite.
+Taking this latency value as a baseline, we now describe three
+image processing scenarios with different levels of difficulty based
+on this Arctic mission’s goals.
+Scenario 1: Real-time imaging. Images are taken with a pe-
+riod of 4.42s, as derived above. The inference has to be performed
+and completed before the next image is captured. This would allow
+for the results of an inference to be used in decision making for the
+next image capture.
+Scenario 2: Arctic region imaging. In this scenario, images
+are taken only while the satellite is over the polar regions relaxing
+the latency requirements. Images are buffered and inference is
+performed in the remainder of the orbit, with the IPU available for
+inference before the subsequent orbit. The capture-inference cycle
+completes in 5739s.
+Scenario 3: Greenland imaging. The images are taken only
+while over Greenland further relaxing the latency requirements.
+As the orbit is solar synchronous with a period of one day. This
+results in subsequent bursts but only when the swathe passes over
+Greenland. Images are again buffered and the capture-inference
+cycle completes in 21600s.
+4
+METHODOLOGY AND SETUP
+Our goal is to characterize the performance of modern low-power
+hardware for deployment as the IPU of the DISCO2 satellite. This
+section presents our experimental methodology and setup to achieve
+this goal.
+4.1
+Devices under Test
+As representative hardware for this study, we picked three devices,
+each based on a different hardware architecture. These choices were
+made based on their physical dimensions, weight, power draw, and
+high performance considering the low power draw limits.
+ARM Cortex-M Mirocontroller, based on the STM32H745
+chip, has the potential of the lowest power draw, while the least
+performant of the choices. As an additional advantage, this chip
+has extensive flight history and, therefore, would be a safe choice
+for deployment in space applications.
+
+We are Going to the Space - Part 1:
+Which device to deploy in a satellite?
+In order to use this device as an on-board IPU, we can use Tensor-
+Flow Lite for Microcontrollers [5], a framework designed to allow
+neural network inference with only 16 KB of memory overhead
+(core runtime) and without the need for an operating system. In
+addition, the framework also provides support for the use of spe-
+cialized neural network kernels, such as CMSIS-NN [12] kernels
+designed specifically for Cortex-M processor cores. The X-CUBE-
+AI [17] framework makes the deployment of the neural networks
+with TensorFlow Lite for microcontrollers even more accessible. It
+provides an intuitive GUI for the libraries or code samples based
+on the provided and trained TensorFlow Lite [2] model.
+Even though this device has support for floating point operations,
+a quantized model was used, in order to achieve better latency
+and, more importantly, lower memory footprint, which is a large
+constraint of this device.
+To simulate the exact performance and power characteristics of
+a flight computer already available for a potential deployment in
+DISCO project, OBC-P3, we used only the Cortex-M7 core of the
+STM32H745 and scaled its clock from 480MHz down to 300MHz.
+Parameters of the OBC-P3 system can be found in the Table 2.
+Processor
+2x ARM Cortex-M7 @ 300 MHz
+SRAM
+384 KB SRAM
+FRAM
+32 KB
+Flash
+2 MB
+Storage
+64 GB eMMC
+Dimensions
+94 x 94 x 13 mm
+Weight
+120 g
+Table 2: Specifications of the OBC-P3 system containing
+the ARM Cortex-M7 MCUs. The dimensions and weight in-
+cludes the enclosure with aluminium shielding.
+NVIDIA Jetson Nano is a portable SoC composed of power-
+efficient ARM CPU and NVIDIA GPU designed for embedded sys-
+tems that require GPU-friendly computations, e.g., image process-
+ing, video encoding/decoding, and machine learning tasks. It is
+the only device in our experiments that supports TensorFlow Core.
+Therefore, the deployment process to this device is equivalent to
+that of servers or desktops. It is also the only evaluated device
+supporting batch sizes larger than one.
+The device can operate at a wattage between 5W - 10W. We
+configured it to operate at 5W by disabling two of the four CPU
+cores. While this setting leads to lower performance, it is necessary
+in order to fit into the power budget outlined in Table 1. Parameters
+of this device can be found in Table 3.
+CPU
+Quad-core ARM A57 @ 1.43 GHz
+GPU
+128-core Maxwell
+GFLOPS
+472
+RAM
+4 GB 64-bit LPDDR4 25.6 GB/s
+Storage
+64 GB SD card
+Dimensions
+100 x 80 x 29 mm
+Weight
+141 g
+Table 3: Specifications of the NVIDIA Jetson Nano. The di-
+mensions and weight are based on the developer kit version
+of this device.
+CoralAI Dev
+Board Mini
+Raspberry Pi 2B
+CPU
+Quad-core Arm
+Cortex-A35 @ 1.5 GHz
+Quad-core ARM
+Cortex-A7 @ 900 MHz
+RAM
+2 GB
+1 GB
+Storage
+8 GB eMMC
+32 GB SD card
+Dimensions
+64 x 48 x 14.6 mm
+85.6 x 56.5 x 17 mm
+Weight
+25.5 g
+45 g
+CoralAI TPU
+TOPS
+Interface
+Dimensions
+Weight
+4
+USB2
+65 x 30 mm
+4.3 g
+Table 4: Specifications of the CoralAI Dev Board Mini and
+Raspberry Pi 2B, as well as the CoralAI TPU chip. Note
+that the CoralAI Dev Board Mini’s physical dimensions and
+weight include the on-board TPU.
+CoralAI TPU is an ASIC, AI accelerator developed by Google
+to accelerate machine learning workloads at the edge. TPU is not
+a standalone device and must be deployed together with a Linux,
+Windows, or macOS system. Even though this device is highly
+specialized, model deployment is performed by compiling a Ten-
+sorFlow Lite model containing a subset of supported layers [1].
+This model must be quantized to an 8-bit integer, as this is only
+supported data type by this device. This compiled model can then
+be used with the TPU’s Python or C++ library. In our experiments,
+we rely on the C++ library.
+We evaluated this device in two formats:
+(1) CoralAI Dev Board Mini, which couples this chip and an SoC
+on a single board.
+(2) CoralAI USB accelerator connected to a Raspberry Pi 2B.
+Even though (1) has the TPU chip on the same board as the SoC,
+the two communicate through the USB2 bus, as in the case of (2).
+In addition, both of the devices run Linux. The parameters of the
+different devices can be found in Table 4.
+4.2
+Metrics
+The metrics to evaluate the suitability of the devices as the on-
+satellite IPUs are based on the requirements Section 3 outlined.
+Latency is reported in seconds per inference of a sample, where
+a sample is a whole image of size 4512 x 4512 pixels or 400 tiles of
+size 224 x 224 pixels. We measure the latency of the neural network
+inference once the data is already in the device’s memory.
+Nominal power draw is the average power draw in mW mul-
+tiplied by the duty cycle, which is the ratio between achieved and
+required latency for each scenario (outlined in Section 3). We use
+an external appliance for CoralAI TPU and the ARM Cortex-M7
+and tegrastats for Jetson Nano to measure this metric.
+Peak power draw is the maximum power draw over the pe-
+riod of inference in mW. It is measured using the same tools as in
+nominal power draw.
+Power consumption is reported per inference of a sample. This
+metric shows the power efficiency of the device and will guide
+the choice of a more suitable device if multiple devices fulfill the
+requirements of a particular scenario.
+
+Bayer, et al.
+Scaling factor
+0.25
+0.5
+1.0
+Device
+Latency (s)
+Power consump-
+tion (mWh)
+Latency (s)
+Power consump-
+tion (mWh)
+Latency (s)
+Power consump-
+tion (mWh)
+ARM Cortex-M7
+118.80
+12.10
+N/A
+N/A
+N/A
+N/A
+CoralAI Dev Board Mini
+3.778
+1.76
+3.991
+1.90
+4.533
+2.40
+Raspberry Pi + CoralAI TPU
+3.106
+1.69
+3.331
+1.93
+3.825
+2.40
+NVIDIA Jetson Nano
+13.900
+10.54
+14.109
+12.74
+18.581
+22.10
+NVIDIA Jetson Nano, batch size 64
+3.529
+3.71
+5.382
+5.94
+10.171
+12.89
+Table 5: Latency and power consumption results with different scaling factors of the model as measured on corresponding
+devices. Batch size of 1 is used if not stated otherwise.
+Accuracy of the deployed model is also measured to show the
+effect quantization has on the model’s predictive performance.
+4.3
+Workload
+To simulate the imaging scenarios of interest (Section 3), we use an
+image classification workload consisting of a 5-class classification
+problem. For this workload, an off-the-shelf MobileNetV1 model
+[11] pre-trained on the ImageNet dataset [6] was chosen for its
+strong predictive power and availability in multiple scaling factors,
+affecting the number of hidden layers in the model. The availability
+of multiple scaling factors is an important factor as the memory of
+some of the devices is highly limited and can therefore fit only the
+smallest of the variants.
+We further fine-tuned the model before deployment on the de-
+vices using the Flowers dataset [15], containing 3670 color images
+of size 224 x 224 pixels belonging to 5 different classes. While the
+weights in the model do not affect the inference latency or power
+required to perform the inference, the model was fine-tuned using
+this dataset because it mimics one of the possible use cases closely
+(in number of classes as well as size of the images) and allows us to
+quantify the effects of the size and precision of the model.
+Since the models running on the Cortex-M7 and CoralAI TPU
+were quantized, rescaling of images was not needed before infer-
+ence. For Jetson Nano, the pixel values must be rescaled to the range
+of [0, 1), or, in other words, divided by 255. An additional layer was
+prepended to the model for this process in order to take advantage
+of the GPU parallelism on Jetson Nano.
+As the size of the images would stress the memory available on
+the evaluated devices, we employed a tiling method for inference.
+Each image was divided into 400 patches of size 224 x 224.1 This
+method results in a positive side effect, where each patch can be
+filtered separately acting as a coarse-grained image segmentation.
+This way only the tiles of interest are sent back to a ground station
+saving us bandwidth.
+The results are reported as an average of 10 inferences on the full
+4512 x 4512 pixel image. The inference on Cortex-M7 and CoralAI
+TPU were performed with batch size of 1 and on Jetson Nano with
+the batch size of 2𝑥, where x is from range of 0-6. All of the devices
+were tested with the scaling factors 0.25, 0.5, and 1.0 of the model,
+with the exception of the ARM Cortex-M7 microcontroller, which
+could not fit the larger models in memory.
+1The image is not evenly divisible; therefore, the borders are disregarded.
+5
+RESULTS
+The results are split into three parts: (1) analysis of latency and
+power draw results with respect to the three scenarios outlined in
+Section 3, (2) impact of quantization on the accuracy of models with
+various scaling factors, (3) impact of batch size on the performance
+of NVIDIA Jetson Nano.
+5.1
+Scenarios
+Table 5 shows the latency achieved and power consumption of the
+devices using the various scaling factors of the model. Furthermore,
+Tables 6 and 7 show the nominal and peak power draw of different
+configurations, respectively. The Sections 5.1.1-5.1.3 discuss the
+suitability of each device for the different scenarios based on the
+results on these tables. The reported latency, power consumption,
+and the peak power draw are independent of the scenarios. Only
+the nominal power draw depends on the use case scenarios, due to
+the duty cycle (see Section 4.2).
+5.1.1
+Scenario 1: Real-time imaging. Real-time imaging is the most
+constrained scenario. Therefore, a high degree of specialization is
+necessary to fulfill the requirements.
+Latency. The pipeline has to achieve a 4.42s latency, which is
+the spacing between images with 50% overlap. There are no passive
+periods. Therefore the latency of the pipeline has to be lower than
+that of the imaging for the pipeline not to get backed up. There are
+multiple configurations that achieved the required latency (Table
+5). These are all the configurations utilizing a TPU and the smallest
+model running on Jetson Nano with batch size of 64.
+Power draw. Out of the configurations that satisfy the latency
+requirements for this scenario, only of the TPU configurations fit
+into the nominal power budget of the satellite (Table 6). Even though
+the Jetson Nano with batch size 64 achieves a latency comparable
+to the TPUs with a scaling factor of 0.25, the power consumption
+per inference is more than twice as high.
+Furthermore, the Raspberry Pi with external TPU module does
+not fit into the power budget when using the largest model, due
+to exceeding the 5W requirements for peak power draw (Table 7).
+NVIDIA Jetson Nano using the largest model with batch size of 64
+also exceeds this peak power requirement. Since the peak power
+draw does not depend on neither the use case scenario nor the duty
+cycle, this conclusion about peak power draw holds for the rest of
+the scenarios as well.
+
+We are Going to the Space - Part 1:
+Which device to deploy in a satellite?
+Scaling factor
+0.25
+0.5
+1.0
+Device
+Power draw (mW)
+Power draw (mW)
+Power draw (mW)
+ARM Cortex-M7
+Scenario 1
+-
+-
+-
+Scenario 2
+-
+-
+-
+Scenario 3
+161
+-
+-
+CoralAI Dev Board Mini
+Scenario 1
+1433
+1546
+-
+Scenario 2
+89
+95
+120
+Scenario 3
+23
+25
+32
+Raspberry Pi + CoralAI TPU
+Scenario 1
+1376
+1571
+1954
+Scenario 2
+85
+97
+120
+Scenario 3
+23
+26
+32
+NVIDIA Jetson Nano
+Scenario 1
+-
+-
+-
+Scenario 2
+529
+639
+1109
+Scenario 3
+141
+170
+295
+NVIDIA Jetson Nano, batch size 64
+Scenario 1
+3021
+-
+-
+Scenario 2
+186
+298
+646
+Scenario 3
+49
+79
+172
+Table 6: Nominal power draw of devices with models of different sizes. The combinations of devices and model sizes, which do
+not fulfil the latency requirements, are omitted as their active duty cycle is greater than 100%. The power draw is highlighted
+in bold when exceeding the power budget.
+Scaling factor
+0.25
+0.5
+1.0
+Device
+Power
+draw (mW)
+Power
+draw (mW)
+Power
+draw (mW)
+Arm Cortex-M7
+389
+N/A
+N/A
+CoralAI Dev Board Mini
+2770
+2930
+4790
+Raspberry Pi + CoralAI
+TPU
+2595
+3405
+5050
+NVIDIA Jetson Nano
+2871
+3546
+4717
+NVIDIA
+Jetson
+Nano,
+batch size 64
+4523
+4684
+5064
+Table 7: Peak power draw of the devices during inference on
+full-size image. Batch size of 1 is used if not stated otherwise.
+The peak power is in bold when exceeding the power budget.
+5.1.2
+Scenario 2: Arctic region imaging. By relaxing the constraints
+and performing the workload equivalent to taking images of only
+the areas above the arctic polar circle, we see more configurations
+passing the requirements necessary to perform such a workload.
+Latency. Since there would be passive imaging periods perform-
+ing the workload for this scenario, the pipeline does not need to be
+lower than the latency of imaging. It can instead buffer the images
+and perform the inference at higher latency, given that the pipeline
+can finish inference of one burst before the next burst of images
+comes, which corresponds to a total latency of 5739 seconds or
+71.74 seconds per image. All the configurations except the ones
+with ARM Cortex-M7 achieve this latency (Table 5).
+Power draw. Because of the large margin between the required
+and achieved latencies for all configurations passing the latency
+requirements, the active duty cycle is relatively low. Therefore, the
+corresponding nominal power draw is well below the required one
+in all cases (Table 6).
+5.1.3
+Scenario 3: Greenland imaging. The least constrained sce-
+nario corresponds to taking images of only Greenland. The low
+constraints mean that all of the devices pass the requirements under
+certain model configurations.
+Latency. The maximum total latency for a burst of images is
+21600 seconds, corresponding to 270 seconds per single image. This
+requirement is easily passed by all of other configurations.
+Power draw. Due to the low active duty cycle, each configura-
+tion passes the nominal power draw requirement (Table 6). There is,
+however, a large gap between the nominal power draw of TPU and
+the rest of the devices. TPU performs the inference much more effi-
+ciently, and its power draw scales more favourably with increasing
+scaling factors in comparison to the other devices.
+Scaling factor
+Precision
+0.25
+0.5
+1.0
+32 bit float
+86.65%
+90.46%
+91.42%
+8 bit integer
+83.24%
+90.32%
+91.01%
+Table 8: Accuracy of the full-precision and quantized Mo-
+bilenetV1 as measured on the Flowers dataset.
+5.2
+Effect of quantization and scaling factor
+Table 8 shows the accuracy of the model with the various scaling
+factors before and after quantization to an 8-bit integer. Models
+with scaling factors 0.5 and 1.0 do not show a significant change in
+predictive performance. We can, however, see a 3% drop in accuracy
+
+Bayer, et al.
+Figure 1: Latency and power consumption at varying model
+scaling factors and batch sizes on the NVIDIA Jetson Nano.
+in the model with the smallest scaling factor. However, this differ-
+ence is acceptable in our case and is outweighed by the benefits of
+lower memory footprint and latency, and higher overall efficiency.
+5.3
+Batch size impact on NVIDIA Jetson Nano
+NVIDIA Jetson Nano is the only device in our evaluation that
+allows doing inference with a batch size higher than 1. Figure 1
+shows the impact of increasing the batch size on latency and power
+consumption of the inference. The most significant difference is
+between batch sizes 1 and 8. The impact of increasing batch size then
+tapers off. We can see that maximizing the batch size can lead to
+45.3-74.6% lower latency and 41.7-64.8% lower power consumption.
+6
+DISCUSSION
+TPU shows the most promising results as it is the only device to
+fulfill the requirements for real-time imaging, which is the most
+constrained scenario. This device is highly specialized for this pur-
+pose, utilizing the systolic array architecture. This architecture is
+purpose-built to perform fast multiply-accumulate operations in a
+highly parallel fashion, which allows performing matrix multiplica-
+tion without the need to load/store intermediate values.
+On the other end of the spectrum was the ARM Cortex-M7 micro-
+controller. Due to its lack of parallelism, this device could only fulfill
+the requirements of the least constrained scenario. Furthermore, the
+microcontroller has a very low amount of memory available even
+after heavy optimizations, such as quantization of the model and a
+stripped-down version of the TensorFlow Core. The low amount of
+memory means that the device cannot hold the full-size image in
+memory. Therefore, in-camera tiling is used, and the results of the
+operations had to be buffered to storage before the device could
+perform inference on the saved tiles.
+NVIDIA Jetson Nano can fulfill the requirements of the real-
+time imaging scenario. However, its nominal power draw exceeds
+the power budget and therefore buffering, similar to the case of
+microcontroller, is used to be able to work at latencies higher than
+those of the imaging. Even though the Jetson Nano could match the
+latency of the devices utilizing a TPU, when doing inference on large
+batches of image tiles using the smallest model, it only could do so
+at the cost of a higher power draw. Even with a batch size of 64, the
+Jetson Nano performs inference with power consumption 112-437%
+higher compared to the devices with TPU. The GPU architecture
+can perform massively parallel operations, but can only perform
+matrix multiplication by loading/storing intermediate results in
+shared memory, which creates inefficiency.
+7
+CONCLUSION
+In this paper, we characterized the performance of three devices
+that are possible candidates to be deployed on a satellite focusing
+on different image analysis scenarios on the satellite. Our results
+demonstrated that the low latency requirements combined with the
+limited budget for power draw, size, and mass necessitate highly
+specialized hardware architectures in this domain. The only device
+that fulfilled these requirements for all scenarios was CoralAI TPU.
+While NVIDIA Jetson Nano could match its performance thanks to
+its GPU, it could only do so at the cost of significantly higher power
+draw, which ultimately led to exceeding the power budget. ARM
+Cortex-M7, in contrast, could only fulfill the requirements of the
+least constrained scenario due to the low degree of parallelism and
+limited memory it has. Even though it provided the lowest peak
+power draw, the high latency of inference using this device led to
+nominal power draw higher than any of the other devices.
+REFERENCES
+[1] 2022. Edge TPU Compiler. https://coral.ai/docs/edgetpu/compiler/
+[2] 2022. TensorFlow Lite. https://www.tensorflow.org/lite/guide
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+2021. Monitoring of Stream Processing Engines Beyond the Cloud: An Overview.
+OJIOT 7, 1 (2021), 71–82.
+[5] Robert David, Jared Duke, Advait Jain, Vijay Janapa Reddi, Nat Jeffries, Jian Li,
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+Rocky Rhodes. 2021. TensorFlow Lite Micro: Embedded Machine Learning for
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+[6] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. 2009. Imagenet:
+A large-scale hierarchical image database. In CVPR. 248–255.
+[7] DISCO 2022. Danish student cubesat program. https://discosat.dk/
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+images/flower_photos.tgz
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+Stefan Fischer. 2022. A SPARQL Benchmark for Distributed Databases in IoT
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+
+SF=25
+SF=50
+SF=100
+25
+20
+Power consumption
+20
+15
+Latency (s)
+(mWh)
+15
+10
+X
+10
+X
+X
+5
+X
+5
+0
+0
+2
+ 4 8 16 32 64
+2
+4816 32 64
+Batch size
+Batch size

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+page_content='We are Going to the Space - Part 1:  Which device to deploy in a satellite?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Robert Bayer  IT University of Copenhagen  roba@itu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='dk  Julian Priest  IT University of Copenhagen  jucp@itu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='dk  Pınar Tözün  IT University of Copenhagen  pito@itu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='dk  ABSTRACT  The shrinkage in sizes of components that make up satellites led to  wider and low cost availability of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' As a result, there has  been an advent of smaller organizations having the ability to deploy  satellites with a variety of data-intensive applications to run on  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' One popular application is image analysis to detect, for exam-  ple, land, ice, clouds, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' However, the resource-constrained nature  of the devices deployed in satellites creates additional challenges  for this resource-intensive application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  In this paper, we investigate the performance of a variety of edge  devices for deep-learning-based image processing in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Our  goal is to determine the devices that satisfy the latency and power  constraints of satellites while achieving reasonably accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Our results demonstrate that hardware accelerators (TPUs, GPUs)  are necessary to reach the latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' On the other hand,  state-of-the-art edge devices with GPUs could have a high power  draw, making them unsuitable for deployment on a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  1  INTRODUCTION  In the last century, most innovation in real-world satellite appli-  cations were only available to the largest countries such as USA  and Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' These innovations led to significant reductions in the  size of the components that make up a satellite and in the cost of  the manufacturing and deployment process of a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This, in  turn, introduced a new CubeSat class of miniature satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Their  format is based on a 10cm cube, with the possibility of combining  multiple modules to create a larger satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This standardization  makes a batch deployment of satellites easier as the format affords  tight configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The reduction in costs that came as a result of  this new deployment method led to the advent of satellites owned  by small or private organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  However, the size of the CubeSat class satellites poses new com-  plex challenges such as the power and thermal constraints as well  as the physical dimensions of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' These CubeSats often  perform resource-intensive tasks, which clashes with the resource-  constrained nature of their format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Image processing and analysis is one class of possible satellite  workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Satellites with this type of workload take large-scale  images, which they have to store and send back to a ground station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  The link between the satellite and a ground station is of limited  bandwidth and short-lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The images must be highly compressed  to send all of them or filtered by quality and areas of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The  images already have a resolution of tens or hundreds of meters  per pixel, and lossy compression would lead to even more loss of  detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Filtering preserves the details of the images, but is more  resource-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  This paper is a step toward understanding the requirements  and limitations of deep-learning-based image filtering systems on  small satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' More specifically, we characterize the requirements  and constraints of an image processing unit (IPU) deployed on a  satellite and outline possible use case scenarios of such a system,  each introducing a different degree of resource constraints and  intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' We then further analyze the performance of multiple  edge devices on these different scenarios and their suitability for  possible deployment on one of the Danish Student CubeSat Project’s  (DISCO) [7] satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  These devices are based on different hardware architectures,  ranging from a microcontroller to a tensor processing unit (TPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  While microcontrollers or more complex CPUs have extensive flight  history (have already been deployed in satellites before) and low  power footprint, they were not built with running deep learning  workloads in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' On the other hand, GPUs have been utilized  more in the past decade in satellites, with one of the leading work-  loads being machine learning, based on the success of GPUs in  terrestrial use cases of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The latest of these archi-  tectures, TPU, has not been extensively researched for deployment  in satellites, even though TPUs were designed to perform fast neural  network inference, promising low latency with high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  The contributions of this study are as follows:  We illustrate the differences between the terrestrial and space  edge IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The edge devices deployed in space have to perform  very resource-intensive workloads with high reliability and do  so in a highly resource-constrained environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  We characterize a set of requirements specific to IPU on a  CubeSat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' We do this by outlining multiple use case scenarios  and showing how they affect the required latency and possible  power draw of such a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  We compare the performance of multiple edge devices based  on different architectures and show the need and the bene-  fit of using highly specialized hardware for machine learning  workloads on satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  2  BACKGROUND  We first introduce DISCO project and survey related work that  specifically targets data processing in satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1  DISCO  The Danish Student Cubesat Program (DISCO) [7] offers Danish  university students the opportunity to design and operate a small  satellite and to gain space flight experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' DISCO is a collaboration  between four universities in Denmark, which will initially launch  three student Cubesats into Low Earth Orbit with the first launch  scheduled for 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  DISCO2 has been designed by students to carry an Earth imag-  ing payload, infrastructure to capture images in space, into a solar  synchronous polar orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The instrument is a collaboration with  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='04954v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='LG]  12 Jan 2023  Bayer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Image process-  ing Unit  Nominal  Power  Peak Power  Design Margin  Nominal Cycle  Peak Cycle  Power  Mass budget  Dimensions  Constraints  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='00 W  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='00 W  5%  20%  20%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='48 W  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='15 Kg  10x70x80 mm  Table 1: List of constraints of the IPU on board of the DISCO2 satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  the Arctic Research Centre at Aarhus University and will support a  range of field research in Greenland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Initially focused on a long term  marine systems and cryosphere monitoring projects, the satellite  will supplement ground based observations with remote sensing  data, providing both good polar coverage and on demand availabil-  ity for the projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  The payload will include one or more high resolution cameras  as well as a dedicated IPU, which will be capable of simple machine  learning applications for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Students will be able  to carry out conventional on-satellite image processing in addi-  tion to running machine learning applications for classification,  discrimination, and feature identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='2  Related Work  The challenge of low bandwidth on small satellites creates the need  for data post-processing to filter the data on the satellite before  sending elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Neural-network-based filtering has been stud-  ied for this purpose in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Specifically, several works have  tested the suitability of small system-on-chip (SoC) devices leverag-  ing the power of GPUs to accelerate machine learning workloads  [3, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Moreover, some works have also explored the use of ASICs,  such as Intel’s Movidius Myriad vision processing units (VPUs)  [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This VPU has even seen real-life deployment [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Evaluation of TPUs for deployment on satellites is limited, how-  ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' To the best of our knowledge, only one work has explored this  option [10], with no real-life deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' DISCO project would  therefore be the first satellite to leverage TPU for onboard deep  learning applications, based on the findings of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  In this work, we aim to characterize the performance of the dif-  ferent flavors of off-the-shelf edge devices to check their suitability  for being deployed on a satellite in the context of DISCO project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Data management and processing for internet-of-things and edge  computing has been important in the data management community  as well [4, 14, 16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' We are complementing these works with a  very specific application focus, which is data-intensive applications  deployed on satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  3  REQUIREMENTS  The requirements and constraints used in this study are based on  the DISCO2 Arctic imaging mission use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The power and mass  constraints of the edge device to be deployed on the satellite are  based on the power and mass budgets developed in the DISCO2 en-  gineering design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The satellite design is a modular 3U Cubesat with  off-the-shelf modules for attitude control, power, communications,  and flight control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The power budget values are typical for a 3U  earth imaging Cubesat of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The resulting payload capacity  is 1U (10x10x10cm) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='3kg and this must include the cameras,  module enclosure, optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' and image processing unit (IPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The full  list of constraints of the IPU is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  The planned orbit of DISCO2 is a solar synchronous polar or-  bit at 550 km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The camera sensor is based on an Alvium  1800 C-2040 and the lens focal length is the maximum that could  be considered for the mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Assuming a 50% overlap between  images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=', capturing the same land area, this provides a minimum  time of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='42s between consecutive images (𝑇𝑖), which was derived  as follows:  𝑇𝑖 = 𝐺𝑆𝐷 ∗𝑊𝑖 ∗ ∩𝑖  𝑣  = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='8495𝑚/𝑝𝑥 ∗ 4512𝑝𝑥 ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5  7585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='16𝑚/𝑠  = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='42𝑠  (1)  , where 𝐺𝑆𝐷 (ground sample distance) is the spatial resolution of  the image, 𝑊𝑖 is the width of image in pixels, ∩𝑖 is the overlap  between images and 𝑣 is the orbital velocity of the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Taking this latency value as a baseline, we now describe three  image processing scenarios with different levels of difficulty based  on this Arctic mission’s goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Scenario 1: Real-time imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Images are taken with a pe-  riod of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='42s, as derived above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The inference has to be performed  and completed before the next image is captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This would allow  for the results of an inference to be used in decision making for the  next image capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Scenario 2: Arctic region imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' In this scenario, images  are taken only while the satellite is over the polar regions relaxing  the latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Images are buffered and inference is  performed in the remainder of the orbit, with the IPU available for  inference before the subsequent orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The capture-inference cycle  completes in 5739s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Scenario 3: Greenland imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The images are taken only  while over Greenland further relaxing the latency requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  As the orbit is solar synchronous with a period of one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This  results in subsequent bursts but only when the swathe passes over  Greenland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Images are again buffered and the capture-inference  cycle completes in 21600s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  4  METHODOLOGY AND SETUP  Our goal is to characterize the performance of modern low-power  hardware for deployment as the IPU of the DISCO2 satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This  section presents our experimental methodology and setup to achieve  this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1  Devices under Test  As representative hardware for this study, we picked three devices,  each based on a different hardware architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' These choices were  made based on their physical dimensions, weight, power draw, and  high performance considering the low power draw limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  ARM Cortex-M Mirocontroller, based on the STM32H745  chip, has the potential of the lowest power draw, while the least  performant of the choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' As an additional advantage, this chip  has extensive flight history and, therefore, would be a safe choice  for deployment in space applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  We are Going to the Space - Part 1:  Which device to deploy in a satellite?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  In order to use this device as an on-board IPU, we can use Tensor-  Flow Lite for Microcontrollers [5], a framework designed to allow  neural network inference with only 16 KB of memory overhead  (core runtime) and without the need for an operating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' In  addition, the framework also provides support for the use of spe-  cialized neural network kernels, such as CMSIS-NN [12] kernels  designed specifically for Cortex-M processor cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The X-CUBE-  AI [17] framework makes the deployment of the neural networks  with TensorFlow Lite for microcontrollers even more accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' It  provides an intuitive GUI for the libraries or code samples based  on the provided and trained TensorFlow Lite [2] model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Even though this device has support for floating point operations,  a quantized model was used, in order to achieve better latency  and, more importantly, lower memory footprint, which is a large  constraint of this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  To simulate the exact performance and power characteristics of  a flight computer already available for a potential deployment in  DISCO project, OBC-P3, we used only the Cortex-M7 core of the  STM32H745 and scaled its clock from 480MHz down to 300MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Parameters of the OBC-P3 system can be found in the Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Processor  2x ARM Cortex-M7 @ 300 MHz  SRAM  384 KB SRAM  FRAM  32 KB  Flash  2 MB  Storage  64 GB eMMC  Dimensions  94 x 94 x 13 mm  Weight  120 g  Table 2: Specifications of the OBC-P3 system containing  the ARM Cortex-M7 MCUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The dimensions and weight in-  cludes the enclosure with aluminium shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  NVIDIA Jetson Nano is a portable SoC composed of power-  efficient ARM CPU and NVIDIA GPU designed for embedded sys-  tems that require GPU-friendly computations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=', image process-  ing, video encoding/decoding, and machine learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' It is  the only device in our experiments that supports TensorFlow Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Therefore, the deployment process to this device is equivalent to  that of servers or desktops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' It is also the only evaluated device  supporting batch sizes larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  The device can operate at a wattage between 5W - 10W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' We  configured it to operate at 5W by disabling two of the four CPU  cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' While this setting leads to lower performance, it is necessary  in order to fit into the power budget outlined in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Parameters  of this device can be found in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  CPU  Quad-core ARM A57 @ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='43 GHz  GPU  128-core Maxwell  GFLOPS  472  RAM  4 GB 64-bit LPDDR4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='6 GB/s  Storage  64 GB SD card  Dimensions  100 x 80 x 29 mm  Weight  141 g  Table 3: Specifications of the NVIDIA Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The di-  mensions and weight are based on the developer kit version  of this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  CoralAI Dev  Board Mini  Raspberry Pi 2B  CPU  Quad-core Arm  Cortex-A35 @ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5 GHz  Quad-core ARM  Cortex-A7 @ 900 MHz  RAM  2 GB  1 GB  Storage  8 GB eMMC  32 GB SD card  Dimensions  64 x 48 x 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='6 mm  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='6 x 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5 x 17 mm  Weight  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5 g  45 g  CoralAI TPU  TOPS  Interface  Dimensions  Weight  4  USB2  65 x 30 mm  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='3 g  Table 4: Specifications of the CoralAI Dev Board Mini and  Raspberry Pi 2B, as well as the CoralAI TPU chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Note  that the CoralAI Dev Board Mini’s physical dimensions and  weight include the on-board TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  CoralAI TPU is an ASIC, AI accelerator developed by Google  to accelerate machine learning workloads at the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' TPU is not  a standalone device and must be deployed together with a Linux,  Windows, or macOS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Even though this device is highly  specialized, model deployment is performed by compiling a Ten-  sorFlow Lite model containing a subset of supported layers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  This model must be quantized to an 8-bit integer, as this is only  supported data type by this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This compiled model can then  be used with the TPU’s Python or C++ library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' In our experiments,  we rely on the C++ library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  We evaluated this device in two formats:  (1) CoralAI Dev Board Mini, which couples this chip and an SoC  on a single board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  (2) CoralAI USB accelerator connected to a Raspberry Pi 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Even though (1) has the TPU chip on the same board as the SoC,  the two communicate through the USB2 bus, as in the case of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  In addition, both of the devices run Linux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The parameters of the  different devices can be found in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='2  Metrics  The metrics to evaluate the suitability of the devices as the on-  satellite IPUs are based on the requirements Section 3 outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Latency is reported in seconds per inference of a sample, where  a sample is a whole image of size 4512 x 4512 pixels or 400 tiles of  size 224 x 224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' We measure the latency of the neural network  inference once the data is already in the device’s memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Nominal power draw is the average power draw in mW mul-  tiplied by the duty cycle, which is the ratio between achieved and  required latency for each scenario (outlined in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' We use  an external appliance for CoralAI TPU and the ARM Cortex-M7  and tegrastats for Jetson Nano to measure this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Peak power draw is the maximum power draw over the pe-  riod of inference in mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' It is measured using the same tools as in  nominal power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Power consumption is reported per inference of a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This  metric shows the power efficiency of the device and will guide  the choice of a more suitable device if multiple devices fulfill the  requirements of a particular scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Bayer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Scaling factor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='0  Device  Latency (s)  Power consump-  tion (mWh)  Latency (s)  Power consump-  tion (mWh)  Latency (s)  Power consump-  tion (mWh)  ARM Cortex-M7  118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='80  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='10  N/A  N/A  N/A  N/A  CoralAI Dev Board Mini  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='778  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='76  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='991  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='90  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='533  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='40  Raspberry Pi + CoralAI TPU  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='106  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='69  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='331  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='93  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='825  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='40  NVIDIA Jetson Nano  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='900  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='54  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='109  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='74  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='581  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='10  NVIDIA Jetson Nano, batch size 64  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='529  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='71  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='382  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='94  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='171  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='89  Table 5: Latency and power consumption results with different scaling factors of the model as measured on corresponding  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Batch size of 1 is used if not stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Accuracy of the deployed model is also measured to show the  effect quantization has on the model’s predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='3  Workload  To simulate the imaging scenarios of interest (Section 3), we use an  image classification workload consisting of a 5-class classification  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' For this workload, an off-the-shelf MobileNetV1 model  [11] pre-trained on the ImageNet dataset [6] was chosen for its  strong predictive power and availability in multiple scaling factors,  affecting the number of hidden layers in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The availability  of multiple scaling factors is an important factor as the memory of  some of the devices is highly limited and can therefore fit only the  smallest of the variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  We further fine-tuned the model before deployment on the de-  vices using the Flowers dataset [15], containing 3670 color images  of size 224 x 224 pixels belonging to 5 different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' While the  weights in the model do not affect the inference latency or power  required to perform the inference, the model was fine-tuned using  this dataset because it mimics one of the possible use cases closely  (in number of classes as well as size of the images) and allows us to  quantify the effects of the size and precision of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Since the models running on the Cortex-M7 and CoralAI TPU  were quantized, rescaling of images was not needed before infer-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' For Jetson Nano, the pixel values must be rescaled to the range  of [0, 1), or, in other words, divided by 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' An additional layer was  prepended to the model for this process in order to take advantage  of the GPU parallelism on Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  As the size of the images would stress the memory available on  the evaluated devices, we employed a tiling method for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Each image was divided into 400 patches of size 224 x 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1 This  method results in a positive side effect, where each patch can be  filtered separately acting as a coarse-grained image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  This way only the tiles of interest are sent back to a ground station  saving us bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  The results are reported as an average of 10 inferences on the full  4512 x 4512 pixel image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The inference on Cortex-M7 and CoralAI  TPU were performed with batch size of 1 and on Jetson Nano with  the batch size of 2𝑥, where x is from range of 0-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' All of the devices  were tested with the scaling factors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='0 of the model,  with the exception of the ARM Cortex-M7 microcontroller, which  could not fit the larger models in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  1The image is not evenly divisible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' therefore, the borders are disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  5  RESULTS  The results are split into three parts: (1) analysis of latency and  power draw results with respect to the three scenarios outlined in  Section 3, (2) impact of quantization on the accuracy of models with  various scaling factors, (3) impact of batch size on the performance  of NVIDIA Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1  Scenarios  Table 5 shows the latency achieved and power consumption of the  devices using the various scaling factors of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Furthermore,  Tables 6 and 7 show the nominal and peak power draw of different  configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='3 discuss the  suitability of each device for the different scenarios based on the  results on these tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The reported latency, power consumption,  and the peak power draw are independent of the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Only  the nominal power draw depends on the use case scenarios, due to  the duty cycle (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1  Scenario 1: Real-time imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Real-time imaging is the most  constrained scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Therefore, a high degree of specialization is  necessary to fulfill the requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The pipeline has to achieve a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='42s latency, which is  the spacing between images with 50% overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' There are no passive  periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Therefore the latency of the pipeline has to be lower than  that of the imaging for the pipeline not to get backed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' There are  multiple configurations that achieved the required latency (Table  5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' These are all the configurations utilizing a TPU and the smallest  model running on Jetson Nano with batch size of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Out of the configurations that satisfy the latency  requirements for this scenario, only of the TPU configurations fit  into the nominal power budget of the satellite (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Even though  the Jetson Nano with batch size 64 achieves a latency comparable  to the TPUs with a scaling factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='25, the power consumption  per inference is more than twice as high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Furthermore, the Raspberry Pi with external TPU module does  not fit into the power budget when using the largest model, due  to exceeding the 5W requirements for peak power draw (Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  NVIDIA Jetson Nano using the largest model with batch size of 64  also exceeds this peak power requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Since the peak power  draw does not depend on neither the use case scenario nor the duty  cycle, this conclusion about peak power draw holds for the rest of  the scenarios as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  We are Going to the Space - Part 1:  Which device to deploy in a satellite?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Scaling factor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='0  Device  Power draw (mW)  Power draw (mW)  Power draw (mW)  ARM Cortex-M7  Scenario 1 Scenario 2 Scenario 3  161 CoralAI Dev Board Mini  Scenario 1  1433  1546 Scenario 2  89  95  120  Scenario 3  23  25  32  Raspberry Pi + CoralAI TPU  Scenario 1  1376  1571  1954  Scenario 2  85  97  120  Scenario 3  23  26  32  NVIDIA Jetson Nano  Scenario 1 Scenario 2  529  639  1109  Scenario 3  141  170  295  NVIDIA Jetson Nano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' batch size 64  Scenario 1  3021 Scenario 2  186  298  646  Scenario 3  49  79  172  Table 6: Nominal power draw of devices with models of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The combinations of devices and model sizes, which do  not fulfil the latency requirements, are omitted as their active duty cycle is greater than 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The power draw is highlighted  in bold when exceeding the power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Scaling factor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='0  Device  Power  draw (mW)  Power  draw (mW)  Power  draw (mW)  Arm Cortex-M7  389  N/A  N/A  CoralAI Dev Board Mini  2770  2930  4790  Raspberry Pi + CoralAI  TPU  2595  3405  5050  NVIDIA Jetson Nano  2871  3546  4717  NVIDIA  Jetson  Nano,  batch size 64  4523  4684  5064  Table 7: Peak power draw of the devices during inference on  full-size image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Batch size of 1 is used if not stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  The peak power is in bold when exceeding the power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='2  Scenario 2: Arctic region imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' By relaxing the constraints  and performing the workload equivalent to taking images of only  the areas above the arctic polar circle, we see more configurations  passing the requirements necessary to perform such a workload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Since there would be passive imaging periods perform-  ing the workload for this scenario, the pipeline does not need to be  lower than the latency of imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' It can instead buffer the images  and perform the inference at higher latency, given that the pipeline  can finish inference of one burst before the next burst of images  comes, which corresponds to a total latency of 5739 seconds or  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='74 seconds per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' All the configurations except the ones  with ARM Cortex-M7 achieve this latency (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Because of the large margin between the required  and achieved latencies for all configurations passing the latency  requirements, the active duty cycle is relatively low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Therefore, the  corresponding nominal power draw is well below the required one  in all cases (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='3  Scenario 3: Greenland imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The least constrained sce-  nario corresponds to taking images of only Greenland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The low  constraints mean that all of the devices pass the requirements under  certain model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The maximum total latency for a burst of images is  21600 seconds, corresponding to 270 seconds per single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This  requirement is easily passed by all of other configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Due to the low active duty cycle, each configura-  tion passes the nominal power draw requirement (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' There is,  however, a large gap between the nominal power draw of TPU and  the rest of the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' TPU performs the inference much more effi-  ciently, and its power draw scales more favourably with increasing  scaling factors in comparison to the other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Scaling factor  Precision  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='0  32 bit float  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='65%  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='46%  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='42%  8 bit integer  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='24%  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='32%  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='01%  Table 8: Accuracy of the full-precision and quantized Mo-  bilenetV1 as measured on the Flowers dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='2  Effect of quantization and scaling factor  Table 8 shows the accuracy of the model with the various scaling  factors before and after quantization to an 8-bit integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Models  with scaling factors 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='0 do not show a significant change in  predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' We can, however, see a 3% drop in accuracy  Bayer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  Figure 1: Latency and power consumption at varying model  scaling factors and batch sizes on the NVIDIA Jetson Nano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  in the model with the smallest scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' However, this differ-  ence is acceptable in our case and is outweighed by the benefits of  lower memory footprint and latency, and higher overall efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='3  Batch size impact on NVIDIA Jetson Nano  NVIDIA Jetson Nano is the only device in our evaluation that  allows doing inference with a batch size higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Figure 1  shows the impact of increasing the batch size on latency and power  consumption of the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The most significant difference is  between batch sizes 1 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The impact of increasing batch size then  tapers off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' We can see that maximizing the batch size can lead to  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='3-74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='6% lower latency and 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='7-64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='8% lower power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  6  DISCUSSION  TPU shows the most promising results as it is the only device to  fulfill the requirements for real-time imaging, which is the most  constrained scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This device is highly specialized for this pur-  pose, utilizing the systolic array architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' This architecture is  purpose-built to perform fast multiply-accumulate operations in a  highly parallel fashion, which allows performing matrix multiplica-  tion without the need to load/store intermediate values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  On the other end of the spectrum was the ARM Cortex-M7 micro-  controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Due to its lack of parallelism, this device could only fulfill  the requirements of the least constrained scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Furthermore, the  microcontroller has a very low amount of memory available even  after heavy optimizations, such as quantization of the model and a  stripped-down version of the TensorFlow Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The low amount of  memory means that the device cannot hold the full-size image in  memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Therefore, in-camera tiling is used, and the results of the  operations had to be buffered to storage before the device could  perform inference on the saved tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  NVIDIA Jetson Nano can fulfill the requirements of the real-  time imaging scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' However, its nominal power draw exceeds  the power budget and therefore buffering, similar to the case of  microcontroller, is used to be able to work at latencies higher than  those of the imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Even though the Jetson Nano could match the  latency of the devices utilizing a TPU, when doing inference on large  batches of image tiles using the smallest model, it only could do so  at the cost of a higher power draw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Even with a batch size of 64, the  Jetson Nano performs inference with power consumption 112-437%  higher compared to the devices with TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The GPU architecture  can perform massively parallel operations, but can only perform  matrix multiplication by loading/storing intermediate results in  shared memory, which creates inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  7  CONCLUSION  In this paper, we characterized the performance of three devices  that are possible candidates to be deployed on a satellite focusing  on different image analysis scenarios on the satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Our results  demonstrated that the low latency requirements combined with the  limited budget for power draw, size, and mass necessitate highly  specialized hardware architectures in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' The only device  that fulfilled these requirements for all scenarios was CoralAI TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  While NVIDIA Jetson Nano could match its performance thanks to  its GPU, it could only do so at the cost of significantly higher power  draw, which ultimately led to exceeding the power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' ARM  Cortex-M7, in contrast, could only fulfill the requirements of the  least constrained scenario due to the low degree of parallelism and  limited memory it has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' Even though it provided the lowest peak  power draw, the high latency of inference using this device led to  nominal power draw higher than any of the other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
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+page_content=' AI expansion pack for STM32CubeMX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
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+page_content=' 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}
+page_content='  SF=25  SF=50  SF=100  25  20  Power consumption  20  15  Latency (s)  (mWh)  15  10  X  10  X  X  5  X  5  0  0  2  4 8 16 32 64  2  4816 32 64  Batch size  Batch size' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9E4T4oBgHgl3EQfNQwq/content/2301.04954v1.pdf'}

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+Investigating the Combination of Planning-Based and
+Data-Driven Methods for Goal Recognition
+Nils Wilken
+[email protected]
+Institute for Enterprise Systems, University of Mannheim
+69118 Mannheim, Germany
+Lea Cohausz
+[email protected]
+Data and Web Science Group, University of Mannheim
+69118 Mannheim, Germany
+Johannes Schaum
+[email protected]
+Institute for Enterprise Systems, University of Mannheim
+69118 Mannheim, Germany
+Stefan L¨udtke
+stefan.l¨[email protected]
+Institute for Enterprise Systems, University of Mannheim
+69118 Mannheim, Germany
+Heiner Stuckenschmidt
+[email protected]
+Data and Web Science Group, University of Mannheim
+69118 Mannheim, Germany
+Abstract
+An important feature of pervasive, intelligent assistance systems is the ability to dy-
+namically adapt to the current needs of their users. Hence, it is critical for such systems
+to be able to recognize those goals and needs based on observations of the user’s actions
+and state of the environment.
+In this work, we investigate the application of two state-of-the-art, planning-based plan
+recognition approaches in a real-world setting. So far, these approaches were only evaluated
+in artificial settings in combination with agents that act perfectly rational. We show that
+such approaches have difficulties when used to recognize the goals of human subjects,
+because human behaviour is typically not perfectly rational. To overcome this issue, we
+propose an extension to the existing approaches through a classification-based method
+trained on observed behaviour data. We empirically show that the proposed extension
+not only outperforms the purely planning-based- and purely data-driven goal recognition
+methods but is also able to recognize the correct goal more reliably, especially when only
+a small number of observations were seen. This substantially improves the usefulness of
+hybrid goal recognition approaches for intelligent assistance systems, as recognizing a goal
+early opens much more possibilities for supportive reactions of the system.
+1. Introduction
+The ultimate goal of smart assistance technologies is to dynamically adapt the infrastructure
+of a building to best meet the needs of their users by observing their behaviour and deducing
+their current needs.
+Identifying users’ goals and intentions based on their current and
+past activities is an important task in this context.
+While there is some work on goal
+recognition in the context of smart assistance systems (Yordanova, Whitehouse, Paiement,
+1
+arXiv:2301.05608v1  [cs.AI]  13 Jan 2023
+
+Mirmehdi, Kirste, & Craddock, 2017; Yordanova, L¨udtke, Whitehouse, Kr¨uger, Paiement,
+Mirmehdi, Craddock, & Kirste, 2019; Kr¨uger, Nyolt, Yordanova, Hein, & Kirste, 2014), so
+far research has mostly focused on recognizing users’ current activities (Helaoui, Riboni,
+& Stuckenschmidt, 2013; Rashidi, Cook, Holder, & Schmitter-Edgecombe, 2010; Hoque &
+Stankovic, 2012; Yao, Nie, Sheng, Gu, Li, & Wang, 2016; Sztyler & Stuckenschmidt, 2017).
+In this paper, we address the problem of identifying user goals as a basis for automatic
+support. For this purpose, we look at the related problem of plan recognition, which is a
+long-standing topic in the Artificial Intelligence community (Kautz & Allen, 1986; Charniak
+& Goldman, 1993). We believe that plan recognition methods are particularly suited for
+this task as they do not only identify the goal a user intends to achieve, but also aim to
+recognize the most probable plan (i.e., ordered sequence of actions) for achieving this goal.
+Knowing such a plan provides us with a better basis for supporting the user.
+In this paper, we investigate the application of two state-of-the-art plan recognition ap-
+proaches that are based on the principle of Plan Recognition As Planning (PRAP) (Ram´ırez
+& Geffner, 2010). More explicitly, the contributions of this paper are:
+• In Section 4, we reveal and analyze some major shortcomings of PRAP approaches
+when applied to real-world scenarios. The main consequence of these shortcomings is
+that some goals can only be identified relatively closely before they are reached, which
+significantly reduces their potential benefits to an intelligent assistance system.
+• As a possible solution to this problem, we propose a hybrid plan recognition method
+in Section 5. The proposed method combines the principle of PRAP with a data-
+driven probabilistic model that captures statistical relations between certain states of
+the environment and goals that can be learned from past observations.
+• Finally, we empirically evaluate the proposed hybrid method in sections 6 and 7 and
+compare its performance to the performances of purely planning-based and purely
+data-driven approaches. The evaluation shows that both approaches can be applied
+to identify the goals of a user in real-world scenarios. Further, the results show that
+using a hybrid goal recognition method leads to a much earlier identification of the
+correct goal while only requiring very small amounts of training data.
+2. Problem Definition
+Probabilistic goal recognition is the problem of inferring a probability distribution over a set
+of intended goals of an observed agent, given a possibly incomplete sequence of observed
+actions and a domain model that describes the domain in which the observed agent acts.
+More formally, the aim of goal recognition approaches is to find a posterior probability
+distribution P(G|O) over all goals g ∈ G, given a sequence of observed actions o.
+This work considers the smart home domain as an example environment for goal recog-
+nition. Figure 1 shows the layout of a smart flat and partial action sequences that sketch a
+simple use case. This use case will be employed to analyze the shortcomings of the investi-
+gated planning-based goal recognition approaches. It is important to note that this use case
+is completely synthetic and does not correspond to a real-world experimental setup. The
+flat has four rooms and a hallway that connects all rooms. In each room, different devices
+2
+
+h1
+h2
+h3
+h4
+Be1
+Be2
+be3
+be4
+ba1
+ba2
+ba3
+ba4
+k1
+k2
+k3
+k4
+l1
+l2
+l3
+l4
+Livingroom
+Bedroom
+Bathroom
+Kitchen
+Shower
+Toilet
+Basin
+Fridge
+Oven
+Bed
+Closet
+A
+Washing
+Machine
+Laundry
+Basket
+T
+T
+T
+T
+A
+A
+A
+A
+A
+Window
+Window
+Window
+Window
+Window
+20°C
+good
+18°C
+very good
+22°C
+good
+19°C
+bad
+Outdoor
+TA
+5°C
+very good
+Sensing functionality
+Acting functionality
+Sensing/Acting
+functionality
+Agent
+Heater
+Heater
+Heater
+Heater
+Possible Goals:
+•
+A (Prepare Meal)
+•
+B (Watch TV)
+•
+C (Use toilet)
+•
+D (Use shower)
+(a)
+h1
+h2
+h3
+h4
+Be1
+Be2
+be3
+be4
+ba1
+ba2
+ba3
+ba4
+k1
+k2
+k3
+k4
+l1
+l2
+l3
+l4
+Livingroom
+Bedroom
+Bathroom
+Kitchen
+Shower
+Toilet
+Basin
+Fridge
+Oven
+Bed
+Closet
+A
+Washing
+Machine
+Laundry
+Basket
+T
+T
+T
+T
+A
+A
+A
+A
+A
+Window
+Window
+Window
+Window
+Window
+20°C
+good
+18°C
+very good
+22°C
+good
+19°C
+bad
+Outdoor
+TA
+5°C
+very good
+Sensing functionality
+Acting functionality
+Sensing/Acting
+functionality
+Agent
+Heater
+Heater
+Heater
+Heater
+Possible Goals:
+•
+A (Prepare Meal)
+•
+B (Watch TV)
+•
+C (Use toilet)
+•
+D (Use shower)
+(b)
+Figure 1: Illustration of an exemplary smart flat and a simple example use case (i.e., “Beer
+Use Case”).
+and furnishings are located. Some of these objects can possibly function as sensor, actuator,
+or a mixture of both, which is indicated by the green and orange dots. Furthermore, it is
+assumed that the current location of the agent can be sensed at all times and that the agent
+can navigate the cells in the flat by moving in all possible directions, including diagonal
+moves.
+Example 1 (Beer Use Case) Figures 1a and 1b roughly sketch parts of a small use case
+in this smart flat, which we will refer to as “Beer Use Case” (BUC) from here on. In
+the BUC, a single agent is initially located in the cell “l3” in the livingroom, moves to the
+fridge, takes out a beer, and moves back to the couch in the livingroom (Fig. 1a). When
+the agent arrives at the couch in the livingroom, she sits down on the couch, opens the beer,
+and drinks it while watching TV. After a while, the agent decides to get another beer from
+the fridge. When the second beer is empty, the agent gets up from the couch, moves back to
+the kitchen, and subsequently via the hallway to the bathroom to use the toilet (Fig. 1b).
+3. Background
+In this work, we investigate the application of two state-of-the-art approaches to plan recog-
+nition to a real-world goal recognition scenario. In contrast to probabilistic goal recognition,
+probabilistic plan recognition not only describes the problem of inferring a probability dis-
+tribution over a set of goals, but also the probability distribution over a set of possible plans
+that an agent might follow to reach it’s intended goal. From a solution to a plan recognition
+problem, the solution of the corresponding goal recognition problem can be derived by only
+considering the goals of the recognized plans. Plan recognition is a long standing research
+area in the Artificial Intelligence community. Recent plan recognition systems mostly rely
+on the Plan Recognition As Planning (PRAP) (Ram´ırez & Geffner, 2009) principle and
+hence, utilize symbolic planning systems to solve plan- and goal recognition problems.
+3.1 Symbolic Planning
+Symbolic planning is based on a symbolic model of the planning domain that defines possible
+actions, their preconditions and effects on the environment. Given a current state and goals
+3
+
+in terms of partial state descriptions, planning methods aim to construct an optimal plan for
+reaching the goals from the current state consisting of a (possibly partial) order of actions to
+be executed. We adopt the formalization of a planning problem from (Ram´ırez & Geffner,
+2010).
+Definition 1 (Planning Problem) A Planning Problem is a tuple P = ⟨F, s0, A, G⟩ where
+F is a set of fluents (boolean statements about properties of the modeled environment),
+s0 ⊆ F and G ⊆ F are the initial state and the goal description and A is a set of actions
+with preconditions Pre(a) ⊆ F and lists of fluents Add(a) ⊆ F and Del(a) ⊆ F that de-
+scribe the effects of an action a in terms of fluents that are added and deleted from the
+current state. Actions have a non-negative cost c(a). A state is described by the subset of
+fluents which are currently considered to be true. A goal state is a state s with s ⊇ G. An
+action a is applicable in a state s if and only if Pre(a) ⊆ s. Applying an action a in a state
+s leads to a new state s′ = (s ∪ Add(a) \ Del(a)). A solution for a planning problem (i.e., a
+plan) is a sequence of applicable actions π = a1, · · · an that transforms the initial state into
+a goal state. The cost of the plan is defined as c(π) = �
+i
+c(ai). A plan is optimal if the cost
+of the plan is minimal.
+This basic model has been extended in different directions. In this paper, we make use
+of two extensions. One allows us to specify goals of form ¬f that claim that a certain
+fluent f is absent in the goal state. The other enables the use of a conditional effect of
+form p → q, where p and q are single fluents. This means that when an action x has such
+a conditional effect, fluent q only becomes true after the execution of x when p was true
+before the execution (Ram´ırez & Geffner, 2010).
+3.2 Plan Recognition As Planning: State-of-the-Art
+As already mentioned, many recent plan- and goal recognition approaches rely onto the
+principle of Plan Recognition as Planning (PRAP), which was first introduced by Ram´ırez
+and Geffner (Ram´ırez & Geffner, 2009). All approaches that follow this principle have in
+common that they utilize concepts from the area of classical planning to compute probability
+distributions over a set of possible plans or goals, respectively.
+Definition 2 (Probabilistic Plan Recognition Problem) A probabilistic plan recog-
+nition problem is a tuple T = ⟨D, G, O, P(G)⟩ where D = ⟨F, s0, A, ∅⟩ is a planning domain,
+G is a set of possible goals g ⊆ F, o = o1, · · · om, where oi ∈ A is a sequence of actions
+that have been observed and P(G) is the prior probability distribution over G. A solution
+to the probabilistic plan recognition problem is the conditional probability of the goals given
+the observation sequence o (i.e., P(G = g|O = o)∀g ∈ G).
+Estimating Goal Probabilities
+Both plan recognition methods that are used in this
+work are based on the idea of using Bayes Rule to compute the posterior probabilities of
+the goals:
+P(G|O) = αP(O|G)P(G)
+(1)
+It is assumed that the prior probabilities P(G = g) of goals g ∈ G are given in the problem
+definition. Hence, the problem of probabilistic goal recognition boils down to the estimation
+4
+
+of P(O|G). Both investigated approaches utilize symbolic planning systems to estimate this
+probability.
+The idea behind this is based on the assumption that agents act perfectly rational and
+hence, use strictly optimal plans (i.e, plans that minimize costs) to achieve their goals.
+Furthermore, it is assumed that the probability of a goal to be the agent’s actual goal can
+be estimated by relating the costs of an optimal plan that includes a given observation
+sequence o and an optimal plan that does not include o, while reaching a given goal g ∈ G.
+This can be done because an optimal plan that does not have to fulfill the requirement of
+including o is, according to the planning domain, a perfectly rational plan from the given
+initial state to a given goal g. Hence, when the costs of an optimal plan that includes o
+are higher, this means that the agent is taking a detour compared to a perfectly rational
+plan. More precisely, Ram´ırez and Geffner (Ram´ırez & Geffner, 2010) propose to calculate
+P(o|g) as follows:
+P(o|g) = α′
+exp{−β∆(g)}
+1 + exp{−β∆(g)}
+(2)
+Where α′ is a normalization factor and ∆(g) = c(o, g)−c(o, g) is the cost difference between
+an optimal plan for g that satisfies o and an optimal plan for g that does not satisfy o. The
+costs c(o, g) and c(o, g) can be computed out of the box using classical planning systems.
+Translating a Plan Recognition Problem into Planning Problems
+The two state-
+of-the-art plan recognition methods used in this paper were proposed by Ram´ırez and
+Geffner (Ram´ırez & Geffner, 2010) (referred to as “RG” from here on) and Vered et al.
+(Vered, Kaminka, & Biham, 2016) (referred to as “GM” (Goal Mirroring) from here on).
+They mainly differ in the way they transform the original planning problem, which is
+necessary to ensure that the computed plans fulfill some necessary requirements.
+To compute the probabilities P(O|G), the RG approach compiles a plan recognition
+problem T = ⟨P, G, o, Prob⟩ into 2|G| planning problems. For each goal g ∈ G the two
+planning problems Po(g) and Po(g) have to be compiled and solved. Classical planning
+systems naturally cannot handle the requirement of satisfying a given sequence of observed
+actions in a computed plan. To ensure that the computed solutions fulfill this requirement,
+the original planning domain D has to be slightly modified.
+Definition 3 (Transformation of the Planning Domain (RG)) For a given planning
+domain D = ⟨F, s0, A⟩ and a given observation sequence o, the transformed domain is de-
+fined as D′ = ⟨F ′, I, A′⟩ with F ′ = F ∪ {poi|oi ∈ (oi)n
+0}, where poi is a new fluent and the
+actions o ∈ A′ that are in o have an additional effect poi when i = 0 and poi−1 → poi has to
+hold otherwise.
+For this transformation it is assumed that no action appears twice in o. When this is
+the case, the corresponding actions are duplicated and renamed to ensure that the order
+of observed actions is unmodified in the resulting plans. Now the costs c(o, g) and c(o, g)
+can be calculated by solving the planning problems Po(g) = ⟨F ′, s′
+0, A′, g ∪ {pon}⟩ and
+Po(g) = ⟨F ′, s′
+0, A′, g ∪ {¬pon}⟩.
+Goal Mirroring
+The main difference between RG and GM is the domain translation
+procedure: While RG adapts the actions in a given planning domain, GM uses a different
+initial state to generate plans that embed o each time a new observation is observed.
+5
+
+Definition 4 (Transformation of Planning Problem (GM)) For a given planning do-
+main D = ⟨F, s0, A⟩ and a given observation sequence o, the transformed domain is defined
+as D′ = ⟨F ′, s′
+0, A′⟩ with F ′ = F, where s′
+0 = s0[[o]] and s[[o]] returns as a result the
+planning state that is obtained when the action sequence o is applied to a planning state s.
+When this transformation is completed, analogously to the RG approach, GM calculates
+the costs c(o, g) and c(o, g). However, in contrast to RG, GM assumes for the calculation
+that an optimal plan from s0 to a goal g can be obtained by concatenating o with a plan
+for g that starts at the adjusted initial state s′
+0. From such a plan, again the costs c(o, g)
+can be determined. Furthermore, GM does not generate plans that strictly do not embed o,
+but instead computes an optimal plan from s0 to each goal and uses the costs of these plans
+analogously to the costs of plans that do not embed o as RG does (i.e., c(o, g)). Apart from
+this, GM uses the same heuristic as RG (i.e., Equation 1) to compute goal probabilities
+P(G|O) from these costs.
+One major benefit of GM compared to RG is that it is expected to be much more
+time efficient in the case of online probabilistic goal recognition. This becomes increasingly
+important with increasing complexity of the involved planning problems.
+Definition 5 (Online Probabilistic Goal Recognition) We define online probabilistic
+goal recognition as a special variant of the probabilistic goal recognition problem defined
+earlier. In online goal recognition, we assume that the observation sequence o is revealed
+incrementally. More explicitly, we introduce the notion of time t ∈ {0, . . . , T}, where T =
+|o|. For every value of t, one probabilistic goal recognition problem R(t) can be induced
+as R(t) = ⟨D, G, ot, Prob⟩ where D = ⟨F, s0, A, ∅⟩ and ot = {oi|0 ≤ i ≤ t, oi ∈ o}. A
+solution to the online probabilistic goal recognition problem are the conditional probabilities
+Pt(G = g|ot); ∀g ∈ G, t ∈ [0, T].
+Hence, in the case of online probabilistic goal recognition, GM solves, due to the different
+transformation procedure, only |G||O| + |G| planning problems instead of 2|G||O| planning
+problems that RG solves.
+4. Case Study: Goal Recognition in the Beer Use Case
+In this section we evaluate the performance of the RG and GM goal recognition approaches
+when applied to the synthetic BUC example (see Section 2). Furthermore, we demonstrate
+and discuss some major limitations of them.
+4.1 Experimental Setup
+For the experiments, we modeled a planning domain DBUC in the Planning Domain Def-
+inition Language (PDDL) (McDermott, Ghallab, Howe, Knoblock, Ram, Veloso, Weld, &
+Wilkins, 1998). The goal set of the corresponding plan recognition problems (see Definition
+2) is defined as GBUC = {gprepare meal, gwatch TV , guse shower, guse toilet}. Further, following
+the approach of Ram´ırez and Geffner (Ram´ırez & Geffner, 2010), we assume uniform prior
+probabilities PBUC(G) for all goals in GBUC.
+Based on this experimental setup, we conducted two experiments E1 and E2 with
+both recognition approaches, which, however, differ in the observation sequences that are
+6
+
+Table 1: Evaluation results for the RG and GM goal recognition approaches when applied
+to E1 and E2 with the LAMA planner in anytime mode. The results for both approaches
+are identical for both, E1 and E2. Each row describes the probabilities P(G|O) for all
+goals G ∈ GBUC for different lengths of O (|O|).
+g1 = gprepare meal, g2 = gwatch TV ,
+g3 = guse shower, g4 = guse toilet.
+(a) Results for E1
+P(G|O)
+|O|
+g1
+g2
+g3
+g4
+28 + 0
+0.25
+0.25
+0.25
+0.25
+28 + 1
+0.319
+0.043
+0.319
+0.319
+28 + 2
+0.331
+0.006
+0.331
+0.331
+28 + 3
+0.063
+0.001
+0.468
+0.468
+28 + 4
+0.009
+0.0
+0.495
+0.495
+28 + 5
+0.002
+0.0
+0.268
+0.73
+28 + 6
+0.001
+0.0
+0.119
+0.88
+(b) Results for E2
+P(G|O)
+|O|
+g1
+g2
+g3
+g4
+0
+0.25
+0.25
+0.25
+0.25
+1
+0.316
+0.052
+0.316
+0.316
+2
+0.329
+0.012
+0.329
+0.329
+3
+0.106
+0.002
+0.446
+0.446
+4
+0.018
+0.0
+0.491
+0.491
+5
+0.003
+0.0
+0.349
+0.648
+6
+0.001
+0.0
+0.192
+0.806
+used to compile the involved planning problems. For experiment E1, the actions in the
+utilized observation sequence represent the entire BUC (see Example 1). For experiment
+E2, to evaluate how much the goal probability estimates depend on information gained
+from the observations of the agent getting and drinking beer, only the last six actions of the
+observation sequence used in E1 are used (i.e., in E2 the observations of the agent getting
+and drinking beer are not included in the observation sequences). The remaining setups are
+similar for both experiments. To solve the planning problems compiled from these setups,
+we used the LAMA (Richter, Helmert, & Westphal, 2008) planner, which is a satisficing
+planner that can be used either in greedy or anytime mode. When used in greedy mode, the
+planner stops immediately when a plan is found, whereas in anytime mode LAMA returns
+the best plan found in a given time window. Here, we used LAMA in anytime mode.
+4.2 Results and Limitations
+The results of the experiments are shown in Table 1. Each row reports the probabilities
+P(G|O) for all g ∈ GBUC for different lengths of O. As the RG and GM approaches are
+based on the same underlying principle and the BUC domain is small enough to be handled
+efficiently by both approaches, the results for the experiments E1 and E2 are identical for
+both of them. Hence, we only report one result table for each experiment. The results of
+the experiments reveal two major limitations of the two planning-based approaches.
+1. In both experiments, the correct goal g4 is only recognized at |O| = 28 + 5 and
+|O| = 5 respectively, i.e., shortly before the goal is actually reached. This timestep
+corresponds to the observation where the agent has moved to the location ba3, which
+is also where the toilet is located. This circumstance significantly reduces the possible
+usefulness for an intelligent assistance system, as the goal is recognized too late to
+provide support through adaptations of the environment.
+7
+
+2. The additional information that is included in the observation sequence used for E1,
+has no impact on the estimated probabilities P(G|O). Instead, the estimates are only
+based on information that are gained from the up to last six actions that are observed
+afterwards. Intuitively, however, the additional information should have an impact
+onto the estimate, because there exists a causal relation between drinking beer and the
+probability that this agent pursues the goal to use a toilet afterwards. Thus, it should
+be possible to recognize the correct goal much earlier in the observation sequence.
+The main reason for these limitations of the two planning-based approaches is that the
+additional actions that are contained in the observation sequence used for E1 are not strictly
+necessary to fulfill the preconditions of any action that is required to reach one of the
+possible goals in an optimal way, i.e., drinking beer is not a necessary requirement to visit
+the bathroom. As a consequence, these observations have the same impact on the estimate
+of P(G|O) for all possible goals, although they might contain valuable information about
+the true probability of P(G|O).
+5. A Hybrid Goal Recognition Approach
+As shown in the previous section, a significant shortcoming of PRAP based approaches is
+that causal relations between goals and observations in the environment cannot be exploited
+in general. This is due to the fact that the costs of plans are used as the only criterion to es-
+timate the probabilities P(O|G). To solve this problem, we propose to combine these PRAP
+approaches with a data-driven probabilistic causal model of agent goals and observations.
+5.1 A Statistical Causal Model of Observations
+We propose to model the relationship between goals and observations via a probability
+distribution P(F1, . . . , FN|G), where F1, . . . , FN are fluents of the planning state st. This
+distribution models how goals (e.g., making a sandwich) affect the probability of specific
+fluents in the planning states (e.g., whether the agent holds a cucumber). In contrast to
+the PRAP models, the idea here is to learn the parameters of P(F1, . . . , Fn|g) from training
+data. This way, the probabilistic model can capture the relations between fluents and goals
+that cannot be captured by planning-based approaches: Specifically, the planning-based
+models cannot capture statistical relations between fluents and goals that are not necessary
+for an optimal plan. For example, in the BUC planning domain, drinking beer is not a
+necessary precondition for visiting the bathroom. Still, drinking beer makes an eventual
+bathroom visit more likely.
+In general, we could use any probabilistic model, like Bayesian Networks, deep generative
+models etc., to represent P(F1, . . . , FN|G). Here, we focus on a Naive Bayes model (NBM),
+i.e., assuming P(F1, . . . , Fn|g) = �n
+i=1 P(Fi|g). The model is visualized in Figure 2. On
+the one hand, the strong independence assumptions between all fluents do not necessarily
+hold in practice. On the other hand, a Naive Bayes model has few parameters (linear in
+the number of variables). Thus, training is possible even when training data is scarce – as
+is often the case for activity sequences of real human subjects.
+Note that we made another simplifying assumption here: The distribution P(F1, . . . , FN|G)
+only models the dependency of the current planning state on the goal g, but not the de-
+8
+
+pendency of the action sequence o on g. More specifically, different observation sequences
+that result in the same planning state could be associated with different goals, but we
+deliberately neglected this information to make the learning problem tractable.
+F_N
+…
+g
+F_1
+Figure 2: Bayesian network representation of the Naive Bayes Model.
+5.2 A Hybrid Goal Recognition Method
+To overcome the shortcomings of the PRAP approaches discussed in Section 4.2, we propose
+to combine it with the proposed probabilistic learned model of P(o|g).
+For the combination of the goal probabilities, we use model stacking, which is a common
+ensemble learning method. In model stacking, a so-called meta-model is used to generate
+a combined estimate from the estimates of heterogeneous base models. In this work, we
+studied two different, manually designed, meta-models to combine the estimate of one of
+the PRAP approaches and the estimate of the NBM.
+It is important to note that there are some major differences between the ways in which
+the two utilized base models estimate the goal probabilities P(G|O). One of these differences
+is that the PRAP models are based purely on manually specified knowledge (in the form
+of the planning domain), whereas the NBM only relies on this manually defined domain
+knowledge to establish its structure but learns its parameters from training data. Another
+major difference between the models is the set of features which are used to estimate the
+probability P(O|G) and hence, also the probability P(G|O). Although the predictions of
+both models are based on an observed action sequence and an observed initial state of the
+environment, they estimate the probability P(O|G) based on different features that can
+be derived from these observations. The planning-based methods use two entire plans to
+estimate the probability P(o|g). In contrast, the NBM model uses only the planning state
+that results from applying the sequence o to the initial state to estimate P(o|g).
+Weighted Sum of Predictions
+As a first meta-model, we use a weighted sum (WS) of
+the two base-model estimates from one of the planning-based approaches and the NBM.
+More formally, this meta-model is defined as follows:
+P(g|o) = wsPs(g|o) + wdPd(g|o)
+(3)
+Ps is the goal probability estimate of one of the symbolic PRAP approaches, ws is the
+weight for the symbolic approach, Pd is the goal probability estimate of the data-driven
+NBM, and wd is the weight of the data-driven NBM.
+Tiebreaking of the PRAP approaches
+The second meta-model, which we refer to
+as tiebreaking (TB), only considers the estimate of the NBM when more than one goal is
+9
+
+Table 2: Evaluation results for E1 for both meta-models. Each row describes the proba-
+bilities P(O|G) for all goals G ∈ GBUC for different lengths of O (|O|). g1 = gprepare meal,
+g2 = gwatch TV , g3 = guse shower, g4 = guse toilet.
+(a) Results for the tiebreaking (TB) meta-model.
+P(G|O)
+|O|
+g1
+g2
+g3
+g4
+28 + 0
+0.131
+0.61
+0.13
+0.13
+28 + 1
+0.287
+0.176
+0.214
+0.323
+28 + 2
+0.293
+0.158
+0.22
+0.33
+28 + 3
+0.146
+0.14
+0.283
+0.431
+28 + 4
+0.005
+0.16
+0.361
+0.474
+28 + 5
+0.002
+0.0
+0.268
+0.73
+28 + 6
+0.0
+0.0
+0.12
+0.88
+(b) Results for the weighted sum (WS) meta-
+model.
+P(G|O)
+|O|
+g1
+g2
+g3
+g4
+28 + 0
+0.131
+0.61
+0.13
+0.13
+28 + 1
+0.287
+0.176
+0.214
+0.323
+28 + 2
+0.293
+0.158
+0.22
+0.329
+28 + 3
+0.146
+0.14
+0.283
+0.431
+28 + 4
+0.005
+0.16
+0.361
+0.474
+28 + 5
+0.037
+0.088
+0.165
+0.71
+28 + 6
+0.037
+0.088
+0.091
+0.785
+assigned with the highest likelihood by the PRAP approach. The intuition behind this
+meta-model is based on the observations from the results of experiments E1 and E2: Here,
+the PRAP approaches never ranked wrong goals as most probable, but not only ranked
+the true goal as most probable.
+When this is the case, we combine the two estimated
+probabilities of the NBM and the planning-based approaches again by taking the weighted
+sum of them. More formally, the meta-model is defined as follows:
+P(g|o) =
+�
+Ps(g|o),
+if | arg maxg∈G Ps(g|o)| = 1
+wsPs(g|o) + wdPd(g|o),
+if | arg maxg∈G Ps(g|o)| > 1,
+(4)
+Hybrid Goal Recognition Performance for E1
+To evaluate whether the proposed
+hybrid method is able to leverage on the additional information contained in the observation
+sequence used for E1, we applied the hybrid approach to E1.
+For the experiments, we
+assumed both weights ws and wd to be 0.5 and, due to the lack of training data for the
+BUC use case, modeled the parameters of the data-driven model manually. The results
+of the experiments are summarized in Table 2. The results show that goal g4 is ranked
+as most probable once one additional observation is made for both meta-models. Before
+this point, g2 is considered to be the most probable goal. Hence, the results show that a
+hybrid goal recognition model is able to leverage on the information that is contained in
+the observations that the agent drank beer. Consequently, it can recognize the true goal
+much earlier than the PRAP approaches for this example. Also interesting to note is that
+the results are identical for both meta-models until the fifth observation is observed. This
+makes sense as the results in Table 1 show that the PRAP approaches are undecided for
+the first four observation steps. Hence, both meta-models use the same weighted sum to
+combine goal probability estimates. For the fifth and sixth observation steps, both meta-
+models estimate the highest probability for the correct goal (i.e., g4). However, the TB
+meta-model estimates slightly higher probabilities for this goal and hence, is slightly more
+confident in g4 being the actual goal of the agent.
+10
+
+6. Evaluation Setup
+This section describes the experimental setup of the empirical evaluation. The empirical
+evaluation aims to achieve the following goals:
+1. Evaluate the performance of the planning-based methods, the NBM, and other well-
+known data-driven techniques, when applied to goal recognition problems in a real-
+world scenario, to determine which methods are best suited to be used in a hybrid
+goal recognition method.
+2. Show that a hybrid probabilistic goal recognition method is able to achieve superior
+performance, compared to purely planning-based and purely data-driven methods.
+3. Investigate how the performance of the hybrid method is affected by increasing the
+number of possible goals.
+We used real-world and artificial datasets for empirically evaluating the goal recognition
+methods. In all experiments, the online goal recognition problem was considered (as defined
+by Definition 5).
+6.1 Datasets
+This section presents the real-world and artificial datasets that were used for evaluation.
+Real-World Kitchen Dataset
+As a real-world data set, we used the CMU-MMAC
+Kitchen Dataset (De la Torre, Hodgins, Montano, Valcarcel, Forcada, & Macey, ) to which
+we will refer to as “CMU” from here on. It contains data from different sources (e.g., video,
+motion capture, etc.) that were recorded by observing different people while cooking one
+out of five recipes. We will consider the different dishes the observed participants might
+cook as possible goals. We first transformed the existing “raw” data into a suitable format
+for our purpose. As a starting point of this transformation, we used the results of a semantic
+annotation project (Yordanova, Kr¨uger, & Kirste, 2018). In this project, planning domains
+in PDDL format and annotated observation sequences were created for three of the five
+recipes (i.e., brownies, eggs, and sandwich). In addition, we created annotations for the
+remaining two recipes (i.e., pizza and salad). Table 3 displays some summarizing statistics
+of the observation sequence lengths per goal. Note that the CMU dataset only includes the
+first five goals. Interesting to note is that the average- and median observation sequence
+lengths substantially differ between the recipes. In addition, the standard deviations of the
+sequence lengths are relatively high. This indicates that the different observed persons used
+significantly different paths to reach one of the goals.
+One limitation of the CMU Dataset is that although it is based on sensor recordings
+of real human participants that were recorded while they were cooking different recipes,
+the general setup that was used during the sensor recordings is still rather artificial and,
+therefore, does not necessarily reflect all aspects of natural behaviour in a cooking scenario.
+Nevertheless, we still think that it is able to provide a solid basis to judge whether the
+investigated goal recognition approaches are able to handle recognition scenarios of real-
+world complexity, which is the aim of this work.
+11
+
+Table 3: Statistics of the observation sequence lengths per recipe in the CMU and artifi-
+cially extended CMU datasets (g1=brownies, g2=eggs, g3=sandwich, g4=salad, g5=pizza,
+g6=bread, g7=briocheBraid, g8=cheeseburger, g9=spaghetti, g10=spinachFetaPastry). The
+original CMU dataset only contains the first five goals.
+g1
+g2
+g3
+g4
+g5
+g6
+g7
+g8
+g9
+g10
+Average
+111.17
+87.08
+57.85
+110.56
+90.40
+69.73
+50.27
+122.96
+67.63
+108.12
+Median
+108.0
+88.0
+58.0
+108.5
+89.5
+70.0
+50.0
+114.0
+65.5
+96.5
+Std. Dev.
+15.43
+17.25
+9.02
+17.94
+15.35
+7.57
+7.61
+27.29
+9.43
+39.01
+Extending the CMU Dataset with Artificially Generated Data
+To evaluate the
+scalability of the proposed hybrid approach to a higher number of goals, we extended the
+data from the CMU dataset with five artificially generated goals.
+In the remainder of
+this work, we will refer to this extended dataset as “ACMU”. The added goals (bread,
+brioche braid, cheeseburger, spaghetti, and spinach feta pastry) were manually defined
+in the planning domain.
+This domain, which was used in both experiments (just with
+different sets of possible goals), contains 3411 different actions and 1627 different fluents.
+To generate artificial observation sequences for these goals, which is required for training
+the data-driven methods, we developed a procedure to sample such observation sequences
+for a given planning problem. The intuition underlying our proposed hybrid approach is the
+assumption that human behavior includes carrying out actions that are not strictly necessary
+in order to reach the current goal (i.e., which are not rational according to the planning
+domain). Thus, the sampling procedure should reflect this intuition. Consequently, it is
+not sufficient to use optimal plans as generated by existing planning systems: These plans
+generally only contain actions that are strictly necessary to reach a given goal. Additionally,
+a planning system will always generate very similar plans when it is presented with the same
+planning problem. As a solution, we propose a sampling algorithm that generates artificial
+observations sequentially. At each step, the algorithm either returns the action used in an
+optimal plan, or a randomly drawn action (where the probability of drawing a certain action
+depends on the goal). More details on the sampling algorithm can be found in Appendix
+A.
+Table 3 presents some summarizing statistics for the artificially generated observation
+sequences for g6 - g10. The average- and median lengths of the artificially generated obser-
+vation sequences are comparable to the sequences that are based on real observed sensor
+data. In addition, there is also a comparable amount of variation among these sequences,
+which is important to reflect the properties of real-world observations.
+Artificial Dataset
+To further investigate the scalability of the proposed hybrid approach
+and make the results better comparable to existing work, we used a well-known artificial
+planning domain, which was commonly used as a benchmark in the existing literature
+(Ram´ırez & Geffner, 2010),(Pereira, Oren, & Meneguzzi, 2020).
+This domain models a
+logistics problem, where different objects have to be delivered to several destinations. In
+contrast to the CMU domain, this domain is much smaller and contains only 356 actions
+and 84 fluents. We will refer to the resulting dataset as “LOG” from here on. As this
+domain is purely synthetic, no real observation sequences existed.
+Hence, we used the
+12
+
+same sampling procedure that was used to extend the CMU dataset to generate artificial
+observation sequences for this domain. Table 4 displays some summarizing statistics for
+the resulting observation sequences. An important difference to the CMU datasets is that
+the average observations sequence length is much smaller in the logistics dataset, which
+is mainly caused by the synthetic nature of this domain. Nevertheless, although this is
+the case, there is still a recognizable amount of variance among the sampled observation
+sequences.
+Table 4: Statistics of the observation sequence lengths per recipe in logistics dataset.
+g1
+g2
+g3
+g4
+g5
+g6
+g7
+g8
+g9
+g10
+Average
+21.33
+22.47
+23.23
+23.43
+21.40
+22.80
+23.67
+22.27
+24.27
+22.37
+Median
+21.0
+21.0
+23.0
+22.0
+21.0
+22.0
+23.0
+22.0
+23.5
+22.0
+Std. Dev.
+2.31
+3.51
+2.64
+3.48
+2.93
+2.51
+2.27
+2.80
+2.32
+4.52
+6.2 Goal Recognition Methods
+In the empirical experiments, we applied different symbolic, data-driven, and hybrid goal
+recognition methods.
+Symbolic Goal Recognition Methods
+We used two state-of-the-art planning-based
+methods: GM and RG, which were presented in Section 4. To solve the planning problems
+that are generated by the two PRAP approaches, we used the MetricFF (Hoffmann, 2003)
+planner. MetricFF is a satisficing planner that supports metric fluents.Following Ram´ırez
+and Geffner (Ram´ırez & Geffner, 2010) and Vered et al. (Vered et al., 2016), we assumed
+equal cost for all actions and used a value of β = 1. As the timeout for the MetricFF
+planner, we used 340 seconds. After this timeout, the planner will be forced to stop and
+the associated planning problem is considered not solvable.
+Data-Driven Goal Recognition Methods
+We evaluated four different data-driven
+methods: The NBM presented earlier in Section 5 and three additional data-driven ap-
+proaches, which were selected following a recent study by Borrajo et al. (Borrajo, Gopalakr-
+ishnan, & Potluru, 2020). Specifically, we used K-Nearest-Neighbors (KNN) (Russell, 2016,
+pp. 738-740), XGBoost (Chen & Guestrin, 2016), and a Long-Short-Term Memory (LSTM)
+network (Hochreiter & Schmidhuber, 1997).
+For experiments with the KNN and XGBoost approaches, we evaluated two different
+data encodings. First, we used a binary encoding of the planning states, consisting of the
+state of all planning fluents. Second, following the work of Borrajo et al. (Borrajo et al.,
+2020), we used a vector encoding of the observed action sequence.
+We performed a grid search to determine the hyper-parameters for the KNN and XG-
+Boost methods. For the planning fluent-based encoding, we used a leaf size of 30 and k = 8
+for KNN and a learning rate of 0.24, minimum child weight of 5, α = 0.42, λ = 1.15, maxi-
+mum depth of 5, and γ = 0.99 for XGBoost. In contrast, for the action vector encoding, we
+used a leaf size of 40 and k = 3 for KNN and a learning rate of 0.31, minimum child weight
+of 2, α = 3.26, λ = 1.03, maximum depth of 3, and γ = 0.96 for XGBoost.
+13
+
+For the LSTM, we also evaluated two different data encodings: A one hot representation
+of both the observed action sequence and the observed state sequence.
+Here, we used
+different vector encodings than for the other data-driven methods because LSTM models,
+in contrast to the other considered methods, are explicitly designed to handle temporal data
+sequences which are better captured by one hot data encodings (Borrajo et al., 2020). For
+both setups, we used the ADAM optimizer with a learning rate of 0.01, a batch size of 32,
+and 100 epochs.
+Hybrid Goal Recognition Methods
+We evaluated the two different meta-models de-
+scribed in Subsection 5.2. For both meta-models, we computed the weight for the NBM
+as wNBM(n, t) =
+a
+1+e−b(t−((cn)+d)) , where n is the number of training examples used to train
+the NBM, t is the number of observations used for goal recognition, and a, b, c, and d
+are fitting parameters. As for the data-driven approaches, we performed a grid search to
+determine the best performing values for a, b, c, and d. Accordingly, we set the parameters
+to a = 0.5, b = −0.15, c = 4, d = 2.5 (CMU), and a = 0.5, b = −0.15, c = 5, d = 1 (ACMU)
+respectively for the CMU datasets. For the LOG dataset, we used the following parameter
+values: a = 0.2, b = −0.15, c = 0, and d = 0. The weight for the PRAP approaches is
+calculated as wPRAP = 1 − wNBM(n) for all datasets.
+6.3 Experimental Design
+To assess the goal recognition performance of the different methods, we used the mean goal
+recognition accuracy. To calculate the accuracies, in contrast to most previous works, we
+did not consider a goal to be recognized correctly if it is part of a set of goals that were
+assigned with the highest likelihood. Instead, we only considered a goal to be recognized
+correctly if it is the only goal that was assigned with the highest probability. We decided for
+this evaluation method as it, in our opinion, better reflects the usefulness of the prediction
+for practical application in an assistance system. If such a system is provided with more
+than one most probable goal, it has to randomly decide for one goal.
+Furthermore, as
+we consider online goal recognition problems in this evaluation, we calculated the mean
+accuracy for different fractions of total observations that were used for goal recognition.
+Here we used relative numbers because the lengths of the involved observation sequences
+substantially differ. Hence, the mean accuracy Acc for a relative number of observations
+λ ∈ [0, 1] is calculated as follows:
+Acc(λ, D) =
+�
+R∈D [R(⌊TRλ⌋) = ˜
+gR]
+|D|
+(5)
+Here, D is a set of online goal recognition problems R, ˜
+gR denotes the correct goal of goal
+recognition problem R, TR is the maximum value of t for online goal recognition problem
+R (i.e., length of observation sequence that is associated with R), and [R(t) = ˜
+gR] equals 1
+if the correct goal is recognized for R(t) and 0 otherwise.
+To evaluate the performance of the symbolic goal recognition methods, we calculated
+Acc(λ, D) for different values of λ and for different domains D. To investigate the per-
+formance of the data-driven approaches in relation to the number of available training
+examples (i.e., number n of complete observation sequences), we used a slightly adjusted
+cross-validation procedure: For a given value of n, we split a set of online goal recognition
+14
+
+0
+1
+2
+3
+4
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60
+65
+70
+75
+80
+85
+90
+95
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+Θ(λ, CMU)
+GM
+RG
+Figure 3: Mean accuracy of the planning-based methods (RG and GM) on the CMU Dataset
+without artifical samples.
+problems D into k partitions, where k = |D|/n. Then, k models were trained, but in con-
+trast to the typical cross-validation procedure, only one of the partitions was used as the
+training set and the remaining partitions were used for validation. In cases where D cannot
+be splitted into k partitions of equal size n, we randomly sampled sequences from the other
+partitions to complete the partitions with a size smaller than n. To assess the performance
+of a data-driven method, we calculated Acc(λ, D) for all k models and, subsequently, took
+the average over these accuracies. For the evaluation of the hybrid methods, we calculated
+the combined estimates for all results obtained from the cross-validation procedure for the
+data-driven approaches and then also took the average of the accuracies of all k models.
+7. Experimental Results
+In the following, we present and discuss the results of the experiments corresponding to
+each of the three evaluation goals defined in Section 6.
+7.1 Symbolic and Data-Driven Goal Recognition
+Symbolic Goal Recognition Results
+We start by comparing the two planning-based
+goal recognition methods on the CMU dataset. Figure 3 shows the average goal recognition
+accuracy of the RG and GM approaches on this dataset.
+It can be seen that the GM
+approach outperformed the RG approach consistently, except for the case when only very
+small fractions of the observation sequences are used. Furthermore, the accuracy of the RG
+approach decreases when more than 20% of the observations are used. The main reason
+for this behavior is the fact that the involved planning problems became too complex to
+be solved optimally, or were not solvable at all in the given time limit. This fact is also
+the reason for the large difference between GM and RG which could not be observed for
+the (much simpler) BUC before: The higher complexity of the planning problem made
+the solutions that are found within the given time limit less optimal. Hence, the results
+show that the compilation process of the RG approach had a much higher impact onto
+the optimality of the solutions than the transformation procedure of the GM approach.
+Through a detailed analysis of the generated plans, we found that this is mainly caused by
+the fact that, in contrast to the GM approach, the RG approach changes the structure of
+the actions space of a planning problem in a way that most planning heuristics are not able
+to deal with.
+15
+
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+Acc(λ, CMU)
+n=1
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+n=3
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+Acc(λ, CMU)
+n=5
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+n=7
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+Acc(λ, CMU)
+n=9
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+n=11
+NBM
+XGBoost (actions)
+XGBoost (states)
+KNN (actions)
+KNN (states)
+Figure 4: Mean accuracy of the data-driven methods (NBM, KNN and XGBoost) on the
+CMU Dataset without additional samples for different sizes of the training set n.
+As the GM approach provided a much better overall performance, in all following ex-
+periments, we only considered the GM approach.
+Data-Driven Goal Recognition Results
+Next, we compare the different data-driven
+goal recognition methods.
+Figure 4 shows the average, cross-validated goal recognition
+accuracies of the NBM, KNN, and XGBoost for the CMU dataset. As the LSTM approach
+did not achieve accuracy values above 25% for any training set size, we did not include the
+results in Figure 4. For KNN and XGBoost, we compare performances of the fluent-based
+and action-based data encodings, as introduced in Section 6.2.
+The results show that all approaches performed much better, especially early in the
+observation sequence, when the planning state-based data encoding was used. This shows
+that, in case of the CMU domain, the symbolic planning states encode more useful informa-
+tion regarding the actual goal of an observed agent than the sequence of observed actions.
+Furthermore, the accuracies of all three methods did not depend strongly on the amount
+of available training data. Interesting to note is that even though the NBM is the model
+with the lowest computational complexity, it was still not outperformed by the (slightly)
+more complex KNN and XGBoost models. Hence, overall, the NBM is the most favor-
+able data-driven model for this scenario, especially in mobile computing scenarios, where
+computational efficiency is of high relevance.
+16
+
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+Acc(λ, CMU)
+n=1
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+n=3
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+Acc(λ, CMU)
+n=5
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+n=7
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+Acc(λ, CMU)
+n=9
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+n=11
+GM
+NBM
+WS
+TB
+Figure 5: Mean accuracy of the Goal Mirroring (GM) and Naive Bayes Model (NBM)
+approaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on
+the CMU Dataset without artifical samples for different sizes of the training set n.
+7.2 Hybrid Goal Recognition
+In this section, we assess the performance of the hybrid goal recognition models (i.e.,
+Weighted Sum (WS) and Tiebreaking (TB)) in comparison to the purely data-driven NBM
+approach and the purely planning-based GM method. Figure 5 shows the average, cross-
+validated goal recognition accuracies of these approaches for the CMU dataset.
+The results show that both hybrid approaches were at least as good as the GM and
+NBM approaches for small training set sizes. For larger training set sizes (i.e., n ≥ 3), the
+TB approach was increasingly outperformed by the NBM early in the recognition process
+(i.e., when only a small fraction of the observations were seen). The reason for this is that
+the TB approach relies strongly on the predictions of the GM approach, which also became
+increasingly outperformed by the NBM early in the observation sequence with increasing
+training set sizes. In contrast, the WS approach was not outperformed by the NBM, but
+reached at least similar performance as the NBM also when only a small fraction of the
+observations were seen. The WS approach was even able to substantially outperform both
+the NBM and the GM approaches early in the observation sequences when, depending on
+the training set size, between 3% and 30% of the observations were used. This effect was
+most prominent when training set sizes between n = 3 and n = 7 were used.
+The results show that for n ≥ 3, the planning-based and data-driven methods com-
+plemented each other well regarding recognition performance. While the NBM approach
+17
+
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+Acc(λ, ACMU)
+n=1
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+n=3
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+Acc(λ, ACMU)
+n=5
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+n=7
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+Acc(λ, ACMU)
+n=9
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+n=11
+GM
+NBM
+WS
+TB
+Figure 6: Mean accuracy of the Goal Mirroring (GM), Naive Bayes Model (NBM) ap-
+proaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on the
+artificially extended CMU Dataset for different sizes of the training set n.
+achieved the best performances early in the observation sequences (i.e., less than 10% - 20%
+of the observations), the GM approach outperformed the NBM later in the observation
+sequences (i.e., more than 10% - 20% of the observations). The hybrid WS approach was
+able to leverage on the strengths of the two individual approaches, constantly performing
+as good or better as each of them.
+7.3 Scalability of Hybrid Goal Recognition
+Evaluating Scalability on Extended Real-World Dataset
+Next, we investigate the
+scalability of the methods, by assessing goal recognition performance when the number of
+goals is increased. Figure 6 shows the cross-valiated mean accuracy of the GM, NBM, TB,
+and WS approaches for the ACMU dataset (i.e., with sampled observation sequences).
+Due to the doubled number of goals, the GM and the NBM approaches achieve a sig-
+nificantly lower recognition accuracy compared to the results for the CMU dataset that has
+not been extended with artificial data. Nevertheless, it can be observed that the recognition
+performance of the GM approach converges towards the GM performance on the not artifi-
+cially extended CMU dataset with an increasing fraction of observations that were used for
+recognition. The results also show that the NBM approach, even when only a small number
+of training examples were used (i.e., n = 3), is able to achieve a better goal recognition
+performance than the GM when less than 5% of the observation sequences were seen. In
+18
+
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+Acc(λ, LOG)
+n=1
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+n=3
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+Acc(λ, LOG)
+n=5
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+n=7
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+Acc(λ, LOG)
+n=9
+0 1 2 3 4 5 101520253035404550556065707580859095
+0.2
+0.4
+0.6
+0.8
+1
+λ (%)
+n=11
+GM
+NBM
+WS
+TB
+Figure 7: Mean accuracy of the Goal Mirroring (GM), Naive Bayes Model (NBM) ap-
+proaches and the two hybrid approaches Weighted Sum (WS) and Tiebreaking (TB) on the
+logistics domain for different sizes of the training set n.
+addition, the results show that the WS approach again performs similarly well or better
+than the two individual approaches.
+The differences between the achieved recognition performances of the GM and NBM
+approaches are even larger early and late in the observation sequences than for the stan-
+dard CMU dataset. This indicates that increasing the number of possible goals makes the
+weaknesses of the individual approaches even more prominent and hence, using a hybrid
+approach that is able to compensate for them is even more favorable. Note that this obser-
+vation only holds when a limited number of training data is available as the performance
+of the NBM naturally will increase when more training data is available.
+Nevertheless,
+it is very common in practice that annotated training examples are scarce as manually
+annotating observation sequences is costly and error-prone.
+In summary, the results show that the hybrid recognition approach still achieves good
+goal recognition performance when the number of possible goals increases. Moreover, the
+results indicate that using a hybrid approach is even more beneficial when the number of
+goals increases, compared to purely data-driven or purely planning-based methods.
+Evaluating Scalability on an Artificial Dataset
+Finally, we further investigate the
+scalability of the methods by applying them to a benchmark plan recognition domain (which
+has simpler plans, but more possible goals than the real-world CMU domain). Figure 7
+shows the mean goal recognition accuracy of the GM, NBM, TB, and WS approaches for
+19
+
+the logistics planning domain. As for the CMU domain, the NBM performed better than
+the GM approach early in the observation sequences (i.e., when less than, depending on
+n, 5% - 20% of the observations were seen) and the GM performed better later in the
+observation sequences. However, for the logistics domain, the NBM only achieved slightly
+better performance than the GM approach.
+Interestingly, in contrast to the experiments with the CMU Dataset, the Tiebreaking
+(TB) approach also constantly performed as good or better than the two individual ap-
+proaches (in addition to WS, as for the CMU dataset). The main reason for this behavior
+is the fact that the assumptions underlying the TB approach hold more firmly for the LOG
+domain: TB assumes that the planning-based approach (GM, in this case) never predicts
+a wrong goal to be most probable, but only predicts multiple, equally likely goals (one of
+which is correct). This assumption only holds if the involved plans are optimal. The logis-
+tics domain, however, has substantially lower complexity than the CMU domain, such that
+the MetricFF planner was able to find more optimal plans in the given time limit. Thus,
+the assumptions of TB hold and TB could achieve better results than for the CMU and
+ACMU datasets. In summary, the results show that our hybrid goal recognition approach
+is also beneficial in artificial planning domains where the number of goals is substantially
+higher than in the investigated real-world domain.
+8. Related Work
+Existing approaches to goal- and plan recognition can be divided into model-based and
+model-free approaches. Model-based approaches typically reason over handcrafted symbolic
+domain models to solve the recognition task. In contrast, model-free approaches treat the
+recognition problem as a classification problem and learn to predict the current user goal
+from data and, thus, are data-driven.
+Early model-based approaches to plan recognition relied on complete plan libraries that
+encode possible user behavior to recognize the current plan from observed user actions
+(Kautz & Allen, 1986; Charniak & Goldman, 1993). However, these approaches require a
+large manual modeling effort, which is infeasible in large domains. To overcome this issue, a
+new class of approaches to plan recognition that no longer required complete plan libraries,
+but only a domain model that defines possible states and actions, was proposed.
+The
+PRAP approaches considered in this work (Ram´ırez & Geffner, 2010; Pereira et al., 2020),
+(Vered et al., 2016) belong to this class. Another example approach that relies on the use
+of classical planning systems is the approach by Sohrabi et al. (Sohrabi, Riabov, & Udrea,
+2016). They propose to use a top-k planner to generate the top-k plans for all possible
+goals in order to obtain which goal a user currently intents to achieve. Nevertheless, most
+of these approaches have, so far, only been evaluated on relatively small, artificial domains,
+and hence, it is not clear whether they are also applicable to real-world scenarios.
+We
+have shown that these PRAP approaches indeed show good performance in a real-world
+setting, but have problems in capturing relations between observations and user goals that
+cannot be properly modeled manually. Some other recent approaches to goal recognition in
+smart environments also belong to this class of approaches (Yordanova et al., 2017, 2019).
+Consequently, they have the same problems as the approaches considered in this work.
+20
+
+In contrast, model-free approaches learn to predict the most probable user goal directly
+from data. Hence, they have the potential to learn the relations between actions and user
+goals that are not properly captured by model-based approaches. In (Albrecht, Zukerman,
+Nicholson, & Bud, 1997), the authors propose to use a BN model to predict the current quest
+of an observed player of a computer game. Recently, approaches that applied deep learning
+methods to goal recognition problems have been proposed (Min, Mott, Rowe, Liu, & Lester,
+2016; Amado, Aires, Pereira, Magnaguagno, Granada, & Meneguzzi, 2018). For example,
+Min et al. (Min et al., 2016) applied a LSTM for player goal recognition in digital games.
+However, model-free approaches usually require large amounts of training data to produce
+reasonable results. In the case of deep learning models, several thousands of annotated
+training examples are required to train the model adequately. Such amounts of training
+data are usually not easily available for real-world scenarios. Regarding this aspect, model-
+based approaches have a clear advantage because they can rely on handcrafted domain
+knowledge. Thus, to benefit from both paradigms’ strengths, we propose a hybrid approach
+that combines a model-based and a model-free method.
+9. Conclusion and Future Work
+In this work, we investigated whether existing plan recognition as planning (PRAP) ap-
+proaches can be applied to solve the online goal recognition problem in a real-world kitchen
+scenario. More explicitly, we conducted several empirical goal recognition experiments on
+the basis of the well-known CMU Kitchen Dataset, which contains observation sequences
+for five possible goals of up to 36 different subjects. We found that such PRAP approaches
+can indeed be used to solve the online goal recognition problems in real-world scenarios.
+Nevertheless, we also revealed and analyzed some major limitations of PRAP approaches
+when applied to such scenarios. As a possible solution, we proposed a hybrid goal recog-
+nition method, which combines a symbolic PRAP approach and a data-driven model. We
+showed that the hybrid approach is able to recognize an agent’s true goal more reliably than
+the PRAP approaches, especially early in an observation sequence (i.e., when only a small
+fraction of the observations were seen). To investigate the scalability of the proposed hybrid
+approach in terms of the number of possible goals, we conducted an experiment based on an
+artificially extended version of the CMU Kitchen Dataset. The results of these experiments
+indicate that the advantages of using a hybrid approach are becoming even more prominent
+with an increasing number of possible goals.
+In summary, we showed that using a hybrid goal recognition method provides a valuable
+improvement compared to state-of-the-art purely symbolic and data-driven goal recognition
+methods. It was found that our proposed hybrid method is able to outperform purely sym-
+bolic and data-driven methods and recognize the correct goal more reliably based on a
+lower number of observations, although only a small number of training examples are used.
+This result substantially improves the usefulness of goal recognition for intelligent assistance
+systems, as recognizing a goal early opens much more possibilities for supportive reactions
+of the system. Furthermore, it is usually very expansive to obtain annotated training ex-
+amples for real-world application scenarios. Hence, being able to provide valuable results
+based on limited numbers of training examples is an important requirement for potential
+goal recognition methods that should be applied to real-world application scenarios. Nev-
+21
+
+ertheless, we still see some potential for improvements of the extended approach in future
+work. One direction is to optimize the procedure that is used to combine the results of the
+two individual approaches. Another direction that we plan to investigate in future work is
+to use more complex tractable probabilistic models, like Sum-Product Networks (Poon &
+Domingos, 2011).
+10. Acknowledgements
+The data used in this paper was obtained from kitchen.cs.cmu.edu and the data collection
+was funded in part by the National Science Foundation [grant number EEEC-0540865]. This
+work was supported by the German Federal Ministry of Education and Research (BMBF)
+[grant number 01lS18079C].
+Appendix A. Sampling Artificial Observation Sequences
+Algorithm 1 summarizes the sampling procedure that is used in this work to sample artificial
+observation sequences. Here, pplanAction is a parameter that specifies the goal-directedness,
+i.e., the probability that the next action is taken from a precomputed optimal plan. When
+this is not the case, the next action is sampled out of the set of actions that are currently
+applicable in the current planning state. These actions are randomly drawn from all actions
+that are applicable in a certain state of the planning domain, following two predefined
+probability models that model the probability that an interaction with a certain object O is
+observed given that we want to reach a goal G (P(O|G)), and the probability that a certain
+kind of action A is observed given that we want to reach goal G (P(A|G)).
+The distribution P(A|G, S) that is used to sample an action at random is defined as
+follows:
+P(A = ai|g, s) ∝
+�
+wi
+if ai applicable in s
+0
+otherwise
+(6)
+That is, only applicable actions can be selected. The weight wi of an action depends on the
+corresponding “action type” AT(ai) and the set of objects OB(ai) with that each action
+interacts. The underlying intuition is the observation that depending on the current goal,
+the agent will choose actions of different action types with higher probabilities than others
+and also interact with certain objects with higher probabilities. Based on this intuition, we
+use randomly initialized weight score distributions W(AT|G) and W(OB|G) to determine
+the weights wi via
+wi = W(AT(ai)|g)
+�
+x∈OB(ai)
+W(x|g),
+(7)
+We initialize the parameters of P(A|G) and P(O|G) randomly and use the MetricFF
+planner to compute the initial plan.
+Each time an action from the sampling model is
+selected, the optimal plan from the resulting state is recomputed.
+22
+
+Algorithm 1 Sample artificial observation sequence for goal g.
+sampledPlan ← ()
+cState ← initialPlanningState
+optPlan ← computeOptimalPlan(cState, g)
+i = 0
+while goal not reached do
+r ← random(0, 1)
+if r < pplanAction then
+sAction ← optPlan.getAction(i)
+cState ← cState.apply(sAction)
+i ← i + 1
+else
+sAction ← sampleActionFromApplicableActions(cState)
+cState ← cState.apply(sAction)
+optPlan ← computeOptimalPlan(cState, g)
+i = 0
+end if
+sampledPlan ← concat(sampledPlan, sAction)
+end while
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+25
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+Few-Shot Image-to-Semantics Translation for Policy
+Transfer in Reinforcement Learning
+Rei Sato∗‡, Kazuto Fukuchi†‡, Jun Sakuma†‡ and Youhei Akimoto†‡
+∗ Graduate School of Science and Technology, University of Tsukuba, Tsukuba, Japan
+[email protected]
+† Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Japan
+[email protected], [email protected], [email protected]
+‡ RIKEN Center for Advanced Intelligence Project
+Abstract—We
+investigate
+policy
+transfer
+using
+image-to-
+semantics translation to mitigate learning difficulties in vision-
+based robotics control agents. This problem assumes two envi-
+ronments: a simulator environment with semantics, that is, low-
+dimensional and essential information, as the state space, and
+a real-world environment with images as the state space. By
+learning mapping from images to semantics, we can transfer a
+policy, pre-trained in the simulator, to the real world, thereby
+eliminating real-world on-policy agent interactions to learn,
+which are costly and risky. In addition, using image-to-semantics
+mapping is advantageous in terms of the computational efficiency
+to train the policy and the interpretability of the obtained policy
+over other types of sim-to-real transfer strategies. To tackle the
+main difficulty in learning image-to-semantics mapping, namely
+the human annotation cost for producing a training dataset, we
+propose two techniques: pair augmentation with the transition
+function in the simulator environment and active learning. We
+observed a reduction in the annotation cost without a decline
+in the performance of the transfer, and the proposed approach
+outperformed the existing approach without annotation.
+Index Terms—deep reinforcement learning, policy transfer,
+sim-to-real
+I. INTRODUCTION
+Deep reinforcement learning (DRL) has been actively stud-
+ied for robot control applications in real-world environments
+because of its ability to train vision-based agents; that is, the
+robot control actions are output directly from the observed
+images [1]–[4]. One of the major advantages of vision-based
+agents in robotics is that camera-captured images can be
+incorporated into the decision-making of the agent without
+using a handcrafted feature extractor.
+However, allowing vision-based robot control agents to
+learn by reinforcement learning in the real-world is challeng-
+ing in terms of risk and cost because it requires a large amount
+of real-world interactions with unstable robots. Reinforcement
+learning involves a learning policy interacting with the envi-
+ronment, and it is theoretically and empirically known that the
+length of the interaction required for training increases with
+the dimension of the state space [5], [6].
+To address the difficulty associated with reinforcement
+learning in a real-world environment, methods have been
+proposed that pre-train a policy on a simulator environment
+This research is partially supported by the JSPS KAKENHI Grant Number
+19H04179, and based on a project, JPNP18002, commissioned by NEDO.
+and transfer it to the real-world environment [7]–[17]. In
+this methodology, policies are learned in a simulator, that
+is, a reinforcement learning environment on a computer that
+mimics the real-world environment. The policy pre-trained in
+the simulator is expected to be the optimal policy in the real-
+world environment.
+However, developing a simulator that imitates the real-world
+environment is not always an easy task. Particularly, because
+the real world provides image observations, a simulator en-
+vironment requires a renderer to generate images as states.
+However, producing a renderer that can generate photorealistic
+images is fraught with financial and technical difficulties.
+In the case that a photorealistic renderer cannot be produced,
+another style of observations must be adopted as states during
+the pre-training of the policy in a simulator environment. Most
+existing approaches substitute photorealistic observations for
+non-photorealistic ones using transfer techniques [7]–[17].
+We investigated a type of transfer strategy called image-
+to-semantics to deal with the absence of a photorealistic
+renderer, which was created by [18]. In this approach, the
+semantics—low-dimensional and essential information of a
+state that represents an image—are employed as a form of state
+observation instead of images in the simulator environment.
+The transfer algorithm consists of two steps: pre-training a
+policy on the simulator environment with semantics as its
+observation, obtaining a mapping from photorealistic images
+to their corresponding semantics, and using the image-to-
+semantics mapping as a pre-processing component of the
+policy in the real-world environment. A semantics-based pre-
+trained policy can be operated in the real-world environment
+using image observations. In addition to being a solution to
+the case without a photorealistic renderer, image-to-semantics
+mapping has advantages in terms of the computational cost
+for policy pre-training in the simulator and the interpretability
+of the acquired policy.
+The crucial part of this approach is obtaining the image-to-
+semantics translation mapping. To the best of our knowledge,
+[18], [19] are the only studies that have dealt with learning
+image-to-semantics translation. We highlight the remaining
+problems of [18], [19]: (1) [19] used a paired dataset, that is,
+multiple pairs of images and corresponding semantics, to train
+the mapping. Considerable human effort is required to make
+arXiv:2301.13343v1  [cs.LG]  31 Jan 2023
+
+a paired dataset because human annotators provide seman-
+tics that represent images. (2) Although the style translation
+method without a paired dataset [18] aims at saving annotation
+cost, its performance is not often satisfactory owing to the low
+approximation quality of the image-to-semantics translation
+mapping, as confirmed in our experiments.
+In this study, we tackled learning image-to-semantics trans-
+lation using a paired dataset; however, we reduced the cost of
+creating a paired dataset using two strategies: pair augmenta-
+tion and active learning. In our experiments, we confirmed the
+following claims: first, compared to [19], we reduced the cost
+of making a paired dataset while preserving the performance
+of the policy transfer. Second, we achieved significantly higher
+performance than [18], in which a paired dataset was not used,
+by using a small paired dataset. For practicality, we conducted
+experiments under the condition that only inaccurate paired
+data can be obtained due to various errors, such as annotation
+errors, and confirmed that the proposed method has a certain
+robustness against errors.
+Our
+code
+is
+publicly
+available
+at
+https://github.com/
+madoibito80/im2sem.
+II. PROBLEM FORMULATION
+A. Markov Decision Process (MDP)
+We defined a vision-based robotics task in the real world;
+that is, the real-world environment is a target MDP: Mτ =
+(Sτ, A, pτ, rτ, γ), where Sτ is a state space, A is an action
+space, pτ : Sτ ×A×Sτ → R is a transition probability density,
+rτ : Sτ × A × Sτ → R is a reward function, and γ ∈ [0, 1] is
+a discount factor. Because we assumed that the target MDP is
+a vision-based task, Sτ consists of images, and each s ∈ Sτ
+contains single or multiple image frames. In standard model-
+free reinforcement learning (RL) settings, agents can interact
+with the environment: they observe st+1 ∼ pτ(· | at, st) and
+reward rt = rτ(st+1, at, st) by performing action at at state
+st, which is internally preserved in the environment at timestep
+t; after the transition, st+1 is stored in the environment.
+However, there are concerns in terms of the risk and cost
+associated with learning a policy through extensive interaction
+with Mτ.
+To reduce the risk and cost of training a policy in the target
+MDP, we pre-trained a policy on a simulator environment,
+called the source MDP: Mσ = (Sσ, A, pσ, rσ, γ). Note that
+the action space A is the same between the two MDPs.
+In contrast, the state space Sσ, the transition probability
+density pσ : Sσ × A × Sσ → R, and the reward function
+rσ : Sσ × A × Sσ → R are different from those of the target
+MDP. We assumed that because we considered robotics tasks,
+the deterministic transition function Trσ(s, a) = s′ ∼ pσ(· |
+a, s) could be defined in the simulator environment and pσ
+resembled a Dirac delta distribution.
+The source state space Sσ corresponded to a semantic space,
+that is, each s ∈ Sσ was semantic information. For example,
+consider a robot-arm grasp task; each s ∈ Sτ is a single or
+multiple image frame showing a robot arm and objects to be
+Action Space
+State Space
+Simulator Env 
+(Source MDP) 
+Policy
+Action Space
+State Space
+Real-World Env 
+(Target MDP) 
+Semantic Space
+Image Space 
+(Photorealistic)
+Image-to- 
+Semantics
+Fig. 1. Illustration of transfer via image-to-semantics. We approximated the
+image-to-semantics translation mapping F as ˆF. Because the action space was
+common to both MDPs, we operated the composite of the source policy πσ
+and approximated image-to-semantics translation mapping ˆF, that is, πσ ◦ ˆF
+in the target MDP.
+grasped. Each s ∈ Sσ consists of semantics such as xyz-
+coordinates of the end-effector and target objects and angles
+of joints.
+The source MDP and target MDP are expected to have some
+structural correspondence. Here, we describe our assumptions
+regarding the relations of the two MDPs. We assumed the
+existence of a function F : Sτ → Sσ satisfying the following
+conditions:
+Transition Condition:
+For all (s′, a, s) ∈ Sτ × A × Sτ,
+pσ(F(s′) | a, F(s)) =
+�
+¯s∈ ¯
+S pτ(¯s | a, s)d¯s, where ¯S = {¯s ∈
+Sτ | F(¯s) = F(s′)}.
+Reward Condition:
+For all (s′, a, s) ∈ Sτ × A × Sτ,
+rσ(F(s′), a, F(s)) = rτ(s′, a, s).
+In the above conditions, F is considered an oracle that takes
+an image and outputs corresponding semantics; that is, F is the
+true image-to-semantics translation mapping. In the transition
+condition, ¯S is a set of images that has common semantics
+F(s′). Imagine the transition from s ∈ Sτ to s′ ∈ Sτ with
+action a ∈ A in the target MDP, the transition condition holds
+F(s′) = Trσ(F(s), a). The reward condition indicates that
+a reward for this transition rτ(s′, a, s) equals the one for a
+transition from F(s) ∈ Sσ to F(s′) ∈ Sσ with the action a
+in the source MDP.
+B. Transfer via Image-to-Semantics
+1) Policy Transfer: The objective of RL is the expectation
+of the discounted cumulative reward:
+J(π; p, r, γ, p0) = Eπ,p,p0 [�∞
+t=0 γtr(st+1, at, st)]
+(1)
+and maximizing it w.r.t. π. Here, π : S × A → R is a policy,
+that is, a conditional distribution of at given st, and p0 is
+the distribution of the initial state s0 over the state space.
+Our objective was to obtain a well-trained policy on the target
+MDP: πτ = arg max¯πτ J(¯πτ; pτ, rτ, γ, pτ
+0).
+Under the situation in which the transition and reward
+conditions mentioned above hold for some F, we can replace
+πτ by πσ ◦ F, where πσ is a well-trained policy on the
+source MDP, that is, πσ = arg max¯πσJ(¯πσ; pσ, rσ, γ, pσ
+0).
+Solving this maximization by RL requires sole interaction
+with Mσ instead of Mτ. As noted, interactions with Mτ
+
+require real-world operations; however, interactions with Mσ
+are performed on the simulator, which is cost-effective.
+Based on this property, we studied the following transfer
+procedure: pre-train πσ on Mσ, approximate F as ˆF, and out-
+put the target agent πσ◦ ˆF. This procedure was investigated by
+[18]. Figure 1 illustrates the transfer via image-to-semantics.
+2) Advantages: The above-mentioned transfer strategy, that
+is, transfer via image-to-semantics, has the following three ad-
+vantages over approaches using a renderer in the source MDP
+shown in Table I. First, a renderer is not required. Existing
+methods that use a renderer generally aim to transfer an agent
+based on non-photorealistic images in a simulator to photoreal-
+istic images in the real world [7]–[17]. Therefore, they require
+the preparation of a renderer on the simulator to generate non-
+photorealistic images as state observations. Transfer via image-
+to-semantics performs similar transfer learning; however, it
+does not require a renderer because the source MDP has
+a semantic space as its state space. This can reduce the
+development cost of the simulator for some tasks. Second,
+because semantics are low-dimensional variables compared to
+images, we can improve the sample efficiency required to train
+the policy πσ on Mσ [5], [6]. Learning vision-based agents
+are generally associated with large computational costs, even
+on a simulator [20], but transfer via image-to-semantics is
+relatively lightweight in this respect and occasionally allows
+a human to design the policy. Third, using semantics as an
+intermediate representation of the target agent contributes to
+its high interpretability because of the low-dimensionality and
+interpretability of semantics. Similar to [19], [21], because the
+real-world agent πσ◦ ˆF can be separated into two components,
+which are independently trained, it is easier to assess than one
+trained in an end-to-end manner.
+C. Resource Strategy
+In this section, in addition to the two MDP environments,
+we define resources that can be used to approximate F.
+1) Transition Function: In the target MDP, the state transi-
+tion result st+1 due to the selected action at can be observed
+only for state st stored inside the environment. In contrast, in
+the source MDP, we assumed that the state transition result
+for any s ∈ Sσ could be observed, replacing the st stored
+inside the environment with s. This is because the actual
+state transition probability pτ in the target MDP is a physical
+phenomenon in the real world, but the state transition rule Trσ
+in the source MDP is a black-box function on the computer.
+2) Offline Dataset: The offline dataset comprised observa-
+tions of the target MDP, that is, T τ = {(st, at, 1end(st+1)) ∈
+Sτ × A × {0, 1}}t, where 1end(st+1) = 1 represents that
+st+1 corresponding to a terminal state; otherwise, 0. Note
+that successive indices in the offline dataset shared the same
+context of the episode, except at the end of the episode. T τ can
+be obtained before training starts and is collected by a behavior
+policy. Because the offline dataset can be reused for any trial
+and be obtained by a safety-guaranteed behavior policy, we
+assumed it could be created at a relatively low cost.
+TABLE I
+RELATED POLICY TRANSFER METHODS FOR OBSERVATION STYLE SHIFT.
+EACH METHOD REQUIRES DIFFERENT RESOURCES: RENDERER, OFFLINE
+DATASET (OFF), AND PAIRED DATASET (PAIR).
+Method
+Renderer
+OFF
+PAIR
+Tobin et al. [7]
+✓
+RCAN [8]
+✓
+DARLA [9]
+✓
+Pinto et al. [10]
+✓
+MLVR [11]
+✓
+Tzeng et al. [12]
+✓
+✓
+GraspGAN [13]
+✓
+✓
+RL-CycleGAN [14]
+✓
+✓
+RetinaGAN [15]
+✓
+��
+MDQN [16]
+✓
+✓
+✓
+ADT [17]
+✓
+✓
+✓
+Zhang et al. [18]
+✓
+CRAR [19]
+✓
+✓
+Ours
+✓
+✓
+We solely used the offline dataset for supervised and unsu-
+pervised learning purposes. If offline reinforcement learning
+is executed, the vision-based agent can be trained directly
+without approximating F. However, training a vision-based
+agent using an offline dataset by reinforcement learning re-
+quires large-scale trajectories in the scope of millions [22]. In
+this study, we considered situations in which the total number
+of timesteps in the offline dataset was limited, for example,
+less than 100k timesteps.
+We did not need to generate reward signals while collecting
+the offline dataset. World models [23] have been studied for
+the procedure: approximate MDP M as
+ˆ
+M using an offline
+dataset of M; train a policy by reinforcement learning by
+interacting with the approximated environment
+ˆ
+M instead
+of interacting with the original environment M. One could
+imagine that we could replace interactions with the target
+MDP by interactions with the approximated one. However, to
+accomplish this, we must observe signals regarding reward in
+the real world while collecting the offline dataset, and we must
+approximate a reward function that is often sparse; both of
+these are not always easy [24]. Therefore, we did not consider
+approximating the target MDP and did not assume the reward
+was contained in T τ.
+3) Paired Dataset: The paired dataset P consisted of mul-
+tiple pairs of target state observations and their corresponding
+source state observations. Let I denote the set of indices
+that indicate the position of the offline dataset. Using the
+true image-to-semantics translation mapping F, we can denote
+P = {(F(si), si) | (si, ai, ei) ∈ T τ, i ∈ I}. Under prac-
+tical situations, querying F equals annotating corresponding
+semantics to the images of the indices I in the offline dataset
+T τ by human annotators. Because of its annotation cost, we
+assumed the size of the paired dataset |I| to be significantly
+smaller than that of the offline dataset, for example, |I| ≤ 100.
+III. RELATED WORK
+We introduced some existing sim-to-real transfer methods
+that use a non-photorealistic renderer on the simulator. Table I
+
+lists the transfer methods that do not require on-policy inter-
+action in the target MDP, assuming vision-based agents. The
+main difficulty tackled by these methods was the absence of
+a photorealistic renderer on the simulator. In the real world,
+images captured by a camera are input to the agent; however,
+generating photorealistic images on the simulator is generally
+difficult because it requires developing a high-quality renderer.
+In [7]–[11], [25], the algorithms learned policies or interme-
+diate representations that were robust to changes in image style
+using a non-photorealistic renderer. Thus, these algorithms
+were expected to perform well even when a photorealistic style
+was applied in a real-world environment. In particular, the
+domain randomization technique has been widely used [7]–
+[10].
+A. Transfer via Image-to-Image Translation
+In contrast to the above methods, [12]–[19] aimed to
+perform style translation mapping among specific styles. To
+accomplish this, these methods required an offline dataset of
+the target MDP. Because these methods followed the principle
+of collection without execution of on-policy interaction, the
+offline dataset could be collected by a safety-guaranteed pol-
+icy. Unsupervised style translation, such as domain adaptation
+[26] and CycleGAN [27], are often used to change the styles
+for state-of-the-art methods [13]–[15], [17], [18], [24], [28].
+Using this translation mapping as a pre-processing function of
+the target agent, the pre-trained policy can determine actions
+in the same image style as the source MDP in the target MDP.
+However, domain adaptation and cycle-consistency [27]
+only have a weak alignment ability [18], and some existing
+methods use paired datasets to properly transfer styles [16],
+[17], [19]. Therefore, these two datasets have been widely
+employed in previous studies and can be assumed to be a
+common setting.
+The similarity of transfer via image-to-semantics and image-
+to-image is that they train style translation mapping ˆF among
+the source and target state spaces that preserves essential
+information; furthermore, the agent is the composite π ◦ ˆF,
+where π is a policy.
+Again, the above methods use a non-photorealistic renderer
+on the simulator. Thus, these methods cannot be compared
+with transfer via image-to-semantics, as explained in Sec-
+tion II-B2.
+B. Learning Image-to-Semantics
+Previous studies have used semantics in the source MDP
+[10], [12], [16]–[18]. An important perspective on the appli-
+cability of these methods to image-to-semantics is whether
+they use a renderer on the simulator, as shown in Table I
+and as discussed in Section II-B2. Because methods using a
+renderer assume that the source state space is an image space,
+image-to-semantics is beyond their scope, and it is not certain
+that their mechanism will be successful in image-to-semantics.
+For example, CycleGAN, which has been successfully used for
+image-to-image learning, failed in image-to-semantics [18]. In
+this regard, we refer to [18], an unpaired method that applies
+the findings from image-to-image to image-to-semantics. In
+addition, [19] is compared as a representative method that uses
+a paired dataset as in this study.
+1) CRAR: We refer to Section 4.4 of CRAR [19] as a
+baseline of image-to-semantics learning. They described the
+following policy transfer strategy: pre-train a source state
+encoder Eσ : Sσ → Z, where Z is a latent space of the
+encoder; train the source policy πσ : Z → A; and train a
+target state encoder Eτ : Sτ → Z with regularization term
+�
+(sσ,sτ )∈P∥Eσ(sσ)−Eτ(sτ)∥2
+2, where P is a paired dataset.
+Then, the target agent is the composite πσ ◦ Eτ : Sτ → A.
+Here, Eτ can be regarded as a style translation mapping. Note
+that they only performed this experiment in the setting where
+Sσ and Sτ are both image spaces; however, it can be applied
+easily where Sσ is the semantic space.
+2) Zhang et al.: We referred to the cross-modality setting of
+their experiment as our baseline for image-to-semantics [18].
+This setting is the same as the transfer via image-to-semantics.
+There remain some challenges in [18], [19]. For [18], the
+human annotation cost was eliminated because they did not
+use a paired dataset. However, the loss function defined by
+[18] for unpaired image-to-semantics style translation will not
+necessarily provide a well-approximated F. Therefore, we
+decided to use a paired dataset to efficiently supervise the
+loss function as performed in [19], but with a paired dataset
+smaller than [19].
+IV. METHODOLOGY
+Our approach approximates the image-to-semantics transla-
+tion F using an offline dataset T τ. Similar to [19], we used
+a paired dataset P = {(F(si), si) | (si, ai, ei) ∈ T τ, i ∈ I},
+which was constructed by querying F(si) to human annotators
+for an image observation of target MDP si ∈ Sτ included in
+T τ. We incorporated two main ideas to reduce the annotation
+cost. Pair augmentation generates an augmented paired dataset
+P′ using an offline dataset T τ. Active learning defines I, that
+is, it selects a subset of T τ to be annotated to construct P
+(Algorithm 2). We present an overall procedure of our method
+in Algorithm 1.
+We assumed that we have an offline dataset T τ, which
+comprises multiple episodes in the target MDP. Let O denote
+the set of indices corresponding to the beginning of an episode
+in T τ, that is, O = {0} ∪ {i | 0 < i < |T τ| and ei−1 =
+1 for (sτ
+i−1, ai−1, ei−1) ∈ T τ}, where ei is the indicator:
+when timestep i is the end of an episode then ei = 1. For each
+i ∈ O, let Ei = {t | 1 ≤ t ≤ min({k | k ≥ 1, ei+k = 1})}.
+Then, for each i ∈ O, a subsequence of T τ starting from
+timestep i and ending at i + |Ei| corresponds to an episode.
+A. Pair Augmentation by Transition Function
+The objective of pair augmentation is to construct artificial
+paired data P′ such that sσ ≈ F(sτ) for (sσ, sτ) ∈ P′ and
+sτ ∈ T τ. Using an augmented paired dataset, we aimed to
+obtain ˆF that approximates F by minimizing the loss
+L( ˆF, P ∪ P′) =
+1
+|P ∪ P′|
+�
+(sσ,sτ )∈P∪P′
+∥sσ − ˆF(sτ)∥2
+2 . (2)
+
+action
+Semantics
+Offline
+Dataset
+action
+action
+action
+F
+Fig. 2. Illustration of pair augmentation. Oracle F generates semantics corresponding to a particular image in the offline dataset. The next state in semantics is
+computed using the transition function Trσ with the current semantics along with the action taken while collecting the offline dataset. This allows us to obtain
+semantics corresponding to the image at the next timestep in the offline dataset without any annotation costs. Augmented pairs with green dual directional
+arrows were stored in P′. In this figure, note that rendered (non-photorealistic) images are shown in the offline dataset, but in reality, camera-captured
+(photorealistic) images are contained.
+Algorithm 1 Overall Procedure
+Require: Source MDP Mσ, Offline dataset T τ, Oracle F
+1: Train source MDP’s policy πσ on Mσ
+2: Train VAE encoder Eτ using T τ
+3: Determine indices I by active learning (Algorithm 2)
+using Eτ, T τ
+4: Create P for I, T τ by oracle (human annotator) F
+5: Create augmented pairs P′ using P, T τ, Trσ ∈ Mσ
+6: Train ˆF by minimizing Equation (2)
+Ensure: Target MDP’s agent πσ ◦ ˆF
+Note that CRAR [19] adopts L( ˆF, P) instead of L( ˆF, P∪P′).
+Our principle is as follows. Let I ⊆ O be a subset of
+indices corresponding to the beginning of the episodes in T τ.
+Suppose we have a paired dataset P constructed by querying
+semantics sσ
+i = F(sτ
+i ) corresponding to images sτ
+i in T τ for
+time index i ∈ I. Although semantics sσ
+i+1 representing an
+image of the next timestep sτ
+i+1 in T τ is unknown, because of
+the transition condition given in Section II-A and deterministic
+transition, it equals sσ
+i+1 = Trσ(sσ
+i , ai), where ai is the action
+taken at timestep i when collecting the offline dataset T τ
+and is included in T τ. In reality, because human annotations
+and state transition contain errors as compared to the truth,
+the generated semantics sσ
+i+1 do not exactly represent the
+image sτ
+i+1. However, even with errors in F and Trσ, it
+is expected that the generation of the above semantics is a
+valuable approximation. By recursively applying the above
+generation, we obtained the augmented paired dataset P′.
+Formally, P′ was constructed as follows: For each index i ∈
+I, we defined a sequence {�sσ
+i+t}t∈Ei as �sσ
+i = sσ
+i (contained
+in P) and �sσ
+i+t = Trσ(�sσ
+i+t−1, ai+t−1) for t ∈ Ei, where
+ai+t−1 are contained in T τ. The augmented paired dataset is
+then P′ = {(�sσ
+i+t, sτ
+i+t)}i∈I,t∈Ei, where sτ
+i+t is contained in
+T τ. Thus, we could construct an augmented paired dataset P′
+of size |P′| = �
+i∈I|Ei| from the paired dataset P of size
+|P| = |I|.
+Figure 2 illustrates the pair augmentation scheme.
+The reason why I was a subset of episode start indices
+O rather than I ⊆ {j | 0 ≤ j < |T τ|, j ∈ Z} was to
+maximize the size of augmented pairs |Ei|. In other words,
+because we could augment sσ
+i = F(sτ
+i ) until the end of the
+episode including sτ
+i , to maximize |P∪P′|, human annotations
+should be conducted at the beginning of an episode of T τ.
+B. Active Learning for Pair Augmentation
+To select episodes for annotation, that is, decide I, we
+incorporated the idea of diversity-based active learning (AL)
+[29]–[31]. Their motivation was to select dissimilar samples
+to effectively reduce the approximation error. Intuitively, if
+P ∪ P′ has many similar pairs, they might have a similar
+effect on training ˆF; this may lead to a waste in annotation
+cost. Therefore, we attempted to select episodes (indexed by
+I ⊂ O) to be annotated to ensure the inclusion of diverse
+pairs.
+We successively selected the episode to annotate, and we
+called each selection step the n-th round. For i ∈ O, let Bi =
+{sτ
+i+t}t∈{0}∪Ei be a set of target state observations present in
+the episode starting at timestep i ∈ O. We referred to it as
+batch. Let In−1 be the set of selected indices before the n-th
+round, and let Sn−1 = �
+k∈In−1 Bk be a set of all the state
+vectors in the episodes selected before the n-th round. Let
+d : Sτ × Sτ → R be some appropriate distance measure. In
+the n-th round, a batch was selected based on the following
+two diversity measures: The inter batch diversity
+finter(Bi, Sn−1) =
+�
+sτ ∈Bi
+min
+sτ
+j ∈Sn−1 d(sτ, sτ
+j )
+(3)
+can evaluate the dissimilarity of Bi and Sn−1. The batch with
+the greatest finter was considered to be the most dissimilar
+
+Algorithm 2 Active Learning
+Require: Trained VAE encoder Eτ, Offline dataset T τ
+1: Initialize I0 = {c}|c∼Uniform(O)
+2: for 1 ≤ n < N do
+▷ n-th round
+3:
+Set Sn−1 = �
+k∈In−1 Bk
+4:
+Measure finter(Bi, Sn−1) for all i ∈ O
+5:
+Pick top b% of indices in terms of finter as Q
+6:
+Measure fintra(Bi) for all i ∈ Q
+7:
+Pick the index c from Q with the greatest fintra
+8:
+Set In = {c} ∪ In−1
+9: end for
+Ensure: Indices IN−1 (with the size of N) as I
+batch against the pre-selected batches. The intra batch diver-
+sity
+fintra(Bi) =
+�
+sτp∈Bi
+�
+sτq ∈Bi
+d(sτ
+p, sτ
+q)
+(4)
+can evaluate the dissimilarity of the states inside Bi. The batch
+with the greatest fintra was considered to contain the most
+diverse states.
+We selected a batch that maximizes the above two diversity
+measures; we performed a bi-objective optimization for selec-
+tion. To avoid overemphasizing one measure over the other,
+we employed two separate single-objective optimizations for
+each measure. In each round, we picked up indices of batches
+with finter in the top b% (b = 10 in our experiments) from
+unselected episodes as Q, and subsequently, selected the batch
+with the greatest fintra from Q. I0 was initialized with the
+episode sampled from O uniformly at random.
+C. Representation Learning Using Offline Dataset
+For d : Sτ × Sτ → R to be a reasonable distance measure
+in the image space, we employed a VAE encoder [32]: Eτ :
+Sτ → Z. It stochastically outputs a latent vector z ∈ Z for
+sτ ∈ Sτ. The distance between two states sτ
+p ∈ Sτ and sτ
+q ∈
+Sτ was given by the Euclidean distance between the mean
+vectors for their latent representations, that is, d(sτ
+p, sτ
+q) =
+∥E[Eτ(sτ
+p)] − E[Eτ(sτ
+q)]∥2. We trained Eτ using all states in
+the offline dataset T τ before performing the active learning
+procedure.
+We used the states contained in �
+i∈I Bi in training ˆF by
+Equation (2); however, the remaining �
+i∈O\I Bi were not
+used. In order to use it, we included Eτ as a feature extractor
+for ˆF by receiving the benefit of representation learning for
+downstream tasks. We modeled ˆF = φ ◦ Eτ, and we trained
+φ by Equation (2), whereas Eτ was fixed.
+V. EXPERIMENTS
+We aimed to verify the following two claims: (1) the
+proposed paired augmentation and AL reduces the annotation
+cost for approximating ˆF while maintaining its performance
+level; and (2) the paradigm with the paired dataset performs
+better than the method without paired datasets.
+A. Evaluation Metrics
+1) Policy Performance (PP): The most important evalua-
+tion metric for ˆF is the expected cumulative reward of the
+target agent using Equation (1):
+PP( ˆF; πσ, Mτ) = J(πσ ◦ ˆF; pτ, rτ, γ, pτ
+0) .
+(5)
+In our experiments, we approximated it by averaging the
+cumulative reward of 50 episodes with γ = 1. This metric
+was commonly used in [18], [19].
+2) Matching Distance (MD): Because our technical contri-
+bution was mainly to approximate F, we used the following
+empirical approximation error:
+MD( ˆF; T , F) =
+1
+|T |
+�
+(sτ ,a,e)∈T
+∥F(sτ) − ˆF(sτ)∥2
+2,
+(6)
+where T is a trajectory collected by a behavior policy in the
+target MDP, which is not used for learning ˆF. Unfortunately, in
+a real-world environment, evaluating Equation (6) for a large
+size of T is challenging because F requires human annotation.
+To enable MD in our experiment, we performed experiments
+using the simulator for both the source MDP and target MDP.
+We adopted the rendered image space as the state space of
+the target MDP. Because both semantics and images were
+generated in the simulator, F was freely available to calculate
+Equation (6). A similar metric to Equation (6) was used in
+[18].
+B. Environment
+We evaluated the proposed approach on three environments.
+1) ViZDoom Shooting (Shooting): ViZDoom Shooting [33]
+is a first-person view shooter task, in which an agent obtains
+64×64 RGB images from the first-person perspective in the
+target MDP. The agent can change its x-coordinate by moving
+left and right in the room and attacking forward (|A| = 3). An
+enemy spawns with a random x-coordinate on the other side
+of the room at the start of the episode and does not move or
+attack. The agent can destroy the enemy by moving to the front
+of it and shooting it; time to destruction is directly related to
+the reward. Semantics are the x-coordinates of the agent and
+the enemy; hence, Sσ is a 2-dimensional space. The maximum
+timesteps is 50 for each episode. The behavior policy to collect
+the offline dataset T τ is a random policy, and T τ consists of
+200 episodes, that is, 10k timesteps in total.
+2) PyBullet KUKA Grasp (KUKA): This is a grasp task
+using PyBullet’s KUKA iiwa robot arm [34]. Success is
+achieved by manipulating the end-effector of the robot arm
+and lifting a randomly placed cylinder. The semantics are the
+xyz-coordinate and the 3-dimensional Euler angle of the end-
+effector and the xyz-coordinate of the cylinder; hence, Sσ is a
+9-dimensional space. We used the rendered 64×64 RGB im-
+ages captured from three different viewpoints simultaneously
+as the state observations in the target MDP. The total timesteps
+per episode is fixed to 40. The behavior policy to collect T τ
+is a random policy, and T τ comprised 250 episodes, that is,
+10k timesteps in total.
+
+TABLE II
+RESULTS OF SHOOTING. MD VALUES WERE SCALED TO 102 FOR
+CONVENIENCE. πσ HAS PP=45.99, AND THE BEHAVIOR POLICY HAS
+PP=16.39.
+Method
+MD
+PP
+|I| = 0
+Zhang et al.
+37.42 ± 6.23
+22.46 ± 4.99
+|I| = 10
+CRAR
+11.00 ± 1.72
+35.16 ± 4.31
+Ours w/o AL
+3.44 ± 0.89
+43.99 ± 1.29
+Ours
+0.16 ± 0.12
+44.73 ± 1.07
+|I| = 50
+CRAR
+2.80 ± 0.70
+42.29 ± 2.12
+Ours w/o AL
+0.06 ± 0.02
+46.02 ± 0.31
+Ours
+0.02 ± 0.00
+45.66 ± 0.34
+3) PyBullet HalfCheetah-v0 (HalfCheetah): This is a Py-
+Bullet version of the HalfCheetah, that is, a task in which a
+2-dimensional cheetah is manipulated by continuous control
+to run faster. The torque of the six joints can be controlled
+(A = [−1, 1]6), and the semantic space is a 26-dimensional
+space. We collected 64×64 images captured from three differ-
+ent viewpoints for two consecutive timesteps and defined Sτ
+as an image space containing a total of 6 frames. The total
+timesteps per episode is fixed to 1000. The behavior policy
+to collect T τ is a random policy, and T τ consists of 100
+episodes, which is 100k timesteps in total.
+In our experiments, information such as xyz-coordinates
+and velocity can be recovered from a combination of multiple
+images by capturing images from multiple viewpoints at
+consecutive times, and such a setup is necessary in practice.
+C. Setting
+We used a 7-layer convolutional neural network and a 4-
+layer fully connected neural network for the VAE encoders
+Eτ and φ, respectively, for both the proposed and existing
+methods. We trained them in gradients using Adam [35].
+The dimensions of the latent space of VAE Z were set
+to 32, 96, and 192 for Shooting, KUKA, and HalfCheetah,
+respectively. For CRAR [19], we uniformly selected indices I
+from {i | 0 ≤ i < |T τ|, i ∈ Z}. For our method, without an
+AL setting, I was selected uniformly and randomly from O.
+For Shooting and KUKA, we used a handcrafted policy instead
+of one trained by RL as πσ. In HalfCheetah, we trained πσ
+using PPO [36].
+D. Results
+Tables II to IV show the results of the image-to-semantics
+learning in the three environments. These tables show the
+results on average±std over five trials. |I| denotes the number
+of paired data, which is the annotation cost. Because of the
+transition and reward conditions, the PP of πσ ◦ F on Mτ
+assimilate to that of πσ on Mσ.
+Note that most image-to-image methods shown in Table I
+cannot be compared with image-to-semantics methods be-
+cause some assumptions cannot be satisfied under image-to-
+semantics settings. One way to speculate on the performance
+of the image-to-image techniques in an image-to-semantics
+CRAR
+Ours w/o AL
+Ours
+Fig. 3. Scatter of the obtained semantics on ViZDoom Shooting with |P| =
+|I| = 10: {F(sτ) | (sσ, sτ) ∈ P} for CRAR, and {F(sτ) | (sσ, sτ) ∈ P∪
+P′} for our method. Each square represents a 2-dimensional semantic space.
+The semantic space shows that both pair augmentation and AL contribute to
+expanding the coverage.
+TABLE III
+RESULTS OF KUKA. PP CORRESPONDS TO GRASP SUCCESS PROBABILITY.
+πσ HAS PP=1.0, AND THE BEHAVIOR POLICY HAS PP=0.048.
+Method
+MD
+PP
+|I| = 0
+Zhang et al.
+0.90 ± 0.12
+0.12 ± 0.08
+|I| = 10
+CRAR
+0.59 ± 0.07
+0.24 ± 0.22
+Ours w/o AL
+0.35 ± 0.05
+0.52 ± 0.19
+Ours
+0.37 ± 0.03
+0.65 ± 0.07
+|I| = 100
+CRAR
+0.32 ± 0.02
+0.52 ± 0.15
+Ours w/o AL
+0.11 ± 0.01
+0.76 ± 0.09
+Ours
+0.12 ± 0.01
+0.90 ± 0.04
+setting is to see Zhang et al. [18]. Zhang et al. used do-
+main adaptation [26], which is commonly used in image-to-
+image learning; thus, their method can be interpreted as a
+representative example in which the techniques cultivated in
+image-to-image are imported to image-to-semantics. Although
+CycleGAN [27] is also widely employed in image-to-image
+learning, along with domain adaptation, they confirmed in their
+experiments that this method did not outperform their method
+in the image-to-semantics setting [18].
+In all cases, compared with the approach of Zhang et al.
+[18], our approaches with and without AL achieved a smaller
+MD and a greater PP. Zhang et al.’s approach is designed to
+learn ˆF without a paired dataset to eliminate the annotation
+cost. However, learning without pairs does not necessarily
+lead to the true image-to-semantics translation mapping, as
+observed in the high MD and low PP in our results. This
+result shows the effectiveness of the paradigm using paired
+data when aiming for higher performance policy transfer while
+compromising the annotation cost to prepare a small number
+of paired data.
+By comparing the results of our approaches with and
+without AL and those of CRAR, we confirmed the efficacy
+of pair augmentation in achieving a smaller MD and higher
+PP. We achieved PP=44.73 ± 1.07 in Shooting with 10 pairs
+using pair augmentation and AL, which is more than the
+PP=42.29±2.12 achieved by CRAR with 50 pairs. In addition,
+in KUKA, we achieved PP=0.65 ± 0.07 in 10 pairs, which
+exceeds PP=0.52 ± 0.15 achieved by CRAR with 100 pairs.
+This means that the annotation cost was reduced by more
+than ×5 and ×10, respectively. This difference is even more
+
+TABLE IV
+RESULTS OF HALFCHEETAH. πσ HAS PP=2735.73, AND THE BEHAVIOR
+POLICY HAS PP=−1230.01.
+Method
+MD
+PP
+|I| = 0
+Zhang et al.
+3.71 ± 0.32
+−1511.97 ± 192.51
+|I| = 10
+CRAR
+1.80 ± 0.09
+−1411.83 ± 144.21
+Ours w/o AL
+0.40 ± 0.04
+596.20 ± 121.43
+Ours
+0.37 ± 0.05
+580.21 ± 71.95
+|I| = 50
+CRAR
+0.88 ± 0.05
+−818.29 ± 300.08
+Ours w/o AL
+0.12 ± 0.01
+878.79 ± 88.52
+Ours
+0.07 ± 0.01
+968.49 ± 99.53
+pronounced in HalfCheetah. This may be because the ratio
+|P ∪ P′|/|P| is the greatest in this environment: CRAR uses
+|P| = |I| paired data, whereas our proposed approach used an
+additional |P′| = 999|I| augmented paired data because the
+number of timesteps per episode was 1000 in this environment.
+A tendency of reduced MD and increased PP was observed
+in the proposed approach with AL compared to that without
+AL. Specifically, AL reduced MD except in KUKA, and
+clearly improved PP in KUKA, while achieving competitive
+PP in the other two environments.
+In Figure 3, we present the effectiveness of our approach
+in Shooting. The proposed AL maximized the diversity in the
+latent space that represents the image space, but the diversity
+was also maximized when this result was visualized in the
+semantic space.
+E. Experiments with Errors
+In this section, we verify the robustness of the proposed
+method against errors in annotation and state transitions.
+1) Annotation Error: In our previous discussion and in
+experiments of Section V-D, we assumed that we could query
+oracle F, that is, true image-to-semantics mapping by human
+annotations. However, because human annotation indicates the
+process of assigning semantics to images by humans, errors
+are expected to occur in the output semantics. Therefore, we
+provided a new experimental setup here: for some sτ ∈ Sτ,
+we can observe F(sτ) + ϵ instead of F(sτ) while creating
+the paired dataset P, where ϵ ∈ Rdim(Sσ) is a random vector
+representing the annotation error.
+2) Transition Error: In reality, the state transition function
+on the simulator Trσ is expected to contain modeling errors.
+For example, environment parameters such as friction coeffi-
+cients and motor torques in the real world cannot be accurately
+estimated in the simulator, and thus, state transitions in reality
+cannot be correctly imitated. Therefore, we provided a new
+experimental setup here: for some (s, a) ∈ Sσ × A, we could
+obtain Trσ(s, a) + ϵ instead of Trσ(s, a) while augmenting
+a paired dataset, where ϵ ∈ Rdim(Sσ) is a random vector
+representing the transition error. Note that when training πσ,
+we used the one without errors in our experiments.
+3) Error Generation: We generated two types of errors by
+adding a random variable ϵ ∈ Rdim(Sσ). Here, we denoted a
+TABLE V
+RESULTS WITH ANNOTATION ERRORS.
+Method
+α
+MD
+PP
+Shooting (|I| = 50)
+CRAR
+0.0
+2.80 ± 0.70
+42.29 ± 2.12
+Ours w/o AL
+0.06 ± 0.02
+46.02 ± 0.31
+CRAR
+0.04
+2.99 ± 0.93
+42.91 ± 1.71
+Ours w/o AL
+0.12 ± 0.05
+45.90 ± 0.54
+CRAR
+0.15
+3.50 ± 1.21
+43.51 ± 2.04
+Ours w/o AL
+0.47 ± 0.07
+44.56 ± 0.59
+CRAR
+0.3
+4.90 ± 0.85
+40.83 ± 2.78
+Ours w/o AL
+1.42 ± 0.15
+42.47 ± 1.74
+KUKA (|I| = 100)
+CRAR
+0.0
+0.32 ± 0.02
+0.52 ± 0.15
+Ours w/o AL
+0.11 ± 0.01
+0.76 ± 0.09
+CRAR
+0.04
+0.33 ± 0.03
+0.58 ± 0.11
+Ours w/o AL
+0.12 ± 0.01
+0.76 ± 0.18
+CRAR
+0.15
+0.32 ± 0.03
+0.53 ± 0.15
+Ours w/o AL
+0.14 ± 0.01
+0.68 ± 0.15
+HalfCheetah (|I| = 50)
+CRAR
+0.0
+0.88 ± 0.05
+−818.29 ± 300.08
+Ours w/o AL
+0.12 ± 0.01
+878.79 ± 88.52
+CRAR
+0.04
+0.91 ± 0.03
+−919.86 ± 328.14
+Ours w/o AL
+0.12 ± 0.01
+787.0 ± 230.32
+CRAR
+0.15
+0.99 ± 0.06
+−833.35 ± 464.76
+Ours w/o AL
+0.15 ± 0.02
+616.63 ± 260.73
+value of the h-th dimension of x ∈ RH as x(h) ∈ R. We sam-
+pled ϵ(h) ∼ N(h), where N(h) is a Gaussian distribution with
+mean µ = 0 and standard deviation σ = α · std[sσ
+(h)]sσ∈T σ.
+Here, std[sσ
+(h)]sσ∈T σ is the sample standard deviation of a
+source trajectory T σ collected by a behavior policy in source
+MDP, and α ≥ 0 is the noise scale.
+For the annotation error, using the semantics sequence
+of augmented pairs {�sσ
+i+t}t∈Ei provided without error, we
+provided the semantics as F(sτ
+i )+¯ϵi for P and {�sσ
+i+t+¯ϵi}t∈Ei
+for P′, where ¯ϵi is a realized random vector with α > 0.
+For the transition error, we defined {�sσ
+i+t + �t
+j=1 ¯ϵi,j}t∈Ei
+for P′, where ¯ϵi,j is the realized random vector. Note that,
+here, we approximated the error generation based on the
+following assumption: Trσ(s, a) = s + Trσ
+∆(a), that is,
+Trσ(s+¯ϵ1, a)+¯ϵ2 = s+Trσ
+∆(a)+¯ϵ1+¯ϵ2. This is an approxi-
+mation simplifying implementation; however, for Shooting and
+KUKA, the above assumption is actually satisfied for almost
+all states and actions.
+4) Results: Here, we analyze the effect of two types of
+errors on P and P′, and understand how this affects the
+approximation of F. Therefore, we do not experiment with
+the method of Zhang et al., which does not utilize paired
+datasets. In addition, to eliminate the effect of the choice of
+I on the generation of P and P′ in the comparison between
+the proposed method and CRAR, we conducted experiments
+under the setting without AL.
+The results with annotation errors are shown in Table V.
+In both CRAR and the proposed method, the semantics of
+the paired data deviated from the true data as the scale of
+the annotation error α increased; thus, we observed that MD
+tends to increase for both methods. Although PP tended to
+decrease only for the proposed method, the proposed method
+achieved better MD and PP than CRAR for the same error
+
+TABLE VI
+RESULTS WITH TRANSITION ERRORS.
+Method
+α
+MD
+PP
+Shooting (|I| = 50)
+CRAR
+0.0
+2.80 ± 0.70
+42.29 ± 2.12
+Ours w/o AL
+0.06 ± 0.02
+46.02 ± 0.31
+Ours w/o AL
+0.01
+0.12 ± 0.04
+45.90 ± 0.51
+Ours w/o AL
+0.04
+0.67 ± 0.12
+44.78 ± 1.16
+Ours w/o AL
+0.1
+3.43 ± 1.00
+43.44 ± 1.60
+KUKA (|I| = 100)
+CRAR
+0.0
+0.32 ± 0.02
+0.52 ± 0.15
+Ours w/o AL
+0.11 ± 0.01
+0.76 ± 0.09
+Ours w/o AL
+0.01
+0.12 ± 0.01
+0.80 ± 0.11
+Ours w/o AL
+0.04
+0.14 ± 0.00
+0.67 ± 0.22
+HalfCheetah (|I| = 50)
+CRAR
+0.0
+0.88 ± 0.05
+−818.29 ± 300.08
+Ours w/o AL
+0.12 ± 0.01
+878.79 ± 88.52
+Ours w/o AL
+0.01
+0.18 ± 0.02
+426.61 ± 221.12
+Ours w/o AL
+0.04
+1.21 ± 0.13
+−523.8 ± 336.51
+scale α. In addition, we confirmed that PP with an annotation
+error for our method remains comparable to the case without
+an annotation error for a certain degree of α. For example,
+in KUKA experiments, the proposed method achieved PP =
+0.76 ± 0.18 with α = 0.04, which was close to PP = 0.76 ±
+0.09 without annotation error. We conclude that the proposed
+pair augmentation is effective in image-to-semantics learning
+even in the presence of annotation errors.
+The results with transition errors are shown in Table VI.
+Note that CRAR does not use Trσ; thus, the result did not
+depend on transition error scale α; the result of CRAR with
+α > 0 matched the result of α = 0. Because the proposed
+pair augmentation scheme used Trσ to generate semantics,
+for larger t ∈ Ei, the variance of error was expected to be
+large; then, the augmented semantics in P′ were far from the
+actual semantics. In fact, we observed an increase in MD and
+a decrease in PP in the proposed method as the scale of α
+increased. In contrast, both MD and PP were better than CRAR
+up to α = 0.04 for Shooting and KUKA, and up to α =
+0.01 for HalfCheetah. This indicates that the proposed pair
+augmentation is effective in reducing the annotation costs up
+to a certain level of transition errors.
+F. Effect of Behavior Policy
+To further reveal the behavior of image-to-semantics meth-
+ods, we evaluated them on HalfCheetah by adopting a low
+performance policy, rather than the random policy, as the
+behavior policy. We pre-trained the low performance policy
+with a small number of iterations using PPO.
+We observed that, compared with Table IV and Table VII,
+the performance of the behavior policy affected the PP of the
+resulting target agent. Here, the PP of the random policy was
+−1230.01 and that of the low performance policy was 822.39.
+Therefore, the PP of our proposed method was improved from
+968.49 ± 99.53 to 1527.37 ± 133.19 when |I| = 50.
+These results indicate that owing to the low performance
+of the random policy, faster-running states, that is, states with
+high velocity, cannot be observed; in other words, the random
+TABLE VII
+RESULTS OF HALFCHEETAH WHEN T τ WAS COLLECTED BY THE LOW
+PERFORMANCE POLICY. THE BEHAVIOR POLICY HAS PP=822.39 AND πσ
+HAS PP=2735.73. THE TRAJECTORY FOR CALCULATING MD IS
+COLLECTED BY THE LOW PERFORMANCE POLICY.
+Method
+MD
+PP
+|I| = 0
+Zhang et al.
+1.03 ± 0.19
+−1241.31 ± 523.06
+|I| = 10
+CRAR
+0.74 ± 0.06
+−1549.82 ± 106.04
+Ours w/o AL
+0.09 ± 0.02
+1117.05 ± 127.93
+Ours
+0.06 ± 0.00
+1145.77 ± 110.29
+|I| = 50
+CRAR
+0.37 ± 0.04
+−1416.37 ± 194.31
+Ours w/o AL
+0.03 ± 0.00
+1393.11 ± 226.39
+Ours
+0.02 ± 0.00
+1527.37 ± 133.19
+policy can only observe a limited state. This limitation could
+lead to an increase in the approximation error of ˆF. This
+implied that image-to-semantics is affected by the performance
+of the behavior policy in some tasks.
+A promising result for the image-to-semantics framework
+is that the target agents obtained by our approach outperform
+the behavior policy. In particular, in Table VII, the PP of
+the behavior policy is 822.39; furthermore, when image-to-
+semantics was performed with 50 annotations (|I| = 50),
+we obtained a PP of 1527.37 ± 133.19. In other words, we
+could achieve a higher performance compared with that of the
+behavior policy using a small number of annotations and the
+image-to-semantics protocol.
+In the previous discussion, we found that the PP achieved by
+the image-to-semantics framework is affected by the quantity
+and quality of paired data, and the region of state space
+comprising the dataset for training ˆF. In fact, as an extreme
+example, ˆF trained using |P|=100k with the trajectories col-
+lected by the optimal policy, achieved PP=2624.65 ± 29.09,
+which is almost identical to PP=2735.73, the performance of
+the optimal source policy. Note that such a near-complete
+policy transfer is already achieved in Shooting and KUKA,
+as shown in Tables II and III.
+VI. CONCLUSION
+In this study, we investigated the image-to-semantics prob-
+lem for vision-based agents in robotics. Using paired data for
+learning image-to-semantics mapping is favorable for achiev-
+ing high-performance policy transfer; however, the cost of
+creating paired data cannot be ignored. This study contributes
+to existing literature by reducing the annotation cost using
+two techniques: pair augmentation and active learning. We
+also confirmed the effectiveness of the proposed method in
+our experiments.
+In future work, we must address the following limitations:
+(1) Experiments have not been conducted using actual robots;
+therefore, it is not known how difficulties specific to actual
+robots will affect the image-to-semantics performance; (2)
+We cannot always freely query Trσ; therefore, it would be
+beneficial to know if we can substitute the one learned using
+source trajectory, similar to [18], [23]; (3) In some cases, the
+
+transition error is too large, and we would like to be able
+to improve the approximation accuracy of ˆF by considering
+performing pair augmentation for {t | t ∈ Ei, t ≤ K} rather
+than Ei. This expectation is because augmented semantics with
+larger t ∈ Ei are inaccurate. Furthermore, we would like to find
+a way to automatically determine such a K.
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+arXiv:2301.11685v1  [math.FA]  27 Jan 2023
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION
+OPERATORS ON TWO DOMAINS
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND MICHAEL SPECKBACHER
+Abstract. We derive eigenvalue estimates for concentration operators asso-
+ciated with the discrete Fourier transform and two concentration domains sat-
+isfying certain regularity conditions. These conditions are met, for example,
+when the discrete domain, contained in a lattice, is obtained by discretization
+of a suitably regular domain in the Euclidean space. As a limit, we obtain
+eigenvalue estimates for Fourier concentration operators associated with two
+suitably regular domains in the Euclidean space. The proof builds on Israel’s
+work on one dimensional intervals: arXiv:1502.04404v1.
+1. Introduction and results
+Fourier concentration operators act by incorporating a spatial cut-off and a
+subsequent frequency cut-off to the Fourier inversion formula. The chief example
+concerns the Fourier transform on the Euclidean space F : L2(Rd) → L2(Rd), the
+cut-offs are given by the indicator functions of two compact domains E, F ⊆ Rd,
+and the concentration operator is
+Sf = χFF −1χEFχFf,
+f ∈ L2(Rd).
+(1.1)
+These operators, and their analogues defined with respect to the discrete Fourier
+transform L2([−1/2, 1/2]d) → ℓ2(Zd) play a crucial role in many analysis problems
+and fields of application [18, 13, 14, 7], such as imaging, where the shapes of E, F
+are dictated by various acquisition constraints.
+The basic intuition is that the concentration operator (1.1) is approximately a
+projection with rank tr(S) = |E| · |F|. The error of such heuristic is encoded by
+the so-called plunge region
+Mε(S) = {λ ∈ σ(S) : ε < λ < 1 − ε},
+ε ∈ (0, 1/2),
+(1.2)
+consisting of intermediate eigenvalues. Asymptotics for the cardinality of Mε(S)
+go back to Landau and Widom [15, 12] for the case of one dimensional intervals
+E = [−a, a], F = [−b, b] and read
+#Mε(S) = c · log(ab) · log( 1−ε
+ε ) + o(log(ab)),
+as ab → ∞,
+(1.3)
+for an explicit constant c that depends on the normalization of the Fourier trans-
+form. The modern spectral theory of Wiener-Hopf operators gives similar asymp-
+totics for concentration operators associated to rather general multi-dimensional
+domains subject to increasing isotropic dilations.
+2010 Mathematics Subject Classification. 47B35, 47A75, 42B35, 42C40.
+Key words and phrases. Discrete Fourier transform, concentration operator, Hankel operator,
+eigenvalue, spectrum.
+The authors gratefully acknowledge support from the Austrian Science Fund (FWF): Y 1199.
+1
+
+2
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+While (1.3) precisely describes the cardinality of the set Mε(S) in the limit
+ab → ∞, the asymptotic is often insufficient for many purposes because of the
+quality of the error terms. Indeed, the error term in (1.3) depends in an unspecified
+way on the spectral threshold ǫ, which precludes applications where ε is let to vary
+with the domains E, F. Such limitations have motivated a great amount of work
+aimed at deriving upper bounds for #Mε(S) that are threshold robust, that is,
+bounds that are effective for concrete concentration domains and explicit in their
+dependence on the spectral threshold [8, 10, 20, 11, 3, 17], significantly improving
+on more classical results in this spirit [19].
+With the exception of [8], the mentioned articles on threshold-robust spectral
+bounds for Fourier concentration operators concern only the one dimensional case,
+because they exploit a connection with a Sturm–Liouville equation which is spe-
+cific of that setting. On the other hand, while [8] studies Fourier concentration
+operators associated with one dimensional intervals, the technique introduced by
+Israel is very general, as it relies on an explicit almost diagonalization of the
+concentration operator. In fact, as we were finishing this work, the preprint [9]
+provided an extension of [8] to higher dimensions (see Sections 1.1 and 1.5).
+In this article we derive upper bounds for the number of intermediate eigen-
+values (1.2) associated with either the continuous or discrete Fourier transforms.
+We obtain estimates that apply to two suitably regular multi-dimensional spatial
+and frequency domains. The proofs build on Israel’s technique [8] and combine it
+with arguments from geometric measure theory and operator theory.
+1.1. The Euclidean space. Given two compact sets E, F ⊆ Rd, the Fourier
+concentration operator S : L2(Rd) → L2(Rd) is defined by (1.1) where F denotes
+the Fourier transform
+Ff(ξ) =
+�
+Rd f(x)e−2πixξ dx.
+(1.4)
+A set E ⊆ Rd is said to have a maximally Ahlfors regular boundary if there exists
+a constant κ∂E > 0 such that
+Hd−1�
+∂E ∩ Br(x)
+�
+≥ κ∂E · rd−1,
+0 < r ≤ Hd−1(∂E)1/(d−1),
+x ∈ ∂E.
+Here, Hd−1 denotes the (d−1)-dimensional Hausdorff measure. The term maximal
+in the definition refers to the range of r for which the estimate is required to hold.
+See Section 2 for more context on Ahlfors regularity. In what follows, we denote
+for short |∂E| = Hd−1�
+∂E).
+In this article we prove the following.
+Theorem 1.1. Let E, F ⊆ Rd, d ≥ 2, be compact domains with maximally
+Ahlfors regular boundaries with constants κ∂E, κ∂F respectively, and assume that
+that |∂E||∂F| ≥ 1. Consider the concentration operator (1.1) and its eigenvalues
+{λn : n ∈ N}.
+Then for every α ∈ (0, 1/2), there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2):
+#
+�
+n ∈ N : λn ∈ (ε, 1 − ε)
+�
+≤ Aα,d · |∂E|
+κ∂E
+· |∂F|
+κ∂F
+· log
+�|∂E||∂F|
+κ∂E ε
+�2d(1+α)+1
+.
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+3
+A closely related result is presented in the recent preprint [9]. For F = [0, 1]d and
+E = rK, where r > 0 is a dilation parameter and K ⊆ Rd is a convex, coordinate
+symmetric domain [9, Theorem 1.1] gives the following bound for ε ∈ (0, 1/2):
+#
+�
+n ∈ N : λn ∈ (ε, 1 − ε)
+�
+≤ Cd · max{rd−1 log(r/ε)3, log(r/ε)3d}.
+(1.5)
+For large r, the right-hand side of (1.5) becomes Od
+�
+rd−1 log(r/ε)3) while Theo-
+rem 1.1 gives the weaker bound Oα,d
+�
+rd−1 log(r/ε)2d(α+1)+1�
+. On the other hand,
+Theorem 1.1 applies to possibly non-convex, non-coordinate-symmetric and non-
+dilated domains E, and other regular domains F besides cubes. (As pointed out
+in [9], when E and F are both cubes, even slightly stronger estimates hold, c.f.
+[9, Theorem 1.2].)
+Our work is in great part motivated by applications where concentration do-
+mains may be non-convex. For example, noise statistics are often estimated from
+those pixels of a square image located outside a central disk, which is assumed to
+contain the signal of interest (see, e.g., [2]). Thus, the need to sample pure noise
+leads one to consider the complement of a disk within a two dimensional square
+as concentration domain (or, more realistically, the set of grid points within that
+domain; see below). Such a domain E is allowed by Theorem 1.1 (and Theorems
+1.2 and 1.3 below) and has moreover a favorable regularity constant κ∂E.
+1.2. Discretization of continuous domains. Theorem 1.1 is obtained by tak-
+ing a limit on a more precise result concerning a discrete setting, which is our
+main focus.
+We consider a resolution parameter L > 0 and define the discrete Fourier
+transform FL : L2((−L/2, L/2)d) → ℓ2(L−1Zd) by
+FLf(k/L) =
+�
+(−L/2,L/2)d f(x)e−2πixk/Ldx,
+k ∈ Zd.
+(1.6)
+We think of L as a discretization parameter for an underlying continuous problem.
+Let us define the discretization at resolution L > 0 of a domain E ⊆ Rd by
+EL = L−1Zd ∩ E.
+Given two compact domains E ⊆ Rd and F ⊆ (−L/2, L/2)d, consider the dis-
+cretized concentration operator T : L2(F) → L2(F) given by
+T = χFF −1
+L χELFL.
+(1.7)
+Our second result reads as follows.
+Theorem 1.2. Let E, F ⊆ Rd, d ≥ 2, be compact domains with maximally
+Ahlfors regular boundaries with constants κ∂E, κ∂F respectively, and assume that
+that |∂E||∂F| ≥ 1.
+Fix a discretization resolution L ≥ |∂E|−1/(d−1) such that F ⊆ (−L/2, L/2)d
+and consider the discretized concentration operator (1.7) and its eigenvalues {λn :
+n ∈ N}.
+Then for every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2):
+#
+�
+n ∈ N : λn ∈ (ε, 1 − ε)
+�
+≤ Aα,d · |∂E|
+κ∂E
+· |∂F|
+κ∂F
+· log
+�|∂E||∂F|
+κ∂E ε
+�2d(1+α)+1
+.
+
+4
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+1.3. The discrete Fourier transform. Finally, we consider a discrete concen-
+tration problem associated with the usual discrete Fourier transform, denoted
+F1 : L2((−1/2, 1/2)d) → ℓ2(Zd)
+for consistency with (1.6).
+Given a finite set Ω ⊆ Zd and F ⊆ (−1/2, 1/2)d, the discrete Fourier concen-
+tration operator T : L2(F) → L2(F) is defined as
+T = χFF −1
+1 χΩF1.
+(1.8)
+The discrete boundary of a set Ω ⊆ Zd is given by
+∂Ω = {k ∈ Ω : min{|j − k| : j ∈ Zd ∖ Ω} = 1}.
+(1.9)
+We say that Ω ⊆ Zd has a maximally Ahlfors regular boundary if there exists a
+constant κ∂Ω such that
+inf
+k∈∂Ω #
+�
+∂Ω ∩ k + [−n/2, n/2)d�
+≥ κ∂Ω · nd−1,
+1 ≤ n ≤ (#∂Ω)1/(d−1), k ∈ ∂Ω.
+(Note the slight notational abuse: though Ω ⊆ Zd ⊆ Rd, the notions of boundary
+and boundary regularity are to be understood in the discrete sense.)
+Our last result reads as follows.
+Theorem 1.3. Let d ≥ 2, Ω ⊆ Zd a finite set with maximally Ahlfors regular
+boundary and constant κ∂Ω. Let F ⊆ (−1/2, 1/2)d be compact with maximally
+Ahlfors regular boundary and constant κ∂F. Assume that #∂Ω · |∂F| ≥ 1, and
+consider the concentration operator (1.8) and its eigenvalues {λn : n ∈ N}.
+Then for every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such that for ε ∈ (0, 1/2):
+#
+�
+n ∈ N : λn ∈ (ε, 1 − ε)
+�
+≤ Aα,d · #∂Ω
+κ∂Ω
+· |∂F|
+κ∂F
+· log
+�#∂��� · |∂F|
+κ∂Ω ε
+�2d(1+α)+1
+.
+1.4. One sided estimates. Finally, we remark that bounds on the number of
+intermediate eigenvalues, as in Theorems 1.1, 1.2 and 1.3, can be equivalently
+formulated in terms of the distribution function
+Nε := {n ∈ N : λn > ε},
+ε ∈ (0, 1).
+Remark 1.4. For example, for ε ∈ (0, 1) under the assumptions of Theorem 1.1
+we have
+��#Nε(S) − |E| · |F|
+�� ≤ Cα,d · |∂E|
+κ∂E
+· |∂F|
+κ∂F
+· log
+�
+|∂E||∂F|
+κ∂E min{ε, 1 − ε}
+�2d(1+α)+1
+.
+(1.10)
+See Section 8 for details.
+1.5. Methods and related literature. We work for the most part with the
+discrete Fourier transform and then obtain consequences for the continuous one
+by a limiting argument. Theorem 1.2 is proved in two steps. We first revisit Israel’s
+argument [8] and adapt it to prove eigenvalue estimates when one of the domains
+is a rectangle and the other is a general multi-dimensional domain (Theorem 4.1
+below). These estimates are slightly stronger than those in Theorem 1.2, and
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+5
+the extra precision is exploited in the subsequent step. We follow the method of
+almost diagonalization with wavepackets, which we achieve, unlike [8], through a
+redundant system (frame) instead of an orthonormal basis.
+The second step is a decomposition, rescaling, and dyadic approximation ar-
+gument, implemented by means of p-Schatten norm estimates for certain Hankel
+operators, and especially by quantifying those estimates as a function of p, as
+p → 0+.
+Our intermediate result, Theorem 4.1, is close in spirit to Theorem 1.2 in [9]
+(which appeared as we were finishing this article). The estimates in [9], formulated
+in the context of the continuous Fourier transform and concerning dilated convex
+domains, are stronger than what follows from Theorem 4.1 in that regime, as
+[9, Theorem 1.2] involves smaller powers of a certain logarithmic factor (see also
+Section 1.1 and (1.5)).
+On the other hand, Theorem 4.1 concerns sufficiently
+regular, non-dilated and possibly non-convex domains, and covers the discrete
+Fourier transform.
+We also mention our recent work on concentration operators for the short-time
+Fourier transform [16], that also makes use of Ahlfors regularity and Schatten
+norm estimates. Though the goals and results are philosophically similar to those
+in the present article, the settings are rather different from the technical point of
+view. Indeed, the arguments used in [16] rely on the rapid off-diagonal decay of
+the reproducing kernel of the range of the short-time Fourier transform, and do
+not seem to be applicable to Fourier concentration operators.
+The remainder of the article is organized as follows.
+Section 2 sets up the
+notation and provides background on boundary regularity. Section 3 revisits the
+technique from [8] and implements certain adaptations. These are used in Section
+4 to prove Theorem 4.1. Theorem 1.2 is proved in Section 5, Theorem 1.1 is proved
+in Section 6, and Theorem 1.3 is proved in Section 7. Remark 1.4 is proved in
+Section 8.
+2. Preliminaries
+2.1. Notation. We shall focus on Theorem 1.2 and set up the notation accord-
+ingly.
+Theorems 1.1 and 1.3 will be obtained afterwards as an application of
+Theorem 1.2.
+We denote cubes by Qa = [−a/2, a/2)d. The Euclidean norm on Rd is denoted
+| · |. For two non-negative functions f, g we write f ≲ g if there exist a constant
+C such that f(x) ≤ Cg(x), and write f ≍ g is f ≲ g and g ≲ f. The implied
+constant is allowed to depend on the dimension d and the parameter α from
+Theorems 1.1, 1.2 and 1.3, but not on other parameters.
+We enumerate the eigenvalues of a compact self adjoint operator L : H → H
+acting on a Hilbert space H as follows:
+λk = inf{∥L − S∥ : S ∈ L(L2(Rd)), dim(Range(S)) < k},
+k ≥ 1.
+(2.1)
+Then {λk : k ≥ 1} ∖ {0} = σ(L) ∖ {0} as sets with multiplicities — see, e.g., [5,
+Lemma 4.3].
+
+6
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+For a set E ⊆ Rd, we write
+Ec
+L = L−1Zd ∖ EL
+and ∂EL for the points in EL which are at distance L−1 of Ec
+L. For L = 1 this is
+consistent with (1.9).
+We will work with the discrete Fourier transform FL : L2((−L/2, L/2)d) →
+ℓ2(L−1Zd) given by (1.6) and reserve the notation Ff or �f for the continuous
+Fourier transform (1.4). Note that if supp(f) ⊆ (−L/2, L/2)d, then FLf(k/L) =
+Ff(k/L) for every k ∈ Zd.
+We also write PE,L = F −1
+L χELFL. For F ⊆ (−L/2, L/2)d we define the operator
+T = TE,F,L : L2(F) → L2(F) by
+T = TE,F,L = χFPE,L
+and let λn = λn(T) denote its eigenvalues as in (2.1). An easy computation shows
+that
+Tt−1E,tF,tL = Mt−1TE,F,LMt,
+t > 0,
+where Mt denotes the dilation operator
+Mtf(x) = f(tx).
+In particular,
+λn(Tt−1E,tF,tL) = λn(TE,F,L),
+n ∈ N.
+(2.2)
+2.2. Boundary regularity. Let us introduce regularity of sets in more generality
+and discuss a few properties.
+An Hd−1-measurable set X ⊆ Rd is said to be lower Ahlfors (d − 1)-regular
+(regular for short) at scale ηX > 0 if there exists a constant κX > 0 such that
+Hd−1�
+X ∩ Br(x)
+�
+≥ κX · rd−1,
+0 < r ≤ ηX,
+x ∈ X.
+Note that if X ⊆ Rd is regular at scale ηX > 0 with constant κX > 0 and
+t > 0, then tX ⊆ Rd is regular at scale ηtX = tηX with constant κtX = κX. By
+differentiation around a point of positive Hd−1-density,
+κX ≤ cd,
+(2.3)
+for any regular X of positive Hd−1-measure. We also mention that if X is regular
+with parameters ηX and κX, then choosing an arbitrary x ∈ X gives
+Hd−1�
+X
+�
+≥ Hd−1�
+X ∩ BηX(x)
+�
+≥ κX · ηd−1
+X .
+(2.4)
+We shall use the following basic result, derived from [4].
+Lemma 2.1. There exists a universal constant Cd > 0 such that for every compact
+set X ⊆ Rd that is regular at scale ηX > 0 with constant κX and every s > 0,
+|X + Bs(0)| ≤ Cd
+κX
+· Hd−1(X) · s ·
+�
+1 + sd−1
+ηd−1
+X
+�
+.
+Proof. From [4, Theorems 5 and 6] it follows that
+Hd−1�
+{x : d(x, X) = r}
+�
+≤ Cd
+κX
+· Hd−1(X) ·
+�
+1 + rd−1
+ηd−1
+X
+�
+,
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+7
+for almost every r > 0, and in addition, |∇d(x, X)| = 1, for almost every x ∈ Rd.
+From this and the coarea formula, it follows that
+|X + Bs(0)| =
+�
+Rd χ[0,s)(d(x, X))dx =
+�
+Rd χ[0,s)(d(x, X))|∇d(x, X)|dx
+=
+� s
+0
+Hd−1�
+{x : d(x, X) = r}
+�
+dr ≤ Cd
+κX
+Hd−1(X)
+� s
+0
+�
+1 + rd−1
+ηd−1
+X
+�
+dr
+≤ Cd
+κX
+Hd−1(X)s
+�
+1 + sd−1
+ηd−1
+X
+�
+.
+□
+Corollary 2.2. For E ⊆ Rd a compact domain with regular boundary at scale
+η∂E ≥ 1 with constant κ∂E and a discretization resolution L ≥ 1, we have
+L−d#EL ≲ |E| + |∂E|
+κ∂EL.
+In particular, for d ≥ 2
+L−d#EL ≲ max{|∂E|d/(d−1), 1}
+κ∂E
+.
+Proof. Recall that QL−1 = L−1[−1/2, 1/2)d and define E′
+L = {m ∈ EL :
+m +
+QL−1 ⊆ E}. From Lemma 2.1, we get
+L−d#EL =
+���
+�
+m∈E′
+L
+m + QL−1
+��� +
+���
+�
+m∈EL∖E′
+L
+m + QL−1
+��� ≤ |E| + |∂E + BL−1√
+d(0)|
+≲ |E| + |∂E|
+κ∂EL.
+Finally, the second inequality follows from the isoperimetric inequality |E| ≲
+|∂E|d/(d−1) and (2.3).
+□
+3. Israel’s argument revisited
+We now revisit the core argument of [8] and make a few adaptations.
+3.1. Israel’s lemma. We need a slight generalization of Lemma 1 in [8], phrasing
+it in terms of frames rather than orthonormal bases. We include a proof for the
+sake of completeness.
+Recall that a frame for a Hilbert space H is a subset of vectors {φi}i∈I for which
+the exist constants 0 < A, B < ∞ — called lower and upper frame bounds —
+such that
+A∥f∥2 ≤
+�
+i∈I
+|⟨f, φi⟩|2 ≤ B∥f∥2,
+f ∈ H.
+Lemma 3.1. Let T : H → H be a positive, compact, self-adjoint operator on a
+Hilbert space H with ∥T∥ ≤ 1 and eigendecomposition T = �
+n≥1 λn⟨·, fn⟩fn.
+Let {φi}i∈I be a frame of unit norm vectors for H with lower frame bound A.
+If I = I1 ∪ I2 ∪ I3, and
+(3.1)
+�
+i∈I1
+∥Tφi∥2 +
+�
+i∈I3
+∥(I − T)φi∥2 ≤ A
+2 ε2,
+
+8
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+then #Mε(T) ≤ 2
+A#I2, where Mε(T) is defined as in (1.2).
+Proof. Let Sε = span{fn :
+λn ∈ (ε, 1 − ε)} and let Pε : H → Sε denote the
+orthogonal projection onto Sε. For f ∈ Sε one has ∥f∥ − ∥(I − T)f∥ ≤ ∥Tf∥ ≤
+(1 − ε)∥f∥, which shows
+ε∥f∥ ≤ ∥Tf∥,
+and
+ε∥f∥ ≤ ∥(I − T)f∥,
+f ∈ Sε.
+Note that T and Pε commute since Sε is spanned by a collection of eigenvectors
+of T. Therefore, by (3.1) we obtain
+�
+i∈I1∪I3
+ε2∥Pεφi∥2 ≤
+�
+i∈I1
+∥TPεφi∥2 +
+�
+i∈I3
+∥(I − T)Pεφi∥2 ≤ A
+2 ε2,
+which implies
+(3.2)
+�
+i∈I1∪I3
+∥Pεφi∥2 ≤ A
+2 .
+Using the frame property we get for f ∈ Sε:
+A∥f∥2 ≤
+�
+i∈I
+|⟨f, φi⟩|2 =
+�
+i∈I
+|⟨f, Pεφi⟩|2.
+Now assume that dim(Sε) ≥ 1 (otherwise the result is trivial), take an orthonormal
+basis {ψk}dim(Sε)
+k=1
+of Sε, and sum the inequality above over all basis elements to
+derive
+A · #Mε(T) = A · dim(Sε) = A
+dim(Sε)
+�
+k=1
+∥ψk∥2 ≤
+dim(Sε)
+�
+k=1
+�
+i∈I
+|⟨ψk, φi⟩|2
+=
+�
+i∈I
+∥Pεφi∥2 ≤
+�
+i∈I2
+∥φi∥2 + A
+2 = #I2 + A
+2 ≤ #I2 + A
+2 #Mε(T),
+where in the second line we used (3.2). This shows that #Mε(T) ≤ 2
+A#I2.
+□
+3.2. Local trigonometric frames. In this section, we construct a tight frame
+that allows us to apply Lemma 3.1.
+Let α > 0, and θ ∈ C∞(R) be such that
+(i) θ(x) = 1, for x ≥ 1, and θ(x) = 0, for x ≤ −1,
+(ii) θ(−x)2 + θ(x)2 = 1, for every x ∈ R,
+(iii) |Dkθ(x)| ≤ Ck
+αk(1+α)k, for all k ∈ N0, all x ∈ R, and a constant Cα > 0.
+See, for example, [8, Proposition 1] or [6, Chapter 1] for the existence of such a
+function.
+Let W > 0. We decompose the interval
+�
+− W
+2 , W
+2
+�
+into disjoint intervals
+Ij = xj +
+W
+3 · 2|j|+1[−1, 1),
+j ��� Z,
+where
+xj = sign(j)W
+2
+�
+1 − 1
+2|j|
+�
+.
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+9
+Note that |Ij| = |I|j|| = 2|I|j|+1| for every j ∈ Z. We will also denote Dj = Ij∪Ij+1.
+Now define
+θj(x) = θ
+�2(x − xj)
+|Ij|
+�
+θ
+�
+−2(x − xj+1)
+|Ij+1|
+�
+.
+We have that θj(x) = 0 for x /∈ Dj, and furthermore by properties (i) and (ii)
+∥θj∥2
+2 =
+�
+Ij
+θ
+�2(x − xj)
+|Ij|
+�2
+dx +
+�
+Ij+1
+θ
+�
+−2(x − xj+1)
+|Ij+1|
+�2
+dx
+= |Ij|
+2
+� 1
+−1
+θ(x)2dx + |Ij+1|
+2
+� 1
+−1
+θ(x)2dx = |Dj|
+2 .
+We define the set of vectors
+(3.3)
+φj,k(x) =
+�
+2
+|Dj| · θj(x) · exp
+�
+2πi xk
+|Dj|
+�
+,
+j, k ∈ Z,
+and note that ∥φj,k∥2 = 1.
+Lemma 3.2. The family {φj,k}j,k∈Z defined in (3.3) forms a tight frame for
+L2(−W/2, W/2) with frame constants A = B = 2.
+Proof. We write fj := f|Ij so that f = �
+j∈Z fj. Since supp(θj) ⊆ Dj = Ij ∪ Ij+1,
+we observe
+�
+j,k∈Z
+|⟨f, φj,k⟩|2 =
+�
+j,k∈Z
+|⟨fj + fj+1, φj,k⟩|2 .
+As
+�
+|Dj|−1/2 exp (2πikx/|Dj|)
+�
+k∈Z is an orthonormal basis for L2(Dj), we find
+that
+�
+k∈Z
+|⟨fj + fj+1, φj,k⟩|2 = 2∥(fj + fj+1)θj∥2
+2 = 2∥fjθj∥2
+2 + 2∥fj+1θj∥2
+2.
+Combining both identities and using property (ii), we conclude
+�
+j,k∈Z
+|⟨f, φj,k⟩|2 = 2
+�
+j∈Z
+�
+∥fjθj∥2
+2 + ∥fj+1θj∥2
+2
+�
+= 2
+�
+j∈Z
+�
+∥fjθj∥2
+2 + ∥fjθj−1∥2
+2
+�
+= 2
+�
+j∈Z
+�
+Ij
+|f(x)|2
+�
+θ
+�2(x − xj)
+|Ij|
+�2
++ θ
+�
+−2(x − xj)
+|Ij|
+�2�
+dx
+= 2
+�
+j∈Z
+∥fj∥2
+2 = 2∥f∥2
+2.
+□
+Let 0 < Wi ≤ L, i = 1, ..., d, and consider �d
+i=1(−Wi/2, Wi/2). Set also
+Wmax := max
+i=1,...,dWi.
+We define a frame for L2� �d
+i=1(−Wi/2, Wi/2)
+�
+via the tensor product
+Φj,k(x) = Φj1,...,jd,k1,...kd(x1, ..., xd) = φj1,k1(x1) · . . . · φjd,kd(xd),
+where each family {φji,ki(xi)}ji,kk∈Z is the frame for L2(−Wi/2, Wi/2) given by
+(3.3). This construction also yields a tight frame with frame bounds equal to 2d.
+
+10
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+3.3. Energy estimates. Consider
+ψj(x) = θj
+�
+|Dj|x + xj −
+W
+3 · 2|j|+1
+�
+,
+x ∈ R, j ∈ Z.
+A straightforward computation shows that ψj is supported on [0, 1] and satisfies
+|Dkψj(x)| ≤ �
+Cα
+kk(1+α)k by property (iii). As shown in [8, Lemma 1] it thus follows
+that |�
+ψj(ξ)| ≤ Aα · exp
+�
+−aα|ξ|(1+α)−1�
+. Since 1 − α ≤ (1 + α)−1, we derive that
+t(1+α)−1 ≥ t1−α − 1 for t ≥ 0. Adjusting the constant Aα, we therefore get
+(3.4)
+|�θj(ξ)| ≤ Aα · |Dj| · exp
+�
+−aα(|Dj| · |ξ|)1−α�
+,
+ξ ∈ R.
+With this at hand, we estimate the decay of
+F(Φj,k)(ξ) = 2d/2
+d
+�
+i=1
+|Dji|−1/2 · �
+θji
+�
+ξi −
+ki
+|Dji|
+�
+,
+ξ ∈ Rd.
+Define
+Mj = diag(|Dj1|, ..., |Djd|) ∈ Rd×d.
+By (3.4) (possibly enlarging Aα) and |ξ|1−α ≤ �d
+i=1 |ξi|1−α, it follows
+|F(Φj,k)(ξ)| ≤ Ad
+α
+d
+�
+i=1
+|Dji|1/2 · exp
+�
+−aα
+��|Dji|ξi − ki
+��1−α�
+≤ Ad
+α · det(Mj)1/2 · exp
+�
+−aα
+��Mj(ξ − ξj,k)
+��1−α�
+,
+(3.5)
+where (ξj,k)i = ki|Dji|−1.
+Consider now a compact domain E ⊆ Rd. Let s ≥ 1 be a parameter that will be
+determined later. For j ∈ Zd fixed, we cover the index set Zd with three subsets
+as follows
+Llow
+j
+:=
+�
+k ∈ Zd : dist(k, MjEc
+L) ≥ s
+�
+;
+Lmed
+j
+:=
+�
+k ∈ Zd : dist(k, MjEL) < s, and dist(k, MjEc
+L) < s};
+Lhigh
+j
+:=
+�
+k ∈ Zd : dist(k, MjEL) ≥ s
+�
+.
+(3.6)
+(Here, dist is associated with the usual Euclidean distance.) We claim that
+Lmed
+j
+⊆
+�
+k ∈ Zd : dist(k, Mj∂EL) < s
+�
+;
+(3.7)
+let us briefly sketch an argument. Fix k ∈ Lmed
+j
+and let k0 ∈ MjL−1Zd minimize
+the distance to k. For any point x ∈ MjL−1Zd with |k − x| < s we can build a
+path of adjacent points in MjL−1Zd from x to k0 such that the distance to k is
+decreasing. In particular, choosing x1 ∈ MjEL and x2 ∈ MjEc
+L at distance less
+than s from k, we can connect x1 and x2 through a path of adjacent points in
+MjL−1Zd that stays at distance less than s from k. Necessarily, one of the points
+in the path must belong to Mj∂EL, which proves (3.7).
+The indices (j, k) with k ∈ Lmed
+j
+, and j satisfying a condition specified below
+(see (3.10)) will play the role of I2 in Lemma 3.1, so we need to estimate #(Lmed
+j
+).
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+11
+Lemma 3.3. Let E ⊆ Rd be a compact domain with regular boundary at scale
+η∂E ≥ 1 and constant κ∂E. Let L ≥ Wmax and s ≥ 1. Then for all j ∈ Zd we have
+#
+�
+k ∈ Zd : dist(k, Mj∂EL) < s
+�
+≲ max{Wmax, 1/η∂E}d−1 · |∂E|
+κ∂E
+· sd.
+Proof. Since for every x ∈ Rd the cube x + Q1 contains one point in Zd,
+#
+�
+k ∈ Zd : dist(k, Mj∂EL) < s} ≲ sd · #
+�
+k ∈ Zd : k ∈ Mj∂EL + Q1}.
+(3.8)
+For x ∈ R, we denote the unique integer in x + [−1/2, 1/2) by x∗. For a set
+X = {x1, ..., xN} ⊆ R and 0 < a ≤ 1, we write yi = axi. It is easy to check that if
+x∗
+i = x∗
+i′, then |y∗
+i − y∗
+i′| ≤ 1. Using this, a straightforward argument shows that
+#
+�
+k ∈ Z : k ∈ aX + [−1/2, 1/2)} ≤ 2#
+�
+k ∈ Z : k ∈ X + [−1/2, 1/2)}.
+From (3.8), if we apply the inequality above componentwise (noting that (Mj)i,i ≤
+Wmax), we obtain for s ≥ 1
+#
+�
+k ∈ Zd : dist(k, Mj∂EL) < s}
+≲ sd · #
+�
+k ∈ Zd : k ∈ Wmax∂EL + Q1}.
+Since for every x ∈ ∂EL there exists x′ ∈ ∂E such that |x − x′| ≤ L−1, and
+Wmax/L ≤ 1, it follows that
+#
+�
+k ∈ Zd : dist(k, Mj∂EL) < s} ≲ sd · #
+�
+k ∈ Zd : k ∈ Wmax∂E + Q3}
+=: sd · #(KWmax).
+(3.9)
+Now let k ∈ KWmax.
+There exists at least one point xk ∈ ∂E such that k ∈
+Wmaxxk + Q3. In particular, we have that xk ∈ W −1
+maxk + Q4/Wmax. Therefore, for
+every k ∈ KWmax we get by regularity of ∂E
+κ∂E · min{W −1
+max, η∂E}d−1 ≤ Hd−1�
+∂E ∩ B1/Wmax(xk)
+�
+≤ Hd−1�
+∂E ∩ xk + Q2/Wmax
+�
+≤ Hd−1�
+∂E ∩ W −1
+maxk + Q6/Wmax
+�
+.
+So,
+κ∂E · min
+�
+W −1
+max, η∂E
+�d−1 · #(KWmax) ≤
+�
+k∈KWmax
+Hd−1�
+∂E ∩ W −1
+maxk + Q6/Wmax
+�
+≲
+�
+k∈Zd
+Hd−1�
+∂E ∩ W −1
+maxk + Q1/Wmax
+�
+= Hd−1(∂E).
+Plugging this estimate into (3.9) completes the proof.
+□
+Next, for a compact domain E ⊆ Rd and a parameter s ≥ 1 we recall the sets
+(3.6), introduce a second auxiliary parameter 0 < δ < 1, and define the following
+covering of Z2d:
+Γlow :=
+�
+(j, k) : min
+i
+|Dji| ≥ δ, k ∈ Llow
+j
+�
+;
+Γmed :=
+�
+(j, k) : min
+i
+|Dji| ≥ δ, k ∈ Lmed
+j
+�
+;
+Γhigh :=
+�
+(j, k) : min
+i
+|Dji| ≥ δ, k ∈ Lhigh
+j
+�
+∪
+�
+(j, k) : min
+i
+|Dji| < δ, k ∈ Zd�
+.
+(3.10)
+
+12
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+Lemma 3.4. Under the conditions of Lemma 3.3, let 0 < δ < 1 and consider the
+set Γmed from (3.10). Then
+#(Γmed) ≲ max{Wmax, 1/η∂E}d−1 · |∂E|
+κ∂E
+· max{log(Wmax/δ), 1}d · sd.
+Proof. By (3.7) and Lemma 3.3 it follows
+#(Γmed) =
+�
+j∈Zd
+min |Dji|≥δ
+#(Lmed
+j
+) ≲
+�
+j∈Zd
+min |Dji|≥δ
+max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+sd.
+In each coordinate, we have that the number of intervals Dji for which |Dji| ≥ δ
+is bounded by C max{log(Wmax/δ), 1}. Hence, we arrive at
+#(Γmed) ≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+max{log(Wmax/δ), 1}dsd.
+□
+Lemma 3.5. Let d ≥ 2, L ≥ max{Wmax, 1}, and E ⊆ Rd be a compact domain
+with regular boundary at scale η∂E ≥ 1 with constant κ∂E and such that |∂E| ≥ 1.
+Let s ≥ 1 and δ ∈ (0, 1) be parameters and consider the sets from (3.10). Then
+there exists a constant c = cα > 0 such that
+�
+(j,k)∈Γlow
+L−d �
+m∈Ec
+L
+|F(Φj,k)(m)|2
+≲ max{Wmax, 1/η∂E}d−1 · |∂E|
+κ∂E
+· exp
+�
+− cs1−α�
+max{log(Wmax/δ), 1}d,
+(3.11)
+and
+�
+(j,k)∈Γhigh
+L−d �
+m∈EL
+|F(Φj,k)(m)|2 ≲max{Wmax, 1/η∂E}d−1
+κ∂E
+·
+�
+|∂E|
+d
+d−1 · δ
++ |∂E| · exp
+�
+− cs1−α�
+· max{log(Wmax/δ), 1}d�
+.
+(3.12)
+Proof. For j ∈ Zd and l ∈ N0 we set
+Llow
+j,l =
+�
+k ∈ Zd : dist(k, MjEc
+L) ∈ [s2l, s2l+1)
+�
+,
+and
+Lhigh
+j,l
+=
+�
+k ∈ Zd : dist(k, MjEL) ∈ [s2l, s2l+1)
+�
+.
+Notice that
+Llow
+j,l ∪ Lhigh
+j,l
+⊆
+�
+k ∈ Zd : dist(k, MjEL) < s2l+1, and dist(k, MjEc
+L) < s2l+1}
+⊆
+�
+k ∈ Zd : dist(k, Mj∂EL) < s2l+1},
+where the last step follows as in (3.7). From Lemma 3.3 we get
+(3.13)
+#(Llow
+j,l ), #(Lhigh
+j,l ) ≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+sd2dl.
+From (3.5) it follows that if k ∈ Llow
+j,l
+L−d �
+m∈Ec
+L
+|F(Φj,k)(m)|2 ≤ L−d �
+m∈Ec
+L
+A2d
+α det(Mj) exp
+�
+− 2aα|Mj(m − ξj,k)|1−α�
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+13
+≤ A2d
+α L−d det(Mj)
+�
+m′∈MjEc
+L
+exp
+�
+− 2aα|m′ − k|1−α�
+≲
+�
+{|x|≥s2l}
+exp
+�
+− 2aα|x|1−α�
+dx
+≲ exp
+�
+− c(s2l)1−α�
+,
+(3.14)
+where c can for example be chosen as aα. A similar argument also shows that for
+k ∈ Lhigh
+j,l ,
+L−d �
+m∈EL
+|F(Φj,k)(m)|2 ≲ exp
+�
+− c(s2l)1−α�
+.
+As Llow
+j
+= �
+l∈N0 Llow
+j,l , it follows from (3.13) and (3.14) that
+�
+(j,k)∈Γlow
+L−d �
+m∈Ec
+L
+|F(Φj,k)(m)|2 =
+�
+j∈Zd
+min |Dji|≥δ
+�
+l∈N0
+�
+k∈Llow
+j,l
+L−d �
+m∈Ec
+L
+|F(Φj,k)(m)|2
+≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+�
+j∈Zd
+min |Dji|≥δ
+�
+l∈N0
+(s2l)d exp
+�
+− c(s2l)1−α�
+≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+�
+j∈Zd
+min |Dji|≥δ
+exp
+�
+− c′s1−α�
+≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+exp
+�
+− c′s1−α�
+max{log(Wmax/δ), 1}d,
+which completes the proof of (3.11). Again, we can use an analogous reasoning
+to show
+�
+j∈Zd
+min |Dji|≥δ
+�
+k∈Lhigh
+j
+L−d �
+m∈EL
+|F(Φj,k)(m)|2
+≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+exp
+�
+− c′s1−α�
+max{log(Wmax/δ), 1}d.
+Now suppose that j ∈ Zd is such that min1≤i≤d |Dji| < δ. For every m ∈ Zd we
+can uniformly bound the subsequent series
+�
+k∈Zd
+exp
+�
+− 2aα|Mj(m − ξj,k)|1−α�
+=
+�
+k∈Zd
+exp
+�
+− 2aα|Mjm − k|1−α�
+≤ C.
+Since det(Mj) = |Dj|, where Dj = Dj1 × ... × Djd, we thus get by (3.5)
+�
+j∈Zd
+min |Dji|<δ
+�
+k∈Zd
+L−d �
+m∈EL
+|F(Φj,k)(m)|2 ≤ C
+�
+j∈Zd
+min |Dji|<δ
+L−d �
+m∈EL
+det(Mj)
+≤ CL−d#EL
+�
+j∈Zd
+min |Dji|<δ
+|Dj|
+
+14
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+≲ |∂E|d/(d−1)
+κ∂E
+�
+j∈Zd
+min |Dji|<δ
+|Dj|,
+where in the last inequality we used Corollary 2.2. Finally,
+�
+j∈Zd
+min |Dji|<δ
+|Dj| ≤
+d
+�
+i=1
+�
+j∈Zd
+|Dji|<δ
+|Dj| ≤
+d
+�
+i=1
+W d−1
+max4δ
+≲ max{Wmax, 1/η∂E}d−1δ,
+where we used that each interval Dji is at most at |Dji| < δ distance from the
+boundary of (−Wi/2, Wi/2). This concludes the proof of (3.12).
+□
+4. General domain vs. rectangle
+In this section, we prove the following variant of Theorem 1.2 for F a rectangle.
+Theorem 4.1. Let L ≥ 1 be a discretization resolution, d ≥ 2, and E ⊆ Rd be
+a compact domain with regular boundary at scale η∂E ≥ 1 with constant κ∂E and
+such that |∂E| ≥ 1. For 0 < Wi ≤ L, i = 1, ..., d, take F = �d
+i=1(−Wi/2, Wi/2)
+and denote Wmax = maxi Wi. For every α ∈ (0, 1/2) there exists Aα,d ≥ 1 such
+that for ε ∈ (0, 1/2):
+#
+�
+n ∈ N : λn ∈ (ε, 1 − ε)
+�
+≤ Aα,d · max{Wmax, 1/η∂E}d−1 · |∂E|
+κ∂E
+· log
+�max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E ε
+�2d(1+α)
+.
+Proof. We adopt all the notation of Section 3. Fix parameters s ≥ 1, δ ∈ (0, 1)
+and consider the sets from (3.10).
+Observe that for f ∈ L2(F) one has
+∥Tf∥2
+2 = ∥χFPE,Lf∥2
+2 ≤ ∥PE,Lf∥2
+2 = L−d �
+m∈EL
+�� �f(m)
+��2.
+and
+∥f − Tf∥2
+2 = ∥χFf − χFPE,Lf∥2
+2 ≤ ∥(I − PE,L)f∥2
+2 = L−d �
+m∈Ec
+L
+�� �f(m)
+��2.
+By Lemma 3.5 it thus follows
+�
+(j,k)∈Γlow
+∥(I − T)Φj,k∥2
+2 +
+�
+(j,k)∈Γhigh
+∥TΦj,k∥2
+2
+(4.1)
+≤ C max{Wmax, 1/η∂E}d−1
+κ∂E
+�
+|∂E| exp
+�
+− cs1−α�
+max{log(Wmax/δ), 1}d
++ |∂E|d/(d−1)δ
+�
+,
+where the constants depend only on α and d.
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+15
+At last, we can now specify the parameters δ and s in order for the sets Γlow, Γmed
+and Γhigh to play the role of I1, I2 and I3 in Lemma 3.1. We take
+δ =
+κ∂E ε2
+C max{Wmax, 1/η∂E}d−1|∂E|d/(d−1) .
+This ensures that
+C max{Wmax, 1/η∂E}d−1|∂E|d/(d−1)
+κ∂E
+δ ≤ ε2.
+(4.2)
+Also, we select s such that
+C max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+exp
+�
+− cs1−α�
+max{log(Wmax/δ), 1}d ≤ ε2.
+(4.3)
+This condition on s is equivalent to
+s ≥
+�1
+c log
+�C max{Wmax, 1/η∂E}d−1|∂E| max{log(Wmax/δ), 1}d
+κ∂E ε2
+��1/(1−α)
+,
+and is satisfied if
+s = Aα,d log
+�max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E ε
+�1/(1−α)
+,
+for an adequate constant Aα,d. Moreover, we can guarantee that s ≥ 1, since by
+(2.4), the term inside the logarithm is ≥ 2. From (4.1), (4.2), (4.3), Lemma 3.1
+and Lemma 3.4,
+#Mε(T) ≤ 21−d#(Γmed)
+≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+max{log(Wmax/δ), 1}dsd
+≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+log
+�max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E ε
+�d/(1−α)+d
+≲ max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E
+log
+�max{Wmax, 1/η∂E}d−1|∂E|
+κ∂E ε
+�2d(1+α)
+. □
+5. Eigenvalue estimates for two domains
+5.1. Schatten quasi-norm estimates. For 0 < p ≤ 1, and ε > 0, define the
+auxiliary function g = gp,ε : [0, 1] → R given by
+g(t) =
+� t − t2
+ε − ε2
+�p
+.
+Note that since χ(ε,1−ε) ≤ g, for a positive operator 0 ≤ S ≤ 1,
+#Mε(S) = tr(χ(ε,1−ε)S) ≤ tr(g(S)) = ∥S − S2∥p
+p
+(ε − ε2)p ,
+where ∥ · ∥p, 0 < p ≤ 1, denotes the Schatten quasi-norm. The next lemma shows
+that upper bounds for the left-hand side of the last inequality can be transferred
+to the right-hand side without much loss.
+
+16
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+Lemma 5.1. Suppose that for a positive operator 0 ≤ S ≤ 1 there are constants
+C, D, a > 0 such that for every ε ∈ (0, 1/2),
+#Mε(S) ≤ C
+�
+D + log(ε−1)
+�a.
+Then, for every 0 < p ≤ 1 there is a constant Ca > 0 such that
+∥S − S2∥p
+p ≤ CaC
+�
+D + p−1�a.
+Proof. By the symmetry of the function h(x) = x − x2 around 1/2, for 0 ≤ x ≤ 1,
+h(x)p =
+� min{x,1−x}
+0
+(hp)′(t)dt =
+� 1/2
+0
+χ(t,1−t)(x)php−1(t)h′(t)dt
+≤
+� 1/2
+0
+χ(t,1−t)(x)ptp−1dt.
+By a monotone convergence argument we get
+∥S − S2∥p
+p ≤
+� 1/2
+0
+Mt(S)ptp−1dt ≤ C
+� 1
+0
+(D + log(t−1))aptp−1dt
+= C
+� ∞
+0
+(D + u/p)ae−udu
+≤ C(D + 1/p)a + C
+� ∞
+1
+(D + u/p)ae−udu
+≤ C(D + 1/p)a + C(D + 1/p)a
+� ∞
+1
+uae−udu
+≤ (1 + Γ(a + 1))C(D + 1/p)a.
+□
+5.2. Decomposition of the domain and Hankel operators. In what follows,
+we let F ⊆ (−L/2, L/2)d be a compact domain with regular boundary at scale
+η∂F = |∂F|1/(d−1) ≥ 1 with constant κ∂F. We construct two auxiliary sets F − ⊆
+F ⊆ F + which will be dyadic approximations of F from above and below by cubes
+of length at least 1. More precisely, let
+F =
+�
+k∈Z
+�
+j∈Jk
+Qk,j
+be a dyadic decomposition of F in piecewise disjoint cubes of the form Qk,j =
+Q2k + 2kj with k ∈ Z and j ∈ Jk ⊆ Zd, that are maximal (they are not contained
+in a larger dyadic cube included in F). We define
+F − =
+�
+k≥0
+�
+j∈Jk
+Qk,j.
+For F + we add cubes of length 1 to fully cover F and intersect them with
+(−L/2, L/2)d. The result is a covering of F that combines the cubes from F −
+with rectangles of maximal side-length ≤ 1. More precisely, define
+V = {v ∈ Zd : (F ∖ F −) ∩ (Q1 + v) ̸= ∅},
+and
+F + = F − ∪
+�
+v∈V
+�
+(Q1 + v) ∩ (−L/2, L/2)d�
+=: F − ∪
+�
+v∈V
+Rv.
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+17
+Note that Theorem 4.1 can be applied to each rectangle in the decomposition
+of F − and F +. This follows from a translation argument and the fact that the
+boundaries of the rectangles have null measure, so we can replace them by their
+interior. We write T ± for TE,F ±,L.
+For a set K ⊆ (−L/2, L/2)d define the Hankel operator on L2((−L/2, L/2)d)
+by
+HK = (I − PE,L)χKPE,L
+and write H± = HF ±. Note that
+(HK)∗HK = PE,LχKPE,L − PE,LχKPE,LχKPE,L = PE,LχKPE,L − (PE,LχKPE,L)2.
+Since PE,LχKPE,L and TK share the same non-zero eigenvalues, for p > 0,
+∥TK − (TK)2∥p
+p = ∥HK∥2p
+2p.
+Recall that for two operators S1, S2 in the p-Schatten class, 0 < p ≤ 1, one has
+∥S1 + S2∥p
+p ≤ ∥S1∥p
+p + ∥S2∥p
+p.
+Lemma 5.2. Let L ≥ 1, d ≥ 2, and E, F ⊆ Rd be compact domains with regular
+boundaries at scales η∂E ≥ 1, η∂F = |∂F|1/(d−1) ≥ 1, with constants κ∂E, κ∂F
+respectively. Assume also that |∂E| ≥ 1 and F ⊆ (−L/2, L/2)d. For ε ∈ (0, 1/2),
+we have
+#Mε(T ±) ≲ |∂E|
+κ∂E
+· |∂F|
+κ∂F
+· log
+�|∂E| max{|∂F|, 1}
+κ∂E ε
+�2d(1+α)+1
+.
+Proof. If k ∈ Z is such that Jk ̸= ∅, then there is a cube of length 2k included in
+F. In particular, the projection of ∂F onto the hyperplane {x1 = 0} contains a
+(d−1)-dimensional cube of length 2k and therefore 2k(d−1) ≤ |∂F|. The maximality
+of the dyadic decomposition of F implies that Qj,k ⊆ ∂F + B√
+d2k+1(0) for j ∈ Jk.
+From Lemma 2.1 and the fact that η∂F = |∂F|1/(d−1), we thus derive
+2dk#Jk ≤ |∂F + B√
+d2k+1(0)| ≲ 2k |∂F|
+κ∂F
+�
+1 + 2k(d−1)
+|∂F|
+�
+≲ 2k |∂F|
+κ∂F
+.
+(5.1)
+Similarly,
+#V ≤ |∂F + B√
+d(0)| ≲ |∂F|
+κ∂F
+.
+(5.2)
+For 0 < 2p ≤ 1, and ε ∈ (0, 1/2), we thus get
+#Mε(T +) ≤ ∥T + − (T +)2∥p
+p
+(ε − ε2)p
+= ∥H+∥2p
+2p
+(ε − ε2)p
+≤ (2/ε)p �
+k≥0
+�
+j∈Jk
+∥HQk,j∥2p
+2p + (2/ε)p �
+v∈V
+∥HRv∥2p
+2p
+≲ ε−p �
+k≥0
+�
+j∈Jk
+∥TQk,j − T 2
+Qk,j∥p
+p + ε−p �
+v∈V
+∥TRv − T 2
+Rv∥p
+p.
+Theorem 4.1 shows that when applying Lemma 5.1 to TQk,j one can take
+C ≲ 2k(d−1)|∂E|
+κ∂E
+,
+and
+D = log
+�
+2k(d−1)|∂E|
+κ∂E
+�
+.
+
+18
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+Similarly, the same holds for TRv with k = 1. Choosing p = log(2)
+�
+2 log(ε−1)
+�−1
+(which ensures that 2p ≤ 1 for every ε ∈ (0, 1/2)) thus yields
+#Mε(T +) ≲ |∂E|
+κ∂E
+��
+k≥0
+�
+j∈Jk
+2k(d−1)log
+�|∂E|2k(d−1)
+κ∂E ε
+�2d(1+α)
++#V ·log
+� |∂E|
+κ∂E ε
+�2d(1+α)�
+≲ |∂E|
+κ∂E
+|∂F|
+κ∂F
+�
+k≥0
+2k(d−1)≤max{|∂F |,1}
+log
+�|∂E|2k(d−1)
+κ∂E ε
+�2d(1+α)
+= |∂E|
+κ∂E
+|∂F|
+κ∂F
+�
+0≤k≤⌊
+log(max{|∂F |,1})
+log(2)(d−1)
+⌋
+�
+log
+� |∂E|
+κ∂E ε
+�
++ (d − 1) log(2)k
+�2d(1+α)
+,
+where in the second to last step we used (5.1), (5.2), and the fact that 2k(d−1) ≤
+|∂F| whenever Jk ̸= ∅. Finally, noting that for C, D, a > 0,
+N
+�
+k=0
+(C + Dk)a ≤
+� N+1
+0
+(C + Dx)adx ≤ (C + D(N + 1))a+1
+D(a + 1)
+,
+we get,
+#Mε(T +) ≲ |∂E|
+κ∂E
+|∂F|
+κ∂F
+log
+�|∂E| max{|∂F|, 1}
+κ∂E ε
+�2d(1+α)+1
+.
+The same argument works for #Mε(T −).
+□
+5.3. The transition index. The following estimate is part of the proof of [1,
+Theorem 1.5] (see also [16, Lemma 4.3]) and allows us to find the index where
+eigenvalues cross the 1/2 threshold. We include a proof for the sake of complete-
+ness.
+Lemma 5.3. For any trace class operator 0 ≤ S ≤ 1,
+(i) λn ≤ 1
+2, for every n ≥ ⌈tr(S)⌉ + max{2 tr(S − S2), 1};
+(ii) λn ≥ 1
+2, for every 1 ≤ n ≤ ⌈tr(S)⌉ − max{2 tr(S − S2), 1}.
+Proof. First notice that if S is an orthogonal projection, then the result holds
+trivially, so we can assume otherwise. In particular, we have that tr(S − S2) > 0.
+Set K = ⌈tr(S)⌉ and write
+tr(S) − tr(S2) =
+∞
+�
+n=1
+λn(1 − λn)
+=
+K
+�
+n=1
+λn(1 − λn) +
+∞
+�
+n=K+1
+λn(1 − λn)
+≥ λK
+K
+�
+n=1
+(1 − λn) + (1 − λK)
+∞
+�
+n=K+1
+λn
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+19
+= λKK − λK
+K
+�
+n=1
+λn + (1 − λK)
+�
+tr(S) −
+K
+�
+n=1
+λn
+�
+= λKK + (1 − λK) tr(S) −
+K
+�
+n=1
+λn
+= tr(S) −
+K
+�
+n=1
+λn + λK(K − tr(S)).
+Hence
+(5.3)
+∞
+�
+n=K+1
+λn = tr(S) −
+K
+�
+n=1
+λn ≤ tr(S) − tr(S2),
+and
+K−1
+�
+n=1
+(1 − λn) = tr(S) −
+K
+�
+n=1
+λn + λK(K − tr(S)) − (1 − λK)(1 + tr(S) − K)
+≤ tr(S) −
+K
+�
+n=1
+λn + λK(K − tr(S)) ≤ tr(S) − tr(S2).
+(5.4)
+Now let j ∈ N such that j ≥ 2(tr(S) − tr(S2)) and consider k = K + j . It follows
+from (5.3) that
+2(tr(S) − tr(S2)) · λk ≤ j · λK+j ≤
+∞
+�
+n=K+1
+λn ≤ tr(S) − tr(S2),
+which shows λk ≤ 1/2 as 0 < tr(S) − tr(S2) < ∞; this proves part (i).
+For part (ii), if 1 ≤ k = K − j ≤ K − 2(tr(S) − tr(S2)) for j ∈ N, then (5.4)
+implies
+2(tr(S) − tr(S2)) · (1 − λk) ≤ j · (1 − λK−j) ≤
+K−1
+�
+n=1
+(1 − λn) ≤ tr(S) − tr(S2),
+yielding λk ≥ 1/2. This completes the proof.
+□
+5.4. Proof of the main result. With all the preparatory work at hand, we are
+ready to prove the main result.
+Proof of Theorem 1.2. Recall from (2.2) that the eigenvalues of the concentration
+operator remain the same if we replace E, F and L with t−1E, tF and tL respec-
+tively.
+We choose t = |∂E|1/(d−1) and notice that t−1E satisfies |∂t−1E| = 1,
+η∂t−1E = t−1η∂E = 1, and κ∂t−1E = κ∂E.
+Furthermore, we also have that tF
+has regular boundary at scale η∂tF = tη∂F = (|∂E||∂F|)1/(d−1) ≥ 1 with constant
+κ∂tF = κ∂F, and tL ≥ 1 by assumption on L.
+Note that for F ′ ⊆ (−tL/2, tL/2)d, the operator T has integral kernel
+K(x, y) = χF ′(x)χF ′(y)
+1
+(tL)d
+�
+k∈(t−1E)tL
+e−2πik(x−y).
+
+20
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+Thus,
+tr(T) =
+�
+K(x, x)dx =
+�
+F ′
+1
+(tL)d
+�
+k∈(t−1E)tL
+1dx = #(t−1E)tL
+(tL)d
+|F ′|.
+On the other hand, from Lemmas 5.1 and 5.2 we have that
+tr
+�
+T ± − (T ±)2�
+≲ |∂t−1E|
+κ∂t−1E
+|∂tF|
+κ∂tF
+log
+�e|∂t−1E||∂tF|
+κ∂t−1E
+�2d(1+α)+1
+= |∂E|
+κ∂E
+|∂F|
+κ∂F
+log
+�e|∂E||∂F|
+κ∂E
+�2d(1+α)+1
+=: CE,F.
+So from Lemma 5.3,
+λn(T +) ≤ 1
+2,
+n ≥
+�#(t−1E)tL
+(tL)d
+|(tF)+|
+�
++ 2CE,F;
+λn(T −) ≥ 1
+2,
+n ≤
+�#(t−1E)tL
+(tL)d
+|(tF)−|
+�
+− 2CE,F.
+By Corollary 2.2 and |∂t−1E| = 1,
+#{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2} ≲
+1
+κ∂E
+|(tF)+ ∖ (tF)−| + CE,F
+≤
+1
+κ∂E
+|∂tF + B√
+d(0)| + CE,F
+≲
+1
+κ∂E
+td−1|∂F|
+κ∂F
++ CE,F ≲ CE,F,
+where in the second to last step we used Lemma 2.1. Since λn(T −) ≤ λn(T) ≤
+λn(T +) for every n ∈ N, again by Lemma 5.2,
+#Mε(T) ≤#{n ∈ N : 1/2 ≤ λn(T −) < 1 − ε} + #{n ∈ N : ε < λn(T +) ≤ 1/2}
++ #{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2}
+≲#Mε(T −) + #Mε(T +) + #{n ∈ N : λn(T −) < 1/2, λn(T +) > 1/2}
+≲|∂E|
+κ∂E
+|∂F|
+κ∂F
+log
+�|∂E||∂F|
+κ∂E ε
+�2d(1+α)+1
+.
+□
+6. The continuous Fourier transform
+In this section we deduce Theorem 1.1 by taking L → ∞ in Theorem 1.2.
+Proof of Theorem 1.1. Fix E and F as in the statement of Theorem 1.1.
+We
+consider a sufficiently large resolution such that L ≥ |∂E|−1/(d−1) and F ⊆
+(−L/2, L/2)d.
+Let SL : L2(Rd) → L2(Rd) be the operator given by
+SLf = TE,F,L(χFf) = χFF −1
+L χELFLχFf,
+f ∈ L2(Rd).
+An easy computation shows that SL and TE,F,L share the same non-zero eigenval-
+ues. Also, recall the operator S from (1.1).
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+21
+Step 1. We show that
+lim
+L→∞ ∥SL − S∥ = 0.
+(6.1)
+Recall that QL−1 = L−1[−1/2, 1/2)d and define the auxiliary set
+ΓL =
+�
+m∈EL
+m + QL−1.
+Note that the symmetric difference E∆ΓL is included in ∂E + BL−1√
+d(0). From
+Lemma 2.1,
+|E∆ΓL| ≤ |∂E + BL−1√
+d(0)| ≲ |∂E|
+κL
+�
+1 + (Lη∂E)−(d−1)� L→∞
+−−−→ 0.
+Using this and setting RL = χFF −1χΓLFχF, for f ∈ L2(Rd) we have
+∥(RL − S)f∥2
+2 ≤ ∥(χΓL − χE)F(χFf)∥2
+2 ≤ |E∆ΓL|∥F(χFf)∥2
+∞
+≤ |E∆ΓL|∥χFf∥2
+1 ≤ |E∆ΓL||F|∥f∥2
+2
+L→∞
+−−−→ 0.
+To prove (6.1), it only remains to show that
+∥RL − SL∥
+L→∞
+−−−→ 0.
+(6.2)
+To this end, let f ∈ L2(Rd) and estimate
+∥SLf − RLf∥2
+2 =
+�
+F
+���
+�
+ΓL
+F(χFf)(w)e2πiwxdw − L−d �
+m∈EL
+F(χFf)(m)e2πimx���
+2
+dx
+=
+�
+F
+���
+�
+m∈EL
+�
+m+QL−1
+F(χFf)(w)e2πiwx − F(χFf)(m)e2πimxdw
+���
+2
+dx
+=
+�
+F
+���
+�
+m∈EL
+�
+m+QL−1
+�
+F
+f(t)
+�
+e2πiw(x−t) − e2πim(x−t)�
+dtdw
+���
+2
+dx
+≲
+�
+F
+� �
+m∈EL
+�
+m+QL−1
+�
+F
+|f(t)||w − m||x − t|dtdw
+�2
+dx
+≲
+�
+F
+� �
+m∈EL
+L−(d+1)
+�
+F
+|f(t)||x − t|dt
+�2
+dx
+≲ (#EL)2L−2(d+1)
+�
+F
+∥f∥2
+2
+�
+F
+|x − t|2dtdx
+≲ L−2max{|∂E|2d/(d−1), 1}
+κ2
+∂E
+|F|2 diam(F)2∥f∥2
+2,
+where in the inequality step we used Corollary 2.2. Hence (6.2) holds.
+Step 2. Since SL and TE,F,L share the same non-zero eigenvalues, the estimates in
+Theorem 1.2 apply also to SL for all sufficiently large L. By the Fischer-Courant
+formula, operator convergence of positive compact operators implies convergence
+of their eigenvalues. Hence, by (6.1), the estimate satisfied by the spectrum of SL
+extends to the spectrum of S.
+□
+
+22
+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
+7. The discrete Fourier transform
+Proof of Theorem 1.3. Let us define E := Ω + Q1. Then Ω = EL for L = 1. Let
+us apply Theorem 1.2 with L = 1 to E, F. We check the relevant hypotheses.
+For each point k ∈ ∂Ω, there exist at least one face and at most 2d faces of the
+cube k + Q1 that are contained in ∂E. Therefore,
+(7.1)
+#∂Ω ≤
+��∂E
+�� ≤ 2d · #∂Ω,
+and consequently
+|∂E||∂F| ≥ #∂Ω · |∂F| ≥ 1.
+Moreover, (7.1) shows that the choice L = 1 satisfies L ≥ |∂E
+��−1/(d−1) as ∂Ω
+contains at least one point.
+Now fix 0 < r ≤
+√
+d·(#∂Ω)1/(d−1) and let us show that ∂E is regular at maximal
+scale. If r < 2
+√
+d, and x ∈ ∂E we clearly have Hd−1�
+∂E ∩ Br(x)
+�
+≳ rd−1 as E is
+a union of cubes of length 1. If r ≥ 2
+√
+d, set n = ⌊r/
+√
+d⌋ and let x ∈ ∂E. There
+exist kx ∈ ∂Ω such that |kx − x| ≤
+√
+d/2. Note that for y ∈ kx + Qn,
+|y − x| ≤
+√
+dn
+2
++
+√
+d
+2
+≤ r
+2 +
+√
+d
+2
+< r.
+Hence, kx + Qn ⊆ Br(x) and therefore,
+Hd−1�
+∂E ∩ Br(x)
+�
+≥ Hd−1�
+∂E ∩ kx + Qn
+�
+≥ #
+�
+∂Ω ∩ kx + Qn
+�
+≥ κ∂Ωnd−1 ≳ κ∂Ωrd−1.
+This shows that ∂E is regular at scale
+√
+d · (#∂Ω)1/(d−1) with constant Cd · κ∂Ω.
+Note that if a set X is regular at scale ηX and constant κX, then it is also regular
+at scale αηX and constant min{1, α1−d}κX, for every α > 0. By (7.1) we therefore
+see that ∂E is regular at scale η∂E =
+��∂E
+��1/(d−1) and constant κ∂E ≍ κ∂Ω.
+The desired estimates now follow by applying Theorem 1.2 to E and F, with
+L = 1.
+□
+8. Proof of Remark 1.4
+First we combine Lemma 5.3, Lemma 5.1 (for p = 1) and Theorem 1.1 to
+conclude that there exist a constant C = Cα,d > 0 such that if
+n ≥ ⌈|E| · |F|⌉ + C |∂E|
+κ∂E
+|∂F|
+κ∂F
+· log
+�e|∂E||∂F|
+κ∂E
+�2d(1+α)+1
+=: C1,
+then λn ≤ 1/2, and if
+n ≤ ⌈|E| · |F|⌉ − C |∂E|
+κ∂E
+|∂F|
+κ∂F
+· log
+�e|∂E||∂F|
+κ∂E
+�2d(1+α)+1
+=: C2,
+then λn ≥ 1/2.
+For ε ∈ (0, 1), define ε0 := min{ε, 1 − ε} ≤ 1/2 and let 0 < τ < ε0. Observe
+that
+{1, ..., ⌊C2⌋} ∖ Mτ(S) ⊆ N1−ε0(S) ⊆ Nε(S) ⊆ Nε0(S) ⊆ {1, ..., ⌈C1⌉} ∪ Mτ(S),
+
+EIGENVALUE ESTIMATES FOR FOURIER CONCENTRATION OPERATORS
+23
+where we understand {1, ..., ⌊C2⌋} to be ∅ if C2 < 1. Consequently,
+C2 − 1 − #Mτ(S) ≤ #Nε(S) ≤ C1 + 1 + #Mτ(S).
+Rearranging the last expression and using Theorem 1.1 for τ gives
+��Nε(S) − |E| · |F|
+�� ≲ |∂E|
+κ∂E
+· |∂F|
+κ∂F
+· log
+�|∂E||∂F|
+κ∂E τ
+�2d(1+α)+1
+.
+Letting τ ր ε0 yields (1.10).
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+FELIPE MARCECA, JOSÉ LUIS ROMERO, AND M. SPECKBACHER
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+Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,
+A-1090 Vienna, Austria
+Email address: [email protected]
+Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,
+A-1090 Vienna, Austria, and Acoustics Research Institute, Austrian Academy of
+Sciences, Wohllebengasse 12-14, Vienna, 1040, Austria
+Email address: [email protected]
+Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,
+A-1090 Vienna, Austria
+Email address: [email protected]
+