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+S.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+1 
+Preprint accepted in WCEAM 2022 Seville 
+Use of survival analysis and simulation to 
+improve maintenance planning of high 
+voltage instrument transformers in the 
+Dutch transmission system 
+Swasti R. Khuntia1, Fatma Zghal1, Ranjan Bhuyan1, Erik Schenkel1, Paul 
+Duvivier2, Olivier Blancke2, Witold Krasny2 
+Abstract This paper describes the use of survival analysis and simulation to model 
+the lifetime of high voltage instrument transformers in the Dutch transmission sys-
+tem. To represent asset aging, the non-parametric Kaplan-Meier method is used to 
+enable the fitting of Weibull distribution. Such an approach is implemented on three 
+different voltage levels, namely 110kV, 150kV, and 220/380kV. Real failure and 
+inspection data is used to achieve a realistic failure model of the instrument trans-
+formers. Failure and maintenance data occurring between 1989 and 2021 have been 
+used for this study. In spite of missing and low-quality data, a rich failure database 
+could still be prepared. This study also offers insights into factors (i.e., voltage level, 
+in-service age) influencing the remaining life from both graphical survival function 
+and parametric Weibull distribution analysis. Based on the derived statistics, future 
+possible maintenance planning scenarios are simulated under a complex system 
+modelling framework in a digital twin enabled platform. Eventually, the scenarios 
+are evaluated in terms of replacement costs (CAPEX), inspection hours, and una-
+vailability hours. 
+1 Introduction 
+TenneT, as European transmission system operator, is facing power supply reli-
+ability challenges that originate in a globally aging infrastructure and increasing 
+complexity of business operations in the context of energy transition. While power 
+transformers, due to the criticality of their function on the grid have been the focus 
+of many studies, concerns have been raised recently on the lack of focus on long-
+term asset management of Instrument Transformers (ITs). ITs play an important 
+ 
+1 S.R. Khuntia (), F. Zghal, R. Bhuyan, E. Schenkel 
+Asset Management Onshore, TenneT TSO B.V., Arnhem, The Netherlands 
+e-mail: [email protected] 
+2 P. Duvivier, O. Blancke, W. Krasny 
+Cosmo Tech, Lyon, France 
+email: [email protected] 
+
+S.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+2 
+Preprint accepted in WCEAM 2022 Seville 
+role in the metering of electrical quantities and protection of other system compo-
+nents. Due to their importance, any unplanned unavailability due to failures can 
+cause considerable outage costs to utilities. Consequently, it is crucial to properly 
+characterize the aging of ITs using statistical approaches that will enable to predict 
+the evolution of the IT population failure over the next years. In addition, it will 
+yield valuable perspectives in terms of optimizing maintenance and replacement 
+policies accordingly. The reliability analysis of ITs is very much dependent on the 
+defined maintenance strategies which will provide a reliable and safe power supply. 
+By definition, asset management involves strategies to explore, identify, plan, in-
+vest, utilize, maintain, replace, and dispose of assets while maximizing their value 
+and performance under some prescribed financial constraint (Khuntia et al., 2016). 
+Since ITs play such an important role, it is expected that statistical failure analysis 
+will give a better insight on actual maintenance planning performance to the asset 
+management team at TenneT. Technically, in the reliability analysis of IT, it is in-
+teresting to identify the independence or dependence of the specific covariates that 
+indicate the operation of the IT. 
+For any kind of data-driven methodology and, in particular, asset reliability char-
+acterization, a robust database is needed, both in terms of volumetry and quality 
+(Balzer and Neumann, 2011). However, it can be argued that there should be a pref-
+erence for robust data and that there are techniques that could be used to cope with 
+data discrepancies. In our case, the historical failure data play an important role in 
+understanding the behavior of ITs. Literature study reveals that explosion is one of 
+the highest reported failure modes. Impact of explosion not only relates to direct 
+cost of IT replacement but also chances of replacement of neighboring equipment 
+damaged in the explosion. CIGRE reports are one of the primary sources for pub-
+licly available failure databases of ITs. Three series of CIGRE reports are available 
+online. The first report was published in 1990 which covered failures of ITs (voltage 
+>72.5kV) in about 15 countries. The survey covered 136033 transformers in the 
+period from 1970 to 1986 (CIGRE, 1990). The second report published results for 
+131207 ITs (voltage > 60kV) in the period from 1985 to 1995 in the year 2009 
+(CIGRE, 2009). The third results of a wider international survey was published in 
+2012. It collected population and failure data for ITs of voltage > 60kV and ex-
+cluded AIS ring current transformers that were in service during the years 2004 to 
+2007 inclusive (CIGRE, 2012). Some other failure investigations were reported 
+(Poljak et al., 2010; Raetze et al., 2012; Tee et al., 2021), where authors focus on 
+reduction of IT explosion and better condition monitoring of ITs. Nonetheless, the 
+truth is that failure is probabilistic in nature, and it needs investigations on the rela-
+tionship with asset data and failure cause. The use of semi-parametric Cox model 
+was reported in (Tee et al., 2021). The authors elaborated the factors influencing the 
+probability of failures through analysis on the lifetime data from both graphical sur-
+vival function plots and semi-parametric Cox model. 
+With the use of Simulation Digital Twin technology from Cosmo Tech, TenneT 
+analyzed various maintenance strategies. The Digital Twin has been calibrated 
+
+S.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+3 
+Preprint accepted in WCEAM 2022 Seville 
+based on the historical failure data that it recorded with statistical technique relying 
+on survival analysis. Literature study shows that survival analysis was used for 
+power transformer reliability studies of around 2000 nos. in the Canadian and 
+around 6000 nos. in the Australian utility (Picher et al., 2014; Martin et al., 2018). 
+Ref. (Picher et al., 2014) described the data of Canadian utility Hydro-Quebec 
+where they adopted a good match using the Kaplan-Meier and Weibull distribution. 
+Finally, the method concluded that Weibull distribution is a better fit and the results 
+looked promising. Similarly, ref. (Martin et al., 2018) followed a similar strategy 
+for Australian data. The authors deduced the choice of Kaplan-Meier or Weibull 
+distribution based on the different voltage classes. In practice, Weibull distribution 
+fitted to empirical failure data are commonly used to calculate life expectancy. 
+However, the challenge in applying such a distribution to electrical assets is that 
+often the root cause of failure is not related to the normal aging of the asset, but 
+rather external factors. The aim of this paper is three-fold: (1) use of real failure data 
+to model a time-varying failure rate based on Weibull parameters obtained from 
+Kaplan-Meier survival analysis, (2) investigate extrapolation methods to maximize 
+value of existing inspection results across IT population, and (3) use digital twin 
+enabled simulation to tune the required resources necessary to realize TenneT’s 
+strategy for considered substation equipment maintenance and renewals. 
+2 Data and Methodology 
+2.1 
+Description of Data 
+As of the date of writing this paper, TenneT owns and maintains a large fleet of 
+ITs in the Dutch high voltage AC network (i.e., 110, 150, 220 and 380kV) as shown 
+in Figure 1(a). It is of interest to see the age profile of the existing population, in 
+terms of years since manufacture because reliability is often related to age. How-
+ever, lifetime data can be complicated as some ITs often extend over several dec-
+ades. At TenneT, the expected design life of an IT is 45 years. This age is affected 
+and reduced, sometimes substantially, depending on the design or utilization of the 
+IT, i.e. its loading or the environment to which it is exposed. In some cases, a good 
+maintenance scheme can even increase the replacement age. Although there is no 
+prescribed replacement age, it is the responsibility of the asset management depart-
+ment to formulate the maintenance policies based on failure history. For this study, 
+failure data was obtained from various sources, starting from failure records, reports 
+to talking to experts. Fortunately, TenneT did not record a high number of major 
+failures since the 1989. A major failure is defined as a sudden explosive event that 
+has caused an immediate emergency system outage or trip. Figure 1(b) lists the fail-
+ure events with respect to manufacturer (coded for confidentiality) and IT age. 
+The failure list was not adequate to come up with a statistical model. In addition, 
+maintenance reports (or work orders) and expert knowledge was used to populate 
+the list and gain utmost information. A work order is a document that provides all 
+
+S.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+4 
+Preprint accepted in WCEAM 2022 Seville 
+the information about a maintenance task and outlines a process for completing that 
+task. In case of IT, corrective work orders are used (the others being periodic 
+maintenance and inspection work orders). Discussion with experts led us to use the 
+work orders when an IT was out of service for any kind of maintenance. Figure 1(c) 
+shows the total recorded failures for the IT population. In the recent years, one ob-
+servation worth noticing is that the number of failures has increased significantly. 
+ 
+(a) 
+ 
+(b) 
+
+10000
+SLI
+8000
+Number of
+6000
+4000
+2000
+0
+110
+150
+220
+380
+Voltage level (kV)5
+Number of ITs
+4
+m
+2
+1
+0
+990
+7
+68
+1
+3
+6
+80
+9
+0
+00
+05
+600
+2
+6
+7
+7
+7
+8
+9
+9
+9
+9
+9
+9
+9
+6
+9
+6
+0
+0
+0
+0
+0
+0
+1
+1
+1
+1
+1
+L
+1
+1
+L
+2
+2
+2
+2
+2
+2
+Year of constructionS.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+5 
+Preprint accepted in WCEAM 2022 Seville 
+ 
+(c) 
+Figure 1 (a) Voltage-based IT population, and (b) Actual failure list until July 2021, 
+(c) Populated failure from work order and expert opinion until July 2021 
+2.2 
+Survival Analysis and Failure Rate Modelling 
+Survival analysis is a statistical technique used to estimate the lifespan of a par-
+ticular population under study. It is an analysis of time-to-event data (Wang et al., 
+2019). One of the widely used survival analysis technique is the Kaplan-Meier 
+(KM) estimate (Bland and Altman, 1998). The KM estimator uses lifetime data to 
+perform survival analysis. Although it is widely used in medical research to gauge 
+the part of patients living for a specific measure of time after treatment, it has been 
+used in the power systems sector to model the survival of electric assets (Martin et 
+al., 2018). The use of KM estimate is supported by two reasons: one is that it does 
+not assume that the data fits a statistical distribution, and second is that it allows the 
+inclusion of censored data (when an IT had not failed by mid-2021). 
+For a population, the survival function 𝑆̂(𝑡) is defined as: 
+𝑆̂(𝑡) = ∏ (1 − 𝑑𝑖
+𝑛𝑖
+)
+𝑖:𝑡𝑖<𝑡
+ 
+where, 𝑡𝑖is the time at least one event happened, 𝑑𝑖 is the number of events that 
+happened at time 𝑡𝑖 and 𝑛𝑖 is the number of individuals known to have survived up 
+to time 𝑡𝑖 (Davidson-Pilon, 2019). In our study, the estimates are calculated for three 
+different voltage levels and 𝑛𝑗 considers observations that occurred between the 
+oldest IT age and mid-2021. An important aspect in survival analysis is considering 
+the censored data. Censoring occurs when the value of an observation is only known 
+
+1000
+SI
+800
+Number of
+009
+400
+200
+YearofconstructionS.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+6 
+Preprint accepted in WCEAM 2022 Seville 
+to some extent. Censored data is often encountered when analysing practical life 
+data, especially in case of electrical power systems where most of the installed 
+equipment is still in-service, and most of the time the exact age of equipment at the 
+moment of failure is unknown (CIGRE, 2017). In this study, a large amount of data 
+falls under the right censored data (suspended data) category. A dataset is termed as 
+right censored or suspended when it is composed of components that did not fail. 
+The term right censored indicates that the event is located to the right of the dataset, 
+which implies that certain components are still operating. In our dataset, we had to 
+deal with right censoring and no left truncation since the year of construction was 
+known to us. Ignoring truncation causes bias in model’s estimation. 
+ 
+ 
+
+1.0
+Weibull
+Kaplan-Meier
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+20
+40
+60
+80
+100
+timeline1.0
+Weibull
+Kaplan-Meier
+0.8
+0.6
+0.4
+0.2
+0.0
+0
+20
+40
+60
+80
+100
+timelineS.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+7 
+Preprint accepted in WCEAM 2022 Seville 
+ 
+Figure 2 Kaplan-Meier estimate of all different voltage levels. 
+The IT dataset was split into three different families, each one with its own deg-
+radation law, based on their voltage level as is shown in Figure 2. A useful statistic 
+in this analysis is calculating the median survival time, which defines the point in 
+time where on average 50% of the population should have failed. For 110kV, the 
+median survival time is 61 years. However, the median survival time for 150, 220 
+and 380kV is infinity because there have been an insufficient number of failures to 
+determine it. In such cases, the two best options are: 
+1. use another quantile (e.g. 0.75) to compare the groups; 
+2. approximate the survival curve by means of a parametric fit and derive the me-
+dian survival time using the model. 
+The second option is chosen in our study since all the three voltages can be mod-
+elled using the parametric fit assuming that failure times have a Weibull distribu-
+tion.  In other words, Weibull distribution is used to parameterize the KM estimate. 
+The Weibull distribution is a widely used method to analyse the statistical features 
+of failure (Rinne, 2008). The probability 𝑓(𝑡) and cumulative density function 𝐹(𝑡) 
+are defined as: 𝑓(𝑡) = 𝛽
+𝑡𝛽−1
+𝜂𝛽 𝑒
+−(𝑡
+𝜂)
+𝛽
+𝑎𝑛𝑑 𝐹(𝑡) = 1 − 𝑒
+−(𝑡
+𝜂)
+𝛽
+; where, 𝑡 is the time, 
+𝛽 is the shape and 𝜂 is the scale parameter. Table 1 shows the different parameters 
+calculated for our study from the corresponding survival function. 
+Table 1 Statistics and Weibull parameters. 
+Voltage (kV) 
+No. of ITs 
+No. of 
+censored 
+β 
+η 
+median 
+110 
+3168 
+255 
+6.67 
+63.79 
+61 
+150 
+10058 
+298 
+6.42 
+74.20 
+infinity 
+220 and 380 
+2982 
+25 
+5.65 
+77.05 
+infinity 
+
+1.0
+0.8
+0.6
+0.4
+0.2
+Weibull
+Kaplan-Meier
+0.0
+0]
+20
+40
+60
+80
+100
+timelineS.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+8 
+Preprint accepted in WCEAM 2022 Seville 
+3 Modelling in Cosmo Tech Asset and Simulations 
+Founded in 2010, Cosmo Tech is a technology company pioneer in the modeling 
+of complex systems (https://cosmotech.com/). Relying on its industry-validated 
+modeling and simulation software platform, Cosmo Tech has developed a solution 
+called Cosmo Tech Asset, henceforth called CTA. CTA allows to build digital twins 
+of asset portfolios with their full complexity such as network dependencies, opera-
+tive strategies, or dynamical resources allocations. 
+3.1 
+Cosmo Tech Asset Platform 
+The different steps involved in the CTA platform are: 
+1. Experiment the CTA platform’s pre-built health assessment methods and com-
+pare the results with internal initiatives. For health assessment, the asset health 
+index is a key simulation variable, and it is described in the next sub-section. 
+2. Demonstrate the calibration of reliability law (using Weibull distribution) for 
+simulations against up-to-date condition of ITs, but also historical IT related 
+data, such as field observation or inspection data and measurement inputs. 
+3. Investigate the functional possibilities that would allow to leverage existing in-
+spection results across ITs using extrapolation methods when applicable, there-
+fore maximize inspection result value. 
+4. Finally, based on the achieved health assessment technique, use the simulation 
+platform to tune the required resources necessary to realize TenneT’s strategy 
+for considered IT maintenance and replacements. 
+3.2 
+TenneT Asset Health Index 
+For health assessment, the TenneT asset health index (AHI) is considered and is 
+shown in Table 1(a) (TenneT, 2021). The AHI is based on asset age and failure 
+probability, and it is used to drive short-term maintenance and long-term replace-
+ment strategies. It provides a consistent way to compare the overall asset health of 
+TenneT's assets. 
+The evaluation of the AHI is based on two metrics: 
+1. probability of failure of IT in the coming years for AHI score of 1 to 6, and 
+2. age of IT for AHI score of 7 to 10. 
+In addition to AHI, the study of IT uses reliability law over which failures are 
+drawn during the simulations. The reliability law corresponds to the KM survival 
+function and the Weibull estimates that are described in section 2. These laws have 
+a cumulative distribution function which represent the probability for a failure to 
+occur before a certain age. And the probability of failure over the next year can be 
+evaluated using the following formula: 
+
+S.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+9 
+Preprint accepted in WCEAM 2022 Seville 
+𝑃(𝑋 < 𝑡 + 3 | 𝑋 > 𝑡) =  1 −  𝑃(𝑋 > 𝑡 + 3 | 𝑋 > 𝑡) 
+                                                        =  1 −
+𝑃(𝑋>𝑡+3 ∩ 𝑋>𝑡)
+𝑃(𝑋 > 𝑡)
+= 1 − 
+𝑃(𝑋 > 𝑡+3)
+𝑃(𝑋 > 𝑡)  
+                                                        = 1 − 
+1 − 𝑃(𝑋 < 𝑡+3)
+1 − 𝑃(𝑋 < 𝑡)   =  1 − 
+1 − 𝐹(𝑡+3)
+1 − 𝐹(𝑡)  
+where, 
+● 
+𝐹is the cumulative distribution function of the reliability law 
+● 
+𝑋 is a random variable representing the occurrence of a failure. 
+ 
+Table 1 (a)TenneT Asset Health Index (AHI) definition, (b) Classification of Resources (FTE: 
+Full Time Employment). 
+ 
+(a) 
+ 
+(b) 
+3.3 
+Simulation 
+The reliability law was used to evaluate the different scenarios for an efficient 
+maintenance planning. A simulation period of 100 years is chosen for this study 
+since it is assumed that the most recent IT replacements will be in operation until 
+the end of this century. Time-based scenario is the current maintenance planning at 
+TenneT. It is compared against a condition-based scenario. Both the scenarios are 
+explained in detail in Table 2. The resources are listed in Table 1(b). 
+Table 2 Different Scenarios under Study. 
+ 
+Condition-based 
+Time-based 
+Replacement 
+220/380kV 
+45 years 
+45 years 
+110/150kV 
+AHI score red or 
+purple 
+45 years 
+Inspections on bay 
+every 3,6,12 months 
+220/380kV 
+No inspections 
+No inspections 
+110/150kV 
+Time-based start-
+ing at 25 years 
+Time-based start-
+ing at 25 years 
+In principle, both scenarios are very similar in the sense that the same simulation 
+model dataset is used. The difference lies in the trigger for the replacement activities 
+of the 110/150kV assets. In fact, in time-based scenario, which represents the cur-
+rent way of working, the trigger is based on the real age of the asset. As soon as the 
+
+AHI Score
+Colour
+Definition
+Purple
+Within 3 years, 80% of chance that the asset is ir-
+reparably damaged
+2
+Purple
+Within 3 years, 50% of chance that the asset is ir-
+reparably damaged
+3
+Purple
+Within 3 years, 20% of chance that the asset is ir-
+reparably damaged
+4
+Red
+Within 7 years, 80% of chance that the asset is ir
+reparably damaged
+5
+Red
+Within 7 years, 50% of chance that the asset is ir-
+reparably damaged
+6
+Red
+Within 7 years, 20% of chance that the asset is ir-
+Orange
+reparably damaged
+7
+Older than 75% of the average age
+8
+Orange
+Between 60% and 75% of the average age
+9
+Older than 5 years old and less than 60% of the av-
+_10
+Green.
+Younger than 5 years old
+erage ageActivity name
+Dura-
+Required
+Material
+Workforce
+Total
+tion (h)
+FTE
+(?) 1503
+() 1503
+(?) 1503
+Inspection
+0.5
+I
+0
+every 3 years
+41.624
+41.624
+Inspection
+1.33
+2
+49.81
+180.18
+229.99
+every 6 years
+Replacement
+IT 110kV
+40
+10
+8211
+35000
+43211
+Replacement
+40
+10
+IT 150kV
+10044
+35000
+45044
+Replacement
+IT 220kV
+40
+10
+15000
+35000
+50000
+Replacement
+IT 380kV
+40
+10
+15000
+35000
+50000S.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+10 
+Preprint accepted in WCEAM 2022 Seville 
+asset reaches 45 years of age, replacement is triggered, and action is performed as 
+resources are unlimited. On the other hand, in the condition-based scenario, the trig-
+ger is based on the apparent age of the asset. The apparent age is an attribute of 
+every asset that reflects its degradation rate and it can be different from the real age 
+of the asset. If the apparent age is higher than the real age, the asset degrades faster 
+than normal. If the apparent age is lower than the real age, the asset degrades slower 
+than normal. When the apparent age of the asset reaches 50 or 54, it means that the 
+asset is reaching AHI score of respectively 6 or 3 that is red or purple (see Table 
+1(a)), and the replacement action is triggered. 
+ 
+Figure 3 Unconstrained Scenarios Simulation. 
+ 
+Figure 4 40 FTE constrained Scenarios Simulation. 
+
+ooFTE-Time-BasedReplacement
+coFTE-Condition-BasedReplacement
+40K
+1.49M
+TOTEX
+0.1M
+1.36M
+20K
+TOTEX
+TOTEX
+0.0M
+OK
+2050
+2100
+2050
+2100
+HR Used (FTE)
+500
+100
+44.87
+40.63
+Av HR
+Av HR
+Used
+Used
+0
+2050
+2100
+2050
+210040FTE-Time-BasedReplacement
+40FTE - Condition-Based Replacemen
+() )
+TOTEX 
+10K
+1.20M
+10K
+1.21M
+5K
+TOTEX
+TOTEX
+OK
+OK
+2050
+2100
+2050
+2100
+HR Used (FTE)
+40
+40
+36.28
+20
+36.19
+20
+AvHR
+Av HR
+Used
+Used
+0
+2050
+2100
+2050
+2100S.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+11 
+Preprint accepted in WCEAM 2022 Seville 
+ 
+Figure 5 60 FTE constrained Scenarios Simulation. 
+From the figures, two conclusions can be made: (1) replacement activities are 
+the major cost driver in the TOTEX (Total Expenses), and (2) Human resources 
+(HR) costs are the major cost driver in the replacement costs. Simulation results 
+show that in case HR availability is restricted, there is no significant difference be-
+tween the time-based and condition-based replacement strategies. In fact, switching 
+to a condition-based strategy might not be beneficial in that case since it comes with 
+change and investments for little to no reward. If HR availability is guaranteed for 
+the foreseeable future, then it is highly beneficial to switch from a time-based re-
+placement strategy to a condition-based strategy as this would contribute to flatten-
+ing the curve. Also, this would represent a lot of work at the beginning to prepare 
+the necessary processes and investments for the new strategy but would lead to sig-
+nificant gains on the long term. 
+4 Conclusion 
+Maintenance planning of high voltage ITs using real data from the Dutch trans-
+mission system operator was illustrated in this study. The study aimed at under-
+standing how digital twin enabled technology along with failure data can help Ten-
+neT to make better future maintenance strategies. The strategies aimed at easing 
+financial decisions related to replacements (in terms of flattening the replacement 
+curve) and unavailability of ITs in the network. Working on real data uncovered 
+several challenges including missing data (both quantity and quality) and outliers. 
+The non-parametric Kaplan-Meier survival analysis helped in parameter estimation 
+of Weibull distribution. TenneT data could be translated to the data format to be 
+used in the digital twin CTA tool, meaning that our data could be easily adapted to 
+other software platforms. It is worth to mention that in this study, both data owner-
+ship as well as data confidence did not hinder the progress. Data confidence was 
+built upon although multiple data sources had to be aligned together. TenneT part-
+nered with Cosmo Tech to build the data ownership philosophy for successful dig-
+ital twin implementation for maintenance planning. 
+
+60FTE-Time-BasedReplacement
+6oFTE-Condition-BasedReplacement
+20K
+TOTEX (C)
+1.44M
+10K
+1.34M
+10K
+TOTEX
+TOTEX
+OK
+OK
+2050
+2100
+2050
+2100
+HR Used (FTE)
+50
+50
+43.35
+40.09
+Av HR
+AvHR
+Used
+Used
+2050
+2100
+2050
+2100S.R.Khuntia - Use of survival analysis and simulation to improve maintenance planning of high 
+voltage instrument transformers in the Dutch transmission system 
+12 
+Preprint accepted in WCEAM 2022 Seville 
+References 
+Balzer, G. and Neumann, C., 2011. Asset simulation and life cycle assessment 
+for gas insulated substation. 
+CIGRE, Germany Bland, J.M. and Altman, D.G., 1998. Survival probabilities 
+(the Kaplan-Meier method). Bmj, 317(7172), pp.1572-1580. 
+CIGRÉ WG 23.07: The paper-oil insulated measurement transformer, CIGRÉ 
+Technical Brochure no. 57, 1990. 
+CIGRÉ SC A3: State of the art of instrument transformers, CIGRÉ Technical 
+Brochure no. 394, 2009. 
+CIGRE Final Report of the 2004 – 2007 International Enquiry on Reliability of 
+High Voltage Equipment Part 4 - Instrument Transformers. Working Group A3.06, 
+2012. 
+CIGRE. Guidelines for the Use of Statistics and Statistical Tools on Life Data, 
+Working Group D1.39, 2017. 
+Davidson-Pilon, C., 2019. lifelines: survival analysis in Python. Journal of Open 
+Source Software, 4(40), p.1317. 
+Khuntia, S.R., Rueda, J.L., Bouwman, S. and van der Meijden, M.A., 2016. A 
+literature survey on asset management in electrical power [transmission and distri-
+bution] system. International Transactions on Electrical Energy Systems, 26(10), 
+pp.2123-2133. 
+Martin, D., Marks, J., Saha, T.K., Krause, O. and Mahmoudi, N., 2018. Investi-
+gation into modeling Australian power transformer failure and retirement statis-
+tics. IEEE Transactions on Power Delivery, 33(4), pp.2011-2019. 
+Picher, P., Boudreau, J.F., Manga, A., Rajotte, C., Tardif, C., Bizier, G., Di 
+Gaetano, N., Garon, D., Girard, B., Hamel, J.F. and Proulx, S., 2014. Use of health 
+index and reliability data for transformer condition assessment and fleet rank-
+ing. A2-101, CIGRE. 
+Poljak, M. and Bojanić, B., 2010. Method for the reduction of in‐service instru-
+ment transformer explosions. European transactions on electrical power, 20(7), 
+pp.927-937. 
+Raetzke, S., Koch, M. and Anglhuber, M., 2012, September. Modern insulation 
+condition assessment for instrument transformers. In 2012 IEEE International Con-
+ference on Condition Monitoring and Diagnosis (pp. 52-55). IEEE. 
+Rinne, H., 2008. The Weibull distribution: a handbook. Chapman and Hall/CRC. 
+Tee, S., Liu, Q., Wang, Z., Hafid, F. and Tournet, P., 2021. Failure investigation 
+and asset management of combined measuring instrument transformers. High Volt-
+age, 6(1), pp.61-70. 
+Wang, P., Li, Y. and Reddy, C.K., 2019. Machine learning for survival analysis: 
+A survey. ACM Computing Surveys (CSUR), 51(6), pp.1-36. 
+

-tAzT4oBgHgl3EQfSvv8/content/tmp_files/load_file.txt

@@ -0,0 +1,382 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf,len=381
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  1  Preprint accepted in WCEAM 2022 Seville  Use of survival analysis and simulation to  improve maintenance planning of high  voltage instrument transformers in the  Dutch transmission system  Swasti R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Khuntia1, Fatma Zghal1, Ranjan Bhuyan1, Erik Schenkel1, Paul  Duvivier2, Olivier Blancke2, Witold Krasny2  Abstract This paper describes the use of survival analysis and simulation to model  the lifetime of high voltage instrument transformers in the Dutch transmission sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' To represent asset aging, the non-parametric Kaplan-Meier method is used to  enable the fitting of Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Such an approach is implemented on three  different voltage levels, namely 110kV, 150kV, and 220/380kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Real failure and  inspection data is used to achieve a realistic failure model of the instrument trans-  formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Failure and maintenance data occurring between 1989 and 2021 have been  used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In spite of missing and low-quality data, a rich failure database  could still be prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' This study also offers insights into factors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', voltage level,  in-service age) influencing the remaining life from both graphical survival function  and parametric Weibull distribution analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Based on the derived statistics, future  possible maintenance planning scenarios are simulated under a complex system  modelling framework in a digital twin enabled platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Eventually, the scenarios  are evaluated in terms of replacement costs (CAPEX), inspection hours, and una-  vailability hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  1 Introduction  TenneT, as European transmission system operator, is facing power supply reli-  ability challenges that originate in a globally aging infrastructure and increasing  complexity of business operations in the context of energy transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' While power  transformers, due to the criticality of their function on the grid have been the focus  of many studies, concerns have been raised recently on the lack of focus on long-  term asset management of Instrument Transformers (ITs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' ITs play an important  1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Khuntia (\uf02a), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Zghal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Bhuyan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Schenkel  Asset Management Onshore, TenneT TSO B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', Arnhem, The Netherlands  e-mail: firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='lastname@tennet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='eu  2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Duvivier, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Blancke, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Krasny  Cosmo Tech, Lyon, France  email: firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='lastname@cosmotech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='com  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  2  Preprint accepted in WCEAM 2022 Seville  role in the metering of electrical quantities and protection of other system compo-  nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Due to their importance, any unplanned unavailability due to failures can  cause considerable outage costs to utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Consequently, it is crucial to properly  characterize the aging of ITs using statistical approaches that will enable to predict  the evolution of the IT population failure over the next years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In addition, it will  yield valuable perspectives in terms of optimizing maintenance and replacement  policies accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The reliability analysis of ITs is very much dependent on the  defined maintenance strategies which will provide a reliable and safe power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  By definition, asset management involves strategies to explore, identify, plan, in-  vest, utilize, maintain, replace, and dispose of assets while maximizing their value  and performance under some prescribed financial constraint (Khuntia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Since ITs play such an important role, it is expected that statistical failure analysis  will give a better insight on actual maintenance planning performance to the asset  management team at TenneT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Technically, in the reliability analysis of IT, it is in-  teresting to identify the independence or dependence of the specific covariates that  indicate the operation of the IT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  For any kind of data-driven methodology and, in particular, asset reliability char-  acterization, a robust database is needed, both in terms of volumetry and quality  (Balzer and Neumann, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' However, it can be argued that there should be a pref-  erence for robust data and that there are techniques that could be used to cope with  data discrepancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In our case, the historical failure data play an important role in  understanding the behavior of ITs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Literature study reveals that explosion is one of  the highest reported failure modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Impact of explosion not only relates to direct  cost of IT replacement but also chances of replacement of neighboring equipment  damaged in the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' CIGRE reports are one of the primary sources for pub-  licly available failure databases of ITs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Three series of CIGRE reports are available  online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The first report was published in 1990 which covered failures of ITs (voltage  >72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='5kV) in about 15 countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The survey covered 136033 transformers in the  period from 1970 to 1986 (CIGRE, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The second report published results for  131207 ITs (voltage > 60kV) in the period from 1985 to 1995 in the year 2009  (CIGRE, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The third results of a wider international survey was published in  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' It collected population and failure data for ITs of voltage > 60kV and ex-  cluded AIS ring current transformers that were in service during the years 2004 to  2007 inclusive (CIGRE, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Some other failure investigations were reported  (Poljak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Raetze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Tee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2021), where authors focus on  reduction of IT explosion and better condition monitoring of ITs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Nonetheless, the  truth is that failure is probabilistic in nature, and it needs investigations on the rela-  tionship with asset data and failure cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The use of semi-parametric Cox model  was reported in (Tee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The authors elaborated the factors influencing the  probability of failures through analysis on the lifetime data from both graphical sur-  vival function plots and semi-parametric Cox model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  With the use of Simulation Digital Twin technology from Cosmo Tech, TenneT  analyzed various maintenance strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The Digital Twin has been calibrated  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  3  Preprint accepted in WCEAM 2022 Seville  based on the historical failure data that it recorded with statistical technique relying  on survival analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Literature study shows that survival analysis was used for  power transformer reliability studies of around 2000 nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' in the Canadian and  around 6000 nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' in the Australian utility (Picher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' (Picher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2014) described the data of Canadian utility Hydro-Quebec  where they adopted a good match using the Kaplan-Meier and Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Finally, the method concluded that Weibull distribution is a better fit and the results  looked promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Similarly, ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' (Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2018) followed a similar strategy  for Australian data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The authors deduced the choice of Kaplan-Meier or Weibull  distribution based on the different voltage classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In practice, Weibull distribution  fitted to empirical failure data are commonly used to calculate life expectancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  However, the challenge in applying such a distribution to electrical assets is that  often the root cause of failure is not related to the normal aging of the asset, but  rather external factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The aim of this paper is three-fold: (1) use of real failure data  to model a time-varying failure rate based on Weibull parameters obtained from  Kaplan-Meier survival analysis, (2) investigate extrapolation methods to maximize  value of existing inspection results across IT population, and (3) use digital twin  enabled simulation to tune the required resources necessary to realize TenneT’s  strategy for considered substation equipment maintenance and renewals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  2 Data and Methodology  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='1  Description of Data  As of the date of writing this paper, TenneT owns and maintains a large fleet of  ITs in the Dutch high voltage AC network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 110, 150, 220 and 380kV) as shown  in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' It is of interest to see the age profile of the existing population, in  terms of years since manufacture because reliability is often related to age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' How-  ever, lifetime data can be complicated as some ITs often extend over several dec-  ades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' At TenneT, the expected design life of an IT is 45 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' This age is affected  and reduced, sometimes substantially, depending on the design or utilization of the  IT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' its loading or the environment to which it is exposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In some cases, a good  maintenance scheme can even increase the replacement age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Although there is no  prescribed replacement age, it is the responsibility of the asset management depart-  ment to formulate the maintenance policies based on failure history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' For this study,  failure data was obtained from various sources, starting from failure records, reports  to talking to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Fortunately, TenneT did not record a high number of major  failures since the 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' A major failure is defined as a sudden explosive event that  has caused an immediate emergency system outage or trip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Figure 1(b) lists the fail-  ure events with respect to manufacturer (coded for confidentiality) and IT age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  The failure list was not adequate to come up with a statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In addition,  maintenance reports (or work orders) and expert knowledge was used to populate  the list and gain utmost information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' A work order is a document that provides all  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  4  Preprint accepted in WCEAM 2022 Seville  the information about a maintenance task and outlines a process for completing that  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In case of IT, corrective work orders are used (the others being periodic  maintenance and inspection work orders).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Discussion with experts led us to use the  work orders when an IT was out of service for any kind of maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Figure 1(c)  shows the total recorded failures for the IT population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In the recent years, one ob-  servation worth noticing is that the number of failures has increased significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  (a)  (b)  10000  SLI  8000  Number of  6000  4000  2000  0  110  150  220  380  Voltage level (kV)5  Number of ITs  4  m  2  1  0  990  7  68  1  3  6  80  9  0  00  05  600  2  6  7  7  7  8  9  9  9  9  9  9  9  6  9  6  0  0  0  0  0  0  1  1  1  1  1  L  1  1  L  2  2  2  2  2  2  Year of constructionS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  5  Preprint accepted in WCEAM 2022 Seville  (c)  Figure 1 (a) Voltage-based IT population, and (b) Actual failure list until July 2021,  (c) Populated failure from work order and expert opinion until July 2021  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='2  Survival Analysis and Failure Rate Modelling  Survival analysis is a statistical technique used to estimate the lifespan of a par-  ticular population under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' It is an analysis of time-to-event data (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' One of the widely used survival analysis technique is the Kaplan-Meier  (KM) estimate (Bland and Altman, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The KM estimator uses lifetime data to  perform survival analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Although it is widely used in medical research to gauge  the part of patients living for a specific measure of time after treatment, it has been  used in the power systems sector to model the survival of electric assets (Martin et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The use of KM estimate is supported by two reasons: one is that it does  not assume that the data fits a statistical distribution, and second is that it allows the  inclusion of censored data (when an IT had not failed by mid-2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  For a population, the survival function 𝑆̂(𝑡) is defined as:  𝑆̂(𝑡) = ∏ (1 − 𝑑𝑖  𝑛𝑖  )  𝑖:𝑡𝑖<𝑡  where, 𝑡𝑖is the time at least one event happened, 𝑑𝑖 is the number of events that  happened at time 𝑡𝑖 and 𝑛𝑖 is the number of individuals known to have survived up  to time 𝑡𝑖 (Davidson-Pilon, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In our study, the estimates are calculated for three  different voltage levels and 𝑛𝑗 considers observations that occurred between the  oldest IT age and mid-2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' An important aspect in survival analysis is considering  the censored data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Censoring occurs when the value of an observation is only known  1000  SI  800  Number of  009  400  200  YearofconstructionS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  6  Preprint accepted in WCEAM 2022 Seville  to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Censored data is often encountered when analysing practical life  data, especially in case of electrical power systems where most of the installed  equipment is still in-service, and most of the time the exact age of equipment at the  moment of failure is unknown (CIGRE, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In this study, a large amount of data  falls under the right censored data (suspended data) category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' A dataset is termed as  right censored or suspended when it is composed of components that did not fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  The term right censored indicates that the event is located to the right of the dataset,  which implies that certain components are still operating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In our dataset, we had to  deal with right censoring and no left truncation since the year of construction was  known to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Ignoring truncation causes bias in model’s estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='0  Weibull  Kaplan-Meier  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='0  0  20  40  60  80  100  timeline1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='0  Weibull  Kaplan-Meier  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='0  0  20  40  60  80  100  timelineS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  7  Preprint accepted in WCEAM 2022 Seville  Figure 2 Kaplan-Meier estimate of all different voltage levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  The IT dataset was split into three different families, each one with its own deg-  radation law, based on their voltage level as is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' A useful statistic  in this analysis is calculating the median survival time, which defines the point in  time where on average 50% of the population should have failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' For 110kV, the  median survival time is 61 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' However, the median survival time for 150, 220  and 380kV is infinity because there have been an insufficient number of failures to  determine it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In such cases, the two best options are:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' use another quantile (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='75) to compare the groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' approximate the survival curve by means of a parametric fit and derive the me-  dian survival time using the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  The second option is chosen in our study since all the three voltages can be mod-  elled using the parametric fit assuming that failure times have a Weibull distribu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In other words, Weibull distribution is used to parameterize the KM estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  The Weibull distribution is a widely used method to analyse the statistical features  of failure (Rinne, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The probability 𝑓(𝑡) and cumulative density function 𝐹(𝑡)  are defined as: 𝑓(𝑡) = 𝛽  𝑡𝛽−1  𝜂𝛽 𝑒  −(𝑡  𝜂)  𝛽  𝑎𝑛𝑑 𝐹(𝑡) = 1 − 𝑒  −(𝑡  𝜂)  𝛽  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' where, 𝑡 is the time,  𝛽 is the shape and 𝜂 is the scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Table 1 shows the different parameters  calculated for our study from the corresponding survival function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Table 1 Statistics and Weibull parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Voltage (kV)  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' of ITs  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' of  censored  β  η  median  110  3168  255  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='67  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='79  61  150  10058  298  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='42  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='20  infinity  220 and 380  2982  25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='65  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='05  infinity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='2  Weibull  Kaplan-Meier  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='0  0]  20  40  60  80  100  timelineS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  8  Preprint accepted in WCEAM 2022 Seville  3 Modelling in Cosmo Tech Asset and Simulations  Founded in 2010, Cosmo Tech is a technology company pioneer in the modeling  of complex systems (https://cosmotech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='com/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Relying on its industry-validated  modeling and simulation software platform, Cosmo Tech has developed a solution  called Cosmo Tech Asset, henceforth called CTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' CTA allows to build digital twins  of asset portfolios with their full complexity such as network dependencies, opera-  tive strategies, or dynamical resources allocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='1  Cosmo Tech Asset Platform  The different steps involved in the CTA platform are:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Experiment the CTA platform’s pre-built health assessment methods and com-  pare the results with internal initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' For health assessment, the asset health  index is a key simulation variable, and it is described in the next sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Demonstrate the calibration of reliability law (using Weibull distribution) for  simulations against up-to-date condition of ITs, but also historical IT related  data, such as field observation or inspection data and measurement inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Investigate the functional possibilities that would allow to leverage existing in-  spection results across ITs using extrapolation methods when applicable, there-  fore maximize inspection result value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Finally, based on the achieved health assessment technique, use the simulation  platform to tune the required resources necessary to realize TenneT’s strategy  for considered IT maintenance and replacements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='2  TenneT Asset Health Index  For health assessment, the TenneT asset health index (AHI) is considered and is  shown in Table 1(a) (TenneT, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The AHI is based on asset age and failure  probability, and it is used to drive short-term maintenance and long-term replace-  ment strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=" It provides a consistent way to compare the overall asset health of  TenneT's assets." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  The evaluation of the AHI is based on two metrics:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' probability of failure of IT in the coming years for AHI score of 1 to 6, and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' age of IT for AHI score of 7 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  In addition to AHI, the study of IT uses reliability law over which failures are  drawn during the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The reliability law corresponds to the KM survival  function and the Weibull estimates that are described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' These laws have  a cumulative distribution function which represent the probability for a failure to  occur before a certain age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' And the probability of failure over the next year can be  evaluated using the following formula:  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  9  Preprint accepted in WCEAM 2022 Seville  𝑃(𝑋 < 𝑡 + 3 | 𝑋 > 𝑡) =  1 −  𝑃(𝑋 > 𝑡 + 3 | 𝑋 > 𝑡)  =  1 −  𝑃(𝑋>𝑡+3 ∩ 𝑋>𝑡)  𝑃(𝑋 > 𝑡)  = 1 −  𝑃(𝑋 > 𝑡+3)  𝑃(𝑋 > 𝑡)  = 1 −  1 − 𝑃(𝑋 < 𝑡+3)  1 − 𝑃(𝑋 < 𝑡)   =  1 −  1 − 𝐹(𝑡+3)  1 − 𝐹(𝑡)  where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' 𝐹is the cumulative distribution function of the reliability law 𝑋 is a random variable representing the occurrence of a failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Table 1 (a)TenneT Asset Health Index (AHI) definition, (b) Classification of Resources (FTE:  Full Time Employment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  (a)  (b)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='3  Simulation  The reliability law was used to evaluate the different scenarios for an efficient  maintenance planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' A simulation period of 100 years is chosen for this study  since it is assumed that the most recent IT replacements will be in operation until  the end of this century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Time-based scenario is the current maintenance planning at  TenneT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' It is compared against a condition-based scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Both the scenarios are  explained in detail in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The resources are listed in Table 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Table 2 Different Scenarios under Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Condition-based  Time-based  Replacement  220/380kV  45 years  45 years  110/150kV  AHI score red or  purple  45 years  Inspections on bay  every 3,6,12 months  220/380kV  No inspections  No inspections  110/150kV  Time-based start-  ing at 25 years  Time-based start-  ing at 25 years  In principle, both scenarios are very similar in the sense that the same simulation  model dataset is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The difference lies in the trigger for the replacement activities  of the 110/150kV assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In fact, in time-based scenario, which represents the cur-  rent way of working, the trigger is based on the real age of the asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' As soon as the  AHI Score  Colour  Definition  Purple  Within 3 years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' 80% of chance that the asset is ir-  reparably damaged  2  Purple  Within 3 years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' 50% of chance that the asset is ir-  reparably damaged  3  Purple  Within 3 years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' 20% of chance that the asset is ir-  reparably damaged  4  Red  Within 7 years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' 80% of chance that the asset is ir  reparably damaged  5  Red  Within 7 years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' 50% of chance that the asset is ir-  reparably damaged  6  Red  Within 7 years,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' 20% of chance that the asset is ir-  Orange  reparably damaged  7  Older than 75% of the average age  8  Orange  Between 60% and 75% of the average age  9  Older than 5 years old and less than 60% of the av-  _10  Green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Younger than 5 years old  erage ageActivity name  Dura-  Required  Material  Workforce  Total  tion (h)  FTE  (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=') 1503  () 1503  (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=') 1503  Inspection  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='5  I  0  every 3 years  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='624  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='624  Inspection  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='33  2  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='81  180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='18  229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='99  every 6 years  Replacement  IT 110kV  40  10  8211  35000  43211  Replacement  40  10  IT 150kV  10044  35000  45044  Replacement  IT 220kV  40  10  15000  35000  50000  Replacement  IT 380kV  40  10  15000  35000  50000S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  10  Preprint accepted in WCEAM 2022 Seville  asset reaches 45 years of age, replacement is triggered, and action is performed as  resources are unlimited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' On the other hand, in the condition-based scenario, the trig-  ger is based on the apparent age of the asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The apparent age is an attribute of  every asset that reflects its degradation rate and it can be different from the real age  of the asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' If the apparent age is higher than the real age, the asset degrades faster  than normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' If the apparent age is lower than the real age, the asset degrades slower  than normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' When the apparent age of the asset reaches 50 or 54, it means that the  asset is reaching AHI score of respectively 6 or 3 that is red or purple (see Table  1(a)), and the replacement action is triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Figure 3 Unconstrained Scenarios Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Figure 4 40 FTE constrained Scenarios Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  ooFTE-Time-BasedReplacement  coFTE-Condition-BasedReplacement  40K  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='49M  TOTEX  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='1M  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='36M  20K  TOTEX  TOTEX  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='0M  OK  2050  2100  2050  2100  HR Used (FTE)  500  100  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='87  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='63  Av HR  Av HR  Used  Used  0  2050  2100  2050  210040FTE-Time-BasedReplacement  40FTE - Condition-Based Replacemen  () )  TOTEX  10K  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='20M  10K  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='21M  5K  TOTEX  TOTEX  OK  OK  2050  2100  2050  2100  HR Used (FTE)  40  40  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='28  20  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='19  20  AvHR  Av HR  Used  Used  0  2050  2100  2050  2100S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='Khuntia - Use of survival analysis and simulation to improve maintenance planning of high  voltage instrument transformers in the Dutch transmission system  11  Preprint accepted in WCEAM 2022 Seville  Figure 5 60 FTE constrained Scenarios Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  From the figures, two conclusions can be made: (1) replacement activities are  the major cost driver in the TOTEX (Total Expenses), and (2) Human resources  (HR) costs are the major cost driver in the replacement costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Simulation results  show that in case HR availability is restricted, there is no significant difference be-  tween the time-based and condition-based replacement strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In fact, switching  to a condition-based strategy might not be beneficial in that case since it comes with  change and investments for little to no reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' If HR availability is guaranteed for  the foreseeable future, then it is highly beneficial to switch from a time-based re-  placement strategy to a condition-based strategy as this would contribute to flatten-  ing the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Also, this would represent a lot of work at the beginning to prepare  the necessary processes and investments for the new strategy but would lead to sig-  nificant gains on the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  4 Conclusion  Maintenance planning of high voltage ITs using real data from the Dutch trans-  mission system operator was illustrated in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The study aimed at under-  standing how digital twin enabled technology along with failure data can help Ten-  neT to make better future maintenance strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' The strategies aimed at easing  financial decisions related to replacements (in terms of flattening the replacement  curve) and unavailability of ITs in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Working on real data uncovered  several challenges including missing data (both quantity and quality) and outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  The non-parametric Kaplan-Meier survival analysis helped in parameter estimation  of Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' TenneT data could be translated to the data format to be  used in the digital twin CTA tool, meaning that our data could be easily adapted to  other software platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' It is worth to mention that in this study, both data owner-  ship as well as data confidence did not hinder the progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' Data confidence was  built upon although multiple data sources had to be aligned together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' TenneT part-  nered with Cosmo Tech to build the data ownership philosophy for successful dig-  ital twin implementation for maintenance planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
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+page_content='06,  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
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+page_content='  Davidson-Pilon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
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+page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
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+page_content=' Modern insulation  condition assessment for instrument transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content=' In 2012 IEEE International Con-  ference on Condition Monitoring and Diagnosis (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
+page_content='  Rinne, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
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+page_content=' Chapman and Hall/CRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
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+page_content=', Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tAzT4oBgHgl3EQfSvv8/content/2301.01239v1.pdf'}
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+Analogical Inference Enhanced Knowledge Graph Embedding
+Zhen Yao1*, Wen Zhang1*, Mingyang Chen2, Yufeng Huang1, Yi Yang4, Huajun Chen2,3,5†
+1School of Software Technology, Zhejiang University
+2College of Computer Science and Technology, Zhejiang University
+3Donghai Laboratory, Zhoushan 316021, China
+4Huawei Technologies Co., Ltd
+5Alibaba-Zhejiang University Joint Institute of Frontier Technologies
+{yz0204, zhang.wen, mingyangchen, huangyufeng, huajunsir}@zju.edu.cn
+[email protected],
+Abstract
+Knowledge graph embedding (KGE), which maps entities
+and relations in a knowledge graph into continuous vector
+spaces, has achieved great success in predicting missing links
+in knowledge graphs. However, knowledge graphs often con-
+tain incomplete triples that are difficult to inductively infer
+by KGEs. To address this challenge, we resort to analogi-
+cal inference and propose a novel and general self-supervised
+framework AnKGE to enhance KGE models with analog-
+ical inference capability. We propose an analogical object
+retriever that retrieves appropriate analogical objects from
+entity-level, relation-level, and triple-level. And in AnKGE,
+we train an analogy function for each level of analogical in-
+ference with the original element embedding from a well-
+trained KGE model as input, which outputs the analogical
+object embedding. In order to combine inductive inference
+capability from the original KGE model and analogical in-
+ference capability enhanced by AnKGE, we interpolate the
+analogy score with the base model score and introduce the
+adaptive weights in the score function for prediction. Through
+extensive experiments on FB15k-237 and WN18RR datasets,
+we show that AnKGE achieves competitive results on link
+prediction task and well performs analogical inference.
+1
+Introduction
+Knowledge graphs (KGs) storing a large number of triples
+in the form of (head entity, relation, tail entity), (h, r, t)
+for short, are popular data structures for representing fac-
+tual knowledge. Many knowledge graph projects such as
+Freebase (Bollacker et al. 2008), WordNet (Miller 1994),
+YAGO (Suchanek, Kasneci, and Weikum 2007) and DB-
+pedia (Lehmann et al. 2015) are significant foundations to
+support artificial intelligence applications. They have been
+successfully used in downstream applications such as word
+sense disambiguation (Bevilacqua and Navigli 2020), ques-
+tion answering (Yasunaga et al. 2021), and information
+extraction (Hu et al. 2021), gaining widespread attention.
+However, most KGs are incomplete, so predicting the miss-
+ing links between entities is a fundamental problem for KGs
+*These authors contributed equally.
+†Corresponding Author.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+called link prediction. One of the common approaches to
+this problem is knowledge graph embedding (KGE) meth-
+ods, which make prediction through a predefined triple score
+function with learnt entity and relation embeddings as input.
+Many KGE models have been proposed like TransE (Bordes
+et al. 2013), DistMult (Yang et al. 2015), RotatE (Sun et al.
+2019) and HAKE (Zhang et al. 2020). These methods have
+gained great success in knowledge graph completion task.
+For most KGE methods, the parametric learning paradigm
+can be viewed as memorization regarding training data as
+a book and predicting missing links as the close-book test
+(Chen et al. 2022), which belongs to inductive inference.
+However, the large knowledge graphs often contain incom-
+plete triples that are difficult to be inductively inferred by
+applying memorization paradigm. Nevertheless, the problem
+may be well solved by using analogical inference method.
+That is because analogical inference is a referential method,
+which retrieves similar solutions to solve new problems,
+similar to an open-book examination. For example, it seems
+that most people could not remember even learn about what
+company Ron Wayne founded. However, if they know that
+Ron Wayne and Steve Jobs are the co-founders, i.e., Steve
+Jobs and Ron Wayne are analogical objects in this context,
+and it is well known that Steve Jobs founded Apple Inc.;
+thus they could analogically infer that Ron Wayne founded
+Apple Inc. .
+In order to enhance KGEs with analogical inference ca-
+pability, there are three problems should be solved: 1) How
+to define the analogical objects of elements given a task? 2)
+How to enable the model to map elements to analogical ob-
+jects? 3) How to combine the original inductive inference
+capability and enhanced analogical inference capability?
+We propose AnKGE, a novel and general self-supervised
+framework, which solves these problems very well and en-
+hances well-trained KGEs with analogical inference capa-
+bility. For problem 1, we think that an analogical object can
+solve the given task well, and inspired by the nearest neigh-
+bor language model (Khandelwal et al. 2020), we propose an
+analogical retriever covering objects of three levels, includ-
+ing entity, relation, and triple level. Specifically, we consider
+the score function of KGEs as the assessment of the quality
+of triples and regrade the replacement triples with the high-
+est scoring as the appropriate analogical objects. For prob-
+arXiv:2301.00982v1  [cs.AI]  3 Jan 2023
+
+lem 2, we trained a projecting function using analogical ob-
+jects as supervision signals. This function projects original
+objects onto appropriate analogical objects. For problem 3,
+we interpolate the analogy score with the base model score
+to combine the original inductive inference capability and
+enhanced analogical inference capability. Moreover, we in-
+troduce the adaptive weight to adjust analogical inference in
+knowledge graph completion task.
+Finally, through link prediction experiments on FB15k-
+237 and WN18RR datasets, we demonstrate the AnKGE is
+significantly compatible and outperforms the other baseline
+models. To the best of our knowledge, AnKGE is the first
+framework to enhance KGEs with analogical inference abil-
+ity.
+In summary, our contributions in this work include:
+• We explore the knowledge graph completion task from
+the analogical inference view. We propose an effective
+retrieval method covering three levels to obtain the ap-
+propriate analogy objects.
+• We propose a novelty analogical inference enhanced
+framework called AnKGE, which could project original
+objects onto appropriate objects for analogical inference.
+To our knowledge, the AnKGE is the first framework of
+knowledge graph embedding to enhance analogical infer-
+ence ability.
+• We conduct experimental evaluations to demonstrate
+the proposed AnKGE is significantly compatible and
+achieves competitive performance on FB15k-237 and
+WN18RR datasets, promising practical applications.
+2
+Related Work
+Knowledge graph embedding
+According to previous
+work (Zhang et al. 2022), the KGE methods can be di-
+vided into two categories based on the scoring function and
+whether a global graph structure is utilized. The first cat-
+egory is the Conventional KGEs (C-KGEs), which apply a
+geometric assumption in vector space for true triples and use
+single triple as input for triple scoring. Conventional KGEs
+use the score function to measure the plausibility of triple.
+TransE (Bordes et al. 2013) is a representative conventional
+KGE method whose score function is ∥h + r − t∥2. What
+is more, there are many variants to improve the performance
+of TransE, such as RotatE (Sun et al. 2019), DistMult (Yang
+et al. 2015) and HAKE (Zhang et al. 2020). The other cate-
+gory is the GNN-based methods, which use representations
+of entities and relations aggregated from their neighbors in
+the graph instead of embedding them for triple scoring to
+capture the graph patterns explicitly. R-GCN (Schlichtkrull
+et al. 2018) is the first GNN framework to model relational
+data. It introduces relation-specific transformations when
+neighbor aggregating. SE-GNN (Li et al. 2022) models three
+levels semantic evidence into knowledge embedding. Note
+that SE-GNN introducing three levels from the semantic ev-
+idence view differs from our three levels analogical objects.
+Enhanced KGE framework
+Recently, some work has
+proposed some frameworks and strategies to improve the
+performance of KGE models, which are called enhanced
+KGE, such as CAKE (Niu et al. 2022), PUDA(Tang
+et al. 2022) and REP (Wang et al. 2022). CAKE is a
+commonsense-aware knowledge embedding framework to
+extract commonsense from factual triples with entity con-
+cepts automatically, which generates commonsense aug-
+ments to facilitate high-quality negative sampling. PUDA
+is a data augmentation strategy to address the false nega-
+tive and data sparsity issue. REP is a post-processing tech-
+nique to adapt pre-trained KG embeddings with graph con-
+text. Our method is designed to enhance a well-trained KGE
+model with analogical inference capability belonging to the
+enhanced KGE framework.
+Analogical inference
+In classic artificial intelligence, ana-
+logical inference was an active research topic. However, the
+early computational model of analogy-making study (Gen-
+tner 1983; Turney 2008) mainly focuses on structure map-
+ping theory and its implementation in the structure map-
+ping engine. Recently, some researchers proposed k-Nearest
+Neighbor language model(kNN-LM) (Khandelwal et al.
+2020), which can directly query training examples at test
+time, also can be considered the analogy inference model
+in the neural language process topic. While effective, these
+models often require retrieval from a large datastore at test
+time, significantly increasing the inference overhead. In the
+field of knowledge graph, the study of analogical inference
+to solve knowledge graph incomplete problem is missing.
+ANALOGY (Liu, Wu, and Yang 2017) is the first method for
+modeling analogical structures in multi-relational embed-
+ding, but the performance is not good. Differ in our method
+uses the nearest neighbor method to perform explicit anal-
+ogy, ANALOGY uses the commutativity constraint of the
+normal matrix to model analogical relations implicitly.
+3
+Analogical Object Retriever
+Before introducing our method, in this section, we firstly in-
+troduce the background of knowledge graph and analogical
+inference, and then we propose the analogical object retriev-
+ers that retrieve appropriate analogical objects from entity-
+level, relation-level, and triple-level. The retrieved analog-
+ical objects will be used as supervision signals with our
+method.
+Background
+A knowledge graph is denoted as G
+=
+(E, R, F), where E represents the set of entities, R repre-
+sents the set of relations, and F = {(h, r, t)} ⊆ E × R × E
+represents the set of triple facts.
+Analogical inference, which has been long researched in
+artificial intelligence, maps the target problem to a known
+source problem that could effectively utilize known knowl-
+edge (Hall 1989). Applying analogical inference into link
+prediction task (h, r, ?) in knowledge graphs, instead of di-
+rectly predicting the tail entity t, we could make predic-
+tion through similar triples that we know, i.e. triples in train
+dataset. We consider similar triples are composed by ana-
+logical objects of (h, r, t). Specifically, we assume that the
+analogy objects may come from three levels: the analogy
+of head entity h part resulting similar triple (h′, r, t)(entity-
+level), the analogy of relation r part resulting similar triple
+
+(h, r′, t) (relation-level) and the analogy of combination pair
+(h, r) part t resulting similar triple (h′, r′, t) (triple-level).
+Thus, we propose three retrievers to obtain different
+level’s analogical objects.
+Entity-Level Retriever
+The retriever is designed based on
+the score function fkge(h, r, t) predefined in a well-trained
+KGE model, where triples with higher scores are assumed
+with higher probability to be true. Inspired by the near-
+est neighbor language model (Khandelwal et al. 2020), we
+replace all possible objects of the triple and regrade the
+replacement triples with highest scoring as the appropri-
+ate analogical objects. Given a triple (h, r, t), entity-level
+retriever retrieves similar true triples (h′, r, t) for entity-
+level analogical inference. For example, we could get the
+answer of (Sergey Brin, found, ?) is Google through
+(Larry Page, found, Google) if we know Sergey Brin and
+Larry Page are co-founders.
+Specifically, in entity-level retriever, we first replace h
+with all entities resulting |E| replacement triples, and then
+regard triples with highest scores measured by the KGE as
+similar triples. And we name the head entity in similar triples
+as analogical objects from entity-level retriever. Thus ana-
+logical object set could be represented as
+Ehrt
+Ne = {hi | Top( {fkge(hi, r, t) | hi ∈ E} )Ne},
+(1)
+where Top(·)k denotes the k elements with top k values
+among all inputs, fkge(·, ·, ·) is the predefined score function
+in KGE model, and hrt denotes a specific triple (h, r, t) as
+input. If not otherwise specified, we omit hrt and use ENe in-
+stead of Ehrt
+Ne for simplicity. Compared to retrieving similar
+triples directly from the train dataset, retrieving according to
+scores from KGEs could help overcome the incompleteness
+of KGs.
+Relation-Level Retriever
+Given (h, r, t), relation-level
+retriever retrieves (h, r′, t) for relation-level analogical in-
+ference, since there are relations with similar contexts in
+KGs. For example, the founder of a company is usually
+the board member. Thus the relation-level analogy object
+of found is board member. Similar to the entity-level re-
+triever, the analogical object set of (h, r, t) from relation-
+level retriever is as follow :
+RNr = {ri | Top( {fkge(h, ri, t) | ri ∈ R} )Nr}.
+(2)
+Triple-Level Retriever
+Given (h, r, t), triple-level re-
+triever retrieves (h′, r′, t) for triple-level analogical infer-
+ence, which is the combination of entity-level and relation-
+level retriever. For instance, Sergey Brin is the founder of
+Google and Sundar Pichai is the CEO of Google. Therefore,
+the triple-level analogical objects of (SergeyBrin, found)
+is (SundarPichai, CEO). Actually, the number of all can-
+didate (h′, r′) pairs is in millions in most knowledge graphs.
+In order to reduce the cost of retrieving candidate pairs and
+inspired by the principle of locality, we often select m en-
+tities and n relations with high triple scores separately, and
+then pair them with each other. Thus the set of analogical
+objects, namely (h′, r′) pairs, from triple-level retriever is
+TNt = {(hi, ri) |
+Top( {fkge(hi, ri, t) | hi ∈ Em, ri ∈ Rn})Nt}.
+(3)
+4
+Methodology
+In this section, we present a novel KGE enhanced frame-
+work called Analogy Enhanced Knowledge Graph Embed-
+ding (AnKGE), which could model the three levels of ana-
+logical inference as introduced in Section 3. Next, we first
+introduce the definition of analogy function (Section 4.1)
+and how to train it by using analogical objects (Section 4.2
+and Section 4.3). Finally, we introduce how to combine the
+original inductive inference capability and enhanced ana-
+logical inference capability in knowledge graph completion
+task. (Section 4.4)
+4.1
+Analogy Function
+Given a well-trained KGE model M = {E, R, fkge, Θ},
+where E, R and fkge are entity embedding table, relation
+embedding table, and score function of the M, and Θ is the
+set of other parameters, AnKGE enhances M with capa-
+bility of analogical inference through a projecting function
+called analogy function f. We train an analogy function for
+each level of analogical inference with the original element
+embedding from E or R in M as input and output the ana-
+logical object embedding to conduct link prediction.
+Specifically, analogy function for relation-level analogical
+inference frel maps an original embedding of a relation r
+in (h, r, t) to the analogical embedding through a relation
+projecting vector vR
+r ∈ Rdr that
+frel(r) = ra = vR
+r ◦ r,
+(4)
+where dr is the relation hidden dimension, ◦ is the element-
+wise product.
+Similarly, the analogy function for entity-level analogical
+inference fent maps an original embedding of an entity h
+in (h, r, t) to the analogical embedding. Considering that an
+entity generally tends to be associated with multiple rela-
+tions, we define fent as:
+fent(h, r) = ha = vE
+h ◦ h + λMtrans × vR
+r ◦ r,
+(5)
+where vE
+h ∈ Rde is the entity projecting vector and de is
+the entity hidden dimension. Mtrans ∈ Rde×dr denotes the
+transformation matrix that enable to make relation r into
+consideration. λ is a weight hyper-parameter.
+Analogy function for triple-level analogical inference ftrp
+outputs the analogical embedding of entity and relation pairs
+through combining embedding of entity-level and relation-
+level according to KGEs as follows:
+ftrp(h, r) = za = gkge (ha, ra) ,
+(6)
+gkge(·, ·) is the function in KGEs that maps a head entity em-
+bedding to the tail entity embedding according to the given
+relation embedding. gkge(·, ·) and fkge(·, ·, ·) of representa-
+tive KGE models are provided in Appendix A.
+4.2
+Analogy Objects Aggregator
+In order to enhance the framework’s robustness for analog-
+ical inference, we make the analogical objects retrieved fol-
+lowing Section 3 as the supervision signals for analogy func-
+tions. Specifically, we make the analogy embedding as intro-
+duced in Section 4.1 to approach the weighted average of the
+analogical objects from KGE model M.
+
+Entity Level
+Relation Level
+Triple Level
+Entity Analogy  
+Embedding 
+Relation Analogy  
+Embedding
+Triple Analogy  
+Embedding
+Analogy Function
+Entity Loss
+close
+Relation Loss
+close
+Triple Loss
+close
+ 
+Training Stage 
+ 
+Testing Stage
+Analogical Retriever
+Link Prediction
+Analogy Score
+Base Model Score
+Score Function: 
+ 
+Entity Embedding:
+ 
+Relation Embedding:
+ 
+ 
+Base KGE Model
+Figure 1: This is the AnKGE structure diagram with TransE as the base model. For simplicity, we set the numbers of three levels
+analogical object are 1. The upper half of figure shows the module of base model. The predefined score function is applied to
+learnt embedding to get the well-trained model. The lower half of figure shows the module of AnKGE. First, AnKGE retrieves
+the analogy objects for training the analogy function. The solid line arrow indicates the AnKGE training process. Then, AnKGE
+remakes the prediction ranking by interpolating analogy score. The dashed line arrow indicates the AnKGE testing process.
+The aggregated embeddings of entity-level and relation-
+level, h+ and r+ respectively, are calculated as follows
+h+ =
+�
+hi∈ENe
+hi S(fkge(hi, r, t)),
+(7)
+r+ =
+�
+ri∈RNr
+ri S(fkge(h, ri, t)),
+(8)
+where S(·) is the softmax function that converts a vector of
+K real numbers into a probability distribution of K possible
+outcomes, which is formulated as S (ci) = eci/�K
+k=1 eck.
+Triple-level aggregated embedding z+ is obtained by the
+firstly aggregating entity and relation embedding separately
+and then calculating combination embedding, which can be
+formulated as:
+z+ =gkge
+�
+z+
+e , z+
+r
+�
+,
+z+
+e =
+�
+(hi,ri)∈TNt
+hi S(fkge(hi, ri, t)),
+z+
+r =
+�
+(hi,ri)∈TNt
+ri S(fkge(hi, ri, t)).
+(9)
+4.3
+Loss Function
+The training goal of the analogy function is to reduce the dis-
+tance between the analogy embedding and aggregated em-
+bedding obtained following Section 4.1 and 4.2 respectively.
+In addition, considering that fkge performs priori on the
+truth value of triples, we take the analogy triple score as an-
+other supervision signal. Therefore, given a pair of analogy
+embedding Xa and aggregated embedding X + of a triple
+embeddings (h, r, t), the loss function is
+L(X,(h, r, t)) =
+logσ
+�
+γ
+��Xa − X +��
+2 − fkge(h, r, t)
+�
+,
+(10)
+where γ is a hyper-parameter of the loss function, σ is the
+sigmoid function. ∥·∥2 is the euclidean norm.
+However, the three levels of analogical inference are not
+equally important for different triples. We add weight pa-
+rameters for each loss of three levels and the final training
+objective is1:
+min Loss =
+�
+(h,r,t)∈F
+�
+βEL(h, (ha, r, t))
++βR L(r, (h, ra, t))
++βT L(z, (ha, ra, t))
+�
+.
+(11)
+As a result, considering the different contributions of three
+level, we introduce βE, βR and βT to adjust gradient de-
+scent. The three levels loss function distribution is positively
+correlated with the score of the analogy triple. Due to page
+limitation, we put the calculation details in Appendix B.
+4.4
+Link Prediction
+For a test triple (h, r, t) in test set Fte, we follow the kNN-
+LM (Khandelwal et al. 2020) and interpolate the analogy
+1During the gradient update, the parameters of the original
+model are frozen.
+
+FB15k-237
+WN18RR
+MRR
+Hit@1
+Hit@3
+Hit@10
+MRR
+Hit@1
+Hit@3
+Hit@10
+Conventional KGE
+TransE (Bordes et al. 2013)
+0.317
+0.223
+0.352
+0.504
+0.224
+0.022
+0.390
+0.520
+ANALOGY (Liu, Wu, and Yang 2017)
+0.256
+0.165
+0.290
+0.436
+0.405
+0.363
+0.429
+0.474
+RotatE (Sun et al. 2019)
+0.336
+0.244
+0.370
+0.524
+0.473
+0.428
+0.491
+0.564
+HAKE (Zhang et al. 2020)
+0.349
+0.252
+0.385
+0.545
+0.496
+0.452
+0.513
+0.580
+Rot-Pro (Song, Luo, and Huang 2021)
+0.344
+0.246
+0.383
+0.540
+0.457
+0.397
+0.482
+0.577
+PairRE (Chao et al. 2021)
+0.348
+0.254
+0.384
+0.539
+0.455
+0.413
+0.469
+0.539
+DualE (Cao et al. 2021)
+0.365
+0.268
+0.400
+0.559
+0.492
+0.444
+0.513
+0.584
+GNN-based KGE
+R-GCN (Schlichtkrull et al. 2018)
+0.249
+0.151
+0.264
+0.417
+-
+-
+-
+-
+A2N (Bansal et al. 2019)
+0.317
+0.232
+0.348
+0.486
+0.450
+0.420
+0.460
+0.510
+CompGCN (Vashishth et al. 2020)
+0.355
+0.264
+0.390
+0.535
+0.479
+0.443
+0.494
+0.546
+SE-GNN (Li et al. 2022)
+0.365
+0.271
+0.399
+0.549
+0.484
+0.446
+0.509
+0.572
+Enhanced KGE
+CAKE (Niu et al. 2022)
+0.321
+0.226
+0.355
+0.515
+-
+-
+-
+-
+PUDA (Tang et al. 2022)
+0.369
+0.268
+0.408
+0.578
+0.481
+0.436
+0.498
+0.582
+REP (Wang et al. 2022)
+0.354
+0.262
+0.388
+0.540
+0.488
+0.439
+0.505
+0.588
+AnKGE-HAKE(ours)
+0.385
+0.288
+0.428
+0.572
+0.500
+0.454
+0.515
+0.587
+Table 1: Link Prediction results on FB15k-237 and WN18RR. The best results are bold and second best results are underline.
+score with base model score to get the final score function:
+Score(h, r, t) = fkge (h, r, t) + λEfkge (ha, r, t) +
+λRfkge (h, ra, t) + λT fkge (ha, ra, t) (12)
+where λ is the adaptive weight parameter, which is dynam-
+ically adjusts analogy weight according to training triples.
+λE is proportional to the number of triples with the same
+(r, t) in the training set. λR is proportional to the number of
+triples with the same (h, t) in the training set. λT is propor-
+tional to the number of triples with the same tail entity in the
+training set. The formula for adaptive weight parameter is2:
+λE = min (| {(hi, r, t) ∈ F} |/Ne, 1) × αE,
+λR = min (| {(h, ri, t) ∈ F} |/Nr, 1) × αR,
+λT = min (| {(hi, ri, t) ∈ F} |/Nt, 1) × αT ,
+(13)
+where αE, αR, αT
+are basic weight hyper-parameters.
+Adaptive weight utilizes the train dataset to determine
+whether test triples are suitable for different levels of ana-
+logical inference. When all levels of analogical inference are
+not suitable, this score function degenerates to the base KGE
+model. In fact, AnKGE remakes the rank of hard-predicted
+triples in the base model by analogical inference to improve
+the prediction performance.
+5
+Experiments
+In this section, we present and analyze the experimental re-
+sults.3 We first introduce the experimental settings in de-
+tail. Then we show the effectiveness and compatibility of the
+2When link prediction, we add reverse relations to expand the
+dataset and predict tail entity only, which is equivalent to the effect
+of predicting both head and tail entities. Each prediction will use all
+entities to replace tail entity. Thus, there is no risk of label leakage.
+3Our code is available at https://github.com/zjukg/AnKGE
+AnKGE with multiple base KGE models. Besides, we fur-
+ther analyze the effect of three levels analogical inference by
+ablation study. Finally, we conduct case study presenting a
+new view for the explanations of knowledge graph inference
+by analogical inference.
+5.1
+Experiments Setup
+Dataset
+We conduct experiments on link prediction task
+on two well-known benchmarks: WN18RR and FB15k-237.
+WN18RR and FB15k-237 are subsets of WN18 and FB15k,
+respectively. Some previous work (Dettmers et al. 2018) has
+indicated the test leakage flaw in WN18 and FB15k, which
+means test triples appear in train dataset with inverse rela-
+tions. WN18RR and FB15k-237 removing inverse relations
+are the modified version. Therefore, we use WN18RR and
+FB15k-237 as the experiment datasets. The statistic details
+of these datasets are summarized in Appendix C.
+Evaluation protocol
+We evaluate the KGE framework
+performance by four frequent evaluation metrics: the recip-
+rocal mean of correct entity ranks in the whole entity set
+(MRR) and percentage of test triples with correct entities
+ranked in top 1/3/10 (Hit@1, Hit@3, Hit@10). For a test
+task (h, r, ?) → t, we replace all entities to create corrupted
+triples. Following the filter setting protocol, we exclude the
+other true triples appearing in train, valid and test datasets.
+Finally, we sort the filter corrupted triples according to the
+triple scores.
+Implementation details
+We train AnKGE framework
+based on four representative KGE models : TransE (Bor-
+des et al. 2013), RotatE (Sun et al. 2019), HAKE (Zhang
+et al. 2020) and PairRE (Chao et al. 2021). We use the grid
+search to select the hyper-parameters of our framework. We
+search the number of analogy objects of three levels Ne,
+
+FB15k-237
+WN18RR
+MRR
+Hit@1
+Hit@3
+Hit@10
+MRR
+Hit@1
+Hit@3
+Hit@10
+TransE
+0.317
+0.223
+0.352
+0.504
+0.224
+0.022
+0.390
+0.520
+AnKGE-TransE
+0.340
+0.245
+0.379
+0.523
+0.232
+0.031
+0.402
+0.526
+RotatE
+0.336
+0.244
+0.370
+0.524
+0.473
+0.428
+0.491
+0.564
+AnKGE-RotatE
+0.366
+0.273
+0.405
+0.546
+0.480
+0.431
+0.499
+0.578
+HAKE
+0.349
+0.252
+0.385
+0.545
+0.496
+0.452
+0.513
+0.580
+AnKGE-HAKE
+0.385
+0.288
+0.428
+0.572
+0.500
+0.454
+0.515
+0.587
+PairRE
+0.348
+0.254
+0.384
+0.539
+0.455
+0.413
+0.469
+0.539
+AnKGE-PairRE
+0.376
+0.281
+0.417
+0.558
+0.462
+0.415
+0.480
+0.556
+Table 2: AnKGE upon different model on FB15k-237 and WN18RR. The better results are bold.
+1
+3
+5
+10
+50
+100
+HAKE Ranking
+100
+50
+10
+5
+3
+1
+AnKGE Ranking
+0
+0
+0
+3
+366
+1727
+2
+9
+19
+332
+5824
+593
+5
+48
+292
+2189
+817
+29
+22
+283
+1448
+546
+186
+3
+352
+3816
+882
+456
+214
+18
+9953
+1250
+257
+143
+158
+15
+0
+200
+400
+600
+800
+1000
+Figure 2: Comparison of the ranking between AnKGE and
+base model on the FB15k-237.
+Nr and Nt ∈ {1, 3, 5, 10, 20}, the basic weight of three
+levels αE, αR and αT ∈ {0.01, 0.05, 0.1, 0.2, 0.3}, learn
+rate α ∈ {1e−3, 1e−4, 1e−5}. The loss function weight
+γ in Equation (10) is set to 10, the transformation matrix
+weight λ in Equation (5) is set to 1 and 0 in FB15k-237 and
+WN18RR respectively. Before training AnKGE, we retrieve
+analogical objects of three levels in train dataset for once.
+In both training and inference processes, AnKGE is ex-
+tended based on the scoring function of the original model.
+Thus, AnKGE has the same model complexity as the origi-
+nal model.
+5.2
+Link Prediction Results
+Main results
+We use HAKE (Zhang et al. 2020) as the
+base model for AnKGE to compare with other baselines.
+Baselines are selected from three categories Conventional
+KGE models including TransE (Bordes et al. 2013), ANAL-
+OGY (Liu, Wu, and Yang 2017), RotatE (Sun et al. 2019),
+HAKE, Rot-Pro (Song, Luo, and Huang 2021), PairRE
+(Chao et al. 2021), and DualE (Cao et al. 2021), GNN-
+based KGE models including R-GCN (Schlichtkrull et al.
+2018), A2N (Bansal et al. 2019), CompGCN (Vashishth
+et al. 2020), and SE-GNN (Li et al. 2022), and Enhanced
+KGE framework including CAKE (Niu et al. 2022), PUDA
+Models
+FB15k-237
+WN18RR
+MRR
+Hit@1
+MRR
+Hit@1
+AnKGE
+0.385
+0.288
+0.500
+0.454
+w/o entity-level
+0.384
+0.288
+0.497
+0.451
+w/o relation-level
+0.349
+0.253
+0.500
+0.455
+w/o triple-level
+0.384
+0.287
+0.499
+0.453
+w/o all
+0.349
+0.252
+0.496
+0.452
+Table 3: Ablation study of three analogy level, where w/o
+means removing the corresponding level in AnKGE.
+(Tang et al. 2022), and REP (Wang et al. 2022).
+The Table 1 summarizes experiment results on FB15k-
+237 and WN18RR. The result of ANALOGY is from code4.
+The result of TransE, RotatE, HAKE and PairRE are from
+our trained model. The base model and AnKGE frame-
+work training details are provided in Appendix D. The
+other results are from the published paper. We can see that
+AnKGE enhances the analogical inference ability of the
+base model HAKE through analogical inference and outper-
+forms the baseline models on most evaluation metrics except
+the Hit@10 metric where results of AnKGE slightly lower
+than PUDA and REP and achieve the second best. Overall,
+AnKGE remakes the rank of hard-predicted triples in HAKE
+by analogical inference, achieving the best results on both
+datasets.
+Compatibility results
+The AnKGE is a framework to en-
+hance the analogical inference ability of KGE models, which
+retrieves analogical objects through fkge predefined in KGE
+models. Theoretically, our framework is applicable to most
+KGE models defining a score function for triples. We chose
+four C-KGE models: TransE, RotatE, HAKE, PairRE as
+base model to validate compatibility. As Table 2 shows,
+AnKGE achieves a significant improvement over the base
+model on all metrics. The MRR metric improves by about
+3% on the FB15k-237. The result demonstrates that AnKGE
+is compatible with a wide range of KGE models. Moreover,
+AnKGE based on HAKE achieves a more significant im-
+provement on FB15k-237 dataset. HAKE makes the entities
+4https://github.com/thunlp/OpenKE
+
+Incomplete triple
+Analogy object
+AnKGE
+Original
+Rank
+Rank
+Entity
+(diencephalon, has part, ?) → hypothalamus
+brain
+5
+25
+(rest, derivationally related form, ?) → breath
+drowse
+6
+38
+(roof, hypernym, ?) → protective covering
+cap
+39
+20
+Relation
+(felidae, member meronym, ?) → panthera
+has part
+5
+17
+(monodontidae, member meronym, ?) → delphinapterus
+hypernym Reverse
+1
+64
+(literary composition, hypernym, ?) → writing
+has part
+88
+18
+Triple
+(ticino, instance hypernym, ?) → swiss canton
+(switzerland, has part)
+8
+54
+(south korea, has part, ?) → inchon
+(port, instance hypernym Reverse)
+1
+31
+(elementary geometry, hypernym, ?) → geometry
+(construct, synset domain topic of)
+39
+12
+Table 4: Analogical inference case Study. The better ranks are blod.
+hierarchical by using the depth of the entity to model differ-
+ent levels of the hierarchy, which is more helpful for analog-
+ical inference.
+Compared with WN18RR, the improvement of the model
+on FB15k-237 is more significant, which we speculate is
+because FB15k-237 has richer relational patterns. So it has
+more improvement in the process of relation-level analogi-
+cal inference. In addition, AnKGE is designed to predict the
+hard-predicted triples. The overall accuracy of FB15k-237
+is lower than WN18RR. Consequently, the boosting effect
+of the model is reflected more obviously.
+5.3
+Model Analysis
+Ranking study
+In order to analyze the improvement ef-
+fect of AnKGE, we compare the ranking results in FB15k-
+237 of the AnKGE-HAKE and original HAKE in Figure 2.
+The horizontal coordinate represents the ranking range of
+the HAKE model, and the vertical coordinate represents the
+ranking range of AnKGE. We found that ranking changes
+are less apparent when the ranking is more significant than
+100, so we selected the triples ranking within 100 and di-
+vided them into six ranking ranges for analysis. The diag-
+onal line represents the unchanged ranking, the lower right
+of the diagonal line represents the AnKGE ranking as better
+than the HAKE ranking, and the upper left represents worse.
+We find some triples with worse rankings, but the number is
+much smaller than those with better rankings. In addition,
+the change in ranking is not so evident as the base model
+ranking increases; the better the base model ranking is, the
+more possible that AnKGE could improve the rankings.
+Ablation Study
+We conduct ablation experiments for the
+analogical inference part of AnkGE. Table 3 shows the re-
+sults of the ablation study for the AnKGE-HAKE on two
+datasets. We can see that the removal of any part makes the
+model less effective, except the relation-level on WN18RR
+dataset. Since there are only 11 relations in WN18RR, it
+is hard to retrieve suitable relation-level analogical objects.
+We explain this in more detail in case study. In addition, the
+WN18RR consists of a lexicon containing contextual words
+that naturally provide entity-level analogical objects, which
+makes the model more effective for entity-level analogical
+inference. The result of FB15k-237 is the opposite. It may be
+because it has rich relationship patterns, making the relation-
+level analogical inference more effective.
+Case Study
+Analogical inference can generate explana-
+tions for predicted triples, which are valuable for real-life
+applications. Our method also provides an analogy view for
+the explanations of knowledge graph inference. As the Table
+4 shows, we provide an intuitive demonstration about ana-
+logical inference. For each level, we select multiple example
+cases from WN18RR test set, and list their corresponding
+analogical objects and prediction results based on RotatE.
+For entity-level, the idea is to retrieve hypernym or hyponym
+as the analogy object. For example, the diencephalon is lo-
+cated in the core of the brain. The fact that hypothalamus is
+part of brain improves the reliability of the people‘s trust on
+predicted result. However, if hyponym entity becomes the
+analogy object, it will generate bad explanations and results.
+For instance, although cap can be regraded as a special type
+of roof, it is not the protective covering. Thus the misleading
+explanation that (cap, hypernym, protective covering)
+downgrades the trustworthiness of the predicting result,
+which ranks the correct answer at 39. For relation-level,
+AnKGE tends to retrieve the conceptually similar rela-
+tions, such as the ( member meronym) and ( has part).
+Nevertheless, there are only 11 relations on WN18RR,
+which makes the AnKGE sometimes retrieve the inappro-
+priate analogy relations. For example, ( hypernym) and
+( has part) are the relations of opposite concepts, which
+leads to bad explanation and worse ranking. For triple-level,
+AnKGE typically focuses on the (h, r) pair structure. As
+proof, ticino is a canton of Switzerland means that triple
+(switzerland, has part, swiss canton) is good explana-
+tion. However, sometimes the (h, r) pair structure varies too
+much leading the misclassification.
+6
+Conclusion
+In this paper, we resort to analogical inference to study the
+knowledge graph completion task. We propose an analogical
+object retriever that retrieves appropriate analogical objects
+from entity-level, relation-level, and triple-level. Then, we
+design a novel and general self-supervised framework to en-
+hance well-trained KGEs with analogical inference capabil-
+ity called AnKGE. Our method achieves competitive results
+on knowledge graph completion task and performs enhanced
+analogical inference ability. Some future directions include
+exploring more analogy patterns and a more general frame-
+work to adapt to the GNN-based KGE.
+
+7
+Acknowledgments
+This work is funded by NSFCU19B2027/91846204.
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+
+Dataset
+|E|
+|R|
+Train
+Valid
+Test
+FB15k-237
+14,541
+237
+272,115
+17,535
+20,466
+WN18RR
+40,493
+11
+86,835
+3,034
+3,134
+Table 5: Statistics of datasets. Train, Valid, and Test denote
+the size of train set, validation set, and test set, respectively.
+A
+KGE Models
+We can divide knowledge graph embedding models into
+two categories: conventional knowledge graph embedding
+models and GNN-based models. Theoretically, the AnKGE
+framework applies to most conventional KGE models defin-
+ing a score function for triples. In order to demonstrate
+the effectiveness and compatibility of AnKGE. We chose
+four representative conventional knowledge graph embed-
+ding models: TransE (Bordes et al. 2013), RotatE (Sun et al.
+2019), HAKE (Zhang et al. 2020) and PairRE (Chao et al.
+2021) as base models for AnKGE. Table 6 exhibits the
+gkge(·, ·) and fkge(·, ·, ·) defined in these knowledge graph
+embedding models. We will introduce the four KGE models,
+respectively.
+TransE
+is the first knowledge graph embedding model
+proposing a geometric interpretation of the latent space. The
+TransE model is inspired by Word2vec vectors, requiring the
+sum of head embedding and relation embedding close to the
+tail embedding. It makes TransE successfully capture the re-
+lations between words through translations. However, due
+to the limit of translation, TransE cannot correctly handle
+N-to-one, one-to-N, and symmetric relations.
+RotatE
+requires the embedding of (h, r, t) belong to Ck,
+and considers the relation embedding as rotation vector in
+a complex latent space. Specifically, the complex compo-
+nent conveys the rotation along that axis, whereas the real
+component always equals 1. RotatE has been demonstrated
+that rotation allows modeling correctly numerous relational
+patterns, such as symmetry, anti-symmetry and inversion.
+However, RotatE cannot model the relation with hierarchy
+pattern.
+HAKE
+is a hierarchy-aware knowledge graph embedding
+model that uses the depth of entity to model different levels
+of the hierarchy. HAKE distinguishes the entities into two
+categories: the entities at different levels of the hierarchy and
+the entities at the same level of the hierarchy, to model the
+semantic hierarchies. Experiments demonstrate that HAKE
+can effectively model the semantic hierarchies in knowledge
+graphs.
+PairRE
+is a method capable of simultaneously encoding
+complex relations and multiple relation patterns. The model
+uses paired relation representations to adjust the margin in
+loss function to fit different complex relations. PairRE cap-
+tures the semantic connection among relation vectors which
+have been demonstrated to encode three important relation
+patterns, symmetry/anti-symmetry, inversion and composi-
+tion.
+B
+Loss Weight
+Considering the different contributions of the three levels,
+we introduce βE, βR and βT to adjust gradient descent. Sim-
+ilarity to the softmax function, firstly, we replace the original
+element embedding with three level aggregated embeddings
+respectively, then calculate the exponential sum of analogy
+triple scores and original triple score, which is formulated
+as:
+T =efkge(h+,r,t) + efkge(h,r+,t)
++ efkge(z+
+e ,z+
+r ,t) + efkge(h,r,t).
+(14)
+The weights of loss function in Equation (11) are the rate of
+T :
+βE = efkge(h+,r,t)/T ,
+βR = efkge(h,r+,t)/T ,
+βT = efkge(z+
+e ,z+
+e ,t)/T .
+(15)
+The highest analogy triple score means mapping original el-
+ement embedding to aggregate embedding is necessary. If
+the original triple score is the highest, the triple should not
+analogy with other objects.
+C
+Datasets
+We evaluate the AnKGE framework on two widely-used
+datasets: WN18RR and FB15k-237. FB15k-237 is from the
+Freebase knowledge graph project, whose design is inspired
+by broadly used information communities such as The Se-
+mantic Web and Wikipedia. FB15k-237 contains informa-
+tion including locations, media, geographical and people.
+WN18RR is from the WordNet knowledge graph project,
+a dataset that characterizes associations between English
+words.WN18RR contains information including symmetric,
+asymmetric and compositional relations. Statistics of these
+datasets are shown in Table 5.
+D
+Implementation Details
+Firstly, we train four KGE models: TransE, RotatE, HAKE
+and PairRE, as base models. In the training stage, we ap-
+ply a widely used negative sampling loss function with self-
+adversarial training:
+L = log σ(γm − fkge(h, r, t))
++
+n
+�
+i=1
+p(h′
+i, r, t′
+i) log σ(fkge(h′
+i, r, t′
+i) − γm),
+where γm is a fixed margin, σ is the sigmoid function,
+(h′
+i, r, t′
+i) is the ith corrupting negative triple for (h, r, t) and
+n is the number of negative triples. Moreover, p(h′
+j, r, t′
+j) is
+the self-adversarial weight for this negative triple. The cal-
+culation of the weight is:
+p(h′
+j, r, t′
+j) =
+exp αtempfkge(h′
+j, r, t′
+j)
+�
+i exp αtempfkge(h′
+i, r, t′
+i)
+which is the probability distribution of negative sampling
+triples, where αtemp is the adversarial temperature of sam-
+pling. When training and testing, we add reverse relations to
+
+Model M
+fkge(h, r, t)
+gkge(h, r)
+Parameters
+TransE
+−∥h + r − t∥1
+h + r
+h, r, t ∈ Rk
+RotatE
+−∥h ◦ r − t∥2
+h ◦ r
+h, r, t ∈ Ck, |ri| = 1
+HAKE
+−∥hm ◦ rm − tm∥2−
+Cat[∥hm ◦ rm∥2,
+hm, tm ∈ Rk,rm ∈ Rk
++,
+λ∥ sin((hp + rp − tp)/2)∥1
+λ∥ sin((hp + rp)/2)∥1]
+hp, rp, tp ∈ [0, 2π)k, λ ∈ Rk
+PairRE
+−∥h ◦ rH − t ◦ rT ∥1
+h ◦ rH
+h, r, t ∈ Rk
+Table 6: The details of knowledge graph embedding models, where ∥ · ∥1 denotes the absolute-value norm, Cat [·] denotes the
+concatenate vector function.
+Dataset
+Model
+Emebdding
+Margin
+Adversarial
+Negative
+Batch Size
+Inverse Relation
+Dimension
+Temperature
+Samples
+FB15k-237
+TransE
+500
+9.0
+1.0
+256
+1024
+True
+RotatE
+500
+9.0
+1.0
+256
+1024
+True
+HAKE
+1000
+9.0
+1.0
+512
+1024
+True
+PairRE
+1500
+6.0
+1.0
+256
+1024
+True
+WN18RR
+TransE
+500
+6.0
+1.0
+256
+2048
+True
+RotatE
+500
+6.0
+0.5
+1024
+512
+True
+HAKE
+500
+6.0
+0.5
+1024
+512
+True
+PairRE
+500
+6.0
+0.5
+1024
+512
+True
+Dataset
+Model
+Entity
+Relation
+Triple
+Entity
+Relation
+Triple
+Cand. Ne
+Cand. Nr
+Cand. Nt
+lambda αE
+lambda αR
+lambda αT
+FB15k-237
+AnKGE-TransE
+1
+1
+3
+0.01
+0.2
+0.02
+AnKGE-RotatE
+1
+1
+5
+0.01
+0.2
+0.05
+AnKGE-HAKE
+1
+1
+5
+0.05
+0.3
+0.1
+AnKGE-PairRE
+1
+1
+3
+0.01
+0.3
+0.05
+WN18RR
+AnKGE-TransE
+1
+1
+20
+0.01
+0.3
+0.3
+AnKGE-RotatE
+1
+1
+3
+0.1
+0.05
+0.1
+AnKGE-HAKE
+1
+1
+3
+0.1
+0.05
+0.1
+AnKGE-PairRE
+1
+1
+3
+0.1
+0.05
+0.2
+Table 7: The hyper-parameter settings of base model and AnKGE over different datasets.
+expand the dataset. Specifically, for a triple (h, r, t), we add
+a new reverse triple (t, r−1, h) in dataset. r−1 represents the
+reverse relation of r. In the link prediction task, the model
+only predicts tail entity, which is equivalent to the effect of
+predicting both head and tail entities.
+Then, we use AnKGE to enhance the well-trained KGEs
+with analogical inference capability. We show that AnKGE
+achieves competitive results on knowledge graph comple-
+tion task and performs enhanced analogical inference abil-
+ity. The loss function weight γ in Equation (10) is set to 10,
+the transformation matrix weight λ in Equation (5) is set to 1
+and 0 in FB15k-237 and WN18RR respectively. We use the
+grid search to select other hyper-parameters, including: en-
+tity candidates Ne, relation candidates Nr triple candidates
+Nt, entity lambda αE, relation lambda αR and triple lambda
+αT . Other experimental settings are the same. The experi-
+ment of AnKGE-TransE on WN18RR is the only exception.
+We use fixed weight parameter instead of adaptive weight
+parameter and cosine similarity instead of euclidean norm.
+In addition, since there is no negative sampling, the mem-
+ory footprint and time cost are lower than the base model,
+which is generally acceptable.
+We implement all the models with PyTorch, and run ex-
+periments on NVIDIA RTX3090 GPUs with 24GB RAM
+and Intel(R) Xeon(R) Silver 4210R CPU @ 2.40GHz with
+40 cores. The hyper-parameter settings of base model and
+AnKGE are shown in Table 7
+

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@@ -0,0 +1,3607 @@
+arXiv:2301.00277v1  [econ.EM]  31 Dec 2022
+Higher-order Refinements of Small Bandwidth Asymptotics for
+Density-Weighted Average Derivative Estimators∗
+Matias D. Cattaneo†
+Max H. Farrell‡
+Michael Jansson§
+Ricardo Masini¶
+January 3, 2023
+Abstract
+The density weighted average derivative (DWAD) of a regression function is a canonical
+parameter of interest in economics. Classical first-order large sample distribution theory for
+kernel-based DWAD estimators relies on tuning parameter restrictions and model assumptions
+leading to an asymptotic linear representation of the point estimator.
+Such conditions can
+be restrictive, and the resulting distributional approximation may not be representative of the
+underlying sampling distribution of the statistic of interest, in particular not being robust to
+bandwidth choices. Small bandwidth asymptotics offers an alternative, more general distribu-
+tional approximation for kernel-based DWAD estimators that allows for, but does not require,
+asymptotic linearity. The resulting inference procedures based on small bandwidth asymptotics
+were found to exhibit superior finite sample performance in simulations, but no formal theory
+justifying that empirical success is available in the literature. Employing Edgeworth expan-
+sions, this paper shows that small bandwidth asymptotics lead to inference procedures with
+demonstrable superior higher-order distributional properties relative to procedures based on
+asymptotic linear approximations.
+Keywords: density weighted average derivatives, Edgeworth expansions, small bandwidth asymp-
+totics.
+∗Prepared for the Conference in Honor of James L. Powell at UC-Berkeley, March 25–26, 2022. We thank the
+conference participants for their comments. Cattaneo gratefully acknowledges financial support from the National
+Science Foundation through grants SES-1947805 and DMS-2210561, Jansson gratefully acknowledges financial support
+from the National Science Foundation through grant SES-1947662, and Masini gratefully acknowledges financial
+support from the National Science Foundation through grant DMS-2210561.
+†Department of Operations Research and Financial Engineering, Princeton University.
+‡Booth School of Business, University of Chicago.
+§Department of Economics, UC Berkeley.
+¶Center for Statistics and Machine Learning, Princeton University.
+
+1
+Introduction
+Identification, estimation and inference in the context of two-step semiparametric models has a long
+tradition in econometrics (Powell, 1994). Canonical two-step semiparametric parameters are finite
+dimensional functionals of some other unknown infinite dimensional parameters in the model (e.g.,
+a density or regression function), a leading example being the density weighted average derivative
+(DWAD) of a regression function (Stoker, 1986).
+This paper seeks to honor the many contri-
+butions of Jim Powell to semiparametric theory in econometrics by juxtaposing the higher-order
+distributional properties of Powell et al. (1989)’s two-step kernel-based DWAD estimator under
+two alternative large sample approximation regimes: one based on the classical asymptotic linear
+representation, and the other based on a more general quadratic distributional approximation.1
+In a landmark contribution, Powell et al. (1989) proposed a kernel-based DWAD estimator and
+obtained first-order, asymptotically linear distribution theory employing ideas from the U-statistics
+literature, along with plug-in standard error estimators, to develop valid inference procedures in
+large samples. This work sparked a wealth of subsequent developments in the econometrics litera-
+ture: Robinson (1995) obtained Berry-Esseen bounds, Powell and Stoker (1996) considered mean
+square error expansions, Nishiyama and Robinson (2000, 2001, 2005) developed Edgeworth expan-
+sions, and Newey et al. (2004) investigated bias properties, just to mention a few contributions.
+The two-step semiparametric estimator in this literature employs a preliminary kernel-based esti-
+mator of a density function, which requires choosing two main tuning parameters (a bandwidth and
+a kernel function), and their “optimal” choices depend on the goal of interest (e.g., point estimation
+vs. inference) as well as the features of the underlying data generating process (e.g., smoothness of
+the unknown density and dimensionality of the covariates).
+Classical first-order distribution theory for kernel-based DWAD estimators has focused on cases
+where tuning parameter restrictions and model assumptions lead to an asymptotic linear representa-
+tion of the two-step semiparametric point estimator (Newey and McFadden, 1994; Ichimura and Todd,
+2007, for overviews), that is, the two-step estimator is approximated by a sample average based
+on the so-called influence function. This approach can lead to semiparametric efficient inference
+procedures in large samples, but the implied distributional approximation may not be “robust”
+to tuning parameter choices and/or model features. More specifically, the limiting distribution
+obtained based on the asymptotic linear representation is invariant to the way that the preliminary
+nonparametric estimators are constructed, and requires potentially high smoothness levels of the
+underlying unknown functions and thus the use of higher-order kernels. At its core, asymptotic
+linear approximations assume away the contribution of additional terms forming the statistic of
+interest, despite the fact that these terms do contribute to the sampling variability of the two-step
+semiparametric estimator and, more importantly, do reflect the effect of tuning parameter choices
+1Jim Powell’s contributions to semiparametric theory are numerous. Honor´e and Powell (1994), Powell and Stoker
+(1996), Blundell and Powell (2004), Aradillas-Lopez et al. (2007), Ahn et al. (2018), and Graham et al. (2023) are
+some of the most closely connected to the our work. These papers employ U-statistics methods for two-step kernel-
+based estimators similar to those considered herein. See Powell (2017) for more discussion and references.
+1
+
+in finite samples.
+Cattaneo et al. (2014a) proposed an alternative distributional approximation for kernel-based
+DWAD estimators that allows for, but does not require, asymptotic linearity.
+The key idea is
+to capture the joint contribution to the sampling distribution of both linear and quadratic terms
+forming the kernel-based DWAD estimator. To operationalize this idea, Cattaneo et al. (2014a)
+introduced an asymptotic experiment where the bandwidth sequence is allowed to vanish at a speed
+that would render the classical asymptotic linear representation invalid because the quadratic term
+becomes first order even in large samples, which they termed “small bandwidth” asymptotics. This
+framework was carefully developed to obtain a distributional approximation that explicitly depends
+on both linear and quadratic terms, thereby forcing a more careful analysis of how the quadratic
+term contributes to the sampling distribution of the statistic.
+Small bandwidth asymptotics inference methods for kernel-based DWAD estimators were found
+to perform well in simulations (Cattaneo et al., 2010, 2014a,b), but no formal justification for its
+finite sample success is available in the literature. Methodologically, this alternative distributional
+approximation leads to a new way of conducting inference (e.g., constructing confidence interval
+estimators) because the original standard error formula proposed by Powell et al. (1989) must be
+modified to make the asymptotic approximation valid across the full range of allowable bandwidths
+(including the region where asymptotic linearity fails). Theoretically, however, the empirical success
+of small bandwidth asymptotics could in principle come from two distinct sources: (i) it could deliver
+a better distributional approximation to the sampling distribution of the point estimator; or (ii) it
+could deliver a better distributional approximation to the sampling distribution of the Studentized
+t-statistic because the standard error formula was modified.
+Employing Edgeworth expansions (Bhattacharya and Rao, 1976; Hall, 1992), this paper shows
+that the small bandwidth asymptotics approximation framework leads to inference procedures with
+demonstrable superior higher-order distributional properties relative to procedures based on asymp-
+totic linear approximations. We study both standardized and Studentized t-statistics, under both
+asymptotic linearity and small bandwidth asymptotic regimes, and show that both standardized
+and Studentized t-statistics emerging from the small bandwidth regime offer higher-order correc-
+tions as measured by the second cummulant underlying their Edgeworth expansions. An immediate
+implication of our results is that the small bandwidth asymptotic framework delivers both a better
+distributional approximation (Theorem 1, standardized t-statistic) and leads to a better standard
+error construction (Theorem 2, Studentized t-statistic). Therefore, our results have both theoretical
+and practical implications for empirical work in economics, in addition to providing a theory-based
+explanation for prior simulation-based findings exhibiting better numerical performance of infer-
+ence procedures constructed using small bandwidth asymptotics relative to those constructed using
+classical distributional approximations.
+The closest antecedent to our work is Nishiyama and Robinson (2000, 2001), who also studied
+Edgeworth expansions for kernel-based DWAD estimators. Their expansions, however, were moti-
+vated by the asymptotic linear approximation to the point estimator, and hence can not be used
+2
+
+to compare and contrast to the distributional approximation emerging from the alternative small
+bandwidth asymptotic regime. Therefore, from a technical perspective, this paper also offers novel
+Edgeworth expansions that allow for different standardization and Studentization schemes, thereby
+allowing us to plug-and-play when comparing the two competing asymptotic frameworks. More
+specifically, Theorem 1 below concerns a generic standardized t-statistic and is proven based on
+Theorem A in the appendix, which may be of independent technical interest due to is generality.
+Theorem 2 below concerns a more specialized class of Studentized t-statistic because establishing
+valid Edgeworth expansions is considerably harder when dealing with Studentization.
+The idea of employing alternative (more general) asymptotic approximation frameworks that do
+not enforce asymptotic linearity for two-step semiparametric estimators has also featured in other
+context such as partially linear series-based, many covariates and many instrument estimation as
+well as certain network estimation settings (Cattaneo et al., 2018a,b; Matsushita and Otsu, 2021),
+as well as other non-linear two-step semiparametric settings (Cattaneo et al., 2013; Cattaneo and Jansson,
+2018; Cattaneo et al., 2019). While our theoretical developments and results focus specifically on
+the case of kernel-based DWAD estimation, their main conceptual conclusions can be extrapolated
+to those settings as well.
+The main takeaway is that employing alternative asymptotic frame-
+works can deliver improved inference with smaller higher-order distributional approximation errors,
+thereby offering more robust inference procedures in finite samples.
+The paper continues as follows. Section 2 introduces the setup and main assumptions. Section
+3 reviews the classical first-order distributional approximation based on asymptotic linearity and
+the more general small bandwidth distributional approximation, along with their corresponding
+choices of standard error formulas. Section 4 presents the main results of our paper. Section 5
+concludes. The appendix is organized in three parts: Appendix A provides a self-contained generic
+Edgeworth expansion for second-order U-statistics, which may be of independent technical interest,
+Appendix B gives the proof of Theorem 1 (standardized t-statistic), and Appendix C gives the proof
+of Theorem 2 (Studentized t-statistic).
+2
+Setup and Assumptions
+Suppose Zi = (Yi, X′
+i)′, i = 1, . . . , n, is a random sample from the distribution of the random
+vector Z = (Y, X′)′, where Y is an outcome variable of interest and X takes value on Rd with
+Lebesgue density f. We consider the density weighted average derivative of the regression function
+g(X) = E[Y |X] given by
+θ := E[f(X)˙g(X)],
+where for any function a we define ˙a(x) :=
+∂
+∂xa(x).
+To save notation, we also define e(X) :=
+f(X)g(X) and v(X) := E[Y 2|X].
+We impose the following conditions on the underlying data
+generating process. Let ∥ · ∥ be the Euclidean norm.
+Assumption 1.
+3
+
+(a) E[|Y |p] < ∞, for some p ≥ 3.
+(b) Σ := E[ψ(Z)ψ(Z)′] is positive definite, where ψ(Z) := 2
+�
+˙e(X) − Y ˙f(X) − θ
+�
+.
+(c) f is (S +1) times differentiable, and f and its (S +1) derivatives are bounded, for 2S > d+2;
+(d) g is (S + 1) times differentiable and its first three derivatives are bounded;
+(e) e and its first (S + 1) derivatives are bounded;
+(f) v is twice diferentiable, and its first two derivatives are bounded, and v ˙f and E[|Y |3|X]f(X)
+are bounded;
+(g) f, gf, ˙gf and vf vanish on the boundaries of their convex supports;
+(h) Cram´er Condition:
+sup
+ν∈Rd:∥v∥=1
+lim sup
+|t|→∞
+|E exp(ιtℓ1/¯σν)| < 1 where ¯σν := ν′Σν.
+Under Assumption 1 and using integration by parts, the DWAD vector can be expressed as
+θ = −2E[Y ˙f(X)],
+which motivates the celebrated plug-in analog estimator of Powell et al. (1989) given by
+�θ = −2 1
+n
+n
+�
+i=1
+Yi�˙f i(Xi),
+�fi(x) =
+1
+n − 1
+n
+�
+j=1,j̸=i
+1
+hd K
+�Xj − x
+h
+�
+,
+where �fi(·) is a “leave-one-out” kernel density estimator for kernel function K : Rd → R and positive
+vanishing (bandwidth) sequence h. For the kernel function, we impose the following conditions.
+Assumption 2.
+(a) K is even, differentiable, and ˙K is bounded;
+(b)
+�
+Rd ˙K(u) ˙K(u)′du is positive definite;
+(c) For some P ≥ 2,
+�
+Rd |K(u)|(1 + ∥u∥P )du +
+�
+Rd ∥ ˙K(u)∥(1 + ∥u∥2)du < ∞
+and
+�
+Rd uaK(u)du =
+
+
+
+
+
+
+
+
+
+1,
+if [a] = 0,
+0,
+if 0 < [a] < P
+µa < ∞,
+if [a] = P,
+where a ∈ Zd
++ is a multi-index.2
+2We employ standard multi-index notation. For a := (a1, . . . , ad) we have (i) [a] := a1+· · ·+ad, (ii) a! := a1! . . . ad!,
+(iii) xa := xa1
+1 . . . xad
+d
+for x ∈ Rd and (iv) q(a)(x) =
+∂[a]q
+∂a1 x1...∂ad xd for smooth enough q : Rd → R.
+4
+
+The estimator �θ can be expressed as a second-order U-statistic with n-varying kernel:
+�θ =
+�n
+2
+�−1
+n
+�
+i<j
+Uij,
+Uij = −
+1
+hd+1 ˙K
+�Xi − Xj
+h
+�
+(Yi − Yj),
+(2.1)
+where �n
+i<j is shorthand notation for �n−1
+i=1
+�n
+j=i+1.
+3
+First-order Theory
+Before presenting our main results concerning the higher-order distributional properties of different
+statistics based on �θ, we overview conventional and alternative asymptotic distributional approx-
+imations, and the variance estimation methods proposed in the literature emerging from those
+distinct approximation frameworks. Limits are taken as h → 0 and n → ∞ unless otherwise noted.
+3.1
+Distributional Approximation
+In a landmark contribution, Powell et al. (1989) studied the first-order large sample distributional
+properties of �θ.
+They showed that, under appropriate restrictions on h and K, the estimator
+�θ is asymptotically linear with (efficient) influence function ψ(z), and thus with semiparametric
+(efficient) asymptotic variance Σ. More precisely, Powell et al. (1989) showed that if Assumptions
+1 and 2 hold, and if nh2 min(P,S) → 0 and nhd+2 → ∞, then
+√n(�θ − θ) =
+1
+√n
+n
+�
+i=1
+ψ(Zi) + oP(1) ⇝ N(0, Σ).
+(3.1)
+This result follows from the U-statistic representation in (2.1) and its Hoeffding decomposition,
+which gives �θ = E[Uij] + ¯L + ¯Q, where
+¯L = 1
+n
+n
+�
+i=1
+Li,
+Li = 2(E[Uij|Zi] − E[Uij]),
+and
+¯Q =
+�n
+2
+�−1
+n
+�
+i<j
+Qij,
+Qij = Uij − E[Uij|Zi] − E[Uij|Zj] + E[Uij],
+both mean zero random vectors. Because E[Uij] = θ + O(hmin(P,S)) and ¯Q = OP(n−1h−(d+2)/2), it
+follows that
+√n(�θ − θ) =
+1
+√n
+n
+�
+i=1
+�
+E[Uij|Zi] − E[Uij]
+�
++ OP
+�√nhmin(P,S) +
+1
+√
+nhd+2
+�
+,
+from which the asymptotic linear representation based on the (efficient) influence function in (3.1)
+is established upon noting that E[∥¯L − �n
+i=1 ψ(Zi)/n∥2] = O(n−1h).
+5
+
+Conceptually, the Hoeffding decomposition and subsequent analysis of each of its terms shows
+that the estimator admits a bilinear form representation in general, which then is reduced to a
+sample average approximation by assuming a bandwidth sequence and kernel shape that makes
+both the misspecification error (smoothing bias) and the variability introduced by ¯Q (“quadratic
+term” term) negligible in large samples. As a result, provided that such tuning parameter choices
+are feasible, the estimator will be asymptotically linear.
+Asymptotic linearity of a semiparametric estimator has several distinct features that may be
+considered attractive from a theoretical point of view (Newey, 1994). In particular, it is a necessary
+condition for semiparametric efficiency and it leads to a limiting distribution that is invariant to the
+choice of the first-step nonparametric estimator entering the two-step semiparametric procedure.
+However, insisting on asymptotic linearity may also have its drawbacks because it requires several
+potentially strong assumptions and leads to a large sample theory that may not accurately represent
+the finite sample behavior of the statistic. In the case of �θ, asymptotic linearity requires P > 2
+unless d = 1, thereby forcing restrictive smoothness conditions (S ≥ P) and the use of higher-order
+kernels or similar debiasing techniques (see, e.g., Chernozhukov et al., 2022, and references therein).
+In addition, classical asymptotic linear theory (whenever valid) leads to a limiting experiment
+which is invariant to the particular choices of smoothing (K) and bandwidth (h) tuning parameters
+involved in the construction of the estimator, and therefore it is unable to “adapt” to changes in
+those choices. As a result, asymptotically linear large sample distribution theory is silent with
+respect to the impact that tuning parameter choices may have on the finite sample behavior of the
+two-step semiparametric statistic.
+To address the aforementioned limitations with classical asymptotic distribution theory, Cattaneo et al.
+(2014a) proposed a more general distributional approximation for kernel-based DWAD estimators
+that accommodates but does not enforces asymptotic linearity. The core idea is to characterize the
+joint asymptotic distributional features of both the linear (¯L) and quadratic ( ¯Q) terms jointly, and
+in the process develop an alternative first-order asymptotic theory that accommodates weaker as-
+sumptions than those imposed in the classical asymptotically linear distribution theory. Formally,
+if Assumptions 1 and 2 hold, and if min(nhd+2, 1)nh2 min(P,S) → 0 and n2hd → ∞, then
+(V[�θ])−1/2(�θ − θ) ⇝ N(0, I),
+(3.2)
+where
+V[�θ] = V[¯L] + V[ ¯Q],
+V[¯L] = 1
+n
+�
+Σ + o(1)
+�
+,
+V[ ¯Q] =
+�n
+2
+�−1
+h−d−2�
+∆ + o(1)
+�
+,
+and ∆ = 2E[v(X)f(X)]
+�
+Rd ˙K(u) ˙K(u)′du.
+This more general distributional approximation was developed explicitly in an attempt to better
+characterize the finite sample behavior of �θ.
+The result in (3.2) shows that the conditions on
+the bandwidth sequence may be considerably weakened without invalidating the limiting Gaussian
+distribution, albeit the asymptotic variance formula may change. Importantly, if nhd+2 is bounded
+6
+
+then �θ is no longer asymptotically linear and its limiting distribution will cease to be invariant with
+respect to the underlying preliminary nonparametric estimator. In particular, if nhd+2 → c > 0
+then �θ is root-n consistency but not asymptotically linear. In addition, because the bandwidth
+is allowed to be “smaller” than usual, the bias of the estimator is controlled in a different way,
+removing the need for higher-order kernels. Interestingly, (3.2) allows for the point estimator to
+not even be consistent for θ, for sufficiently small bandwidth sequences.
+Beyond the aforementioned technical considerations, the result in (3.2) can conceptually be
+interpreted as a more refined first-order distributional approximation for the standarized statistics
+(V[�θ])−1/2(�θ − θ), which by relying on a quadratic approximation (i.e., capturing the stochastic
+contributions of both ¯L and ¯Q) it is expected to offer a “better” distributional approximation.
+The idea of standarizing a U-statistic by the joint variance of the linear and quadratic terms
+underlying its Hoeffding decomposition can be traced back to the original paper of Hoeffding
+(1948, p.
+307).
+Furthermore, the asymptotic distribution theory proposed by Cattaneo et al.
+(2014a) can be viewed as highlighting the well known trade-off between robustness and efficiency in
+two-step semiparametric settings: �θ is semiparametric efficient if and only if nhd+2 → ∞, while it
+seems possible to construct more robust inference procedures under considerably weaker conditions
+that would not be semiparametric efficient. Simulation evidence reported in Cattaneo et al. (2010,
+2014a,b) corroborated those conceptual interpretations numerically, but no formal justification is
+available in the literature. Theorem 1 below will offer the first theoretical result in the literature
+highlighting specific robustness features of the distributional approximation in (3.2) by showing that
+such approximation has a demonstrably smaller higher-order distributional approximation error.
+3.2
+Variance Estimation
+Based on the asymptotically linear distributional approximation in (3.1), Powell et al. (1989) also
+proposed the following variance estimator
+�Σ = 1
+n
+n
+�
+i=1
+�Li�L′
+i,
+�Li = 2
+�
+1
+n − 1
+n
+�
+j=1,j̸=i
+Uij − �θ
+�
+,
+and proved its consistency (i.e., �Σ →P Σ) under the same bandwidth sequences (nh2 min(P,S) → 0
+and nhd+2 → ∞) required for asymptotic linearity. This result justifies employing the Studentized
+statistic
+�Σ−1/2√n(�θ − θ) ⇝ N(0, I)
+(3.3)
+for inferences purposes, that is, to construct a confidence interval for θ and smooth trasformations
+thereof, or to carry out statistical hypothesis testing in the usual way.
+However, motivated by their alternative asymptotic approximation, Cattaneo et al. (2014a) showed
+that
+1
+n
+�Σ = 1
+n[Σ + oP(1)] + 2
+�n
+2
+�−1
+h−d−2[∆ + oP(1)],
+7
+
+which implies that the consistency result �Σ →P Σ is valid if and only if nhd+2 → ∞; otherwise,
+�Σ is in general asymptotically upwards biased relative to V[�θ] in (3.2). Because �Σ is asymptoti-
+cally equivalent to the jackknife variance estimator of �θ, Cattaneo et al. (2014b) also noted that
+the asymptotic bias of �Σ is a result of a more generic phenomena underlying jackknife variance
+estimators studied in Efron and Stein (1981).
+See also Matsushita and Otsu (2021) for related
+discussion.
+To conduct asymptotically valid inference under the more general small bandwidth asymptotic
+regime, Cattaneo et al. (2014a) proposed several “debiased” variance estimators, including the
+following
+�V = 1
+n
+�Σ −
+�n
+2
+�−1
+h−d−2 �∆,
+�∆ = hd+2
+�n
+2
+�−1 n−1
+�
+i=1
+n
+�
+j=i+1
+UijU ′
+ij,
+and show that �∆ →P ∆ under the same bandwidth sequences (nh2 min(P,S) → 0 and n2hd → ∞)
+required for (3.2) to hold. The estimator �∆ is asymptotically equivalent to the debiasing procedure
+proposed in Efron and Stein (1981). This result justifies employing the Studentized statistic
+�V −1/2(�θ − θ) ⇝ N(0, I)
+(3.4)
+for more “robust” inferences purposes relative to those constructed using (3.3).
+Heuristically, robustness manifests in two distinct ways. First, the underlying Gaussian distribu-
+tional approximation holds under weaker bandwidth restrictions and does not require asymptotic
+linearity, thereby making the limiting distribution explicitly depend on tuning parameter choices.
+Second, the new standard error formula �V is derived from the more general small bandwidth ap-
+proximation and make explicit the contribution of terms regarded as higher-order by classical large
+sample distributional approximations.
+While not reproduced here to conserve space, the in-depth Monte Carlo evidence reported in
+Cattaneo et al. (2010, 2014a,b) also showed that employing inference procedures based on (3.4) lead
+to large improvements in terms of “robustness” to bandwidth choice and other tuning inputs, when
+compared to classical asymptotically linear inference procedures based on (3.3). Theorem 2 below
+will study those two feasible statistics and show formally that the distributional approximation
+(3.4) has demonstrably smaller higher-order errors than the distributional approximation (3.3).
+4
+Higher-order Distribution Theory
+We present Edgeworth expansions for scalar standarized and studentized statistics based on �θν −θν
+with �θν := ν′�θ and θν := ν′θ, where ν ∈ Rd is a fixed non-random vector. Considering scalar
+statistics substantially simplify the developments and proofs without affecting the main conceptual
+and theoretical takeaways. The sequence ϑ will first be non-random, thereby allowing us to inves-
+tigate the role of classical distributional approximations based on asymptotic linearity vis-`a-vis the
+more general distributional approximations based on small bandwidth asymptotics for standarized
+8
+
+statistics. Then, the sequence ϑ will be taken to be random based on the two alternative variance
+estimators introduced in the previous section, thereby allowing us to investigate the role of variance
+estimation on the performance of distributional approximations for Studentized statistics.
+4.1
+Distributional Approximation
+Our first theorem offers a valid Edgeworth expansion for the sampling distribution function
+Fϑ(t) := P
+� �θν − θν
+ϑ
+≤ t
+�
+,
+t ∈ R,
+with precise characterization of the first three cummulants determining the leading errors in dis-
+tributional approximation of the Studentized statistic. Define the following key quantities:
+β := 2(−1)P �
+[k]=P
+µk
+k! E
+�
+g(X) ∂k
+∂Xk ν′ ˙f(X)
+�
+,
+σ2 := V[�θν],
+κ1 := E[ν′ψ(Z)3],
+κ2 := 4E[δ(Z) ˙η(Z)] − 8E[δ(Z)2]θν + 4θ3
+ν,
+where δ(Z) := ν′ψ(Z)/2 + θν and η(Z2) = limn→∞ E[δ(Z1)ν′U12|Z2].
+Theorem 1 (Standardized). Suppose Assumptions 1 and 2 hold. If √nhP → 0 and nhd+2 → ∞,
+then for any positive non-random sequence ϑ such that ϑ/σ → 1,
+sup
+t∈R
+��Fϑ(t) − Gϑ(t)
+�� = O(Rn) + o(n−1/2)
+with
+Gϑ(t) := Φ(t) − φ(t)
+�β
+ϑhP +
+�σ2
+ϑ2 − 1
+�
++ κ1 + κ2
+6n2ϑ3 (t2 − 1)
+�
+,
+and Rn := nh2P +
+�
+(log n)3
+nhd+2
+�3/2
++ hd/3+1
+nhd+2 +
+�
+hd/9+2/3
+nhd+2
+�3/2
+, where Φ and φ are the c.d.f. and p.d.f. of
+a standard Gaussian distribution. Furthermore, if (log n)3
+nhd+2 → 0, then Rn = o
+�√nhP +
+1
+nhd+2
+�
+.
+This theorem is proven by verifying the high-level conditions of a result in Appendix A es-
+tablishing a valid Edgeworth Expansion for a generic class of U-statistics with n-varying kernels,
+which may be of independent theoretical interest.
+Specifically, Theorem A.1 and its corollary
+A.1 improve on Jing and Wang (2003) by allowing for n-varying kernels under more general con-
+dition suitable for the semiparametric problem of interest herein. Theorem 1 also improves on
+Nishiyama and Robinson (2000, Theorem 1) in two respects: (i) it allows for a generic standard-
+ization scheme ϑ instead of their specific choice
+�
+ν′Σν/n; and (ii) it presents a valid Edgeworth
+expansion with precise error rates with respect to the bandwidth. These improvements enable us
+to compare the two different distributional approximations of interest, (3.1) vs. (3.2).
+The main conclusion in Theorem 1 follows the expected logic underlying Edgeworth Expansions:
+β
+ϑhP , σ2
+ϑ2 −1 and κ1+κ2
+6n2ϑ3 capture, respectively, the standardized bias, variance and higher moments of
+9
+
+the statistic. Inspection of these terms lead to interesting implications for large sample distribution
+theory, in particular leading to a sharp contrast between distribution theory based on asymptotic
+linear representations vis-`a-vis alternative asymptotics, each with either fixed-bandwidth or leading
+asymptotic variance standardization. More specifically, we can consider four distinct standarization
+schemes: from first-order asymptotic linear theory (3.1) we have
+ϑ2
+AL := V[ν′ ¯L] = 1
+nV[ν′Li]
+and
+˘ϑ2
+AL := 1
+nν′Σν,
+while from small bandwidth distribution theory (3.2) we have
+ϑ2
+SB := V[�θν] = σ2
+and
+˘ϑ2
+SB := 1
+nν′Σν +
+�n
+2
+�−1
+h−d−2ν′∆ν.
+The standardizations ϑAL and ϑSB correspond to those constructed using the pre-asymptotic variance
+of the point estimator, each justified according to the asymptotic regime considered (asymptotic
+linear and small bandwidth, respectively). In contrast, the standardizations ˘ϑAL and ˘ϑSB correspond
+to employing the leading term only in the large sample approximation of the pre-asymptotic variance
+of the point estimator, again keeping only those terms that are justified by the asymptotic regime
+considered. That is, ϑAL = ˘ϑAL + o(n−1) and ϑSB = ˘ϑSB + o(n−1) under the assumptions of Theorem
+1. For comparison, Nishiyama and Robinson (2000, Theorem 1) used ˘ϑAL.
+Employing Theorem 1 we can now compare the different approaches to standardization and
+their associated errors generated in the distributional approximation. Firstly, it is easy to see that
+employing ˘ϑAL and ˘ϑSB will generate larger distributional approximation errors relative to their pre-
+asymptotic counterparts, ϑAL and ϑSB, respectively. See the proof in the appendix for exact rates,
+which are not reproduced here to conserve space. The main conceptual message is that one should
+always employ variance formulas that capture the full variability of the statistic whenever possible,
+as opposed to employing those that capture only the leading variability in large samples.
+See
+Calonico et al. (2018, 2022) for closely related results in the context of nonparametric kernel-based
+density and local polynomial regression estimation and inference.
+Secondly, and more importantly for our purposes, Theorem 1 shows that even if the full finite-
+sample variance of the point estimator is captured for standardization purposes, it is still crucial to
+incorporate the variability of both the linear and quadratic terms. More precisely, setting ϑ = ϑAL
+then σ2
+ϑ2 − 1 = O(n−1h−d−2), while setting ϑ = ϑSB implies that σ2
+ϑ2 − 1 = 0. As a consequence,
+our first main result shows that employing the pre-asymptotic variance of the statistic, which is
+naturally justified by the more general asymptotic distributional approximation (3.2), leads to the
+smallest error in the distributional approximation of the sampling distribution of the standardized
+statistic. This result thus provides theory-based evidence in favor of employing small bandwidth
+asymptotics for kernel-based DWAD methods whenever the goal is to minimize errors of inference
+procedures relying on large sample Gaussian approximations.
+The methodological implications of our first theoretical result can be illustrated by analyzing the
+10
+
+coverage error of standardized confidence intervals. According to Theorem 1, for any α ∈ (0, 1), a
+100(1 − α)% two-sided confidence interval based on asymptotic linearity satisfy
+P
+�
+θν ∈
+��θν ± Φ1−α/2ϑAL
+��
+= 1 − α +
+KAL
+nhd+2 + o
+�√nhP + n−1h−d−2 + n−1/2�
+,
+where Φα = Φ−1(α), and KAL = 2Φ1−α/2φ(1 − α/2)n−1h−d−2(σ2/ϑ2
+AL − 1) = O(1 + h2), with the
+exact form of the leading terms described in the appendix. On the other hand, under the conditions
+in Theorem 1, a 100(1− α)% two-sided confidence intervals based on small bandwidth asymptotics
+satisfy
+P
+�
+θν ∈
+��θν ± Φ1−α/2ϑSB
+��
+= 1 − α + o
+�√nhP + n−1h−d−2 + n−1/2�
+,
+implying a smaller coverage error distortion in large samples.
+The above coverage error comparison is conceptually useful, but it does not directly translate to
+practice because the confidence intervals are infeasible. To complement the results in this section,
+we consider next the implications of constructing variance estimators and hence study feasible
+(Studentized) inference procedures.
+4.2
+Variance Estimation
+We study the role of Studentization and thus obtain valid Edgeworth expansion for the sampling
+distribution functions
+FAL(t) := P
+� �θν − θν
+�ϑAL
+≤ t
+�
+,
+�ϑAL := 1
+nν′�Σν
+and
+FSB(t) := P
+� �θν − θν
+�ϑSB
+≤ t
+�
+,
+�ϑSB := 1
+nν′�Σν −
+�n
+2
+�−1
+h−d−2ν′ �∆ν.
+Crucially, the estimators �Σ and �∆ target the total variability nV[¯L] = V[Li] and
+�n
+2
+�
+hd+2V[ ¯Q] =
+hd+2V[Qij], respectively, and not just their leading quantities Σ and ∆. Therefore, in light of the
+results reported in the previous section, we do not explicitly consider na¨ıve plug-in estimators of
+˘ϑAL and ˘ϑSBA such as
+2
+n2
+�n
+i=1(ν′[�˙e(Xi) − y�˙f(Xi) − �θ])2 for the former, where �˙e(x) and �˙f(x) are
+plug-in nonparametric estimators of ˙e(x) and ˙f(x), respectively. These alternative Studentization
+schemes will lead to larger higher-order distributional approximation errors when compared to �ϑAL
+and �ϑSB.
+Theorem 2 (Studentized). Suppose Assumptions 1 and 2 hold with p ≥ 8. If √nhP → 0 and
+nhd+2/(log n)9 → ∞, then
+sup
+t∈R
+��FAL(t) − GAL(t)
+�� = o(rn)
+with
+GAL(t) := Φ(t) − φ(t)
+�√nhP β
+ν′Σν
+−
+1
+nhd+2
+ν′∆ν
+ν′Σν t −
+1
+√n6(ν′Σν)3
+�
+κ1(2t2 + 1) + κ2(t2 + 1)
+��
+,
+11
+
+and
+sup
+t∈R
+��FSB(t) − GSB(t)
+�� = o(rn)
+with
+GSB(t) := Φ(t) − φ(t)
+�√nhP β
+ν′Σν
+−
+1
+√n6(ν′Σν)3
+�
+κ1(2t2 + 1) + κ2(t2 + 1)
+��
+,
+where rn := √nhP + n−1h−d−2 + n−1/2
+This theorem shows that employing Studentization based on small bandwidth asymptotics offers
+demonstrable improvements in terms of distributional approximations for the resulting feasible t-
+test. The main practical implication of our second result can again be illustrated by analyzing
+the coverage error of Studentized confidence intervals. According to Theorem 2, and as it was
+the case for stdentized confidence intervals, a 100(1 − α)% two-sided confidence intervals based on
+asymptotic linearity satisfy
+P
+�
+θν ∈
+��θν ± Φ1−α/2 �ϑAL
+��
+= 1 − α +
+1
+nhd+2 2Φ1−α/2φ(1 − α/2)ν′∆ν
+ν′Σν + o(rn),
+while, under the conditions in Theorem 2, a 100(1 − α)% two-sided confidence intervals based on
+small bandwidth asymptotics satisfy
+P
+�
+θν ∈
+��θν ± Φ1−α/2 �ϑSB
+��
+= 1 − α + o(rn),
+implying a smaller coverage error distortion in large samples. This result provides a theoretical
+justification to the simulation evidence reported in Cattaneo et al. (2014a,b, 2010) where feasible
+confidence intervals based on small bandwidth asymptotics were shown to offer better finite sample
+performance in terms of coverage error than their counterparts based classical asymptotic linear
+approximations.
+5
+Conclusion
+Employing Edgeworth expansions, we study the higher-order properties of two alternative first-order
+distributional approximations and their associated inference procedures (e.g., confidence intervals)
+for the kernel-based DWAD estimator of Powell et al. (1989). We showed that small bandwidth
+asymptotics not only give demonstrable better distributional approximations than asymptotic linear
+approximations, but also justify employing a variance estimator for Studentization purposes that
+also improves the distributional approximation.
+The main take away from our results is that
+in two-step semiparametric settings and related problems, alternative asymptotic approximations
+that capture higher-order terms ignored by classic asymptotic linear approximation can deliver
+better distributional approximations and, by implication, better inference procedures with improved
+performance in finite samples.
+While beyond the scope of this paper, it would be of interest to develop analogous Edgeworth
+12
+
+expansions for non-linear two-step semiparamtric procedures developed using alternative asymp-
+totic approximations and resampling methods (Cattaneo et al., 2013; Cattaneo and Jansson, 2018;
+Cattaneo et al., 2019). For the special case of kernel-based DWAD estimators (a linear two-step
+kernle-based semiparametric estimator), Nishiyama and Robinson (2005) present results that could
+be contrasted with those obtained under under small bandwidth asymptotics (Cattaneo et al.,
+2014b). We relegate such developments for future research due to the substantial amount of addi-
+tional technical work required.
+A
+Edgeworth Expansion for Second-Order U-Statistic
+Consider the sequence of maps (un : Rd × Rd → R, n ∈ N) where u := un is symmetric in terms
+of the permutation of its two arguments for every n ∈ N. Given a random sample Z1, . . . , Zn for
+n ≥ 2 of the random variable Z taking values on Rd, the object of interest in the second order
+U-statistics with an n-varying kernel given by
+¯U :=
+�n
+2
+�−1
+n
+�
+1≤i<j≤n
+u(Zi, Zj).
+(A.1)
+We drop the subscript n to simplify notation. By the Hoeffding decomposition,
+¯U − θ
+ϑ
+= B + L + Q,
+where B := (Eu(Z1, Z2) − θ)/ϑ, L :=
+1
+ϑn
+�n
+i=1 ℓi and Q := 1
+ϑ
+�n
+2
+�−1 �n
+1≤i<j≤n qij, where ℓi := ℓ(Zi)
+and qij := q(Zi, Zj) with ℓ(Z1) := 2[Eu(Z1, Z2|Z1) − Eu(Z1, Z2)] and q(Z1, Z2) := u(Z1, Z2) −
+ℓ(Z1)/2 − ℓ(Z2)/2 − Eu(Z1, Z2). Given the decomposition above,
+σ2 := V[ ¯U] = 1
+nσ2
+ℓ +
+�n
+2
+�−1
+σ2
+q,
+(A.2)
+where σ2
+ℓ := Eℓ2
+1 and σ2
+q := Eq2
+12.
+We establish a valid third-order Edgeworth expansion for the the sampling distribution of the
+centered and standardized version of ¯U:
+F(t) := P
+� ¯U − θ
+ϑ
+≤ t
+�
+,
+t ∈ R,
+(A.3)
+where θ ∈ R and ϑ > 0 are non-random.
+Theorem A.1. Let the following conditions hold:
+(a) E
+�
+(ℓ1/σℓ)3�
+= O(1) and E[|q12|]2+δ < ∞, and σℓ > 0
+(b)
+σq
+√nσℓ → 0 and σ
+ϑ → 1.
+13
+
+(c) lim sup
+n→∞
+lim sup
+|t|→∞
+|E exp(ιtℓ1/σℓ)| < 1.
+Then, supt∈R |F(t) − G(t)| = O(E) + o(n−1/2) where G is the distribution function with character-
+istic function
+χG(t) := eιtB− t2
+2
+
+1 +
+9
+�
+j=2
+(ιt)j γj
+
+ ,
+with ι := √−1,
+γ2 = 1
+2
+�
+σ2
+ϑ2 − 1
+�
+,
+γ3 =
+1
+6ϑ3n2
+�
+Eℓ3
+1 + 6Eℓ1ℓ2q12
+�
+,
+γ4 =
+1
+4ϑ2
+�
+σ2
+ϑ2 − 1
+� �n
+2
+�−1
+σ2
+q
+γ5 =
+1
+12n2ϑ5
+��n
+2
+�−1
+(Eℓ3
+1)σ2
+q + 6ϑ2 � σ2
+ℓ
+ϑ2n − 1
+�
+Eℓ1ℓ2q12
+�
+γ6 =
+1
+6ϑ6n4
+�
+(Eℓ3
+1)Eℓ1ℓ2q12 + 12
+�n
+2
+�−2�n
+4
+�
+[Eℓ1ℓ2q12]2
+�
+,
+γ7 = 0,
+γ8 =
+1
+4ϑ6n4
+� σ2
+ℓ
+ϑ2n − 1
+� �n
+2
+�−2�n
+4
+�
+[Eℓ1ℓ2q12]2 ,
+γ9 =
+1
+12ϑ9n6
+�n
+2
+�−2�n
+4
+�
+Eℓ3
+1 [Eℓ1ℓ2q12]2 ,
+and
+E :=
+�
+log n
+n3/2σℓ
+�2+δ
+Π2+δ(n) +
+�
+(log n)
+4+δ
+2+δ σ2
+q
+nσ2
+ℓ
+�2+δ
+2
++
+�
+log n
+nσℓ
+�2+δ
+Π2+δ(log n)
++
+1
+σ4
+ℓ nE|ℓ2
+1ℓ2q12| +
+1
+σ5
+ℓ n3/2 E|ℓ2
+1ℓ2
+2q12| +
+1
+σ2
+ℓ n2 E|ℓ1q2
+12| +
+1
+σ5
+ℓ n3/2 E|ℓ1ℓ2ℓ3q13q23|
++
+1
+σ7
+ℓ n3/2 (Eℓ1ℓ2q12)(E|ℓ2
+1ℓ2q12|) +
+1
+σ8
+ℓ n2 (Eℓ1ℓ2q12)(E|ℓ2
+1ℓ2
+2q12|),
+with Π2+δ(m) := E| �[m]−1
+i=1
+�n
+j=i+1 qij|2+δ for real m > 1 and [·] denoting the floor operator.
+Corollary A.1. Let the assumptions of Theorem A.1 hold. If B → 0, then
+sup
+t∈R
+|F(t) − G(t)| = O
+�
+B2 + E
+�
++ o(n−1/2),
+with
+χG(t) := e− t2
+2
+
+1 + Bιt +
+9
+�
+j=2
+� ιt
+ϑ
+�j γj
+
+ .
+Remark A.1. Lemma A.2 below gives the following simpler bound
+Π2+δ(m) ≲ (nmσ2
+q)(2+δ)/2 ∨ mn1+δ/2E
+�
+(E(q2
+12|Z1))1+δ/2�
+∨ nmE|q12|2+δ,
+where ≲ denotes bounded up to a fixed constant, and a ∨ b = max{a, b}.
+⌟
+14
+
+Remark A.2. We can invert the characteristic function above to obtain a close form for F using
+the fact that for non-negative integer k,
+1
+2π
+�
+R exp (−ιtx − t2/2)(ιt)kdt = Hk(x)φ(x), where Hk(x) is
+the k-th order Hermite polynomial (e.g., H0(k) = 1, H1(x) = x, H2(x) = x2 − 1, H3(x) = x3 − 3x).
+Therefore, the distribution function of χG(t) from Corollary A.1 is
+G(x) = Φ(x) − φ(x)
+
+
+9
+�
+j=1
+γjHj−1(x)
+
+ .
+⌟
+Remark A.3. To compare to Jing and Wang (2003), let u(·, ·) not dependent on n, θ = Eu(Z1, Z2),
+ϑ2 = σ2
+ℓ /n, and E|q12|2+δ bounded. Then, E = o(n−1/2) and χG(t) = exp(−t2/2)
+�
+1 − ικ3t3
+6√n
+�
++
+o(n−1/2), giving
+G(x) = Φ(x) − φ(x)
+1
+6√n
+�
+E
+�
+ℓi
+σℓ
+�3
++ 6Eℓ1ℓ2q12
+σ3
+ℓ
+�
+(x2 − 1).
+⌟
+A.1
+Proof of Theorem A.1
+Let χF denote the characteristic function F and g be the density of G. Using the well-known
+“smoothing inequality” (Bhattacharya and Rao, 1976; Hall, 1992), we write
+ρ(F, G) ≤ 1
+π
+�� υ
+−υ
+����
+χF (t) − χG(t)
+t
+���� dt + 24 supx∈R |g(x)|
+υ
+�
+,
+υ > 0
+where ρ is the Kolmogorov distance. We set v = √n log n and split the range of integration into
+“low” frequencies and “high” frequencies. By the triangle inequality,
+ρ(F, G) ≲ I1 + I2 + I3 + I4 +
+1
+√n log n,
+(A.4)
+where
+I1 :=
+�
+|t|≤log n
+����
+χF(t) − χG(t)
+t
+���� dt,
+I2 :=
+�
+log n<|t|≤c√n
+����
+χF(t)
+t
+���� dt,
+I3 :=
+�
+c√n<|t|≤√n log n
+����
+χF (t)
+t
+���� dt,
+I4 :=
+�
+|t|>log n
+����
+χG(t)
+t
+���� dt;
+Moreover, c > 0 is a fixed constant to be specified later.
+We now bound each of these integrals in turn. We use extensively the fact hat
+������
+exp(ιx) −
+2
+�
+j=0
+(ιx)j
+j!
+������
+≤ |x|2+δ
+,
+∀δ ∈ [0, 1].
+(A.5)
+15
+
+Also, define for ψ(t) := E exp(ιtℓ1) for t ∈ R where σℓ is positive by Assumption (a).
+Bound for I1
+We start by decomposing χF(t) = E exp
+�
+ιt( ¯U−θ
+ϑ )
+�
+= exp(ιtb)χL+Q(t) where χL+Q(t) := E exp(ιtL) exp(ιtQ).
+Use (A.5) to expand the second exponential in χL+Q(t) to write
+χL+Q(t) = E exp(ιtL)[1 + ιtQ − 1
+2(tQ)2 + O((tQ)2+δ)].
+(A.6)
+Since ℓ1, . . . , ℓn is a i.i.d sequence (for a given n ≥ 2), the first term in (A.6) can be written as
+E exp(ιtL) = E exp
+�
+ιt
+ϑn
+n
+�
+i=1
+ℓi
+�
+= E
+n
+�
+i=1
+exp
+�ιtℓi
+ϑn
+�
+=
+n
+�
+i=1
+E exp
+�ιtℓi
+ϑn
+�
+= ψn
+� t
+ϑn
+�
+.
+For the second term in (A.6), we have
+E exp(ιtL)ιtQ = ιt
+ϑ
+�n
+2
+�−1 �
+i<j
+E
+n
+�
+k=1
+exp
+� ιt
+ϑnℓk
+�
+qij
+= ιt
+ϑ
+�n
+2
+�−1 �
+i<j
+E
+n
+�
+k̸=i,j
+exp
+� ιt
+ϑnℓk
+�
+exp
+� ιt
+ϑn(ℓi + ℓj)
+�
+qij
+= ιt
+ϑ
+�n
+2
+�−1 �
+i<j
+n
+�
+k̸=i,j
+E exp
+� ιt
+ϑnℓk
+�
+E exp
+� ιt
+ϑn(ℓi + ℓj)
+�
+qij
+= ιt
+ϑ ψn−2 � t
+ϑn
+�
+E exp
+� ιt
+ϑn(ℓ1 + ℓ2)
+�
+q12.
+Similarly, for the third term in (A.6), we use
+E exp(ιtL)(ιtQ)2 =
+�
+ιt
+ϑ
+�n
+2
+�−1�2
+×
+
+�
+i<j
+E
+n
+�
+k=1
+exp
+� ιt
+ϑnℓk
+�
+q2
+ij
++
+�
+i<j=k<l
+E
+n
+�
+m̸=i,j,l
+exp
+� ιt
+ϑnℓm
+�
+qijqjl
++
+�
+i<j<k<l
+E
+n
+�
+m̸=i,j,k,l
+exp
+� ιt
+ϑnℓm
+�
+qijqkl
+
+
+=
+�
+ιt
+ϑ
+�n
+2
+�−1�2
+×
+�
+ψn−2 � t
+ϑn
+� �n
+2
+�
+E exp
+� ιt
+ϑn(ℓ1 + ℓ2)
+�
+q2
+12
++ ψn−3 � t
+ϑn
+� �n
+3
+�
+E exp
+� ιt
+ϑn(ℓ1 + ℓ2 + ℓ3)
+�
+q12q23
++ψn−4 � t
+ϑn
+� �n
+4
+� �
+E exp
+� ιt
+ϑn(ℓ1 + ℓ2)
+�
+q12
+�2�
+.
+16
+
+For the last in (A.6), we have
+|E exp(ιtL)(tQ)2+δ| ≤ E|tQ|2+δ =
+�
+|t|
+ϑ
+�n
+2
+�−1�2+δ
+E
+���
+�
+i<j
+qij
+���
+2+δ
+= O
+�� |t|
+ϑn2
+�2+δ
+Π2+δ(n)
+�
+.
+Using the last four displays, we simplify (A.6) to
+χL+Q(t) = ψn � t
+ϑn
+�
++ ψn−2 � t
+ϑn
+�
+�
+ιt
+ϑ E exp( ιt
+ϑn(ℓ1 + ℓ2))q12 + (it)2
+2ϑ2
+�n
+2
+�−1
+E exp( ιt
+ϑn(ℓ1 + ℓ2))q2
+12
+�
++ 1
+2
+�
+ιt
+ϑ
+�n
+2
+�−1�2
+ψn−3 � t
+ϑn
+� �n
+3
+�
+E exp( ιt
+ϑn(ℓ1 + ℓ2 + ℓ3))q13q23
++ 1
+2
+�
+ιt
+ϑ
+�n
+2
+�−1�2
+ψn−4 � t
+ϑn
+� �n
+4
+� �
+E exp( ιt
+ϑn(ℓ1 + ℓ2))q12
+�2
++ O
+��
+|t|
+ϑn2
+�2+δ
+Π2+δ(n)
+�
+.
+(A.7)
+We now expand the exponentials inside the expectation and collect terms. For notation brevity,
+write a := ιt
+ϑn. For the first one, we have
+E exp(a(ℓ1 + ℓ2))q12 = E
+�
+exp(aℓ1) − 1
+��
+exp(aℓ2) − 1
+�
+q12
+= E
+��
+exp(aℓ1) − 1 − aℓ1
+��
+exp(aℓ2) − 1 − aℓ2
+�
+q12
++aℓ1
+�
+exp(aℓ2) − 1 − aℓ2
+�
+q12 + aℓ2
+�
+exp(aℓ1) − 1 − aℓ1
+�
+q12 + a2ℓ1ℓ2q12
+�
+= a2Eℓ1ℓ2q12 + O
+�
+|a|3E|ℓ2
+1ℓ2q12| + |a|4E|ℓ2
+1ℓ2
+2q12|
+�
+,
+for the second term we have
+E exp(a(ℓ1 + ℓ2))q2
+12 = σ2
+q + E
+�
+exp(a(ℓ1 + ℓ2)) − 1
+�
+q2
+12 = σ2
+q + O(|a|E|ℓ1q2
+12|),
+and for the third term we have
+E
+3
+�
+i=1
+exp(aℓi)q13q23 = E
+3
+�
+i=1
+�
+exp(aℓi) − 1
+�
+q13q23 = O(|a|3E|ℓ1ℓ2ℓ3q13q23|).
+Plugging the above expansions back into (A.7) yields
+χL+Q(t) = ψn � t
+ϑn
+�
++ ψn−2 � t
+ϑn
+�
+�
+(ιt)3
+ϑ3n2 Eℓ1ℓ2q12 + (it)2
+2ϑ2
+�n
+2
+�−1
+σ2
+q
+�
++ ψn−4 � t
+ϑn
+� 1
+2
+(ιt)6
+ϑ6n4
+�n
+2
+�−2�n
+4
+�
+[Eℓ1ℓ2q12]2
+17
+
++ O
+�
+ψn−2 � t
+ϑn
+� �
+t4
+ϑ4n3E|ℓ2
+1ℓ2q12| +
+|t|5
+ϑ5n4E|ℓ2
+1ℓ2
+2q12| +
+|t|3
+ϑ2n3E|ℓ1q2
+12|
+��
++ O
+�
+ψn−3 � t
+ϑn
+� |t|5
+ϑ5n4E|ℓ1ℓ2ℓ3q13q23|
+�
++ O
+�
+ψn−4 � t
+ϑn
+� �
+|t|7
+ϑ7n5(Eℓ1ℓ2q12)(E|ℓ2
+1ℓ2q12|) +
+|t|8
+ϑ8n6(Eℓ1ℓ2q12)(E|ℓ2
+1ℓ2
+2q12|)
+��
++ O
+��
+|t|
+ϑn2
+�2+δ
+Π2+δ(n)
+�
+.
+(A.8)
+From the Edgeworth expansion theory for sum off i.i.d random variables (Bhattacharya and Rao,
+1976; Hall, 1992), we have for |t| ≤ δ∗√n for some small enough δ∗ > 0
+ψn
+�
+t
+σℓ
+√n
+�
+= exp
+�
+− 1
+2t2�
+�
+1 − ιt3
+6√nE
+� ℓ1
+σℓ
+�3�
++ o
+�(|t|3 + t6)
+√n
+exp(−t2/4)
+�
+.
+Let αk := σℓ
+√n−k
+ϑn
+for k ∈ {0, 2, 3, 4}. Since αk ≍ 1 by assumption, where ≍ denotes proportional
+up to a fixed finite positive constant, we obtain
+ψn−k
+� t
+ϑn
+�
+= ψn−k
+�
+αkt
+σℓ
+√
+n − k
+�
+= exp
+�
+− 1
+2 (αkt)2� �
+1 − ι(αkt)3
+6
+√
+n − kE
+�ℓ1
+σℓ
+�3�
++ o
+�(|t|3 + t6)
+√n
+exp(−(αkt)2/4)
+�
+.
+A first-order Taylor expansion yields
+exp(−(αkt)2/2) = exp(−t2/2)
+�
+1 − (α2
+k − 1)t2
+2 + O(p(t)(α2
+k − 1)2)
+�
+,
+and plugging it back in the previous expression, we have
+ψn−k
+� t
+ϑn
+�
+= exp
+�
+− t2
+2
+� �
+1 − (α2
+k − 1)t2
+2 − ι(αkt)3
+6
+√
+n − kE
+�ℓ1
+σℓ
+�3�
++ O
+�
+(α2
+k − 1)2p(t) exp (−t2/2)
+�
++ o
+�(|t|3 + t6)
+√n
+exp(−(αkt)2/4)
+�
+.
+Use the fact that α2
+k = α2
+0(1 − k/n) =
+�
+σℓ
+ϑ√n
+�2
++ O(n−1) to conclude that
+ψn−k
+� t
+ϑn
+�
+= exp
+�
+− t2
+2
+� �
+1 −
+� σ2
+ℓ
+ϑ2n − 1
+� t2
+2 −
+ιt3
+6ϑ3n2 Eℓ3
+1
+�
++ O
+�� σ2
+ℓ
+ϑ2n − 1
+�2
+p(t) exp (−t2/2)
+�
++ o
+�(|t|3 + t6)
+√n
+exp(−t2/4)
+�
+,
+(A.9)
+for |t| ≤ δ∗√n.
+18
+
+Combine (A.8) and (A.9) to conclude that, for |t| ≤ δ∗√n,
+χL+Q(t) = exp
+�
+− t2
+2
+� �
+1 −
+� σ2
+ℓ
+ϑ2n − 1
+� t2
+2 −
+ιt3
+6ϑ3n2 Eℓ3
+1
+�
+×
+�
+1 + (it)2
+2ϑ2
+�n
+2
+�−1
+σ2
+q + (ιt)3
+ϑ3n2 Eℓ1ℓ2q12 + 1
+2
+(ιt)6
+ϑ6n4
+�n
+2
+�−2�n
+4
+�
+[Eℓ1ℓ2q12]2
+�
++ O
+�
+exp
+�
+− t2
+2
+� �
+1 +
+� σ2
+ℓ
+ϑ2n − 1
+�
+t2 +
+� σ2
+ℓ
+ϑ2n − 1
+�2
+p(|t|) + |t|3
+√n
+�
+R(t)
+�
++ o
+�
+exp
+�
+− t2
+4
+� �
+|t|3+t6
+√n
+�
+R(t)
+�
++ O
+��
+|t|
+ϑn2
+�2+δ
+Π2+δ(n)
+�
+,
+(A.10)
+where
+R(t) :=
+t4
+ϑ4n3 E|ℓ2
+1ℓ2q12| +
+|t|5
+ϑ5n4 E|ℓ2
+1ℓ2
+2q12| +
+|t|3
+ϑ2n3 E|ℓ1q2
+12| +
+|t|5
+ϑ5n4E|ℓ1ℓ2ℓ3q13q23|
++ |t|7
+ϑ7n5 (Eℓ1ℓ2q12)(E|ℓ2
+1ℓ2q12|) +
+|t|8
+ϑ8n6 (Eℓ1ℓ2q12)(E|ℓ2
+1ℓ2
+2q12|).
+After some rearrangement, the first term in (A.10) becomes
+�χL+Q(t) := exp
+�
+− t2
+2
+�
+P(t) = exp
+�
+− t2
+2
+�
+
+1 +
+9
+�
+j=2
+� ιt
+ϑ
+�j γj
+
+ ,
+where
+P(t) := 1 + (ιt)2
+2
+�
+σ2
+ϑ2 − 1
+�
++
+(ιt)3
+6ϑ3n2
+�
+Eℓ3
+1 + 6Eℓ1ℓ2q12
+�
++ (ιt)4
+4ϑ2
+�
+σ2
+ϑ2 − 1
+� �n
+2
+�−1
+σ2
+q
++
+(ιt)5
+12ϑ5n2
+��n
+2
+�−1
+(Eℓ3
+1)σ2
+q + 6ϑ2 � σ2
+ℓ
+ϑ2n − 1
+�
+Eℓ1ℓ2q12
+�
++
+(ιt)6
+6ϑ6n4
+�
+(Eℓ3
+1)Eℓ1ℓ2q12 + 12
+�n
+2
+�−2�n
+4
+�
+[Eℓ1ℓ2q12]2
+�
++ 1
+4
+(ιt)8
+ϑ6n4
+� σ2
+ℓ
+ϑ2n − 1
+� �n
+2
+�−2�n
+4
+�
+[Eℓ1ℓ2q12]2
++ 1
+12
+(ιt)9
+ϑ9n6
+�n
+2
+�−2�n
+4
+�
+Eℓ3
+1 [Eℓ1ℓ2q12]2 .
+Since �χL+Q(t) = exp(−ιtb)χG(t), we have under Assumption (a) and (b)
+|χF (t) − χG(t)| = O
+�
+exp
+�
+− t2
+4
+�
+R(t) +
+�
+|t|
+ϑn2
+�2+δ
+Π2+δ(n)
+�
+.
+19
+
+Therefore,
+I1 = O
+��
+|t|≤log n
+|t|−1 exp(−t2/4)R(t)dt + Π2+δ(n)
+(ϑn2)2+δ
+�
+|t|≤log n
+|t|1+δdt
+�
+= O
+�
+R(1) +
+�log n
+ϑn2
+�2+δ
+Π2+δ(n)
+�
+.
+Bound for I2
+For 1 ≤ m < n, define Qm := 1
+ϑ
+�n
+2
+�−1 �m
+i=1
+�n
+j=i+1 qij. Using (A.5) we can write
+|χF (t)| = |χL+Q(t)| ≤
+�����E exp(ιt(L + Q − Qm))
+2
+�
+k=0
+(itQm)k
+k!
+)
+����� + |t|2+δE|Qm|2+δ.
+Exploiting the fact that Q − Qm is only a function of Xm+1, . . . , Xn, we have
+|E exp(ιt(L + Q − Qm)| ≤ |ψ
+� t
+ϑn
+�
+|m.
+For the second term
+|E exp(ιt(T − Qm)Qm| = 1
+ϑ
+�n
+2
+�−1
+������
+m
+�
+i=1
+n
+�
+j=i+1
+E exp(ιt(L + Q − Qm)qij
+������
+≲
+1
+ϑn2 |ψ
+� t
+ϑn
+�
+|m−2mnE|q12|.
+Similarly, using the fact that
+�
+ϑ
+�n
+2
+�
+Qm
+�2
+=
+m
+�
+i=1
+n
+�
+j=i+1
+q2
+ij, +
+m
+�
+i=1
+n
+�
+j=i+1
+m
+�
+k=1,
+k̸=i,j
+qijqjk +
+m
+�
+i=1
+n
+�
+j=i+1
+m
+�
+k=1,
+k̸=i,j
+n
+�
+l=k+1,
+l̸=i,j
+qijqkl,
+we conclude for k ∈ {0, 1, 2},
+|E exp(ιt(T − Qm))Qk
+m| ≲ |ψ
+� t
+ϑn
+�
+|m−2k
+�
+mn
+ϑ
+�n
+2
+�−1�k
+E|q12|k.
+Finally, using the fact that ϑ = O(σℓ/√n) by Assumption (b) and combining the last displays,
+|χF (t)| ≲
+2
+�
+k=0
+�|t|m
+√n
+�k
+|ψ
+� t
+ϑn
+�
+|m−2kE
+���q12
+σℓ
+���
+k
++ |t|2+δE|Qm|2+δ,
+(A.11)
+for 1 ≤ m < n and δ ∈ [0, 1].
+20
+
+By the triangle inequality followed by (A.5), we have
+|ψ
+� t
+ϑn
+�
+| − |1 −
+t2
+2(ϑn)2 σ2
+ℓ | ≤ |ψ
+� t
+ϑn
+�
+− (1 −
+t2
+2(ϑn)2 σ2
+ℓ)| ≤ 1
+6
+|t|3
+(ϑn)3 E|ℓ1|3.
+For |t| ≤
+√
+2ϑn
+σℓ
+we have |1 −
+t2
+2(ϑn)2 σ2
+ℓ| = 1 −
+t2
+2(ϑn)2 σ2
+ℓ hence
+|ψ
+� t
+ϑn
+�
+| ≤ 1 −
+t2
+2(ϑn)2 σ2
+ℓ + 1
+6
+|t|3
+(ϑn)3 E|ℓ1|3;
+|t| ≤
+√
+2ϑn
+σℓ .
+Assumption (b) together with (A.2) implies that
+σℓ
+√nϑ → 1 as n → ∞. Then, we can find a N1 ∈ N
+such that
+�
+5/6 ≤
+σℓ
+√nϑ ≤ (6/5)1/3 for n ≥ N1. Also, Assumption (a) implies the existence of a
+constant C > 0 and N2 ∈ N such that E|ℓ1/σℓ|3 ≤ C for n ≥ N2. Then, for |t| ≤ c√n where
+c := (
+√
+2/(6/5)1/3) ∧ (5/(12C)) and n ≥ N0 := N1 ∨ N2, we have
+|ψ
+� t
+ϑn
+�
+| ≤ 1 − t2
+n
+�
+1
+2
+� σℓ
+ϑ√n
+�2
+− |t|E|ℓ1/σℓ|3
+6√n
+� σℓ
+ϑ√n
+�3�
+≤ 1 − t2
+3n ≤ exp(− t2
+3n).
+(A.12)
+For log n < |t| ≤ c√n, set m = [15n log n
+t2
+] + 1 = O(n), then plug in (A.12) to conclude that
+|ψ
+� t
+ϑn
+�
+|m−2k ≲ exp(− t2m
+3n ) ≲ n−5. Combining this last bound with (A.11), we obtain
+|χF (t)| ≲
+2
+�
+k=0
+|t|k
+n5−k E
+���q12
+σℓ
+���
+k
++ |t|2+δE|Qm|2+δ,
+(A.13)
+for |t| ≤ c√n and n ≥ N0. Then,
+I2 ≲
+2
+�
+k=0
+1
+n5−k−k/2
+E|q12|k
+σk
+ℓ
++
+�√n log n
+n
+�2+δ �
+σq
+σℓ
+�2+δ
+log n
+≲ 1
+n
+σ2
+q
+nσ2
+ℓ
++ log n
+�
+(log n)σ2
+q
+nσ2
+ℓ
+� 2+δ
+2
+.
+Therefore, since
+σ2
+q
+nσ2
+ℓ = o(1) by Assumption (b), we conclude
+I2 = o(n−1) + O
+
+
+
+
+
+(log n)
+4+δ
+2+δ σ2
+q
+nσ2
+ℓ
+
+
+2+δ
+2
+
+
+
+ .
+Bound for I3 and I4
+Under Assumption (c), for sufficient large n, we may find a b > 0 such that for |t| > c√n
+|ψ
+� t
+ϑn
+�
+| ≤ 1 − b < exp(−b),
+21
+
+where c > 0 is define just before (A.12). Set m = [4 log n
+b
+]+1, then nm ≲ n log n and |ψ
+� t
+ϑn
+�m−s | ≲
+n−4 for sufficient large n and s ∈ {1, 3, 4, 5}. Use these upper bounds on (A.11) to conclude that
+|χF (t)| ≲ n−4 �
+1 + |t| log nE|q12/σℓ| + t2(log n)2 σ2
+q
+σ2
+ℓ
+�
++ |t|2+δ(n log n)1+δ/2E|q12|2+δ,
+(A.14)
+for sufficient large n and |t| > c√n. Then,
+I3 = o(n−1/2) + O
+�
+(n log n)1+δ/2E|q12|2+δ
+�
+c√n≤|t|≤√n log n
+|t|1+δdt
+�
+= o(n−1/2) + O
+��log n
+n
+�2+δ
+Π2+δ(log n)
+�
+.
+Finally,
+I4 =
+�
+|t|>log n
+|t|−1 exp(− t2
+2 )
+������
+1 +
+9
+�
+j=2
+�it
+ϑ
+�j γj
+������
+dt
+≤ C
+�
+t>log n
+t−1 exp(− t2
+2 )dt +
+9
+�
+j=2
+|γj|
+ϑj
+�
+t>log n
+tj−1 exp(− t2
+2 )dt,
+where the first integral is o(n−1) and the second is o(1). Therefore,
+I4 = o
+
+n−1 +
+9
+�
+j=2
+|γj|
+ϑj
+
+ .
+The proof is complete.
+A.2
+Auxiliary Lemmas
+Lemma A.1. Let n ≥ 2, 1 ≤ l ≤ m < n and p ≥ 2 then
+E
+������
+m
+�
+i=l
+n
+�
+j=i+1
+qij
+������
+p
+≤ Cp(n − l)p/2 max
+l<j≤n E
+������
+(m∧j)−1
+�
+i=l
+qij
+������
+p
+≤ Kp [(n − l)(m − l)]p/2 E|q12|p,
+where Cp :=
+�
+8(p − 1)(1 ∨ 2p−3)
+�p and Kp is a constant only depending on p.
+Proof. The double summation on the left-hand side can be written as �n
+j=l+1 ξj where ξj :=
+�(m∧j)−1
+i=l
+qij. Notice that {ξj, Fj} is m.d.s when Fj is the σ-algebra generated by {X1, . . . , Xj} for
+j ≥ 1 and F0 is trivial. Then by Dharmadhikari et al. (1968) followed by a trivial bound
+E
+������
+m
+�
+i=l
+n
+�
+j=i+1
+qij
+������
+p
+= E
+������
+n
+�
+j=l+1
+ξj
+������
+p
+≤ Cp(n − l)p/2−1
+n
+�
+j=l+1
+E|ξj|p ≤ Cp(n − l)p/2 max
+l<j≤n E |ξj|p .
+22
+
+Lemma A.2. For p ∈ [2, ∞) there exist a constant Cp only depending on p such that for S ⊆
+{(i, j) : 1 ≤ i < j ≤ n}
+E
+�����
+�
+S
+qij
+�����
+p
+≤ Cp
+�
+|S|p/2�
+Eq2
+12
+�p/2 ∨ sE
+�
+(E(q2
+12|Z1))p/2�
+∨ |S|E|q12|p�
+,
+where |S| denotes the cardinality of the set S, s := si ∨ sj with si := �
+(i,·)∈S
+��
+(·,j)∈S 1
+�p/2
+and
+sj := �
+(·,j)∈S
+��
+(i,·)∈S 1
+�p/2
+.
+Proof. Combining Proposition 2.1 with expression (2.18) in Gin´e et al. (2000), we obtain the in-
+equality above for the decoupled version of qij, defined as �qij := q(Z(1)
+i
+, Z(2)
+j
+) where Z(j)
+i
+: 1 ≤ i ≤ n,
+1 ≤ j ≤ 2 are i.i.d. Finally, we can apply the decoupling inequalities in de la Pe˜na and Montgomery-Smith
+(1995) to obtain the result at the expense of increasing the constant without altering the order of
+the upper bound. For further details, see section 2.5 in Gin´e et al. (2000).
+B
+Proof of Theorem 1 (Standardized Edgeworth Expansion)
+We apply Corollary A.1 with u(Zi, Zj) = ν′Uij in (2.1) and δ = 1. We assume throughout that
+Assumptions 1 and 2 hold.
+Condition (a) in Theorem A.1 is verified by direct calculations as
+in Cattaneo et al. (2010, 2014a,b). Condition (b) in Theorem A.1 is verified because (A.2) gives
+σ2 = 1
+nV[ν′Li] +
+�n
+2
+�−1V[ν′Qi,j], which implies
+σ2
+ℓ = ν′Σν + O(hP )
+and
+σ2
+q =
+1
+hd+2 [ν′∆ν + h2ν′Vν] + o(h−d),
+with V given in Cattaneo et al. (2010). These results imply σ2
+q = o(nσ2
+ℓ) if (and only if) nhd+2 → ∞.
+Therefore, we take ϑ ≍ σ ≍ 1/√n. Condition (c) in Theorem A.1 holds by assumption.
+The additional condition B → 0 in Corollary A.1 holds if (and only if, when β ̸= 0) √nhP → 0.
+To see this, using integration by parts, E[U12|Z1] =
+�
+Rd ν′ ˙e(X1 + uh)K(u)du − Y1
+�
+Rd ν′ ˙f(X1 +
+uh)K(u)du. Then, repeated Taylor series expansions and integration by parts give E[u(Z1, Z2)|Z1] =
+δ(Z1) + hP (−1)P �
+[k]=P
+µk
+k! δ(1+k)(z) + o(hP ).
+In turn, this result implies that E[u(Z1, Z2)] =
+θν + hP β + o(hP ). As a consequence, B = (E[�θν] − θν)/ϑ = hP β/ϑ + o(√nhP ). See Cattaneo et al.
+(2010, 2014a,b) for details.
+Law of iterated expectations, integration by parts, and Taylor series expansions give
+E[ℓ3
+1] = κ1 + O(hP ).
+Proceeding analogously, because E[ℓ1ℓ2q12] = E[ℓ2E[ℓ1q12|Z2]] = E[ℓ2ℓ1U12] and E[ℓ2ℓ1U12] =
+4E[E[U12|Z1]E[U12|Z2]U12] − 8E[U12]E[E[U12|Z1]2] + 4E[U12]3, we have E[E[U12|Z1]E[U12|Z2]U12] =
+23
+
+E[δ(Z) ˙η(Z)] + O(hP ), E[E[U12|Z1]2] = E[δ(Z1)2] + O(hP ), and E[U12] = θ + O(hP ) and E[U12]3 =
+θ3 + O(hP ). Collecting these results, we verify
+E[ℓ1ℓ2q12] = κ2 + O(hP ).
+These results imply γ3 = O(n−2), γ4 = O(n−3h−d−2), γ5 = O(n−2), γ6 = O(n−1), γ8 = O(n−3),
+γ9 = O(n−7/2).
+It remains to bound E. First, by standard results E|q12|3 = O(h2d+3), so
+Π3(m) = O
+� (mn)3/2
+h(3/2)(d+2) ∨
+mn3/2
+h(3/2)(d+2) ∨ mn
+h2d+3
+�
+= O
+� mn
+hd+2
+�3/2
+.
+Second, using the results in Cattaneo et al. (2014b, Supplemental Appendix), we have E|ℓ2
+1ℓ2q12| =
+O(h−2d/3−1), E|ℓ2
+1ℓ2
+2q12| = O(h−2d/3−1), E|ℓ1q2
+12| = O(h−d−2), E|ℓ1ℓ2ℓ3q13q23| = O(h−4d/3−2). Thus,
+collecting all the bounds, we verify:
+E = O
+��(log n)3
+nhd+2
+�3/2
++ hd/3+1
+nhd+2 +
+�hd/9+2/3
+nhd+2
+�3/2�
+= o
+�
+1
+nhd+2
+�
+This completes the proof.
+C
+Proof of Theorem 2 (Studentized Edgeworth Expansion)
+For any estimated scale �ϑ and nonrandom centering ϑ, we have
+�θν − θν
+�ϑ
+=
+�θν − θν
+ϑ
+�
+1 −
+�ϑ2 − ϑ2
+2ϑ2
++
+�ϑ + 2ϑ2
+2ϑ2 �ϑ
+(�ϑ2 − ϑ2)2
+�ϑ2 + ϑ2
+�
+.
+Recall that �θν is a second-order U-statistic satisfying the H-decomposition (�θν −θν)/ϑ = B + ¯L/ϑ+
+¯Q/ϑ. Using standard results for Edgeworth expansions (Bhattacharya and Rao, 1976; Hall, 1992),
+sup
+t∈R
+���P
+� �θν−θν
+�ϑ
+≤ t
+�
+− G(t)
+��� ≤ E + R1 + R2 + R3 + O
+� rn
+log n
+�
+,
+where
+E := sup
+t∈R
+�����P
+��
+1 −
+�ϑ2 − ϑ2
+2ϑ2
+��
+ν′ ¯L/ϑ + ν′ ¯Q/ϑ
+�
++ B ≤ t
+�
+− G(t)
+����� ,
+B = ν′(E[U12] − θ)/ϑ, G denoting a distribution function later to be set to either GAL or GSB as
+appropriate, and
+R1 := P
+������
+�ϑ + 2ϑ2
+2ϑ2 �ϑ
+�����
+(�ϑ2 − ϑ2)2
+�ϑ2 + ϑ2
+> C
+rn
+(log n)2
+�
+,
+R2 := P
+������
+�θν − θν
+ϑ
+����� > C log n
+�
+,
+24
+
+R3 := P
+������
+�ϑ − ϑ
+ϑ
+B
+����� > C
+√nhP
+log n
+�
+,
+with C denoting a generic constant, which can take different values in different places. The term
+E will give the Edgeworth expansion upon setting �ϑ and ϑ appropriately, while the terms R1–R3
+capture higher-order remainders.
+Variance Estimators
+The estimators �ϑ2
+AL and �ϑ2
+SB are linear combinations of U-statistics as follows:
+�ϑ2
+AL = 1
+nν′�Σν = 2
+�n
+2
+�−1
+¯W1 + 4
+n
+n − 2
+n − 1
+¯W2 − 4
+n
+�θ2
+ν
+and
+�n
+2
+�−1
+h−d−2ν′ �∆ν =
+�n
+2
+�−1
+¯W1
+with
+�θν =
+�n
+2
+�−1 �
+i<j
+(ν′Uij),
+¯W1 =
+�n
+2
+�−1 �
+i<j
+(ν′Uij)2,
+¯W2 =
+�n
+3
+�−1 �
+i<j<k
+Wijk,
+Wijk = (ν′Uij)(ν′Uik) + (ν′Uij)(ν′Uh,jk) + (ν′Uik)(ν′Uh,jk)
+3
+.
+See Lemmas 3.1.1 and 3.1.2 in the Supplemental Appendix of Cattaneo et al. (2014b) for a proof.
+Thus, for c ∈ R, we consider the following generic (debiased when c = 1) Studentization:
+�ϑ2
+c := (2 − c)
+�n
+2
+�−1
+¯W1 + 4
+n[1 + o(n−1)] ¯W2 − 4
+n
+�θ2
+ν.
+In particular, �ϑ2
+AL = �ϑ2
+0 and �ϑ2
+SB = �ϑ2
+1. The centering considered in the literature is �ϑ2
+c is
+ϑ2
+c := c
+�n
+2
+�−1
+E[ ¯W1] + 4
+nE[ ¯W2] − 4
+n(E[�θν])2,
+which implies that ϑ2
+0 = ϑ2
+AL and ϑ2
+1 = ϑ2
+SB + o(n−1).
+The underlying U-statistics have the following mean square convergence rates:
+E[(�θν − E[�θν])2] = O(n−1 + n−2h−d−2),
+E[( ¯W1 − E[(ν′U12)2])2] = O(n−1h−2d−4 + n−2h−3d−4),
+E[( ¯W2 − E[(E[ν′U12|Z1])2])2] = O(n−1 + n−2h−d−4 + n−3h−2d−4),
+The proof is given in Cattaneo et al. (2014b, Supplemental Appendix): see Lemma 3.1.3 for the first
+25
+
+two results, and Lemma 3.1.4 for the third result. (Note that while the statement of those lemmas
+gives convergence rates in probability, the proof mean square convergence rates.) Therefore,
+E
+�
+(�ϑ2
+c − ϑ2
+c)2�
+≤ Cn−4E[( ¯W1 − E[(ν′U12)2])2] + Cn−2E[( ¯W2 − E[(E[ν′U12|Z1])2])2]
++ Cn−2E[( ¯U − E[ν′U12])2]
+= O(n−3 + n−4h−d−4).
+Similar long calculations as in Cattaneo et al. (2014b, Supplemental Appendix) show that:
+E[(�θν − E[�θν])4] = O(n−2 + n−4h−d−4 + n−5h−2d−4 + n−6h−3d−4),
+E[( ¯W1 − E[(ν′U12)2])4] = O(n−2h−4d−8 + n−4h−6d−8 + n−5h−6d−8 + n−6h−7d−8),
+E[( ¯W2 − E[(E[ν′U12|Z1])2])4] = O(n−2 + n−4h−d−8 + n−5h−2d−8 + n−6h−3d−8),
+which gives
+E
+�
+(�ϑ2
+c − ϑ2
+c)4�
+≤ Cn−8E[( ¯W1 − E[(ν′U12)2])4] + Cn−4E[( ¯W2 − E[(E[ν′U12|Z1])2])4]
++ Cn−4E[( ¯U − E[ν′U12])4]
+= O(n−6 + n−8h−d−8).
+Consequently, for the remainder of the proof we set �ϑ2 = �ϑ2
+c and ϑ2 = ϑ2
+c .
+Bounds for R1–R3
+For n large enough, and using Markov inequality,
+R1 ≤ P
+�
+(�ϑ2
+c − ϑ2
+c )2 >
+Crn
+n2(log n)2
+�
++ o(rn) ≤ Cn4(log n)4r−2
+n E
+�
+(�ϑ2
+c − ϑ2
+c)4�
++ o(rn)
+= n5(log n)4O(n−6 + n−8h−d−8) + o(rn) = o(rn).
+Using Theorem A.1 and Corollary A.1, it follows that a valid Edgeworth expansion holds for
+�θν−θν
+ϑc
+, which implies that
+R2 = 1 − P
+� �θν − θν
+ϑ
+≤ C log n
+�
++ P
+� �θν − θν
+ϑ
+≤ −C log n
+�
+= 1 − Φ(C log(n)) + Φ(−C log(n)) + C φ(log n) log n
+nhd+2
++ o(rn) = o(rn),
+by properties of the Gaussian distribution.
+Finally, Markov inequality implies
+R3 ≤ Cn(log n)2E
+�
+(�ϑ2
+c − ϑ2
+c)2�
+= n(log n)2O(n−3 + n−4h−d−4) = o(rn).
+26
+
+Therefore, R1 + R2 + R3 = o(rn).
+Expansion for E
+We consider E = ρ( ˘Fc, Gc), where
+˘Fc(t) := P
+��
+1 −
+�ϑ2
+c − ϑ2
+c
+2ϑ2c
+��ν′ ¯L
+ϑc
++ ν′ ¯Q
+ϑc
+�
++ Bc ≤ t
+�
+,
+Bc := E[�θν] − θν
+ϑc
+,
+and
+Gc(t) := Φ(t) − φ(t)
+�√nhP β
+ν′Σν
+− 1 − c
+nhd+2
+ν′∆ν
+ν′Σν t −
+1
+√n6(ν′Σν)3
+�
+κ1(2t2 + 1) + κ2(t2 + 1)
+��
+.
+Recall that, in particular, c = 0 corresponds to AL implementation and c = 1 corresponds to SB
+implementation (i.e., GAL(t) = G0(t) and GSB(t) = G1(t)). Then, applying the smoothing inequality
+as in Theorem A.1,
+ρ( ˘Fc, Gc) ≲ ˘I1 + ˘I2 + ˘I3 + ˘I4 +
+1
+√n log n,
+where
+˘I1 :=
+�
+|t|≤log n
+����
+χ ˘Fc(t) − χGc(t)
+t
+���� dt,
+˘I2 :=
+�
+log n<|t|≤c√n
+����
+χ ˘Fc(t)
+t
+���� dt,
+˘I3 :=
+�
+c√n<|t|≤√n log n
+����
+χ ˘Fc(t)
+t
+���� dt,
+˘I4 :=
+�
+|t|>log n
+����
+χGc(t)
+t
+���� dt.
+The last three integrals above can be upper bounded following the same arguments used in the
+proof of Theorem A.1 to conclude that ˘I2 + ˘I3 + ˘I4 = o(√nhP + n−1h−d−2 + n−1/2). The first
+integral, ˘I1, is analyzed by expanding χ ˘Fc(t) by generalizing the proof of Theorem A.1 to account
+for the contribution from Studentization to the sampling distribution of the linearized version of
+the statistic ( ˘Fc).
+First, by (A.5) we write
+χ ˘Fc(t) = exp(ιtBc)E exp(ιt�Uc) =
+�
+1 + ιtBc + O(t2B2
+c )
+�
+E exp(ιt�Uc),
+(C.1)
+where
+�Uc =
+�
+1 −
+�ϑ2
+c − ϑ2
+c
+2ϑ2c
+� �ν′ ¯L
+ϑc
++ ν′ ¯Q
+ϑc
+�
+From (C.8) below, we have
+−
+�ϑ2
+c − ϑ2
+c
+2ϑ2c
+= Hc + Tc
+(C.2)
+27
+
+with
+Hc := −
+�n
+2
+�−1 1 − c
+2ϑ2c
+E[q2
+12] −
+1
+2nϑ2c
+1
+n
+n
+�
+i=1
+��
+ℓ2
+i − E[ℓ2
+i ]
+�
++ 4E[ℓiqij|Zi]
+�
+−
+2
+nϑ2c
+�n
+2
+�−1 �
+i<j
+E[qijqik|Zj, Zk],
+where we define ℓi := ν′Li and qij := ν′Qij, and Tc := −Vc/(2ϑ2
+c ) with Vc is given in (C.8). Next,
+applying (A.5) repeatedly, we are left with
+E exp(ιt�Uc) = E exp
+�
+ιt
+�
+ν′ ¯L
+ϑc + ν′ ¯Q
+ϑc
+��
++ ιtEHc ν′ ¯L
+ϑc exp(ιt ν′ ¯L
+ϑc ) + O (E1(t)) ,
+(C.3)
+where E1(t) = |t|E|Tc(ν′ ¯L)+(Hc +Tc)(ν′ ¯Q)|+t2(E(Hc(ν′ ¯L))2 +E|Hc(ν′ ¯L)(ν′ ¯Q)|. The first term was
+expanded in the proof of Theorem A.1, as it corresponds to the standardized version of the statistic.
+The second term can be expanded analogously (see, e.g., Appendix B-(a) in Nishiyama and Robinson
+(2001)):
+E
+�
+Hc ν′ ¯L
+ϑc exp(ιtν′ ¯L/ϑc)
+�
+(C.4)
+= −[ψ
+�
+t
+nϑc
+�
+]n−1
+�
+1 − c
+2
+ιt
+ϑ2c
+�n
+2
+�−1
+E[q2
+12] + O
+�
+|t|
+n2hd+2 +
+t2
+n3/2hd+2 + |t|hP
+hd+2
+��
+− [ψ
+�
+t
+nϑc
+�
+]n−1
+�
+1
+2ϑ3c n2 (Eℓ3
+1 + 4Eℓ1ℓ2q12) + O
+�
+|t|
+n
+��
+− [ψ
+�
+t
+nϑc
+�
+]n−2
+� (ιt)2
+2ϑ3c n2 (Eℓ3
+1 + 4Eℓ1ℓ2q12) + O
+�
+t2+|t|3
+n
++
+t4
+n3/2
+��
+− [ψ
+�
+t
+nϑc
+�
+]n−3 �
+O
+�
+|t|3+|t|
+n
++
+t2
+n3/2hd+2 +
+t6
+n3hd+2 +
+|t|5
+n5/2hd+2 + t4+|t|3
+n2hd+2
+��
+.
+(C.5)
+Combine (A.9), (A.10), (C.3), and (C.4) to obtain
+E exp(ιt�Uc) = exp
+�
+− t2
+2
+� �
+1 + (ιt)2
+2
+� E[ℓ2
+1]
+ϑ2c n − 1
+�
++ (ιt)3
+6ϑ3c n2 Eℓ3
+1 + O(E2(t)) + o(E3(t))
+�
+×
+�
+1 + (ιt)2
+2ϑ2c
+�n
+2
+�−1
+E[q2
+12] + (ιt)3
+ϑ3cn2 Eℓ1ℓ2q12
+− 1 − c
+2
+(ιt)2
+ϑ2c
+�n
+2
+�−1
+E[q2
+12] −
+�ιt + (ιt)3
+ϑ3cn2
+� �Eℓ3
+1
+2
++ 2Eℓ1ℓ2q12
+�
++ O(E4(t))
+�
++ O(E1(t)),
+(C.6)
+where E2(t) and E3(t) are the last two rates appearing in (A.9) respectively. Also, proceeding as in
+Nishiyama and Robinson (2001),
+E4(t) = o
+�t2 + t10
+nhd+2 + t2 + t6
+√n
+�
+.
+28
+
+Combine (C.6) with (C.1) and expand the product to obtain
+χ ˘Fc(t) = exp
+�
+− t2
+2
+�
+
+1 +
+3
+�
+j=1
+(ιt)j˘γc,j
+
+ + O(E5(t)),
+where
+˘γc,1 :=
+�βhP
+ϑc
+− Eℓ3
+1/2 + 2Eℓ1ℓ2q12
+ϑ3cn2
+�
+,
+˘γc,2 := −(1 − c)
+�n
+2
+�−1 Eq2
+12
+2ϑ2c
+,
+˘γc,3 := −
+1
+6n2ϑ3c
+(2Eℓ3
+1 + 6Eℓ1ℓ2q12),
+and
+E5(t) :=
+�
+e−t2/2 |t|3
+√n + o(n−1/2(t6 + |t|3)e−t2/4)
+� �
+t2
+n2hd+2 + |t|3+|t|
+√n
++ E4(t)
+�
++ e−t2/2 �
+|t|√nhP + t2h2P + |t|√nh2P � �
+t2
+n2hd+2 + |t|3+|t|
+√n
++ E4(t)
+�
++ (|t|√nhP + t2nh2P )
+�
+e−t2/2 |t|3
+√n + o(n−1/2(t6 + |t|3)e−t2/4)
+�
++ (|t|√nhP + t2nh2P )
+�
+e−t2/2 |t|3
+√n + o(n−1/2(t6 + |t|3)e−t2/4)
+� �
+t2
+n2hd+2 + |t|+|t|3
+√n
++ E4(t)
+�
++ (|t| + t2√nhP + |t|3nh2P )
+�
+E|Tc ¯L| + E|(Hc + Tc) ¯Q|
+�
++ (t2 + |t|3√nhP + t4nh2P )
+�
+E(Hc ¯L)2 + E|Hc ¯L ¯Q|
+�
+.
+We showed in the proof of Theorem 1 that Eℓ3
+1 = κ1+O(hP ) = o(1) and Eℓ1ℓ2q12 = κ2+O(hP ) =
+o(1), and hence
+χ ˘Fc(t) = exp
+�
+− t2
+2
+� �
+1 + ιt
+�
+βhP
+ϑc − κ1/2+2κ2
+ϑ3c n2
+�
+− (ιt)2
+�n
+2
+�−1 Eq2
+12
+2ϑ2c
+−
+(ιt)3
+6n2ϑ3c (2κ1 + 6κ2)
+�
++ O(E5(t)) + o
+�
+exp
+�
+− t2
+2
+�
+|t|+|t|3
+√n
+�
+.
+Note that the first term is the characteristic function of G. Finally, we bound the moments ap-
+pearing in E5(t): E|Tc ¯L|, E|(Hc +Tc) ¯Q|, E(Hc ¯L)2, and E|Hc ¯L ¯Q|. Holder’s inequality combined with
+the theorem assumptions give
+E|Tc ¯L| ≤
+�
+E|Tc|2E|¯L|2 = O(n−1h−(d+2)/2)
+E|(Hc + Tc) ¯Q| ≤
+�
+E|Hc + Tc|2E| ¯Q|2 = O
+�
+(n−1/2 + n−1h−d−2)(n−1/2h−d/2−1)
+�
+E(Hc ¯L)2 = O(n−1 + n−2h−2d−4)
+E|Hc ¯L ¯Q| = O
+�
+(n−1/2 + n−1h−d−2)(n−1/2h−d/2−1)
+�
+.
+29
+
+Therefore, if (log n)9/(nhd+2) → 0,
+˘I1 :=
+�
+|t|≤log n
+|χ ˘Fc(t) − χGc(t)|
+|t|
+= o(√nhP + n−1h−d−2 + n−1/2).
+The proof is finalized.
+C.1
+Alternative Decomposition of �ϑc
+Let uij = ν′Uij and following Callaert and Veraverbeke (1981) with S2
+N given in the their main
+Theorem, we have
+S2
+N := n2(n − 1)
+(n − 2)2 �ϑ2
+AL = 4(n − 1)
+(n − 2)2
+n
+�
+i=1
+(ν′�Li/2)2
+=
+8
+(n − 1)(n − 2)2
+
+�
+i<j
+(uij − Eu12)2 +
+n
+�
+i=1
+�
+j<k,j̸=i
+(uij − Eu12)(uik − Eu12)
+
+
+− 4n(n − 1)
+(n − 2)2 (�θ − Eu12).
+Define gi := E[ℓjqij|Zi] and use the fact that uij − Eu12 = ℓi/2 + ℓj/2 + qij to further decompose
+S2
+N = 1
+n
+n
+�
+i=1
+ℓ2
+i + 4gi −
+�n
+2
+�−1
+n
+�
+i<j
+ℓiℓj + 2
+�n
+2
+�−1
+n
+�
+i<j
+�
+(ℓi + ℓj)qij − gi − gj
+�
+− 4
+n
+�n − 1
+2
+�−1
+n
+�
+i=1
+ℓi
+n
+�
+j<k,j̸=i
+qjk +
+4
+n − 2
+�n − 1
+2
+�−1
+n
+�
+i=1
+n
+�
+j<k,j̸=i
+qijqik
+− 4n(n − 1)
+(n − 2)2
+
+
+�n
+2
+�−1 �
+i<j
+qij
+
+
+2
++
+4n
+(n − 2)2
+�n
+2
+�−1 �
+i<j
+q2
+ij.
+(C.7)
+The first term, we center on its expectation
+1
+n
+n
+�
+i=1
+ℓ2
+i + 4gi = E[ℓ2
+1] + 1
+n
+n
+�
+i=1
+�
+ℓ2
+i − E[ℓ2
+1]
+�
++ 4gi.
+Define ϕij := E[qkiqkj|Zi, Zj], and for the fifth term we write
+4
+n − 2
+�n − 1
+2
+�−1
+n
+�
+i=1
+n
+�
+j<k,j̸=i
+qijqik = 4
+�n − 1
+2
+�−1 �
+i<j
+ϕij +
+4
+n − 2
+�n − 1
+2
+�−1
+n
+�
+i=1
+n
+�
+j<k,j̸=i
+(qijqik−ϕjk).
+Finally, for the last term
+4n
+(n − 2)2
+�n
+2
+�−1 �
+i<j
+q2
+ij =
+4n
+(n − 2)2
+�n
+2
+�−1 �
+i<j
+�
+q2
+ij − ϕii − ϕjj + E[q12]2]
+30
+
++
+8
+(n − 2)2
+n
+�
+i=1
+�
+ϕii − E[q2
+12]
+�
++
+4n
+(n − 2)2 E[q2
+12]
+Plug the last three displays back into (C.7) to conclude
+S2
+N = E[ℓ2
+1] +
+4n
+(n − 2)2 E[q2
+12] + 1
+n
+n
+�
+i=1
+�
+(ℓ2
+i − E[ℓ2
+1]) + 4gi
+�
++ 4
+�n − 1
+2
+�−1 �
+i<j
+ϕij + Sc,
+where Sc collection the remaining terms.
+Next, we have
+�n
+2
+�−1 �
+i<j
+u2
+ij =
+�n
+2
+�−1 �
+i<j
+(qij − ℓi/2 − ℓj/2 − Eu12)2
+= Eq2
+12 +
+�n
+2
+�−1 �
+i<j
+(q2
+ij − Eq2
+ij) −
+�n
+2
+�−1
+n
+�
+i=1
+(ℓi/2)2 − (Eu12)2
+−
+�n
+2
+�−1 �
+i<j
+qij(ℓi + ℓj) + 2Eu12
+�n
+2
+�−1 �
+i<j
+(qij + ℓi/2 + ℓj/2)
++ 1
+2
+�n
+2
+�−1 �
+i<j
+ℓiℓj
+= Eq2
+12 + Qc,
+where Qc is by definition.
+We have
+�ϑc = �ϑ2
+AL − c
+�n
+2
+�−1
+h−d−2ν′ �∆ν
+= (n − 2)2
+n2(n − 1)S2
+N − c
+�n
+2
+�−1
+
+
+�n
+2
+�−1 �
+i<j
+u2
+ij
+
+
+= (2 − c)
+�n
+2
+�−1
+E[q2
+12] + 1 + o(1)
+n
+E[ℓ2
+1] + 1 + o(1)
+n2
+n
+�
+i=1
+�
+(ℓ2
+i − E[ℓ2
+1]) + 4gi
+�
++ 1 + o(1)
+n
+4
+�n
+2
+�−1 �
+i<j
+ϕij + 1 + o(1)
+n
+Sc + c
+�n
+2
+�−1
+Qc,
+which gives the following simplified expression for the class of Studentizations:
+�ϑc = (2 − c)
+�n
+2
+�−1
+E[q2
+12] + 1
+nE[ℓ2
+1] + 1
+n2
+n
+�
+i=1
+�
+(ℓ2
+i − E[ℓ2
+1]) + 4gi
+�
++ 1
+n4
+�n
+2
+�−1 �
+i<j
+ϕij + Vc,
+(C.8)
+where Vc is by definition.
+31
+
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+34
+

39FAT4oBgHgl3EQfEhxj/content/tmp_files/2301.08422v1.pdf.txt

@@ -0,0 +1,944 @@
+A vision-based autonomous UAV inspection framework for
+unknown tunnel construction sites with dynamic obstacles
+Zhefan Xu, Baihan Chen, Xiaoyang Zhan, Yumeng Xiu, Christopher Suzuki, and Kenji Shimada
+Abstract—Tunnel construction using the drill-and-blast method
+requires the 3D measurement of the excavation front to evaluate
+underbreak locations. Considering the inspection and measure-
+ment task’s safety, cost, and efficiency, deploying lightweight
+autonomous robots, such as unmanned aerial vehicles (UAV),
+becomes more necessary and popular. Most of the previous works
+use a prior map for inspection viewpoint determination and do
+not consider dynamic obstacles. To maximally increase the level
+of autonomy, this paper proposes a vision-based UAV inspection
+framework for dynamic tunnel environments without using a
+prior map. Our approach utilizes a hierarchical planning scheme,
+decomposing the inspection problem into different levels. The
+high-level decision maker first determines the task for the robot
+and generates the target point. Then, the mid-level path planner
+finds the waypoint path and optimizes the collision-free static
+trajectory. Finally, the static trajectory will be fed into the low-
+level local planner to avoid dynamic obstacles and navigate to the
+target point. Besides, our framework contains a novel dynamic
+map module that can simultaneously track dynamic obstacles
+and represent static obstacles based on an RGB-D camera.
+After inspection, the Structure-from-Motion (SfM) pipeline is
+applied to generate the 3D shape of the target. To our best
+knowledge, this is the first time autonomous inspection has been
+realized in unknown and dynamic tunnel environments. Our
+flight experiments in a real tunnel prove that our method can
+autonomously inspect the tunnel excavation front surface.
+Index Terms—Field Robotics, Motion and Path Planning, Per-
+ception and Autonomy, Robotics and Automation in Construction
+I. INTRODUCTION
+Drilling and blasting is a common tunnel construction and
+excavation method. The main cycle of this method includes
+steps such as drilling for explosives, blasting, measuring
+underbreaks, and spraying concrete. Among these steps, mea-
+suring underbreaks in the tunnel excavation front is dangerous
+for workers because of the potential falling rocks. With the
+emergence of lightweight unmanned aerial vehicles, the robot
+becomes suitable for handling measurement and inspection
+tasks as it can avoid potential human dangers and inspect un-
+reachable locations. Consequently, an autonomous inspection
+framework is essential to improve the safety and efficiency of
+underbreaks measurement and tunnel construction.
+There are two main challenges of autonomous UAV in-
+spection in tunnel environments. First, since the tunnel en-
+vironments under construction are changing with time, it is
+unlikely to have update-to-date maps of huge construction
+Zhefan Xu, Baihan Chen, Xiaoyang Zhan, Yumeng Xiu, Christopher
+Suzuki, and Kenji Shimada are with the Department of Mechanical Engineer-
+ing, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA, 15213,
+USA., [email protected]
+Fig. 1.
+Illustration of UAV navigating and inspecting the excavation front
+in the tunnel environment. (a) The tunnel under construction. (b) The target
+inspection area (the excavation front). (c) The robot navigates toward the
+inspection target and avoids obstacles. (d) The robot inspects the target area.
+vehicles and equipment nearby the excavation front. In this
+way, the robot should be able to navigate from arbitrary
+positions in the tunnel towards the excavation front area
+(i.e., the end of the tunnel) based on the onboard sensing.
+Previous works of the sampling-based unknown exploration
+[1][2][3][4] can make the robot successfully navigate and map
+unknown environments with the onboard sensor and applies
+this exploration method to the unknown tunnel inspection [5].
+However, because their approaches only utilize the explored
+map information to randomly sample viewpoints, the output
+trajectory could be zigzag and over-conservative, making
+navigation less efficient. The second challenge comes from
+the moving workers and machines in tunnels, as the robot
+should track them and avoid them safely. Even though some
+recent research [6][7][8] has investigated the UAV dynamic
+obstacle avoidance problems, their local planning strategies
+without global path fusion make them insufficient for complex
+inspection tasks in tunnel environments, which contain com-
+plicated static structures and unpredictable dynamic obstacles.
+To solve these issues, this paper proposes a vision-based
+autonomous UAV inspection framework for unknown and
+dynamic tunnel environments. We develop a small, lightweight
+quadcopter with an RGB-D camera for safely sharing and
+operating with vehicles, equipment and workers in the tun-
+nel. The proposed approach utilizes the hierarchical planning
+method decomposing the entire inspection planning into high,
+mid, and low levels. The current task is determined at the high
+planning level to generate the goal position for navigation and
+arXiv:2301.08422v1  [cs.RO]  20 Jan 2023
+
+b
+Excavation Front
+Target Inpection Area
+Inspection
+Navigation
+(d)
+Cexploration. Then, the mid-level planner will find and optimize
+a smooth trajectory toward the goal based on the static obstacle
+information from the incrementally built map. Finally, at the
+low level, our vision-aided gradient-based planner is applied to
+locally optimize the trajectory for avoiding dynamic obstacles.
+In addition, we propose a novel dynamic map representation
+that can simultaneously represent static and track dynamic ob-
+stacles. The example tunnel environment and the autonomous
+robot using the proposed method are shown in Fig. 1. The
+main contributions and novelties of this work are:
+• Hierarchical inspection framework: This paper applies
+a hierarchical scheme to solve the autonomous inspection
+problem based on different planning layers.
+• Depth-based 3D dynamic map: Our method utilizes
+depth images to detect and track dynamic obstacles and
+update the occupancy information of static environments.
+• Gradient-based dynamic obstacle avoidance: We pro-
+pose a gradient-based B-spline trajectory optimization to
+avoid dynamic obstacles in real time.
+• Tunnel experiments with 3D reconstruction: The entire
+system is verified with a customized quadcopter in a
+tunnel with 3D reconstruction results of the target surface.
+II. RELATED WORK
+This section first discusses the recent trends and approaches
+in construction site inspection by autonomous UAVs. Then,
+relevant works on the key challenges of tunnel inspection (i.e.,
+exploration and dynamic obstacle avoidance) are reviewed.
+There are mainly two categories of construction site and
+building inspection methods: model-based and non-model-
+based methods. For the model-based methods, the inspection
+target model is usually available, and the planner generates a
+set of optimal viewpoints based on the provided model. In
+[9], the target bridge is first partitioned into surfaces with
+inspection nodes, and their GTSP solver is then applied to
+find optimal paths for inspection. Similarly, some works use
+the BIM model to find viewpoints of interest (VPI) and
+solve the path-planning problem using the TSP-based method
+[10][11]. However, the target model can be unavailable for
+tunnel inspection, so the robot can only rely on the onboard
+sensors. In this way, the reactive methods are proposed for
+unknown tunnel navigation using the lidar points measurement
+[12][13]. Their methods can navigate tunnels of arbitrary
+shapes but do not consider obstacle avoidance. Bendris et al.
+[5] utilize the sampling-based method to generate viewpoints
+for unknown exploration and inspection. Their method can
+successfully avoid static obstacles but might not be safe for
+dynamic obstacles due to the long replanning time. Besides,
+their random sampling strategy in the explored area can
+lead to zigzag and over-conservative paths for navigation. In
+[14], it proposes a 3D reconstruction method for UAV tunnel
+inspection without the path-planning strategy.
+The unknown exploration problem can be viewed as deter-
+mining a series of informative viewpoints [15]. Yamauchi [16]
+first uses the frontier exploration approach, allowing robots to
+visit the map boundary to gain environment information. Later
+in [17], it extends the frontier exploration to high-speed UAVs.
+Some approach [18] applies the information-theoretic method
+to evaluate the information gains of viewpoints. Considering
+the limited computation power of lightweight UAVs, the
+sampling-based methods [1][2][3][4] have been preferred in
+recent years. In [1], their RH-NBV planner grows an RRT with
+the information gains stored in each node. The robot will then
+follow the highest gain branch in a receding horizon manner.
+Selin et al. [2] combine the RH-NBV with frontier exploration,
+further improving the exploration efficiency. To save and reuse
+the computation in each planning iteration, Schmid et al. [3]
+adopt the RRT* algorithm with the rewiring to incrementally
+build the tree. With a similar incremental sampling idea in [4],
+it proposes a PRM-based method for exploration and obstacle
+avoidance in dynamic environments.
+Dynamic obstacle avoidance problem still remains open in
+recent years. In the reactive-based methods, the robots directly
+generate control velocities to avoid obstacles. Khatib [19]
+constructs the artificial potential field to find the velocity for
+obstacle avoidance and navigation, and Berg et al. [20] use
+linear programming to optimize velocities based on Velocity
+Obstacle [21]. These methods require less computation than
+the trajectory-based methods but might lead to more myopic
+performance. The trajectory-based methods are more prevalent
+in UAV planning in recent years. Some [22][23][24][25] use
+the model predictive control scheme to generate collision-
+free trajectories based on the kinematic constraints. In [8],
+it utilizes the B-spline optimization to generate collision-free
+trajectory with vision aided, and Chen et al. [26] evaluate
+trajectory risks using their dual-structure particle map.
+III. PROBLEM DESCRIPTION
+In an unknown tunnel space, Vt ∈ R3, with a straight tunnel
+centerline C of a finite length, there exists an excavation front
+(i.e., the target wall for inspection) at the end of the tunnel.
+Inside the tunnel space Vt, there are different sizes of static
+obstacles Ostatic and dynamic obstacles Odynamic. A UAV with
+an onboard depth camera is deployed for the inspection task.
+Without a prior map M, the robot needs to first navigate
+toward the excavation front area from an arbitrary position in
+the space Vt, then generate an inspection path to collect RGB
+images of the inspection target, and finally return to the start
+location. During the forward navigation and returning period,
+the robot should avoid all static obstacles Oall
+static and dynamic
+obstacles Osensor
+dynamic in its sensor range. The final output of the
+entire system should be the 3D shape of the inspection target
+reconstructed using the collected RGB images.
+IV. PROPOSED METHOD
+The proposed inspection framework has three main compo-
+nents shown in Fig. 2: visual perception, hierarchical planning,
+and data post-processing. The visual perception step processes
+the sensor measurements from the onboard depth camera and
+the inertial measurement unit (IMU). The localization module
+runs the visual-inertial odometry (VIO) algorithm with the
+EKF fusion to get robot state estimation. Besides, the dynamic
+
+Fig. 2. System framework for autonomous inspection. Our proposed framework contains three parts: visual perception, hierarchical planning, and data post-
+processing. In the visual perception step, the localization module applies the visual-inertial odometry with EKF fusion for state estimation. The dynamic map
+module builds the static voxel map and tracks dynamic obstacles based on depth images. In the hierarchical planning section, the high-level and mid-level
+planners use the static voxel map to generate the static trajectory. Then, the low-level planner uses the dynamic obstacle information to optimize the output
+trajectory for execution. The final data post-processing step takes the images collected from the inspection stage to reconstruct the target model for analysis.
+map module utilizes depth images to track dynamic obstacles
+and update the occupancy information for static obstacles
+using the voxel map, which will be further discussed in
+Sec.IV-B. After the perception step, the hierarchical planning
+section generates collision-free trajectories for the robot to
+achieve the entire inspection task. Sec. IV-A will introduce the
+logic of our hierarchical planning for the tunnel inspection and
+the task decision maker in the high-level planner. Then, the
+obstacle avoidance based on the mid-level trajectory planner
+and low-level dynamic planner will be covered in Sec. IV-C.
+After finishing the inspection task, the data post-processing
+step, mentioned in Sec. IV-D, takes the collected target images
+to perform 3D reconstruction to obtain the target model.
+A. Hierarchical Planning and High-level Task Planner
+Since our inspection problem consists of multiple compli-
+cated procedures, applying only one planner cannot efficiently
+accomplish the entire task. There are mainly three stages
+of the inspection: (a) approaching the inspection target (i.e.,
+the end of the tunnel), (b) collecting target images, and (c)
+returning to the start location. Based on the inspection stages,
+we decompose the problem into the following abstract tasks:
+ST = {Forward, Explore, Inspect, Return},
+(1)
+where the Forward task aims at approaching the inspection
+target, the Explore task helps the robot gain local map in-
+formation for navigation, the Inspect task mode generates the
+path for collecting target images, and the Return task mode
+navigates the robot back to the starting location. During the
+inspection process, the robot constantly alternates the task
+mode using the proposed task planning algorithm (Alg. 1). For
+each abstract task, the task planner generates the corresponding
+goal positions and passes them to the lower-level planners for
+path planning and trajectory optimization.
+In the beginning stage of task planning (Alg. 1), the task
+planner sets the robot to the Forward task mode as the robot
+needs first to approach the tunnel end (Line 1). The task
+planner runs at a certain replanning frequency to select the
+current task mode for the robot. Before the robot arrives at
+the inspection location, the Forward mode (Lines 7-12) lets
+the robot generate a forward goal with a distance l from the
+current robot position for navigation. Since, at this stage, the
+robot does not have a complete environment map and can
+only rely on the partially built from its flight, it will first try
+using the partial map to perform local obstacle avoidance to
+achieve the forward goal (Line 9). Suppose the lower-level
+planner fails to find a collision-free trajectory due to the lack
+of environmental knowledge. In that case, the task planner
+will switch the current task to the explore mode to increase the
+local map information (Lines 10-12). In the Explore mode, the
+planner first samples to get the best viewpoints with the highest
+sensor information gain in the current map then uses the lower-
+level planner to generate a feasible trajectory for exploration,
+and finally switches back to the previous task mode (Lines 13-
+16). For the information gain evaluation, refers to [1][2][3][4]
+for further details. At the start of each replanning iteration, the
+algorithm checks whether the robot has reached the inspection
+target (Lines 4-6). If the robot detects the inspection target
+wall, the planner will enter the Inspect mode and generate
+a zigzag path for collecting target images. However, when
+the built map around the target is not detailed enough for
+the inspection path generation, the planner will switch to
+the explore mode again to increase the explored map range
+(Lines 17-21). After finishing collecting images, the planner
+will enter the Return mode and navigate back to the start
+position (Lines 24-27). Note that in the returning step, the
+robot has already had a sufficient informative map for static
+obstacles, incrementally built from the forward and explore
+step, to generate a global trajectory to the origin directly.
+
+Localization Module
+High-level Task Planner
+Navigation
+Forward
+Visual Inertial
+ PX4 EKF Fusion
+Goal
+Task Decision
+Odometry
+Explore
+Maker
+Exploration
+Viewpoint
+Backward *
+Dynamic Map Module
+Inspect
+Inspection
+Return Position
+Dynamic Obstacle
+Dynamic Obstacle
+Location
+Proposal Detection
+Tracking
+★ Target Position
+Static Occupancy
+Mid-level Static Planner
+Map Update
+RRT* Waypoint
+Minimum Snap
+Planner
+raiectoryplanner
+Obstacle
+Static Trajectory
+Bounding
+Low-level Dynamic Planner
+Boxes
+3D Reconstruction Module
+Vision-aided B-spline
+Collision Cheking
+Correspondence
+Incremental
+Trajectory Planner
+& Replanning
+Search
+ReconstructionAlgorithm 1: High-level Task Planning Algorithm
+1 Tcurr ← Forward Mode ;
+▷ initial task
+2 Ct ← false ;
+▷ termination condition
+3 while not Ct do
+4
+Icond ← reachInspectionTarget();
+5
+if Icond then
+6
+Tcurr ← Inspect Mode;
+7
+if Tcurr ≡ Forward Mode then
+8
+Pgoal ← getForwardGoal();
+9
+σtraj, success ← lowerLevelPlanner(Pgoal);
+10
+if not success then
+11
+Tcurr ← Explore Mode;
+12
+Tprev ← Forward Mode;
+13
+else if Tcurr ≡ Explore Mode then
+14
+Pgoal ← getBestViewpoint();
+15
+σtraj ← lowerLevelPlanner(Pgoal);
+16
+Tcurr ← Tprev;
+17
+else if Tcurr ≡ Inspect Mode then
+18
+σtraj, success ← getInspectionPath();
+19
+if not success then
+20
+Tcurr ← Explore Mode;
+21
+Tprev ← Inspect Mode;
+22
+else
+23
+Tcurr ← Return Mode;
+24
+else if Tcurr ≡ Return Mode then
+25
+Pgoal ← getReturnGoal();
+26
+σtraj ← lowerLevelPlanner(Pgoal);
+27
+Ct ← isInspectionComplete();
+B. Perception and 3D Dynamic Mapping
+This section introduces our proposed 3D dynamic map for
+navigating dynamic environments, as shown in Fig. 3d. Our
+dynamic map adopts a hybrid method to represent environ-
+ments by using the occupancy voxels for static obstacles and
+the bounding boxes for dynamic obstacles. For static obstacles,
+we predefine a static voxel map size (i.e., maximum voxel
+numbers) based on the environment and store the occupancy
+information of each voxel in an array with the preserved
+length. This allows our planners to access the occupancy
+information with O(1) time complexity. For the occupancy
+information update of each voxel, as most static occupancy
+mapping algorithm does, we apply the classic Bayesian filter
+with the Markov assumption:
+lt(x) = log p(x|zt)
+p(¯x|zt) + log p(¯x)
+p(x) + lt−1(x),
+(2)
+where lt(x) is the log odds for the voxel being occupied. By
+applying Eqn. 2, we can update the occupancy information
+(i.e., log odds) for each voxel by recursively adding the inverse
+sensor model log p(x|zt)
+p(¯x|zt) with the predefined prior log p(¯x)
+p(x).
+Besides, since dynamic obstacles can also be mapped into the
+static voxel map, which can lead to noisy voxels, we iterate
+through each detected dynamic obstacle bounding box and set
+all voxels inside the dynamic regions to be free.
+Fig. 3. Illustration of the proposed 3D dynamic map. (a) A person walks in
+front of the robot in the RGB camera view. (b) The person is detected as a
+dynamic obstacle in the depth image. (c) The detection results in the U-depth
+map for obstacle widths and thicknesses. (d) The 3D dynamic map shows the
+dynamic obstacle as a bounding box and static obstacles as the voxel map.
+The dynamic obstacles are detected and tracked using the
+depth image and represented by axis-aligned 3D bounding
+boxes. There are mainly three steps in the proposed method:
+region proposal detection, map-depth fusion and dynamic
+obstacle filtering. In the region proposal detection step, we use
+the method mentioned in [6] to generate the U-depth map, as
+shown in Fig. 3c, by constructing a histogram of the depth
+values using the depth image. The vertical axis from top to
+bottom of the U-depth map represents the depth range of the
+user-defined bin width. Intuitively, the U-depth map can be
+viewed as a top-down view image. Inspired by [6][24], we
+apply the line grouping method to detect the obstacle regions
+in the U-depth map. With these detection results, we can obtain
+the widths and thicknesses of obstacles and then further find
+the corresponding heights in the original depth image as shown
+in Fig. 3b. After this step, we can get the “region proposal
+bounding boxes” for dynamic obstacles by applying coordinate
+transformation into the map frame. Since the region proposals
+are only the rough detection results, our second step, map-
+depth fusion, inflates those region proposals locally with a
+ratio λ and then searches occupied voxels from the static voxel
+map to get the refined bounding boxes of obstacles. With the
+refined bounding boxes, the dynamic obstacle filtering method
+is applied to identify and track dynamic obstacles. First, we
+utilize the Kalman filter to track and compute the velocity of
+each obstacle bounding box with the linear propagation model:
+pk+1
+o
+= pk
+o + vk
+o(tk+1 − tk),
+vk
+o = pk
+o − pk−1
+o
+tk − tk−1
+,
+(3)
+where pk+1
+o
+is the predicted obstacle position in the next time
+step and vk
+o is the previously estimated velocity. Then, we
+identify those bounding boxes with velocities greater than the
+threshold Vth as the dynamic obstacles. Finally, we remove
+the bounding boxes with jerky motions using the obstacles’
+history velocities, considering the detection noises that make
+static obstacles shake back and forth slightly.
+
+Dynamic Obstacle
+Dynamic Obstacle
+Camera Pose
+Dynamic Obstacle
+-Robot
+Voxel Map (static)
+(d) 3D Dynamic MapC. Navigation and Obstacle Avoidance
+When a goal position is determined by the high-level task
+planner, the mid-level static planner first finds a smooth
+trajectory considering static obstacles. Then, using this static
+trajectory, the low-level dynamic planner optimizes a collision-
+free trajectory based on static and dynamic obstacles at a
+certain replanning frequency. For the mid-level static planner,
+we apply the RRT* planner to find the waypoint path and use
+the minimum snap-based polynomial optimization with corri-
+dor constraints [27][28] for trajectory generation. To achieve
+fast replanning for dynamic obstacle avoidance, the low-level
+planner adopts our gradient-based trajectory optimization. The
+B-spline trajectory with order k over a time knot vector can
+be parameterized as a series of control points:
+ˆS = {P1, P2, P3, ..., PN−1, PN},
+Pi ∈ R3,
+(4)
+where the optimization variable set S contains the N−2(k−1)
+intermediate control points Pi. With the trajectory optimization
+variables, we can write the objective function as follows:
+Ctotal(S) = αcontrol · Ccontrol + αsmooth · Csmooth
++αstatic · Cstatic + αdynamic · Cdynamic,
+(5)
+and the weighted sum has four costs to minimize: the control
+limit cost, the smoothness cost, the static collision cost, and
+the dynamic collision cost. The control limit cost ensures the
+trajectory has feasible velocities and accelerations. The control
+points for velocity Vi and acceleration Ai are computed by:
+Vi = Pi+1 − Pi
+δt
+, Ai = Vi+1 − Vi
+δt
+,
+(6)
+where δt is the time step. We use the L2 norm to penalize the
+infeasible velocities and accelerations:
+Ccontrol =
+�
+i
+||Vi − vmax||2
+2
+λvel
++ ||Ai − amax||2
+2
+λacc
+,
+(7)
+in which vmax and amax are the maximum velocity and accel-
+eration limits. The λ terms are the unit normalization factor.
+Note that the control limit costs are zero for velocities and
+acceleration that are less than the limits. The smoothness cost
+tries to reduce the jerk (i.e., the third derivative to position)
+of the trajectory using the following equations:
+Csmooth =
+�
+i
+||Ji||2
+2,
+Ji = Ai+1 − Ai
+δt
+.
+(8)
+The static collision cost is computed based on the proposed
+circle-based guide-point method shown in Fig. 4a. The initial
+trajectory is shown as the blue dot line with the brown collision
+control points. To calculate the costs for those collision control
+points, we first search a collision-free path (purple dots and
+lines in Fig. 4a) using A* or Dijkstra to bypass the static
+obstacle. If there are N collision control points, we cast a ray
+for the collision control point of sequence order n with the
+angle
+180
+n+1 degree. Note that the angle is between the casting
+ray (dot blue arrow) and the line connecting the first and
+last collision control points. The guide points Pguide are the
+intersection points of the casting ray with the searched path.
+The algorithm is circle-based because the direction angles
+sweep a semi-circle. With the associated guide points for each
+collision control point, we design the total static collision cost
+based on experiments as a clipped cubic penalty function:
+Cstatic =
+�
+i
+�
+max
+�
+dsafe − signDist(Pi, Pi
+guide), 0
+��3
+,
+(9)
+where dsafe is the user-defined safe distance, and the signed
+distance function defines the positive and negative distance as
+the control point outside and inside the obstacle. Intuitively,
+we penalize the control points with small or negative distances
+to obstacles, and the static collision costs are zero for control
+points with a distance greater than the safe distance.
+Since the dynamic obstacles are moving, it is unreliable to
+only use the current detected information for cost computation.
+So, we propose the receding horizon distance field to estimate
+the dynamic collision cost with future predictions shown in
+Fig. 4b. In this figure, the dynamic obstacle with left moving
+velocity Vo is represented as the blue circle with the center
+O and the radius r. We apply linear prediction to get the
+obstacle’s future position C with the prediction horizon k
+time step. Since the reliability of future prediction decreases
+with the increasing prediction time, we linearly decrease the
+obstacle size to zero at the final predicted position C in the
+receding horizon manner. So, we can obtain the collision
+region as the combination of a polygon region AOBC and
+a circular region enclosed by the arc >
+AEB, line AO, and line
+BO. When the control point Pi,p is inside the polygon region,
+we draw a red line through the control point Pi,p perpendicular
+to the line AC intersecting at point D. The distance di to the
+safe area (outside the collision region) can be computed as:
+∆di = ||D − O
+′||2 − ||Pi,p − O
+′||2.
+(10)
+On the other hand, when the control point Pi,c is inside the
+circular region, the distance di to the safe area is:
+∆di = r − ||Pi,c − O0||2.
+(11)
+For the control points Pi,out that are outside both polygon and
+circular regions, we set the distance di to the safe area to zero.
+So, with the distance to the safe area, we can use the following
+equation to compute the final dynamic collision cost:
+Cdynamic =
+�
+i
+�
+max(∆di, 0)
+�3
+.
+(12)
+For both static and dynamic collision costs, the gradients can
+be computed using the chain rule with Eqn. 9 and Eqn.12.
+D. Inspection and 3D Reconstruction
+After finishing the entire inspection task, the data post-
+processing module applies the Structure-from-Motion (SfM)
+to reconstruct the 3D shape of the inspection target from
+the collected target images. When the robot has reached
+the inspection target, it first explores the local area until
+having enough map information about the target. Then, in
+our implementation, the robot generates a zigzag pattern path
+
+Fig. 4. Illustration of the collision cost in our B-spline optimization. (a) The
+static collision cost is calculated using the proposed circle-based guide points
+(red dots). (b) The dynamic collision cost is obtained by the receding horizon
+distance field, which considers the future predictions of the obstacle positions.
+fully covering the target wall and collects the color images
+during the flight. Our SfM pipeline for reconstruction is based
+on COLMAP [29]. The algorithm first extracts the features
+of each image using a numerical descriptor. Since our input
+images are from the streaming of an RGB camera, the second
+step utilizes sequential matching to find the correspondence in
+different images. Finally, from an initial corresponding image
+pair, the algorithm incrementally reconstructs the 3D shape of
+the inspection target by triangulating new points.
+V. RESULT AND DISCUSSION
+A. Implementation Details
+We conduct simulation experiments and physical flight tests
+in dynamic tunnel environments to evaluate the proposed
+method’s performance. The simulation environments are based
+on ROS and Gazebo. For the physical experiments, we visited
+a tunnel under construction in Japan and applied our cus-
+tomized quadcopter (Fig. 5) to test the proposed framework.
+The quadcopter is equipped with an Intel RealSense D435i
+stereo camera, a PX4-based flight controller, and an NVIDIA
+Jetson Xavier NX onboard computer. The weight is ∼1.5kg
+with a 15-minute flight duration. We adopt the visual-inertial
+odometry (VIO) algorithm [30] for robot state estimation. All
+of the perception and planning computations are performed
+within the onboard computer. The color images are collected
+during the inspection stage with the RealSense D435i camera,
+and the data post-processing for 3D reconstruction is com-
+pleted using the desktop with an NVIDIA RTX 3080 GPU.
+B. Evaluation of Navigation and Obstacle Avoidance
+The navigation and obstacle avoidance in the forward task
+(i.e., approaching the tunnel end) is the most challenging and
+time-consuming part of the entire inspection process since the
+environment is cluttered and unknown. So, to evaluate the
+performance of forward navigation and obstacle avoidance, we
+prepared 5 simulation environments containing different static
+and dynamic obstacles, with one example environment shown
+Fig. 5. The customized autonomous quadcopter for inspection experiments.
+Fig. 6. Illustration of an example simulation tunnel environment in Gazebo.
+In the forward task, the robot needs to navigate from the tunnel start (left
+side) to the tunnel end (right side) and avoid static and dynamic obstacles.
+in Fig. 6. For benchmarking, we select the sampling-based
+planning methods (SBP) [1][5] and the dynamic exploration
+planning (DEP) method [4] with modifications to the tunnel
+environments. Besides, we also include our method without
+using the dynamic map (mentioned in Sec. IV-B) to compare
+the obstacle avoidance performance. In each experiment, we
+let the robot navigate from the start of the tunnel to the end
+of the tunnel. We run 10 experiments in each environment
+of different obstacles and record the average navigation time,
+the average replanning time for dynamic obstacle avoidance,
+and the collision rate over all experiments. Note that we set
+the navigation time and replanning time of the sampling-based
+planning methods (SBP) [1][5] to 100% for comparison. The
+collision rate is calculated by the number of experiments with
+collisions divided by the total number of experiments.
+From the results in Table I, one can see that our method
+has the second least navigation time, which is 81.69% of the
+sampling-based planning (SBP) method, and takes almost the
+same amount of time as its non-dynamic-map version. The
+dynamic exploration planning (DEP) method uses less time
+than the sampling-based method and longer time than our
+method. From our observations, both the SBP and the DEP
+generate their trajectories inside the explored regions, which
+is over-conservative, leading to more stop-and-go behavior. On
+the contrary, since our planner adopts a hierarchical scheme,
+the task planner first tries using the more aggressive local
+planner for obstacle avoidance by planning in the unknown
+regions and only applies the conservative exploration planner
+when the local planning fails. This task-switching behavior
+hugely reduces the navigation time. For the replanning time,
+our method takes only 1.16% of the time compared to the
+SBP and significantly less than the DEP. This huge difference
+in the replanning speed is mainly due to our computationally
+lightweight gradient-based trajectory optimization and the long
+
+Tunnel Start
+Static Obstacle
+Tunnel End
+Robot
+Dynamic Obstacle
+Inspection Targetcomputation time in the information gain evaluation of the
+SBP and the DEP. For the collision rate, it is shown that our
+method has no collision among all experiment runs, and both
+the SBP and our method without the dynamic map have a high
+collision rate (around 30%). The DEP has a lower collision
+rate than the SBP since it utilizes an incremental roadmap for
+faster dynamic obstacle avoidance but still has more collisions
+than our method. Comparing our method with and without the
+dynamic map shows that the dynamic map version has a much
+lower collision rate by using dynamic obstacle information.
+TABLE I
+THE BENCHMARK OF THE NAVIGATION TIME, THE REPLANNING TIME,
+AND THE COLLISION RATE BY RUNNING 50 SIMULATION EXPERIMENTS.
+Methods
+Nav. Time
+Replan. Time
+Collision Rate
+SBP [1][5]
+100 ± 0%
+100%
+30.00%
+DEP [4]
+92.80 ± 3.01%
+54.30%
+24.00%
+Ours w/o DM
+81.06 ± 4.40%
+1.20%
+32.00%
+Ours
+81.69 ± 3.66%
+1.16%
+0.00%
+C. Evaluation of Dynamic Obstacle Tracking
+We measure the average tracking errors in positions, ve-
+locities, and obstacle sizes shown in Table II to evaluate the
+dynamic obstacle detection and tracking performance. The
+ground truth states of the obstacles can be easily obtained
+in the simulation experiments, and we apply the OptiTrack
+motion capture system in the physical tests to obtain the
+ground truth states. We let two persons walk within the motion
+capture area, compare the tracking results from the robot
+and the motion capture system, and use the average value
+differences as tracking errors. From Table II, one can see that
+the position errors are 0.09m and 0.19m in simulation and
+physical tests, respectively. The position errors in the physical
+tests are larger than in simulation tests due to the image’s
+noises from the depth camera. Similarly, the camera noises
+also make the velocity errors in physical tests greater than the
+simulations’. The size errors are similar in both simulation and
+physical tests. In the experiments, to account for the tracking
+errors in the positions, velocities, and sizes, we increase the
+safety distance to obstacles by a self-defined size r, and our
+experiment results prove that our dynamic obstacle tracking
+system can let successfully avoid moving obstacles.
+TABLE II
+MEASUREMENT OF THE DETECTION AND TRACKING ERRORS.
+Errors
+Simulation Tests
+Physical Tests
+Position Error (m)
+0.09
+0.19
+Velocity Error (m/s)
+0.10
+0.21
+Size Error (m)
+0.25
+0.25
+D. Physical Flight Tests
+To evaluate and verify the proposed framework, we ran
+flight tests in a tunnel under construction in Japan, shown in
+Fig. 1 and 7. In each flight test, the robot starts at 20 meters
+Fig. 7. The physical inspection flight test in a tunnel under construction in
+Japan. The bottom shows the Rviz of the obstacles and the trajectory.
+in front of the tunnel excavation front and navigates toward
+the inspection area. Note that there are static and dynamic
+obstacles (i.e., walking workers) on the robot’s way to its target
+location shown at the top of Fig. 7. The corresponding Rviz
+visualization is shown at the bottom of Fig. 7, and one can
+see that the robot can generate a collision-free trajectory for
+navigation. After reaching the inspection area, the robot will
+follow the zigzag path to inspect the tunnel excavation front
+shown in Fig. 1d and collect RGB images for further 3D re-
+construction. During the navigation period, the robot’s velocity
+is maintained at 1.0m/s. The results show that our framework
+can complete the entire inspection task autonomously.
+E. Evaluation of 3D Reconstruction
+The final output of our framework is the 3D shape of
+the tunnel excavation front shown in Fig. 8. To obtain the
+results, we run the SfM-based reconstruction mentioned in
+Sec. IV-D with 294 color images of 640x480 resolution. The
+total processing time is 30 minutes using an NVIDIA RTX
+3080 GPU, and the minimum number of images required for
+this experiment is 60 images which take only 5 minutes for
+reconstruction. In Fig. 8, the first row shows the reconstruction
+results from different views, and the second row visualizes
+the error heatmap from the comparison with the ground truth
+model. Note that we use the Topcon laser scanner to obtain
+the ground truth model of the inspection target. The red
+and blue portion of the heatmap represents the high and
+low reconstruction error values. The average error of the
+reconstruction model is 5.38cm with a standard deviation of
+7.96cm. The third row shows the heatmap comparison with
+the tunnel CAD model, the designed shape for the tunnel.
+From the heatmap, the workers can identify the yellow and red
+regions as the locations for concrete spraying and excavation.
+VI. CONCLUSION AND FUTURE WORK
+This paper presents a vision-based autonomous UAV in-
+spection framework for tunnel environments. The proposed
+framework adopts a hierarchical planning scheme to solve
+the complicated inspection problem using different planning
+layers. Our depth-based 3D dynamic map can represent static
+
+Dynamic Obstacle
+Trajectory
+Static Obstacle
+RobotFig. 8.
+The 3D reconstruction results of the excavation front of the tunnel
+under construction in Japan. The first row shows the 3D reconstruction model
+from different views. The second row visualizes the error heatmap obtained
+from the comparison of the laser-scanned ground truth. The third row presents
+the heatmap comparison of the reconstruction model with the CAD model.
+obstacles and track dynamic obstacles simultaneously. The
+experiment results prove that our framework can make the
+quadcopter safely navigate toward the inspection target to
+perform the inspection and return to the origin. The final
+3D reconstruction results obtained from our SfM-based data
+post-processing pipeline have a low error compared to the
+ground truth. For future work, we want to apply learning-based
+methods to classify dynamic obstacles for better performance.
+VII. ACKNOWLEDGEMENT
+The authors would like to thank TOPRISE CO., LTD and
+Obayashi Corporation for their financial support in this work
+and for providing a tunnel construction site for the flight tests.
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+Peculiar properties in quasi-normal spectra from loop quantum
+gravity effect
+Guoyang Fu1,∗ Dan Zhang 2,† Peng Liu 3,‡ Xiao-Mei Kuang1,4,§ and Jian-Pin Wu1,4¶
+1 Center for Gravitation and Cosmology,
+College of Physical Science and Technology,
+Yangzhou University, Yangzhou 225009, China
+2
+Key Laboratory of Low Dimensional Quantum
+Structures and Quantum Control of Ministry of Education,
+Synergetic Innovation Center for Quantum Effects and Applications,
+and Department of Physics, Hunan Normal University, Changsha, Hunan 410081, China
+3 Department of Physics and Siyuan Laboratory,
+Jinan University, Guangzhou 510632, P.R. China and
+4 Shanghai Frontier Science Center for Gravitational Wave Detection,
+Shanghai Jiao Tong University, Shanghai 200240, China
+Abstract
+We investigate the quasi-normal mode (QNM) spectra for scalar and electromagnetic fields over
+a covairant loop quantum gravity black hole (LQG-BH). For the fundamental modes, the LQG
+effect reduces the oscillations in the scalar field, however it induces stronger oscillations in the elec-
+tromagnetic field, comparing to the classical case. Under the scalar field perturbation, the system
+enjoys faster decaying modes with more oscillations than the electromagnetic field. Some peculiar
+phenomena emerge in the scalar field’s QNM spectra with high overtones for the angular quantum
+numbers l > 0. It is that the LQG-BH has a larger real part of QNM with high overtones than
+the Schwarzschild black hole (SS-BH). Such an anomalous phenomenon results in the oscillation
+of the scalar field in the LQG-BH to be nearly identical to that in the SS-BH. Therefore, the high
+overtone modes of the scalar field in LQG-BH play an important role in the modes with l > 0.
+This anomalous phenomenon, however, does not occur in the electromagnetic field’s QNM spectra.
+∗[email protected]
+†[email protected]
+‡[email protected]
+§[email protected]
+¶[email protected]
+1
+arXiv:2301.08421v1  [gr-qc]  20 Jan 2023
+
+I.
+INTRODUCTION
+A non-perturbative and background-independent technique, loop quantum gravity (LQG)
+[1–4], provides a scenario for quantizing space-time structure. This approach has been suc-
+cessfully applied to quantize symmetry reduced cosmological space-times, known as loop
+quantum cosmology (LQC) [5–12]. Effective LQC theory can be constructed by incorporat-
+ing two key quantum gravity effects, namely the inverse volume correction and the holonomy
+correction, which can be achieved using both the canonical approach [13–18] and the path
+integral perspectives [19–25]. The quantum gravity effects in LQC can be connected to
+low-energy physics, resulting in a solvable cosmological model for studying quantum gravity
+effects. In particular, the big bang singularity in classical general relativity (GR) is suc-
+cessfully avoided by the quantum gravity effects [5–12, 26–31], which instead result in a
+non-singular big bounce even at the semi-classical level [32, 33].
+Following the same idea in LQC [5–12], several effective black holes (BH) models with
+LQG corrections have been constructed. Up to date, most of effective LQG-BHs are im-
+plemented through the input of the holonomy correction; see, for example, [34–44] and
+references therein. A common feature of LQG-BHs is that the singularity is replaced by a
+transition surface between a trapped and an anti-trapped region, which can be understood
+as the interior region of black hole and white hole.
+The heart of the holonomy correction is the phase space regularisation technique called
+polymerisation [45]. Because of this, the effective LQG-BH with holonomy correction is also
+known as the polymer BH. The basic idea behind polymerisation is the replacement of the
+conjugate momentum p with their regularised counterpart sin(¯λp)/¯λ, where ¯λ is a quantity
+known as polymerisation scale, which is linked to the area-gap. Depending on whether the
+polymerization scale is constant or phase space dependent function, the polymer BHs are
+classified into two basic types:
+• µ0-type scheme
+In this scheme, the polymerization scale is assumed to remain constant over the whole
+phase space [34–38]. This approach has the drawback that the final result is reliant on
+the fiducial structures, which are introduced in the construction of the classical phase
+space. In addition, even in the low-curvature regimes, significant quantum effects may
+manifest, making these models unphysical.
+2
+
+• ¯µ-type scheme
+The polymerization scale in ¯µ-type scheme is chosen to be a function of the phase
+space [39–42] such that the dependency on fiducial structures is removed, though in
+this scheme significant quantum corrections near the horizon may still survive.
+Further, to fix the problems mentioned above, the authors in [46, 47] came up with an
+generalized version of the µ0-scheme in which the polymerization scale relies on the black hole
+mass such that it stays the same only along the dynamical trajectories. This effective model
+does remove the problems described above, namely, the dependence on fiducial structures
+and the large quantum effects that emerge in the low-curvature regime. This model, however,
+also suffers from mass amplifications when crossing the transition surface. More recently,
+A. Ashtekar, J. Olmedo, and P. Singh proposed an improved version of the generalized
+µ0-scheme [48, 49], which is now known as the AOS model. The polymerization scale in
+this model is thought to depend on the effective Hamiltonian itself. This model is quite
+interesting since it removes not only the drawbacks of the µ0-scheme, but also the mass
+amplifications in the primitive version of the generalized µ0-scheme [46, 47].
+However, the covariance in the aforementioned models is usually broken [50–53]. Follow-
+ing the idea of the anomaly-free polymerization in [54], the authors in [55, 56] construct a
+covariant model of a spherically symmetric BH with holonomy correction. The polymeriza-
+tion scale ¯λ is a constant in this model, and it is related to a fundamental length scal r0
+by a constant of motion m. The resulting geometry corresponds to a singularity-free inte-
+rior region and two asymptotically flat exterior regions of equal mass. It is expected that
+this covariant model can alleviate certain long-standing concerns in the LQG community.
+Therefore, it certainly deserves further exploration.
+In this paper, we will mainly study the properties of the quasi-normal modes (QNMs)
+of a probe scalar field and a probe Maxwell field over this covariant polymer BH. As we all
+know, during the ringdown phase of binary system coalescence, the BH emits the gravita-
+tional waves (GWs) with typical discrete frequencies, i.e., quasi-normal frequencies (QNFs).
+According to [57], QNFs encode decaying scales and damped oscillating frequencies . Cer-
+tainly, quantum effects have the imprints in the QNM spectra, which are expected to be
+detected in GW observations. Also, conversely, GW detection will serve as an important
+criterion for the correctness of candidate quantum gravity theories.
+3
+
+Our paper is organized as follows. In section II, we present a brief discussion on the
+effective potentials of scalar and Maxwell fields over the covariant LQG-BH. Section III
+is dedicated to the properties of the QNM spectra. Then, we further study the ringdown
+waveform in section IV. We present the conclusions and discussions in section V. Appendixes
+A and B present the detailed derivation of the wave equations and the QNMs in the eikonal
+limit.
+II.
+SCALAR FIELD AND MAXWELL FIELD OVER A COVARIANT LQG-BH
+In [55, 56], the authors proposed a novel effective LQG black hole model with holonomy
+correction that is covariant. The exterior geometry of this covariant LQG black hole is given
+by
+ds2 = −f(r)dt2 +
+1
+g(r)f(r)dr2 + r2dΩ2 ,
+f(r) = 1 − 2m
+r ,
+g(r) = 1 − r0
+r .
+(1)
+Thanks to the quantum gravity effects, a new length scale r0 is introduced
+r0 = 2m
+¯λ2
+1 + ¯λ2 ,
+(2)
+where the parameter ¯λ is a dimensionless constant related to the fiducial length of the
+holonomies and m is a constant of motion. The length scale r0 define a minimum area r2
+0
+of this model [55, 56]. In the ¯λ → ∞ limit, this new length scale vanishes, i.e., r0 = 0,
+restoring the classical Schwarzschild black hole is recovered. In addition, as we move to the
+low curvature regions, the quantum gravity effects die off. Without loss of generality, we
+shall set m = 1/2 through this paper, which leads to the horizon located at rh = 1.
+We focus on the perturbations of the massless scalar field Φ and electromagnetic field Aµ
+over this LQG black hole and study their response. We write down the covariant equations
+for the test scalar field and electromagnetic field as follows:
+1
+√−g(gµν√−gΦµ),ν = 0 ,
+(3)
+1
+√−g(gαµgσν√−gFασ),ν = 0 ,
+(4)
+where Fασ = ∂αAσ − ∂σAα is the field strength of the Maxwell field. After the separation of
+variables, the aforementioned equations can be packaged into the Schr¨odinger-like form (for
+4
+
+more details, see Appendix A)
+∂2Ψ
+∂r2
+∗
++ (ω2 − Veff)Ψ = 0 ,
+(5)
+where r∗ is the tortoise coordinate and Veff is the effective potentials:
+Veff = f(r)l(l + 1)
+r2
++ 1 − s
+r
+d
+dr∗
+f(r)
+�
+g(r) ,
+(6)
+with l being the angular quantum numbers. s = 0 and s = 1 correspond to the scalar field
+and electromagnetic field, respectively. Figs.1 and 2 demonstrate the effective potentials as
+a function of r∗ for scalar and electromagnetic fields with different l and r0. It is found that
+both effective potentials are positive, indicating the LQG black hole is stable under scalar
+and electromagnetic perturbations. Furthermore, we would like to compare the differences
+in effective potentials between scalar and electromagnetic fields. It is easy to find that for
+the electromagnetic field (s = 1), the second term in Eq.(6) vanishes, such that all the peaks
+of the effective potentials Vel have the same height for different r0 (see Fig.2). However, for
+the scalar field, i.e., s = 0, the second term in Eq.(6) survives and the height of the effective
+potential Vs depends on r0. In particular, with increasing r0, the height of Vs decreases
+(Fig.1). The shape of the effective potentials shall definitely results in different properties
+of the QNMs.
+r0=0
+r0=1/4
+r0=1/2
+r0=3/4
+-10
+10
+20
+r*
+0.02
+0.04
+0.06
+0.08
+0.10
+Vs(r)
+l=0
+r0=0
+r0=1/4
+r0=1/2
+r0=3/4
+-10
+10
+20
+r*
+0.1
+0.2
+0.3
+0.4
+Vs(r)
+l=1
+FIG. 1: The effective potentials Vs(r∗) of the scalar field for different r0 with fixed l.
+5
+
+r0=0
+r0=1/4
+r0=1/2
+r0=3/4
+0.0
+0.5
+1.0
+1.5
+0.275
+0.280
+0.285
+0.290
+0.295
+0.300
+-10
+10
+20
+r*
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Vel(r)
+l=1
+r0=0
+r0=1/4
+r0=1/2
+r0=3/4
+0.0
+0.5
+1.0
+1.5
+0.86
+0.87
+0.88
+0.89
+0.90
+-10
+10
+20
+r*
+0.2
+0.4
+0.6
+0.8
+Vel(r)
+l=2
+FIG. 2: The effective potentials Vel(r∗) of the electromagnetic field for different r0 with fixed l.
+III.
+QUASI-NORMAL MODES
+In this section, we investigate the QNMs spectra and specially focus on the effects from
+quantum gravity corrections. The nature of determining the QNMs is to solve the eigenvalue
+problem. To this end, we will impose a purely outgoing wave at infinity and purely ingoing
+wave at the horizon:
+horizon: ∂tΨ − ∂r∗Ψ = 0,
+infinity: ∂tΨ + ∂r∗Ψ = 0.
+(7)
+By solving Eq.(5) with the aforementioned boundary conditions, numerous techniques have
+been developed to determining the QNMs spectra, such as the WKB method [58–63],
+Horowitz-Hubeny method [64], continued fraction method [65], asymptotic iteration method
+[66–68], pseudo-spectral method [69, 70], and so on. In this paper, we will solve the eigen-
+value problem using the pseudo-spectral method. For more applications of pseudo-spectral
+method in determining the QNMs in the black hole physics, we can refer to [71–80] and
+references therein. It is convenient to work in the Eddington-Finkelstein coordinate, which
+makes the wave equation (5) is linear in the frequency. To achieve this goal, it is direct to
+make a transformation as
+r → 1/u and Ψ = e−iωr∗(u)ψ .
+(8)
+6
+
+Then, the wave equation (5) turns into the following form:
+ψ′′(u) +
+�
+f ′(u)
+f(u) + g′(u)
+2g(u) +
+2iω
+u2f(u)
+�
+g(u)
+�
+ψ′(u)
+−1
+u
+�
+2iω
+u2f(u)
+�
+g(u)
++
+Veff(u)
+u3f(u)2g(u) + f ′(u)
+f(u) + g′(u)
+2g(u)
+�
+ψ(u) = 0.
+(9)
+Combining with the boundary conditions (7), one can solve Eq.(9) by the pseudo-spectral
+method.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0r0
+0.17
+0.18
+0.19
+0.20
+0.21
+0.22
+Reω
+l=0, n=0
+0.2
+0.4
+0.6
+0.8
+r0
+-0.20
+-0.18
+-0.16
+-0.14
+Imω
+l=0, n=0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0r0
+0.570
+0.575
+0.580
+0.585
+Reω
+l=1, n=0
+0.2
+0.4
+0.6
+0.8
+r0
+-0.18
+-0.16
+-0.14
+Imω
+l=1, n=0
+FIG. 3: QNFs as a function of r0 for the scalar field perturbation.
+Now, we evaluate the QNM spectra for various values of the free parameter r0 to explore
+the LQG effects on these spectra, as well as the differences between them and those of the
+Schwarzschild black hole (r0 = 0). First, we focus on the fundamental modes and show the
+QNF as a function of r0 in Fig.3 for the scalar field and Fig.4 for the electromagnetic field.
+The main properties are presented as what follows.
+• For scalar field, the real parts of the QNF, Reω decreases with increasing r0 (left plots
+in Fig.3).
+It means that the LQG effect reduces the oscillations in comparison to
+7
+
+the Schwarzschild black hole. By contrast, Reω of the electromagnetic field exhibits
+an inverse tendency.
+That is, when r0 increases, so does Reω.
+As a result, the
+LQG effect produces stronger oscillations in the electromagnetic field than that of the
+Schwarzschild black hole.
+• Whether for scalar or electromagnetic field, the imaginary part of QNF Imω al-
+ways lives in the lower half-plane, and their absolute values are less than that of
+the Schwarzschild black hole. Therefore, the system is stable in the presence of scalar
+or electromagnetic field perturbations, and the LQG effect results in faster decaying
+modes.
+• When we fix r0, the scalar field has larger absolute values of Reω or Imω than the
+electromagnetic field (see Figs.3 and 4). It indicates that in comparison to the elec-
+tromagnetic field, the system under scalar field perturbation enjoys faster decaying
+modes with greater oscillations.
+0.0
+0.2
+0.4
+0.6
+0.8
+r0
+0.500
+0.505
+0.510
+0.515
+0.520
+0.525
+Reω
+l=1, n=0
+0.2
+0.4
+0.6
+0.8
+r0
+-0.18
+-0.16
+-0.14
+-0.12
+Imω
+l=1, n=0
+0.0
+0.2
+0.4
+0.6
+0.8
+r0
+0.920
+0.925
+0.930
+Reω
+l=2, n=0
+0.2
+0.4
+0.6
+0.8
+r0
+-0.18
+-0.16
+-0.14
+-0.12
+Imω
+l=2, n=0
+FIG. 4: QNFs as a function of r0 for the electromagnetic field perturbation.
+We also work out the QNM spectra with high overtones. Table I shows the results for
+8
+
+scalar field. For l = 0, we see that the LQG-BH has a lower Reω than the Schwarzschild
+black hole (SS-BH). This is consistent with the fundamental mode’s behavior. The most
+striking difference occurs at the case with l > 0. To demonstrate this, we define the difference
+in QNFs between SS-BH (r0 = 0) and LQG-BH (r0 ̸= 0) as follows:
+δω = ωLQG − ωSS .
+(10)
+Observing the left plot in Fig.5 (also see the second column in Table I), Reω with high
+overtones is larger for the LQG-BH than the SS-BH for l > 0. It indicates that the overtones
+may play an important role in the modes with l > 0. We shall testify this point by the time
+evolution of the field.
+We also briefly address the properties of the QNMs for the electromagnetic field. Table
+II shows the QNM spectra for the electromagnetic field. It is found that for all modes, Reω
+for the LQG-BH is always smaller than that of the SS-BH. It differs from the scalar field,
+where Reω with high overtones is larger for the LQG-BH than the SS-BH. This discrepancy
+also results in the corresponding difference in the evolutions of the scalar field and the
+electromagnetic field, which will be addressed below.
+l = 0
+l = 1
+n
+ω (r0 = 0)
+ω (r0 = 1/2)
+ω (r0 = 0)
+ω (r0 = 1/2)
+0 0.220910-0.209792i 0.200799-0.164527i 0.585872-0.195320i 0.579649-0.158265i
+1 0.172223-0.696106i 0.160438-0.534669i 0.528897-0.612515i 0.547089-0.487735i
+2 0.151564-1.202642i 0.126266-0.921455i 0.459079-1.080267i 0.502191-0.843179i
+3 0.142272-1.705216i 0.076951-1.314823i 0.406517-1.576596i 0.461392-1.216781i
+4 0.134739-2.211987i 0.053109-1.808749i 0.370218-2.081524i 0.426854-1.597881i
+5 0.129639-2.712112i 0.098367-2.213623i 0.344154-2.588236i 0.395550-1.981462i
+TABLE I: The QNM spectra for the scalar field perturbation with different n, l, and r0.
+9
+
+r0=1/100
+r0=1/10
+r0=1/2
+1
+2
+3
+4 n
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+δReω
+l=1
+r0=1/100
+r0=1/10
+r0=1/2
+1
+2
+3
+4 n
+0.1
+0.2
+0.3
+0.4
+0.5
+δImω
+l=1
+FIG. 5: The difference of QNFs of scalar field for the mode with l = 1 between the LQG-BH and
+SS-BH.
+l = 1
+l = 2
+n
+ω (r0 = 0)
+ω (r0 = 1/2)
+ω (r0 = 0)
+ω (r0 = 1/2)
+0 0.496527-0.184975i 0.513377-0.152855i 0.915191-0.190009i 0.924716-0.155637i
+1 0.429031-0.587335i 0.476434-0.474099i 0.873085-0.581420i 0.901934-0.472399i
+2 0.349547-1.050375i 0.427459-0.825862i 0.802373-1.003175i 0.862549-0.803573i
+3 0.292353-1.543818i 0.385422-1.197325i 0.725190-1.460397i 0.816205-1.152343i
+4 0.253105-2.045090i 0.351646-1.576036i 0.657473-1.943219i 0.770903-1.515796i
+5 0.224562-2.547950i 0.322640-1.956797i 0.602986-2.439431i 0.730157-1.888692i
+TABLE II: The QNM spectra for the electromagnetic field perturbation with different system
+parameters n, l and r0.
+Finally, we will discuss the properties of the QNMs in the eikonal limit (l → ∞). In [81],
+Cardoso et al have demonstrated that, in the eikonal limit, QNMs may be connected with the
+behavior of null particle trapped on the unstable circular geodesic of the spacetime, which
+have been validated in most static, spherically symmetric, asymptotically flat spacetime.
+The Reω is determined by the angular velocity Ωc at the unstable null geodesic [82–86],
+whereas the Imω is connected to the Lyapunov exponent λ [87, 88].
+In the LQG-BH
+background, we can calculate the QNMs in the eikonal limit, which is given by
+ω = Ωcl − i
+�
+n + 1
+2
+�
+|λ| .
+(11)
+10
+
+For the detailed calculation, we can refer to Appendix B. It is found that as SS-BH, the
+angular velocity Ωc is completely determined by the black hole mass:
+Ωc =
+1
+3
+√
+3m .
+(12)
+Therefore, the Reω is independent of the LQG parameter r0. While the Lyapunov exponent
+λ is given by
+λ =
+�
+− r2
+c
+f(rc)
+� d2
+dr2
+∗
+f(r)
+r2
+� ���
+r=rc ,
+(13)
+where rc is the radius of the photon sphere. Obviously, the Lyapunov exponent is affected
+by the LQG correction. Left plot in Fig.6 shows the Lyapunov exponent λ as a function of
+r0. We see that the Lyapunov exponent decreases with r0 increasing. Correspondingly, the
+absolution value of Imω is suppressed by the the LQG effect (see the right plot in Fig.6).
+0.2
+0.4
+0.6
+0.8
+1.0r0
+-0.18
+-0.16
+-0.14
+-0.12
+-|λ|/2
+38.60
+38.65
+38.70
+38.75
+-0.20
+-0.18
+-0.16
+-0.14
+-0.12
+Re ω
+Im ω
+l=100,n=0
+FIG. 6: Left plot: The Lyapunov exponent λ as a function of the LQG corrected parameter r0.
+Right plot: The QNFs for different r0 for large l.
+We notice that since the real part of QNF is independent of the LQG parameter r0 in
+the eikonal limit. Therefore, we expect that as l increases, the difference in Reω between
+LQG-BH and SS-BH will be suppressed and vanish. Fig.7 validates this argument that as l
+increases, the difference rapidly decreases and goes to zero.
+11
+
+90
+92
+94
+96
+98
+-0.000100
+-0.000098
+-0.000096
+-0.000094
+-0.000092
+20
+40
+60
+80
+100l
+-0.020
+-0.015
+-0.010
+-0.005
+δReω n=0
+90 92 94 96 98
+0.00040
+0.00041
+0.00042
+0.00043
+0.00044
+20
+40
+60
+80
+100 l
+-0.010
+-0.005
+0.005
+0.010
+0.015
+δReω n=1
+FIG. 7: The difference of QNFs of scalar field between the LQG-BH and SS-BH. Left plot is for
+n = 0 and right plot for n = 1.
+IV.
+RINGDOWN WAVEFORM
+In this section, we will study the time evolution of the scalar and electromagnetic per-
+turbations, which help us to further know the total contributions from overtones. Here, we
+will use the finite difference method (FDM) technics to implement the dynamical evolution.
+For more details on the FDM, we can refer to Refs.[73, 89–92] and references therein. To
+this end, we write the wave equation in difference form as
+−(Ψi+1,j − 2Ψi,j + Ψi−1,j)
+△t2
++ (Ψi,j+1 − 2Ψi,j + Ψi,j−1)
+△r2
+∗
+− VjΨi,j + O(△t2) + O(△r2
+∗) = 0 ,
+(14)
+where △t and △r∗ are the time and radial intervals, respectively, wihch are defined by
+t = i△t and r∗ = j△r∗. The Vj is the discrete form of the effective potential (6). Then, the
+iterate formula is derived as:
+Ψi+1,j = −Ψi−1,j + △t2
+△r2
+∗
+(Ψi,j+1 + Ψi,j−1) + (2 − 2 △t2
+△r2
+∗
+− △t2Vj)Ψi,j .
+(15)
+Notice that the Courant-Friedrichs-Lewy (CFL) condition for instability requires that
+△t/△r∗ < 1.
+Using the iterate formula (15) with the initial Gaussian distribution
+Ψ(r∗, t < 0) = 0 and Ψ(r∗, t = 0) = exp − (r∗−a)2
+2b2
+, one can obtain the ringdown profiles.
+In general, there are three different stages in time-evolution profile: initial outburst,
+quasinormal ringing, which depends only on the black hole’s characteristics and is very
+important for GW observations [57, 93–95], and the late tail, which exhibits the power-law
+12
+
+behavior for the asymptotically flat spacetimes or exponential behavior for asymptotically
+de-Sitter spacetimes. We will focus on the properties of the latter two stages in this section.
+Schwarschild BH
+r0=1/4
+r0=1/2
+0
+50
+100
+150
+200
+0.001
+0.010
+0.100
+1
+10
+100
+t
+|Ψs|
+l=0
+Schwarschild BH
+r0=1/4
+r0=1/2
+0
+50
+100
+150
+200
+10-9
+10-7
+10-5
+0.001
+0.100
+10
+t
+|Ψs|
+l=1
+FIG. 8: The time evolution of the scalar field |Ψs(r)| for different r0 with fixed l (left plot for l = 0
+and right plot for l = 1).
+Left plot in Fig.8 shows the time-domain profile for the scalar field perturbation with
+l = 0. In comparison to the SS-BH, the oscillation is slightly weaker and the decay becomes
+slower during the intermediate time. Recalling that both the real and imaginary parts of
+the fundamental QNF all reduce as the LQG corrected parameter r0 enhances. It suggests
+that in the scalar evolution of LQG-BH with l = 0 the fundamental QNMs dominate over
+the ones with high overtones, which is consistent with the case of SS-BH. At asymptotically
+late-times, quasinormal ringing is suppressed, and it follows the same power-law tail as
+Ψ(t) ∼ t−(2l+3) for both LQG-BH and SS-BH [96–98].
+However, for multipoles l > 0, we observe some peculiar behavior that differs from that
+of l = 0. Carefully observing the right plot in Fig.8, we find that the slope of quasinormal
+ringing is smaller for the LQG-BH than for the SS-BH. This observation is consistent with
+the QNFs, which show that the absolute value of Imω for all discrete overtones is smaller
+for the LQG-BH than for the SS-BH (see the right column in Table I). Nevertheless, the
+oscillation for the LQG-BH is nearly coincide with that for the SS-BH (the right plot in
+Fig.8 and also see Fig.9). Recalling that for l > 0, the LQG-BH has a lower Reω for the
+fundamental mode than the SS-BH, whereas the case is reversed for high overtones.
+It
+means that the contribution from the high overtone modes reduces the difference between
+the LQG-BH and SS-BH time-domain profiles. That is, the high overtone modes in the Reω
+13
+
+of the LQG-BH play an important role in determining the oscillation of the time evolution
+of the scalar field.
+r0=0
+r0=1/100
+r0=1/10
+r0=1/4
+0
+10
+20
+30
+40
+50
+-30
+-20
+-10
+0
+10
+20
+30
+40
+t
+|Ψs|
+l=1
+r0=0
+r0=1/100
+r0=1/10
+r0=1/4
+10
+20
+30
+40
+50
+60
+70
+10-4
+0.001
+0.010
+0.100
+1
+10
+t
+|Ψs|
+l=1
+FIG. 9: The time evolution of the scalar field |Ψs(r)| for different r0 with fixed l. Notice that the
+right plot is the semi-log plot.
+Schwarschild BH
+r0=1/2
+0
+50
+100
+150
+200
+10-6
+0.001
+1
+t
+|Ψel|
+l=1
+Schwarschild BH
+r0=1/2
+0
+50
+100
+150
+200
+10-9
+10-6
+0.001
+1
+t
+|Ψel|
+l=2
+FIG. 10: The semi-log plot of the time evolution of the electromagnetic field |Ψel(r)| for different
+r0 with fixed l.
+We also study the time evolution of the electromagnetic field, as seen in Fig.10. We
+observe that the slop of quasinormal ringing is smaller for the LQG-BH than the SS-BH,
+which is similar to the scalar field. However, unlike in the case of scalar field, the oscillations
+of the LQG-BH and the SS-BH are not the same. It is expected because for l > 0, the
+anomalous phenomena seen in the QNMs of scalar field that Reω with high overtones is
+larger for the LQG-BH than the SS-BH does not happen for electromagnetic field.
+14
+
+V.
+CONCLUSION AND DISCUSSION
+As the rapid development of the GW detection technics, it is expected to detect the
+quantum gravity effect. To extract substantial information from GW detectors, one must
+thoroughly know the main features and behaviors of QNM for LQG-BH. As the first step,
+we investigate the QNM for scalar and electromagnetic fields over the covariant LQG-BH
+proposed in [55, 56]. The QNM spectra for scalar and electromagnetic fields share some
+common features. But they also exhibit many different features and behaviors.
+First, we focus on the fundamental modes. It is found that the system is always stable
+under scalar field or electromagnetic field perturbations, and the LQG effect results in faster
+decaying modes. The difference is that the LQG effect reduces the oscillations in the scalar
+field, however it enhances oscillations in the electromagnetic field.
+In addition, we find
+that the system under the scalar field perturbation enjoys faster decaying modes with more
+oscillations than the electromagnetic field.
+Some peculiar phenomena emerge in the scalar field QNM spectra with high overtones
+for l > 0. It is that the LQG-BH has a larger ωR with high overtones than the SS-BH. Such
+an anomalous phenomenon results in the oscillation of the scalar field in the LQG-BH to be
+nearly identical to that in the SS-BH. Therefore, the high overtone modes of the scalar field
+in LQG-BH play an important role in the modes with l > 0. This anomalous phenomenon,
+however, cannot be observed in the electromagnetic field’s QNM spectra.
+Finally, we comment some open questions deserving further exploration.
+• It would be interesting to extend our investigation to the Dirac field and see if the
+peculiar property still emerges in the QNM spectra.
+• It is definitely interesting and valuable to further study the QNM spectrum of the
+gravity perturbations. It provides us a platform for detecting quantum gravity effects
+using the GW detector. In addition, we can examinate if the isospectrality still holds
+in this LQG-BH model.
+• In [99], the anomalous decay rate of QNMs of a massive scalar field is observed. De-
+pending on how large the mass of the scalar field is, the decay timescales of the QNMs
+either grow or decay with increasing angular harmonic numbers. This anomalous be-
+havior is seen in much larger class models beyond a simple massive scalar field, see
+15
+
+[100–104] and references therein. It will interesting to see how the LQG effect affects
+this anomalous behavior.
+• We can also construct an effective rotating LQG-BH solution using the Newman-Janis
+algorithm, starting with this spherical sysmetric LQG-BH, and study the LQG effects
+on its QNM spectrum and shadow, allowing us to constrain the LQG parameters using
+the GW detector and the Event Horizon Telescope (EHT).
+We plan to investigate these questions and publish our results in the near future.
+Acknowledgments
+This work is supported by National Key R&D Program of China (No. 2020YFC2201400),
+the Natural Science Foundation of China under Grants No. 11905083, the Postgraduate Re-
+search & Practice Innovation Program of Jiangsu Province under Grant No. KYCX20 2973,
+the Postgraduate Scientific Research Innovation Project of Hunan Province, the Science and
+Technology Planning Project of Guangzhou (202201010655), the Fok Ying Tung Education
+Foundation under Grant No. 171006, the Natural Science Foundation of Jiangsu Province
+under Grant No.BK20211601. J.-P.W. is also supported by Top Talent Support Program
+from Yangzhou University.
+Appendix A: Wave equations
+In this appendix, we will derive the wave equations for the scalar and electromagnetic
+fields in detail. First, we shall provide a generic version of the wave equation in a static
+spherically symmetric spacetime. The cases of scalar field and electromagnetic field are then
+discussed in detail.
+Because the spacetime is static spherically symmetric, we can separate variables using
+the spherical function and write the radial equation in the form
+(K(r)S(r)ˆΨ′(r))′ +
+�
+ΛF(r) + K(r) ω2
+S(r)
+�
+ˆΨ(r) = 0 ,
+(A1)
+where ˆΨ is the radial part of the wave function, the coefficient functions {K , F , S} only
+depend on the radial coordinate r, and Λ is the separation constant. After introducing the
+16
+
+tortoise coordinate r∗ and redefining the wave function as
+dr∗
+dr =
+1
+S(r) ,
+ˆΨ(r) =
+Ψ
+�
+K(r)
+,
+(A2)
+Eq.(A1) can be recasted into the following form
+d2Ψ(r∗)
+dr2
+∗
++ (ω2 − Veff(r∗))Ψ(r∗) = 0 .
+(A3)
+The above formula provides a general transformation from the usual wave equation to its
+Schr¨odinger-like counterpart.
+In the following, we will go over the specific form of the wave equations for scalar
+and electromagnetic fields.
+For the scalar field equation, we perform the separation as
+Φ(t, r, θ, φ) = �
+l,m ˆΨ(r)e−iωtYlm(θ, φ), where Ylm(θ, φ) is the spherical harmonics. When
+the particular form of the LQG-BH background (1) is substituted into the wave equation
+(A1), one obtains
+�
+r2f(r)
+�
+g(r)ˆΨ′(r)
+�′
++
+�
+r2ω2
+f(r)
+�
+g(r)
+− l(l + 1)
+�
+g(r)
+�
+ˆΨ(r) = 0 .
+(A4)
+We can read off the coefficient functions by comparing Eq.(A4) to Eq.(A1)
+K(r) = r2 ,
+S = f(r)
+�
+g(r) .
+(A5)
+The Schr¨odinger-like version of the wave equation is then easily given as
+∂2Ψ
+∂r2
+∗
++ (ω2 − Vs)Ψ = 0 ,
+(A6)
+Vs = f(r)l(l + 1)
+r2
++ 1
+2r
+d
+drf(r)2g(r) .
+(A7)
+For the electromagnetic field, we can expand the gauge field Aµ in vector spherical har-
+monics [105, 106],
+Aµ(t, r, θ, φ) =
+�
+l,m
+�
+�
+�
+�
+�
+�
+�
+�
+������
+0
+0
+alm(r)
+sin θ ∂φYlm
+−alm(r) sin θ∂θYlm
+�
+������
++
+�
+������
+plm(r)Ylm
+hlm(r)Ylm
+klm(r)∂θYlm
+klm(r)∂φYlm
+�
+������
+�
+�
+�
+�
+�
+�
+�
+e−iωt ,
+(A8)
+where the first term is the odd (axial) perturbation and second term is even (polar) pertur-
+bation. Then, in the following, we will show how to derive the odd perturbation equation
+and even perturbation equation.
+17
+
+When we switch on the odd electromagnetic field perturbation, we can explicitly write
+down the radial equation as
+�
+f(r)
+�
+g(r)a′
+lm(r)
+�′
++
+�
+ω2
+f(r)
+�
+g(r)
+− l(l + 1)
+r2�
+g(r)
+�
+alm(r) = 0 ,
+(A9)
+It is easy to find that K = 1 and S = f(r)
+�
+g(r). Thus, we have
+Vodd = f(r)l(l + 1)
+r2
+,
+(A10)
+where Ψ = alm(r).
+For the even perturbation of the electromagnetic field, the radial equation becomes
+p′′
+lm(r) + q(r)p′
+lm(r) + iω (h′
+lm(r) + q(r)hlm(r)) +
+l(l + 1)
+r2f(r)g(r)(plm(r) + iωklm(r)) = 0 ,
+−iωp′
+lm(r) + ω2hlm(r) + l(l + 1)
+r2
+f(r)(−hlm(r) + k′
+lm(r)) = 0 ,
+(A11)
+where q(r) = 2
+r + g′(r)
+2g(r). After introducing a new variable
+ˆΨ(r) = −p′
+lm(r) − iωhlm(r) ,
+(A12)
+Eq.(A11) can be reduced to
+(r4f(r)g(r)3/2 ˆΨ′(r))′ +
+�
+r4ω2�
+g(r)
+f(r)
+− l(l + 1)r2
+�
+g(r) + 1
+2J(r)
+�
+ˆΨ(r) = 0
+(A13)
+where J(r) = r2�
+g(r)(rf ′(r)(4g(r) + rg′(r)) + f(r)(4g(r) + r(6g′(r) + rg′′(r)))). Thus, the
+coefficient functions are K = r4�
+g(r) and S = f(r)
+�
+g(r) and then we have
+Veven = f(r)l(l + 1)
+r2
+.
+(A14)
+We find that the effective potentials for odd and even electromagnetic field perturbations are
+the same. Therefore, we will use Vel to signify the effective potential of the electromagnetic
+field rather than Vodd and Veven.
+Appendix B: QNMs in the eikonal limit
+In this appendix, we will show the connection between the QNMs in the eikonal limit and
+the behavior of null particle trapped on the unstable circular geodesic. For a null particle,
+18
+
+the Lagrange is1
+L(x, ˙x) = 1/2gµν ˙xµ ˙xν .
+(B1)
+We start with the spherically symmetric geometry (1). Thanks to the symmetry, one can
+only consider the geodesics in the equatorial plane: θ = π/2. Then the Lagrangian (B1)
+becomes
+2L = −f(r)˙t2 +
+˙r2
+f(r)g(r) + r2 ˙φ2 ,
+(B2)
+where the dot represents the derivative with respect to the affine parameter τ.
+In this
+system, there are two constants of the motion, which are
+Pt = −f(r)˙t = −E ,
+Pφ = r2 ˙φ = L .
+(B3)
+Using the canonical transform and combining the above equations (B2) and (B3), we have
+the following reduced Hamiltonian system:
+2H = E ˙t +
+˙r2
+f(r)g(r) + L ˙φ .
+(B4)
+Since the Hamiltonian H satisfies the constraint H = 0, we have
+˙r2 + Veff = 0 ,
+(B5)
+where the effective potential is
+Veff = g(r)
+�L2
+r2 f(r) − E2
+�
+,
+(B6)
+Because ˙r2 > 0, the photon can only emerge in the area of negative potential. When the
+angular momentum is small, the photon will fall from infinity into the black hole. However,
+for the large angular momentum, the photon will escape the bondage of the black hole and
+go back to infinity. Therefore, the critical circular orbit for the photon can be derived from
+the unstable conditions
+Veff = 0 ,
+∂Veff
+∂r
+= 0 ,
+∂2Veff
+∂r2
+< 0 .
+(B7)
+1 For the calculation details of the geodesic of a null particle, please refer to [81, 82, 107, 108].
+19
+
+From the above conditions, we can obtain the equation for the critical radius rc
+2fc(r) = rcf ′
+c(r) .
+(B8)
+Correspondingly, we have the critical impact parameters bc:
+bc = L
+E =
+rc
+�
+fc(r)
+.
+(B9)
+Then, the shadow radius Rs and Lyapunov exponents λ can be calculated as follows:
+Rs =
+�
+ζ2 + η2 = bc = 3
+√
+3m ,
+(B10)
+λ =
+�
+V ′′
+eff
+2˙t2 =
+�
+− r2
+c
+f(rc)
+� d2
+dr2
+∗
+f(r)
+r2
+� ���
+r=rc ,
+(B11)
+where {ζ , η} are the celestial coordinates. We find that the shadow radius reduces to the
+one of SS-BH [109, 110]. It means that the LQG effect doesn’t change the shape of the
+shadow. However, the LQG correction affects the Lyapunov exponent λ.
+On the other hand, we shall use the first order WKB approximation to obtain the analytic
+form of the QNMs in the eikonal limit (l → ∞). In this limit, the last term of the effective
+potential (6) can be ignored, resulting in the following form of the effective potential
+V∞(r) = f(r) l2
+r2 .
+(B12)
+Reminding that the potential (B6) and (B12) are the same. Therefore, in the eikonal limit,
+the QNMs may be obtained by the multiples of the frequency and the instability timescale
+of the unstable circular null geodesic [81]:
+ω = Ωcl − i(n + 1
+2)|λ| ,
+(B13)
+where Ωc is the angular velocity and can be worked out as
+Ωc =
+˙φ
+˙t = 1
+bc
+.
+(B14)
+[1] C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge, UK (2004).
+[2] T. Thiemann, Modern canonical quantum general relativity, gr-qc/0110034.
+20
+
+[3] A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A Status
+report, Class. Quant. Grav. 21 (2004) R53, [gr-qc/0404018].
+[4] M. Han, W. Huang, and Y. Ma, Fundamental structure of loop quantum gravity, Int. J.
+Mod. Phys. D 16 (2007) 1397–1474, [gr-qc/0509064].
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+In Quest of Ground Truth: Learning Confident Models and Estimating
+Uncertainty in the Presence of Annotator Noise
+Asma Ahmed Hashmi
+Artem Agafonov
+Aigerim Zhumabayeva
+Mohammad Yaqub
+Martin Takáˇc
+Mohamed bin Zayed University of Artificial Intelligence (MBZUAI)
+Masdar City, Abu Dhabi, UAE
+https://mbzuai.ac.ae/
+Abstract
+The performance of the Deep Learning (DL) models de-
+pends on the quality of labels. In some areas, the involvement
+of human annotators may lead to noise in the data. When
+these corrupted labels are blindly regarded as the ground
+truth (GT), DL models suffer from performance deficiency.
+This paper presents a method that aims to learn a confident
+model in the presence of noisy labels. This is done in conjunc-
+tion with estimating the uncertainty of multiple annotators.
+We robustly estimate the predictions given only the noisy
+labels by adding entropy or information-based regularizer
+to the classifier network. We conduct our experiments on a
+noisy version of MNIST, CIFAR-10, and FMNIST datasets.
+Our empirical results demonstrate the robustness of our
+method as it outperforms or performs comparably to other
+state-of-the-art (SOTA) methods. In addition, we evaluated
+the proposed method on the curated dataset, where the noise
+type and level of various annotators depend on the input
+image style. We show that our approach performs well and
+is adept at learning annotators’ confusion. Moreover, we
+demonstrate how our model is more confident in predicting
+GT than other baselines. Finally, we assess our approach for
+segmentation problem and showcase its effectiveness with
+experiments.
+1. Introduction
+Real world data is replete with noisy labels. Since the
+labeling process of large-scale datasets is costly and time-
+consuming, researchers often resort to less expensive options,
+such as internet inquiries and crowdsourcing to circumvent
+this issue [32,38]. Unfortunately, these methods are viable
+in producing datasets with incorrect labels. Smaller datasets
+are also vulnerable to the presence of corrupted labels. In
+this case, usually the labelling process is either challenging
+or the annotators have divergent opinions [3,21]. In medical
+imaging, for example, it is imperative to procure annotations
+from the clinical experts. However, it is not only expensive
+to obtain annotated data, but it also suffers from high inter-
+reader variability among domain’s experts [17,20].
+Deep Neural Networks (DNN) noticeably suffer a degen-
+eration in performance when trained on noisy labels. To
+combat this issue, various algorithms have been devised to
+adapt to the presence of noisy labels without compromis-
+ing on the performance of DNNs. Sample Selection meth-
+ods [10,12,22,34,40] started to gain momentum recently;
+these methods involve a two network, Student-Teacher, for
+learning from noisy labels. It uses a small loss trick to sam-
+ple clean instances for additional training by its peer network.
+While these methods aid in selecting the clean samples, the
+small loss trick does not perform well when the loss distri-
+bution of true-labelled and false-labelled examples overlap
+substantially.
+In the instance when there is a significant level of dispute
+in the labels by the annotators, conventional training meth-
+ods that consider such labels as "the truth" result in models
+with low predictive ability. Tanno et al [31] proposed an
+algorithm that jointly estimates the annotators’ confusion
+and the underlying label distribution. The annotators’ con-
+fusion is represented by a stochastic transition probability
+matrix. In their approach, the loss function is augmented by
+adding a regularization term that is the trace of annotators’
+confusion matrix. However, the caveat is that this regular-
+ization may still penalize in instances when the annotator is
+not confused, therefore it will not learn the true annotator’s
+noise distribution. Furthermore, there is no incentive in the
+training process to enforce the classifier network to predict
+the class probabilities.
+Our work is inspired by [31, 41], with a motivation to
+make our model confident in its predictions while also jointly
+1
+arXiv:2301.00524v1  [cs.CV]  2 Jan 2023
+
+estimating annotator’s confusion in the presence of noisy
+labels. We explored entropy and information regularizer
+techniques to encourage our classifier to make confident
+predictions about each class.
+Problem Statement. In this paper, we focus on supervised
+learning problem with noisy labels. We assume that each
+object xn, n = 1, . . . , N is assigned with a set of noisy
+labels {˜y(r)
+n }R
+r=1, where ˜y(r)
+n
+is a label given to the object xn
+by annotator R. Here N denotes the total number of samples
+in the data, and R denotes the total number of annotators.
+The main goal is to construct an algorithm that learns the
+distribution of true labels p(y|x) and to make confident pre-
+dictions about the classes. This is achieved in conjunction
+with estimating annotator’s noise.
+To achieve this, we use the classifier-annotator approach
+[31]. We jointly train two neural networks: classifier and
+annotator. The first network, the classifier, aims to learn
+the ground truth/class true label. So it outputs the class
+probability vector ˆpθ(x). The second network learns each
+annotator’s confusion matrix U(x), which represents the
+likelihood of the annotator being wrong in the class markup
+for a given input.
+However, it is not enough to minimize the loss between
+matrix-vector product ˆUψ(x)ˆpθ(x) and annotator’s label ˜y
+due to various reasons. First of all, there is no evidence why
+annotator and classifier neural networks will learn confusion
+matrix and class probabilities. There are infinite number
+of pairs ( ˆUψ(x), ˆpθ(x)) that approximate ˜y well. Without
+a modification of loss functions, it may turn out that they
+just learn some features of inputs. Secondly, we want to be
+confident in the predictions of the classifier. In evaluation
+mode, this model will be used to make real-time predictions.
+It is important to train the model in a such a way that it
+makes confident and true predictions.
+To tackle the aforementioned problems, we penalize the
+classifier network for uncertainty. We propose two regular-
+ization techniques based on Shannon’s entropy and infor-
+mation. Our methodology for classification is summarized
+in Figure 1. Moreover, we apply our methodology for seg-
+mentation problem. In this case we make predictions and
+estimate the confusion pixel-wise.
+Contributions. The main contributions of our paper are
+outlined as follows:
+1. Learning the ground truth label. Our approach is capa-
+ble of disentangling the GT from the annotation noise. We
+distinguish the noise through the usage of the annotator-
+classifier methodology. We enforce the classifier network
+to learn class probabilities, not some features of the in-
+put, by regularizing its output via Shannons’ entropy and
+information-based regularizer.
+2. Learning confident model. Our choice of regularization
+technique is enforcing the classifier network to make con-
+vincing predictions about the respective classes. This
+CLASSIFIER 
+ANNOTATOR 2
+Input
+Annotators'
+confusion
+matrices
+Annotators'
+class
+probabilities
+ Class
+probabilities
+Negative log
+likelihood loss
+ANNOTATOR 1
+ANNOTATOR 3
+or
+Regularization
+Regularization
+Figure 1. Model architecture. We consider the problem with 4
+classes and 3 annotators. Architecture consists of two neural net-
+works: 1) classifier network predicts class probabilities ˆpθ(x), 2)
+annotator NNs predict confusion matrix U (r)(x) for each annotator
+r. Matrix-vector product U (r)(x)ˆpθ(x) estimates the annotator’s
+prediction. Note, that 2nd annotator tends to confuse classes 1 and
+2. To jointly train two neural networks we minimize regularized
+negative log likelihood loss (NLL). We propose two options for
+regularization: information-based regularizer and entropy.
+has various befitting practical applications in different
+domains, including medical imaging. We use our regular-
+izer to push the predicted probabilities of the first network
+to be closer to 1 or 0 and to make the model to distinguish
+between classes better.
+3. Competitive numerical experiments.
+We have per-
+formed extensive numerical experiments that compares
+our algorithm with other SOTA baselines. We conducted
+experiments on MNIST, CIFAR-10 and FMNIST datasets
+to gauge the performance of our algorithm in the exis-
+tence of noisy labels. The noisy labels were simulated
+using pairflip and symmetric noise.
+Our experiments showed that our algorithm outperforms
+all the evaluated baselines for the higher noise levels such
+as pairflip 45% and symmetric 50%. For smaller noise
+rates, we perform at par with [10,31,34,35,40]. Moreover,
+we show better results than in annotator-classifier setup
+with trace regularizer proposed in [30,31]. Moreover, we
+conduct experiments for segmentation, where our model
+also shows better accuracy and confidence compared to
+trace regularization [41].
+4. Curated dataset. We have also executed experiments for
+a curated dataset, where noise type and level for various
+annotators depend on input image style. The proposed
+approach with the choice of our regularizer results in
+more confident model compared to the one without the
+regularizer. Moreover, we show that our approach is able
+to learn true annotators’ confusion.
+5. Open code. Our code is available online. Our implemen-
+tation includes a suite that easily allows researchers to
+compare their approach against all benchmarks consid-
+ered in this paper.
+Organization. The remainder of the paper is organized as
+2
+
+follows. Related works are described in Section 2. Section 3
+presents the methodology and probabilistic model behind it.
+In Section 4 we describe the proposed regularizers. Numeri-
+cal experiments are provided in Section 5 (additional experi-
+ments are provided in Appendix C). Section 6 is dedicated
+to the segmentation problem, and concluding remarks and
+potential future research directions are given in Section 7.
+2. Related Literature
+Learning with noisy labelled training data has been an
+active area of research for some time. Various algorithms
+have been introduced and have shown resistance to noise
+during training. We highlight the core research being done
+in this domain.
+2.1. Classification
+Noise Transition Matrix/Loss Correction. Loss correc-
+tion approach using noise transition matrix, T, is a crucial
+branch that is used in deep learning systems. The goal of loss
+correction is for training on noisy labels with the corrected
+loss to be roughly equivalent to training on clean labels with
+the original loss. The majority of the early approaches deal-
+ing with noisy labels relied on estimating a noise transition
+matrix to figure out how labels switch across classes.
+Patrini et al. [25] introduced two different approaches for
+loss correction using a stochastic matrix T that delineates
+the probability of a class being flipped with another under a
+certain noise. This two loss correction approaches, namely,
+forward correction and backward correction. The backward
+procedure corrects the loss by multiplying the loss with
+inverse transition matrix T −1; while the forward procedure
+corrects the network predictions by multiplying it with T.
+Hendrycks et al. [11] suggested Gold Loss Correc-
+tion(GLC) based on Forward Correction to address extreme
+noise. The transition matrix cannot be accurately predicted
+by solely noisy data when there is significant noise present.
+The main driver is the assumption that a limited portion of
+the training data is reliable and accessible.
+Sukhbaatar et al. [30] demonstrated a method of forward
+loss correction by introducing a stochastic matrix that quan-
+tifies label corruption, and cannot be calculated without ac-
+cessing the true labels. In order to include learning about the
+label noise, forward loss correction involves adding a linear
+layer to the model’s end and adjusting the loss as necessary.
+Through the use of soft and hard bootstrapping, Reed
+et al. added the concept of consistency to the prediction
+objective [26]. The soft version is identical to softmax re-
+gression with minimum entropy regularization, whereas the
+hard version adjusts regression targets by employing MAP
+estimation. This bootstrapping process, intuitively, gives
+the learner the opportunity to contest an inconsistent train-
+ing label and re-label the training data to enhance the label
+quality. Whereas Goldberger et al. [9] made use of the
+expectation-maximization (EM) algorithm to determine the
+optimal network and noise parameters. The use of transition
+matrices has been investigated further [4,7,36].
+Multi-Network Learning.
+Multi-network training fre-
+quently employs collaborative learning and co-training.
+Therefore, the sample selection procedure is governed by
+the mentor network in the case of joint learning and the
+peer network in the case of co-training. These algorithms
+can be defined as learning to teach methods, and they com-
+prise of a student and teacher network. The responsibility
+of the teacher network is to select more informative sam-
+ples for enhanced student network training. Malach et al.
+proposed decoupling method that simultaneously trains two
+DNNs while only updating parameters on examples/samples
+in cases when the two classifiers disagree [22].
+MentorNet [12] selects clean instances to guide the train-
+ing of the student network after it has trained the teacher
+network first. Co-teaching [10] and [40] also employ two
+DNNs, but each DNN selects a certain number of small-loss
+examples and feeds them to its peer DNN for additional
+training. Co-teaching+ additionally utilize the disagreement
+strategy of Decouple. In comparison, JoCoR [34] reduces
+the diversity of two networks by means of co-regularization,
+making the predictions of the two networks more similar.
+Robust Regularization. Tanno et al. [31] showcased a
+method for simultaneously learning the individual annotator
+model and the underlying true label distribution. Each anno-
+tator’s confusion is represented by a confusion matrix, which
+is estimated in conjunction with the classifier predictions.
+The algorithm comprised of a loss function include a trace
+regularization term. Menon et al. [23] suggests a composite
+loss-based gradient clipping for label noise robustness. It is
+expected that clipping would provide noise robustness, given
+that one does not place excessive trust in any single sample.
+Robust early-learning [35] distinguishes between critical and
+non-critical parameters for fitting clean and corrupted labels,
+respectively. Then, only non-critical updates are penalized
+with a different update rule.
+Other Deep Learning/Statistical Methods.
+DivideMix
+[18] is a framework that splits the training data into a la-
+beled set with clean samples and an unlabeled set with noisy
+samples-samples that comprise of noisy labels; it trains the
+model on both the labeled and unlabeled data in a semi-
+supervised approach. Kun Yi et al [39] proposed a prob-
+abilistic end-to-end noise correction in labels (PENCIL)
+framework. This method only uses noisy labels to initialize
+label distributions; the label distributions get updated by an
+iterative correction of the noisy labels. Consequently, la-
+bel distributions are used in the calculation of the network
+loss instead of the noisy labels. Xia et al. [35] suggested a
+robust early-training method to diminish the side effect of
+noisy labels prior to early stopping. This helps with improv-
+ing the memorization of clean labels. The parameters are
+3
+
+split into critical and non-critical parameters. Each of these
+parameters are updated with a different update rule.
+2.2. Segmentation
+Several strategies have been developed to solve the is-
+sue of annotator-related bias for segmentation in medical
+imaging. We review some prominent work in the field.
+Inter-reader variability among annotators gave promi-
+nence to Simultaneous Truth and Performance level Es-
+timation (STAPLE) [33] algorithm that uses expectation-
+maximization method to merge segmentations from various
+annotators into estimating a single ground truth. There are
+several algorithms that drew their inspiration from STAPLE
+framework such as [1,2,13,14,29]. These methods are reflec-
+tive of generative modelling of annotator’s behaviour. Here
+the latent variables are the true labels which are unobserved,
+and the confidence/expertise of various annotators.
+Mirikharaji et al. [24] provides a sample re-weighting
+strategy that considers the expertise level of annotators. This
+strategy gives greater weights in the loss function for the
+samples annotated by professionals. To disengage annotator
+bias, Tanno et al. [41] uses two coupled CNNs. Similar
+to [31], the CNN for segmentation estimates the label distri-
+bution, while the CNN for annotation is representative of the
+annotator bias using a confusion matrix.
+Annotation distribution learning has been another active
+area that has inspired pioneer work of probabilistic U-Net
+(PU-NET) [15]. This method given an input, examines the
+problem of learning a distribution over segmentations. This
+proposed architecture is a generative segmentation model
+which is an integration of U-Net [27] and conditional varia-
+tional autoencoders (VAE), and is effective in developing an
+extensive number of conceivable hypotheses/segmentation
+results.
+3. Methodology
+3.1. Probabilistic Model For Noisy Labels
+Let X denote the space that contains a set of input data
+X := {x1, . . . , xn}. Each of these objects x in the input
+data are assigned a corresponding label y such that Y :=
+{y1, y2, . . . , yN} ⊆ Y, where Y is the space of labels.
+We synthetically induce noise in our original label set Y
+to corrupt the clean labels. There are multiple different ways
+through which we create the noisy labels for our data, namely
+symmetric and pairflip noise types. In Section 5.1 and in the
+Appendix, we discuss in details about the mainstream noise
+types that we used to create noisy labels for the datasets that
+we utilized in this paper.
+We denote the set of noisy labels given by annotator r that
+labels objects from the set X as ˜Y (r) = {˜y(r)
+1 , . . . , ˜y(r)
+N },
+where r = 1, . . . , R. Our objective is to jointly estimate
+annotator noise as a function of input x, as well as to esti-
+mate the distribution for latent GT label from noisy dataset,
+D = {X, ˜Y (1), . . . , ˜Y (R)}. In our architecture we add an
+entropy/information-based regularization term with the main
+loss function. The goal is to enforce our algorithm to make
+confident predictions while also learning the true labels.
+Following the strategy of [31, 41], we would now demon-
+strate how to set up a probabilistic model for data that has
+been annotated by multiple sources.
+To model annotator-specific characteristics, there are
+some pivotal factors that are to be considered. In modelling
+multiple annotators, it is common to assume that annotators
+exercise their independence according to their expertise and
+experience in labelling an input data point xi. The precision
+of annotator’s labeling may depend on the properties of the
+data point itself. Thus, we do not assume that annotators
+are equally competent (or incompetent) at labeling all the
+data; rather, it depends on the input they observe. This can
+be represented as a probabilistic model for random variables
+x, y, and ˜y. Following the work of [31,38], we describe the
+joint conditional distribution of our probabilistic model as:
+P( ˜Y (r), Y |X) = �N
+i=1p(yi|xi) �R
+r=1 p(˜y(r)
+i
+|xi, yi).
+Here p(yi|xi) represents the distribution for the clean
+labels of the data samples.
+Conditional distribution
+p(˜y(r)
+i
+|xi, yi) signifies that the model estimates a noisy ver-
+sion of clean labels , represented as ˜y(r)
+i
+for each annota-
+tor r. This makes intuitive sense as the noisy labels are
+not only conditional on true latent labels, but also on the
+input data. It is likely for the annotators to label some
+instances of data xi with more precision than other sam-
+ples. Since the annotators’ noise is dependent on the sam-
+ple x, this allows us to model noisy label distribution as
+p(˜y(r) = j|y = i, x) =: u(r)
+j,i (x). We denote by U(x) a
+C ×C confusion matrix [U]j,i(x) = uj,i(x), where C repre-
+sents the number of classes for the true labels, y ∈ [1, ..., C].
+Now using the confusion U(x), we can show the probability
+that input data x, labelled as i originally, is mislabelled as j
+in the set of noisy data:
+p(˜y = j|x) = �C
+i=1p(˜y = j|y = i, x) · p(y = i|x)
+= �
+iuji(x) · p(y = i|x).
+(1)
+To represent the joint probability distribution of noisy la-
+bels using the confusion matrix of each annotator r, we can
+simplify (1) as:
+p(˜y(1), ..., ˜y(R)|x) =
+R
+�
+r=1
+C
+�
+y=1
+u(r)
+˜y(r),y(x) · p(y|x).
+3.2. Jointly Optimizing the two Networks to esti-
+mate the Ground Truth and Confusion
+We minimize negative log-likelihood (NLL) to jointly
+optimize the parameters θ and ψ of classification and
+annotator networks respectively.
+Given the data that
+4
+
+comprises of training inputs and noisy labels, we would
+minimize the negative log likelihood between the ob-
+served noisy labels and predictions from annotator la-
+bel distribution as follows: − log p(�Y (1), ..., �Y (R)|X) =
+�N
+i=1
+�R
+r=1 NLL( ˆU (r)
+ψ (xi)ˆpθ(xi), ˜y(r)
+i
+), where ˆpθ(xi) is
+the output of the classification network and ˆU (r)
+ψ (x) is the
+output of annotator’s r network. Minimizing this loss func-
+tion alone can cause several problems. Firstly, it does not
+ensure that predictions of the classification network will be
+class probabilities. It can learn some feature of inputs in
+order to minimize the NLL loss between the pipeline output
+ˆU (r)
+ψ (x)ˆpθ(x) and corrupted labels ˜y(r). Secondly, there is
+no guarantee that annotator matrices ˆU (r)
+ψ (x) are correctly
+learned to distinguish the noise from true labels. It can also
+learn some uninterpretable features of inputs x, such that
+ˆU (r)
+ψ (x)ˆpθ(x) is close to ˜y(r).
+To tackle these problems, we would add a regulariza-
+tion term that is attached to the base classifier which helps
+in estimating the true class probabilities of the predicted
+ground truth. The main loss, NLL loss is then jointly op-
+timized with a regularization R(ˆpθ(x)). We propose two
+options for regularization: entropy regularization R(p) =
+− �
+i pi(x) log pi(x) and information-based regularization
+R(p) = − log maxi(pi). The combined loss is then given
+as:
+− log p(�Y (1), ..., �Y (R)|X) =
+N
+�
+i=1
+R
+�
+r=1
+NLL( ˆU (r)
+ψ (x)ˆpθ(xi), ˜y(r)
+i
+)+λ 1
+N
+N
+�
+i=1
+R(ˆpθ(xi)).
+Our classification network helps with learning the features
+of our data and gives us an estimate of the ground truth. The
+outputs of this network are probabilities with dimension
+B × C, where B is the batch-size and C is the number of
+classes. Ideally we desire that the predictions we get from
+the classifier network are forced to give us 1 for the most
+probable class and 0 elsewhere.
+We take the sum of this regularization term for the number
+of batch samples and then multiply it with a regularization
+parameter λ and then taking its average for the batch sam-
+ples.
+4. Confident Regularization
+In this section, we will explain in detail the motivation
+for the choice of our regularizer. We used entropy and infor-
+mation based regularizer with the first network to enhance
+the predictions of our model in learning the ground truth.
+4.1. Entropy Regularizer
+Entropy is regarded as a measure for gauging uncertainty.
+The higher the entropy, the more disordered the state. Shan-
+non et al. [28] mathematically described entropy as:
+R(p) := −�
+ipi log pi = E[− log p].
+where pi denotes the i-th class probability. It is to be noted
+that entropy is a feasible choice as it a smooth function. So
+when pi is 0, the function is still differentiable, since 0 log 0
+= limpi→0 pi log pi.
+4.2. Information Regularizer
+We evaluated our experiments on another regularizer
+which resembles the information part of entropy. The motiva-
+tion behind using this regularizer is the same as the entropy
+regularizer mentioned in Section 4.1. This regularizer is
+expressed as :
+R(p) = min
+i (− log pi).
+(2)
+This regularizer would also push the classifier to make
+confident predictions. The caveat in using this regularizer is
+that it becomes undefined when pi = 0. To counter that, we
+modified the regularizer function to:
+R(p) = − log(max
+i
+pi).
+Also, we would show that we achieve similar results when
+we compare it with entropy regularizer. The advantage of
+using entropy regularizer is that it’s a smooth function unlike
+(2).
+4.3. Motivation for Confident Regularizarion
+As we mentioned before, it is not enough to minimize the
+loss between ˆUψ(x)ˆpθ(x) and ˜y. Indeed, let U be the true
+confusion of the annotator and Pθ(x) the confusion matrix
+of classifier. Ideally, we want Pθ(x) = I and ��Uψ(x) = U.
+However, there can be a lot of pairs ( ˆUψ(x), Pθ(x)) that
+satisfy ˆUψ(x)ˆpθ(x) = U. Therefore, we add an entropy
+regularizer, to enforce Pθ converge to I.
+Theorem 1. Assume that classifier is confident, i.e. ˆpθ = ei
+if y = i, where ei is a basis vector of i-th coordinate. Then,
+minimizing NLL loss between ˆU (r)
+ψ (x)ˆpθ(x) and ˜y(r) over
+ˆU (r)
+ψ (x) we get
+[ ˆU (r)
+ψ (x)]i,j = p(˜y(r) = j|y = i, x).
+In Tables 1, Table 2, and Table 3, we show the perfor-
+mance of our algorithm with entropy and information-based
+regularizers on CIFAR-10, MNIST, and FMNIST datasets
+for symmetric and pairflip noise types. The theoretical com-
+parison between the two regularizers is further discussed in
+Appendix ??.
+5. Classification Experiments
+5.1. Implementation Details
+In this section we describe implementation details. We
+used a convolutional neural network (CNN) as a classifier
+5
+
+model which estimates the ground truth. The predictions
+of the classifier network are multiplied to the outputs of a
+fully connected annotator network that learns the confusion
+of the noisy labels. True labels are never introduced to the
+model during training. In our experiments, we synthetically
+introduce noise to the training data; we chose various noise
+rates, such as 20%, 30%, 45% and 50%.
+We evaluate the performance of our algorithm for the clas-
+sifier network as this aids in estimating the GT via making
+confident predictions about the true class. We are partic-
+ularly interested in the performance of the classifier, as in
+evaluation stage this network will be used separately to make
+predictions.
+Baselines. We compare our algorithm with the following ap-
+proaches: (i) Co-teaching [10], which simultaneously trains
+two DNN models, with each network selecting the batch
+of data for the other, based on the instances with a small
+loss. (ii) Co-teaching+ [40], also employs samples with
+small loss, but with disagreement about predictions. This is
+the selection criteria for the networks to pick data for each
+other. (iii) JoCoR [34], extends on the idea of [10,40], but
+uses co-regularization to minimize the diversity of the two
+networks, thus bringing the predictions of the two networks
+closer together. (iv) Robust Early-learning (CDR) [35], cat-
+egorizes the critical and non-critical parameters for clean
+and noisy label fitting, respectively. Different update rules
+are applied to update these parameters. (v) Annotator Con-
+fusion (Trace) [31] is a regularized approach that assumes
+the existence of various annotators to simultaneously learn
+the individual annotator model and the underlying true label
+distribution, using only noisy observations.
+Datasets.
+We used the standard benchmark datasets:
+MNIST [8], FMNIST [37], and CIFAR-10 [16] to demon-
+strate the effectiveness of our methodology.
+Types of Noise. The noise types, used in the experiments,
+are described below.
+1. Pairflip Noise. The pairflip noise involves swapping the
+labels of two adjacent categories/classes based on a preset
+ratio. [19]
+2. Symmetric Noise. In symmetric noise, a portion of the
+original labels are retained, while the remainder are uni-
+formly reassigned to all other categories [21]. This noise
+type is intended to imitate the random noise in the actual
+world, which is typically the result of random web crawl-
+ing or manual annotation errors. It does not consider the
+similarities between classes.
+5.2. Comparison with State-Of-The-Arts
+Results on MNIST. We used the same backbone architec-
+ture to compare our algorithm against the baselines. Table 1
+shows the performance comparison of our algorithm with
+the other methods. We see that for a smaller noise rate such
+as 20% and 30%, which is evidently the least challenging
+case, all algorithms seem to show comparable performance
+above 97% for both pairflip and symmetric noise. However,
+when noise rates increases to 45% or above, there seems to
+be a distinct contrast in the performance of other algorithms,
+as the accuracy of some methods visibly decline to below
+90% in the case of pairflip noise. Our method achieves an
+accuracy of 99.10% for pairflip 45% noise using entropy
+regularizer, followed closely by the Trace method with an
+accuracy of 97.95%. Whereas Co-teaching, JoCoR and CDR
+achieves the test accuracy of 87.63%, 85.86% and 87.04%
+respectively. For symmetric-50% noise, we got test accu-
+racy of 98.94% with information regularizer, with Trace and
+CDR following closely behind at 98.87% and 97.72% re-
+spectively. The results of the our methodology with entropy
+and information-based regularizers are comparable.
+Results on CIFAR-10. Table 2 shows the test accuracy
+results on CIFAR-10 dataset. Our algorithm performs dis-
+tinctly superior when noise gets extreme; we achieve 80.03%
+accuracy for symmetric 50% noise with information regu-
+larizer, surpassing all the baselines. For pairflip 45%, we
+distinctly outperform all the baselines by a considerable mar-
+gin. We acquired an accuracy of 83.43%, which is about 8%
+better than Trace method. All the other baselines acquired
+accuracy of less than 70%. Hence, our algorithm clearly
+outperforms all the baselines. This reinforces that for higher
+noise ratios, our algorithm consistently gives better perfor-
+mance as entropy and information regularization strategy
+helps the model to be more certain in its predictions.
+Our algorithm still surpasses in performance when the
+noise rate is small for symmetric and pairflip noise types.
+For symmetric noise 20% and 30%, we achieved an accuracy
+of 84.22% and 83.85% respectively. The other algorithms
+contested close for symmetric 20% noise by accomplishing a
+test accuracy of 82.86% for Trace, 82.82% for Co-teaching,
+81.12% for JoCoR, 81.01% for CDR, and with Co-teaching+
+settling with an accuracy of 79.51%.
+In pairflip noise 20% and 30%, we again outperform other
+methods by accomplishing an accuracy of 84.92% and 84.5%
+in the given sequence. Here, CDR and Trace follow closely
+behind with an accuracy of 82.89% and 83.86% respectively
+for pairflip 20% noise. For pairflip 30%, CDR attained an
+accuracy of 82.08%, while Trace achieved 83.15%.
+Results on FMNIST. The experimental results of our algo-
+rithm compared with other baselines is shown in Table 3.
+Our algorithm has shown robust performance across most
+baselines.
+We see comparable performance among all the algorithms
+when the noise rate is 20% and 30% for both symmetric and
+pairflip noise types. We see distinguishing performance
+when the noise gets extreme.
+At symmetric 50% noise, we perform about 12% better
+than Co-teaching+ algorithm, while we outperformed CDR
+6
+
+Table 1. Test accuracy (%) on MNIST dataset.
+Noise rate
+Ours-Inf
+Ours-Ent
+Co-tea.
+Co-tea.+
+JoCoR
+Trace
+CDR
+symmetric 20%
+99.20
+99.48
+99.01
+98.88
+98.82
+99.16
+98.97
+symmetric 30%
+99.11
+99.09
+98.78
+98.38
+98.40
+99.01
+98.75
+symmetric 50%
+98.94
+98.93
+92.24
+95.26
+96.83
+98.87
+97.72
+pairflip 20%
+99.08
+99.55
+98.84
+98.59
+98.89
+99.13
+98.88
+pairflip 30%
+98.94
+99.54
+98.57
+97.95
+98.56
+99.08
+98.50
+pairflip 45%
+98.77
+99.10
+87.63
+71.36
+85.86
+97.95
+87.04
+Table 2. Test accuracy (%) on CIFAR-10 dataset
+Noise rate
+Ours-Inf
+Ours-Ent
+Co-tea
+Co-tea+
+JoCoR
+Trace
+CDR
+symmetric 20%
+84.22
+84.00
+81.82
+79.51
+82.12
+82.86
+81.01
+symmetric 30%
+83.85
+83.26
+80.69
+79.29
+80.95
+80.45
+78.90
+symmetric 50%
+80.03
+79.64
+75.74
+73.19
+76.60
+77.82
+69.68
+pairflip 20%
+84.92
+84.78
+81.17
+79.59
+81.86
+83.86
+82.89
+pairflip 30%
+84.36
+84.54
+79.53
+77.83
+79.52
+83.15
+82.08
+pairflip 45%
+83.43
+81.23
+59.04
+47.72
+67.59
+75.88
+58.56
+Table 3. Test accuracy (%) on FMNIST dataset.
+Noise rate
+Ours-Inf
+Ours-Ent
+Co-tea
+Co-tea+
+JoCoR
+Trace
+CDR
+symmetric 20%
+90.67
+90.79
+90.48
+88.69
+91.88
+90.61
+88.69
+symmetric 30%
+91.35
+90.34
+90.36
+88.50
+91.33
+89.64
+87.38
+symmetric 50%
+89.51
+89.49
+89.37
+77.96
+89.21
+88.94
+85.36
+pairflip 20%
+90.90
+90.77
+90.68
+89.12
+91.37
+90.40
+90.01
+pairflip 30%
+90.38
+90.65
+90.11
+89.06
+89.67
+90.33
+88.78
+pairflip 45%
+89.37
+89.02
+78.86
+52.61
+88.10
+89.08
+64.63
+by about 4%. We surpassed other baselines by small margins
+for this instance of noise. For pairflip 45%, we performed
+significantly better than Co-teaching+ and CDR algorithms
+which achieved accuracy of 52.61% and 64.63% respectively.
+Trace algorithm comes second in performance with 89.08%
+accuracy, followed closely by JoCoR at 88.10%.
+Two network architectures such as Co-teaching, Co-
+teaching+, and JoCoR suffers in performance when the
+noise level increase in both symmetric and pairflip noise
+types. Trace comes closer in comparison with our algorithm,
+but we outperform it in all experiments. Both entropy and
+information-based regularizers perform at par compared to
+each other.
+5.3. Curated Dataset
+We have assembled the dataset that is based on MNIST,
+where noise level depends on input image style for vari-
+ous annotators. Three type of image styles were simulated
+by performing morphological transformations (in particu-
+lar, thinning and thickening) on the original images, using
+Morpho-MNIST software [5]. In addition to the noise types
+described in Section 5.1, asymmetric and pairflip with per-
+mutation where applied. In the latter, the ordered label
+categories were first permuted randomly and labels of two
+adjacent categories after permutation were swapped based
+on a preset ratio. Asymmetric noise is a block matrix trans-
+formation, where a portion of original labels are retained
+and the remainder is uniformly reassigned to closest four
+categories. The type and level of noises applied to original
+labels are provided in Table 4.
+For a dataset consisting three different type of images
+(original, thin, and thick) and three different annotators (Ta-
+ble 4), we compare (i) classifier model without annotators
+and regularization, (ii) our approach without regularization,
+Table 4. Annotator Information for three different styles (MNIST).
+Annotators
+Original
+Thin
+Thick
+Annotator 1
+symmetric 80%
+asymmetric 40%
+pairflip 95%
+Annotator 2
+pairflip with permutation 40%
+symmetric 95%
+asymmetric 70%
+Annotator 3
+pairflip 60%
+pairflip with permutation 40%
+symmetric 80%
+0
+2
+4
+6
+8
+10
+Epoch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Training Accuracy
+MNIST
+Model w/o annotators and =0 
+Our approach ( =0)
+Our approach ( =0.01, m=2)
+(a) Training accuracy
+0
+2
+4
+6
+8
+10
+Epoch
+10
+3
+10
+2
+10
+1
+100
+Training Entropy
+MNIST
+Model w/o annotators and =0 
+Our approach ( =0)
+Our approach ( =0.01, m=2)
+(b) Training entropy
+Figure 2. Accuracy and entropy for Curated MNIST training data.
+0
+2
+4
+6
+8
+10
+Epoch
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Testing Accuracy
+MNIST
+Model w/o annotators and =0 
+Our approach ( =0)
+Our approach ( =0.01, m=2)
+(a) Testing accuracy
+0
+2
+4
+6
+8
+10
+Epoch
+10
+3
+10
+2
+10
+1
+100
+Testing Entropy
+MNIST
+Model w/o annotators and =0 
+Our approach ( =0)
+Our approach ( =0.01, m=2)
+(b) Testing entropy
+Figure 3. Accuracy and Entropy for Curated MNIST testing data.
+Image
+Original
+Our (λ=0.01, m=2)
+Our (λ=0)
+Thin
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator1 Thin
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator1 Thin
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator1 Thin
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Original
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator2 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator2 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator2 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Thick
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator3 Thick
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator3 Thick
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator3 Thick
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Figure 4. Original and Predicted confusion for different Annotators
+using different models: our approach with regularizer (λ = 2,
+m=1.5) and without it (λ = 0). (MNIST).
+and (iii) our approach with information-based regularization.
+Each annotator NN has similar architecture as in classifier
+model and takes images as an input. Everything else is the
+same as described in Section 3.
+The result of experiments can be seen in Figures 2 and 3.
+Our approach is more accurate and confident compared to
+the classifier model. The accuracy of our approach with reg-
+ularizer is higher and more confident than the model without
+the regularizer. This observation can be seen for both train-
+ing and testing data. The proposed approach is able to learn
+the annotators’ confusion. Predicted confusion matrices for
+each annotators and different image types are provided in
+7
+
+6Table 5. Test DICE (%) and entropy evaluation on MNIST dataset
+for segmentation.
+Metrics
+Ours-Inf
+Trace
+DICE
+96.97
+96.62
+Entropy
+0.0453
+0.0696
+Figure 4. More results can be found in Appendix C.
+6. Segmentation Experiments
+We explored the performance of our algorithm with
+information-based regularization for segmentation.
+The
+whole approach is the same as for classification, but pre-
+dictions would now be pixel-wise. Holistically, we followed
+the same idea in both the settings of classification and seg-
+mentation. The inputs of the model are the original images
+from MNIST with a Gaussian noise. Annotators (thin, thick,
+fractured) were simulated using morphological transforma-
+tions [5] as mentioned in Section 5.3. The details for the
+MNIST segmentation dataset are provided in Appendix B.2.
+For our method, we used the same model architecture as
+in [41], which is implemented as U-Net [27] with multiple
+output layers: the first is for prediction of true segmentation
+and the second is for predictions of noisy segmentations.
+We compared our method, which has information-based
+regularizer, with the trace-regularized approach [41].
+6.1. Results
+Table 5 shows the accuracy of the our method in com-
+parison with the trace, and it also highlights the entropy
+calculated for these two methods. It can be seen that our
+model achieves better DICE similarity score and is more con-
+fident. Furthermore, in Figure 5, we visualised the results
+of the predictions for our method. It is clearly demonstrated
+that for an input image with Gaussian noise, our algorithm
+is able to produce excellent predictions about the true seg-
+mentation, with given annotators, as it matches closely the
+GT image.
+7. Discussion & Conclusion
+In this research, we proposed an approach of jointly train-
+ing a two network model in a confident way. We improve
+classification/segmentation network by attaching regulariza-
+tion term (Information and Entropy) to make assured pre-
+dictions. Moreover, our algorithm also learns annotators’
+noise and separate it from the true labels under extreme
+noisy supervision. We evaluated our algorithm on the stan-
+dard datasets such as CIFAR-10, FMNIST and MNIST. In
+comparison with other state-of-the-arts, our method secured
+mature robust results. In classification task, we outperformed
+all baselines for extreme noise levels such as pairflip 45%
+and symmetric 50%. For smaller noise levels, we achieved
+comparable performance with SOTAs. In segmentation prob-
+lem, we achieved better DICE similarity score than [41]. We
+also show that prediction of classifier/segmentation model
+are more confident compared to other baselines. This demon-
+strates the effectiveness of our algorithm in making confident
+robust predictions about the true class labels/ground truth.
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+
+A. Proof of Theorem 1
+Proof. Given samples x, y = i we want to minimize
+1
+R
+R
+�
+r=1
+E˜y|x,y
+�
+l( ˆU (r)
+ψ (x)pθ(x), ˜y)
+�
+= 1
+R
+R
+�
+r=1
+C
+�
+j=1
+p(˜y = j|x, y = i)l( ˆU (r)
+ψ (x)ei, ˜y)
+= − 1
+R
+R
+�
+r=1
+C
+�
+j=1
+p(˜y = j|x, y = i) log
+[ ˆU (r)
+ψ (x)]j,i
+C
+�
+j=1
+[ ˆU (r)
+ψ (x)]j,i
+w.r.t. ˆU (r)
+ψ (x), r = 1, . . . , R. Since ˆU (r)
+ψ (x) is a stochastic
+matrix, we have
+C�
+j=1
+[ ˆU (r)
+ψ (x)]j,i = 1. Taking the derivative
+over [ ˆU (r)
+ψ (x)]j,i, we get
+[ ˆU (r)
+ψ (x)]j,i = p(˜y = j|x, y = i).
+B. Experimental Details
+B.1. Classification Datasets
+MNIST: dataset comprise of 60,000 samples for training,
+and 10,000 data samples reserved for testing. The number
+of classes in the dataset is 10.
+CIFAR-10: The CIFAR-10 dataset contains 60,000 color
+images in 10 classifications, with 6000 images each class.
+There are 50,000 training and 10,000 test images. The
+dataset is divided into five training batches and one test
+batch, each contains 10,000 images. The test batch is a col-
+lection of exactly 1,000 data samples randomly selected from
+each class. The training batches comprise of the remaining
+images in a random order, however it’s likely that certain
+training batches may have more images from one class than
+another.
+FMNIST: Fashion-MNIST is a dataset of article images
+from Zalando. It consists of a training set of 60,000 instances
+and a test set of 10,000 instances. Each instance is a 28 × 28
+grayscale image with a label from one of ten classes.
+B.2. Segmentation Datasets
+MNIST.
+We also use the dataset used by [41]; synthetic
+noisy annotations were created based on the assumed GT to
+demonstrate the effectiveness of the method in a hypothetical
+setting where the GT is known. We apply the morphological
+alterations (such as thinning, thickening, fractures, etc.) to
+the ground-truth (GT) segmentation labels using the Morpho-
+MNIST software [6], we mimic a group of five annotators
+with a variety of distinguishing features/transformations. In
+particular, the first annotator near accurately segments the
+image ("good-segmentation"), and look similar to the GT.
+The second tends to over-segment ("thick-segmentation"),
+the third tends to under-segment ("thin-segmentation"), the
+fourth is prone to a combination of over-segmentation and
+small fractures ("fracture-segmentation").
+In a multi-class scenario, we first select a target class and
+then perform morphological operations on the provided GT
+mask to produce 4 different types of synthetic noisy labels:
+over-segmentation, under-segmentation, fracture segmenta-
+tion, and good segmentation. Through the use of simulated
+annotators, we derive labels to create training data. How-
+ever, the good segmentation remain remain latent and are
+not included during training of our algorithm.
+B.3. Types of Noise
+Figure 6 shows the an example of noise transition ma-
+trices for pairflip 20% and symmetric 50% noise types. In
+addition, Figure 7 signifies the noise labels distributions
+for CIFAR-10 dataset for pairflip 45% and symmetric 50%
+noise types; this distribution of the label noise is used in the
+training process.
+B.4. Fine-tuning/Training
+In this section, we would now further elaborate on the
+experimental details for each dataset that we used to validate
+our algorithm against.
+MNIST.
+We used a LeNet model as a classifier for our
+backbone network. For the annotator network, we have a
+linear layer of size C × C, C denotes the number of classes.
+This linear layer represents our annotator confusion matrices,
+and we apply a softmax layer to it to make it a stochastic
+matrix along a certain dimension. We fine-tuned our model
+for a combination of learning rates, α = [0.01, 0.001, 0.0001,
+0.000001, 0.0016, 0.008, 0.0064, 0.005], and about 50 dif-
+ferent lambda values, λ, for our regularizer hyper-parameter.
+We started with a very small value of λ = 0.001506746, and
+slowly increased it exponentially (geometric progression)
+per epoch with a rate, r = 1.18; we trained the model for
+70 epochs and used Adam as an optimizer. In addition, the
+experiments were regulated to assess the performance of the
+model when the confusion matrix is initialized as an identity.
+These fine-tunings are done across all 2 different types of
+noise described in Section 5.1 with the respective noise rate
+that is associated with each of the noise types.
+CIFAR-10.
+For CIFAR-10, we used ResNet-18 as our
+backbone network for the classifier. The annotator network
+remains unchanged (still has one linear layer that represents
+the confusion matrices of class C × C that are stochas-
+tic). For this dataset, we fine-tuned the model for an assort-
+ment of learning rates, such as α = [0.001, 0.00064, 0.0016,
+0.000001, 0.005, 0.008, 0.0016, 0.00064]. We ran the model
+for 150 epochs; the hyperparameter λ for our regularizer was
+slowly increased exponentially again with a rate, r= 1.11.
+11
+
+0.8 0.2
+0
+0
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+0.8
+Pairflip, ε = 20%
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+(a)
+0.5
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
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+0.06
+0.5
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+0.06
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+0.06
+0.06
+0.5
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.5
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.5
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.5
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.5
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.06
+0.5
+Symmetric, ε = 50%
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+(b)
+Figure 6. Noise Transition matrices for Pairflip and Symmetric noise.
+y
+y
+2713
+2287
+0
+0
+0
+0
+0
+0
+0
+0
+0
+2846
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+2780
+2220
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+0
+0
+0
+0
+0
+0
+0
+0
+2701
+ CIFAR-10, Pairflip-45%
+0
+500
+1000
+1500
+2000
+2500
+(a)
+y
+y
+2562
+282
+257
+264
+274
+292
+272
+268
+264
+265
+287
+2486
+273
+285
+271
+290
+272
+284
+271
+281
+308
+272
+2492
+285
+284
+267
+256
+308
+284
+244
+260
+238
+280
+2560
+266
+264
+270
+302
+280
+280
+264
+286
+288
+289
+2461
+275
+309
+272
+277
+279
+280
+258
+275
+285
+263
+2531
+291
+259
+275
+283
+279
+297
+273
+277
+311
+271
+2465
+270
+271
+286
+278
+245
+272
+308
+250
+306
+279
+2523
+281
+258
+266
+268
+275
+287
+287
+282
+298
+230
+2548
+259
+271
+266
+277
+252
+287
+282
+278
+273
+280
+2534
+ CIFAR-10, Symmetric-50%
+500
+1000
+1500
+2000
+2500
+(b)
+Figure 7. Confusion matrix between clean (y) and noisy labels (˜y) of CIFAR-10 dataset for (a) Pairflip-45% and (b) Symmetric-50% noise.
+However, the starting value this time is λ= 3.0517578125e-
+05. We used a standard batch-size, BS=128. We used the
+standard augmentations of random crop of size 32 × 32 and
+horizontal random flipping. These are the standard augmen-
+tations that have been used across all the baselines that we
+have evaluated. The remaining settings remain the same as
+described in MNIST above.
+FMNIST.
+We kept the same settings of CIFAR-10, such as
+ResNet-18 model and batch-size of 128 for FMNIST dataset.
+The model was again fine-tuned for the same set of hyperpa-
+rameters. However, the starting value of λ= 6.103515625e-
+05, and it was increased exponentially with a rate of r=1.12.
+We also retained the same set of augmentations that we used
+in CIFAR-10 dataset.
+C. Additional experimental results
+In our earlier experiments, we kept the same type of noise
+and noise levels across all the number of annotators in the
+annotator network. This is usually not representative of
+the noise in the real world data, as it is possible that each
+annotator would be independent in the way it is confused
+about labelling and annotating the data (subject to their own
+biases). Therefore, we confuse each annotator with different
+types and levels of noise. Table 6 shows the test accuracy of
+the classifier network on CIFAR-10, FMNIST and MNIST
+datasets for different types of noise for each annotators. We
+achieved comparable results with an accuracy of 84.12%,
+91.12% and 98.97% for CIFAR-10, FMNIST and MNIST
+respectively. It’s particularly notable that the accuracy of the
+classifier network remains at par even with using high level
+noise, such as pairflip 45% and symmetric 50% for two of
+the three annotators.
+Table 6. Test accuracy (%) with three different annotators (Annota-
+tor1: Pairflip 45%, Annotator2: Symmetric 20%, Annotator3: 50%)
+representing different noise types and noise levels on CIFAR-10,
+FMNIST and MNIST datasets.
+CIFAR-10 FMNIST MNIST
+84.12
+91.62
+98.97
+12
+
+0
+10
+20
+30
+40
+50
+60
+epoch
+20
+40
+60
+80
+100
+test accuracy
+MNIST, Pairflip 45%
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR
+(a) Pairflip-45%
+0
+10
+20
+30
+40
+50
+60
+epoch
+20
+40
+60
+80
+100
+test accuracy
+MNIST, Symmetric 30%
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR
+(b) Symmetry-30%
+0
+10
+20
+30
+40
+50
+60
+epoch
+20
+40
+60
+80
+100
+test accuracy
+MNIST, Symmetric 50%
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR
+(c) Symmetry-50%
+Figure 8. Test accuracy (%) vs. number of epochs on MNIST dataset.
+0
+20
+40
+60
+80
+100
+120
+140
+epoch
+10
+20
+30
+40
+50
+60
+70
+80
+test accuracy
+(CIFAR-10, Pairflip 45%)
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR
+(a) Pairflip-45%
+0
+20
+40
+60
+80
+100
+120
+140
+epoch
+10
+20
+30
+40
+50
+60
+70
+80
+test accuracy
+CIFAR-10, Symmetric 30% 
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR
+(b) Symmetry-30%
+0
+20
+40
+60
+80
+100
+120
+140
+epoch
+10
+20
+30
+40
+50
+60
+70
+80
+test accuracy
+CIFAR-10, Symmetric 50%
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR
+(c) Symmetry-50%
+Figure 9. Test accuracy (%) vs. epochs on CIFAR-10 dataset.
+MNIST
+In Figure 8, we highlight the test accuracy vs.
+number of epochs. We can see clearly that for symmetric
+noise types, all algorithms gave comparable performance.
+It can be clearly seen that for symmetric noise, our test
+accuracy starts to decline a bit. This could be alleviated
+with an early stopping criteria which was not incorporated in
+these experiments. For pairflip 45%, the test accuracy starts
+to increase and stabilize in the later epochs of the experiment
+and transcends all the baselines.
+CIFAR-10
+Figure 9 shows the illustrative results of test
+accuracy vs. number of epochs. In all the three plots, it can
+be clearly seen that our algorithm performs at par with the
+other algorithms, but the performance gets robustly superior
+in the extreme noise type of pairflip 45%. This shows that
+our method is particularly robust again harder noise as it is
+able to make confident predictions.
+FMNIST
+Figure 10 gives an illustrative result of test accu-
+racy vs. number of epochs on FMNIST dataset. It showcases
+the test performance of our algorithm in comparison with
+other baselines. We can see that for all noise instances, our
+algorithm performs at par with the high achieving method
+like JoCoR. We perform considerably better against sample
+selection methods like Co-teaching and Co-teaching+, as
+well as against other method like CDR in the instance of
+pairflip 45%.
+In addition, Figure 11 highlights the confusion matrices
+13
+
+(Cifar-10. Symmetric 30%)
+90
+80
+M
+70
+accuracy
+60
+50
+test
+40
+30
+20
+10
+0
+25
+50
+75
+100
+125
+150
+epoch
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR(Cifar-10. Symmetric 30%)
+90
+80
+M
+70
+accuracy
+60
+50
+test
+40
+30
+20
+10
+0
+25
+50
+75
+100
+125
+150
+epoch
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoRof the true class and the predicted class by the classifier net-
+work of our algorithm. We show the confusion matrices plots
+for two extreme noise types pairflip 45% and symmetric 50%
+for all the datasets used. It is clearly seen that the confu-
+sion matrices are diagonally dominant thus highlighting the
+robust performance of our method.
+MNIST Curated Dataset
+In Figure 12 we demonstrate
+annotators’ confusion using our algorithm on the curated
+MNIST dataset that showcases different image styles of
+Original, Thin and Thick. The strength of the regularizer,
+λ=0.01, is increased by the multiplicative scalar m=2 ev-
+ery epoch. Figures 13, 14 and 15 highlights the original
+and predicted confusion of annotator 1, annotator 2 and an-
+notator 3 using our approach with the regularizer and the
+non-regularized approach (that is, when λ=0).
+MNIST Segmentation
+In Figure 16, the results of the
+annotators’ (Thin, Thick and Fractured) predictions are vi-
+sualised for our algorithm. The results demonstrate that
+our algorithm has produced good prediction results for the
+annotators.
+14
+
+0
+20
+40
+60
+80
+100
+epoch
+20
+40
+60
+80
+test accuracy
+FMNIST, Pairflip 45%
+(a) Pairflip-45%
+0
+20
+40
+60
+80
+100
+epoch
+20
+40
+60
+80
+test accuracy
+FMNIST, Symmetric 30%
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR
+(b) Symmetry-30%
+0
+20
+40
+60
+80
+100
+epoch
+20
+40
+60
+80
+test accuracy
+FMNIST, Symmetric 50%
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoR
+(c) Symmetry-50%
+Figure 10. Results of test accuracy vs. number of epochs on FMNIST dataset.
+(a)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Predicted class
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+True class
+972
+2
+1
+0
+0
+0
+2
+1
+2
+0
+0
+1123
+6
+1
+0
+1
+1
+0
+3
+0
+1
+0
+1004
+21
+1
+0
+0
+4
+1
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+1
+0
+1
+998
+3
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+0
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+0
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+2
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+0
+879
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+0
+1
+1
+2
+0
+1
+1
+3
+948
+1
+1
+0
+0
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+6
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+1
+0
+0
+1011
+3
+1
+2
+0
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+6
+2
+1
+1
+1
+951
+8
+5
+2
+0
+4
+9
+3
+0
+3
+0
+983
+MNIST, Pairflip 45%
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Predicted class
+True class
+975
+0
+2
+0
+0
+0
+0
+1
+2
+0
+0
+1129
+1
+1
+1
+0
+2
+1
+0
+0
+1
+0
+1026
+0
+1
+0
+0
+3
+1
+0
+1
+0
+3
+997
+0
+5
+0
+2
+2
+0
+1
+0
+0
+0
+969
+0
+4
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+1
+6
+1
+0
+0
+5
+0
+884
+1
+0
+0
+1
+4
+4
+0
+1
+2
+2
+943
+0
+2
+0
+0
+1
+7
+1
+0
+0
+0
+1013
+1
+5
+0
+0
+4
+1
+2
+1
+1
+2
+962
+1
+1
+2
+0
+0
+3
+3
+0
+3
+1
+996
+MNIST, Symmetric 50%
+0
+200
+400
+600
+800
+1000
+0
+200
+400
+600
+800
+1000
+(b)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Predicted class
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+True class
+780
+18
+7
+27
+7
+3
+139
+15
+3
+1
+0
+935
+35
+23
+4
+0
+2
+1
+0
+0
+17
+1
+735
+46
+124
+2
+74
+1
+0
+0
+2
+2
+1
+920
+60
+3
+6
+5
+1
+0
+2
+0
+18
+13
+903
+23
+40
+1
+0
+0
+1
+0
+0
+0
+0
+982
+3
+10
+0
+4
+68
+0
+27
+27
+81
+3
+512 279
+3
+0
+0
+0
+0
+0
+0
+6
+2
+961
+1
+30
+1
+0
+0
+1
+4
+2
+1
+2
+976
+13
+44
+0
+0
+0
+0
+5
+1
+22
+2
+926
+FMNIST, Pairflip 45%
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Predicted class
+True class
+782
+3
+24
+17
+6
+3
+159
+0
+6
+0
+2
+976
+1
+13
+3
+0
+3
+0
+2
+0
+16
+1
+861
+7
+59
+1
+52
+0
+3
+0
+24
+6
+11
+888
+30
+0
+38
+0
+3
+0
+0
+0
+107
+25
+800
+0
+64
+0
+4
+0
+0
+0
+0
+0
+0
+967
+1
+25
+0
+7
+72
+1
+70
+27
+60
+0
+757
+0
+13
+0
+0
+0
+0
+0
+0
+18
+0
+972
+0
+10
+1
+1
+1
+3
+2
+3
+5
+2
+982
+0
+0
+0
+0
+0
+0
+10
+0
+50
+0
+940
+FMNIST, Symmetric 50%
+0
+200
+400
+600
+800
+0
+200
+400
+600
+800
+(c)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Predicted class
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+True class
+891
+13
+21
+23
+5
+2
+6
+4
+20
+15
+13
+918
+5
+6
+1
+2
+7
+1
+3
+44
+28
+1
+651 208
+34
+21
+38
+10
+4
+5
+21
+1
+16
+760
+49
+86
+42
+13
+5
+7
+15
+1
+22
+66
+773
+41
+24
+48
+9
+1
+9
+2
+6
+210
+29
+691
+24
+20
+5
+4
+9
+1
+13
+49
+19
+11
+890
+3
+2
+3
+15
+3
+6
+51
+17
+39
+3
+790
+72
+4
+67
+13
+2
+18
+2
+3
+1
+0
+795
+99
+52
+62
+6
+12
+0
+2
+7
+5
+13
+841
+CIFAR-10, Pairflip 45%
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Predicted class
+True class
+641
+26
+45
+33
+24
+20
+18
+9
+155
+29
+20
+752
+17
+15
+10
+20
+12
+11
+82
+61
+62
+14
+551
+68
+59
+69
+43
+27
+87
+20
+35
+27
+60
+469
+57
+157
+50
+39
+88
+18
+19
+10
+42
+33
+640
+55
+44
+54
+91
+12
+22
+19
+34
+115
+30
+593
+20
+51
+95
+21
+20
+10
+30
+43
+26
+36
+756
+18
+44
+17
+27
+14
+14
+39
+34
+66
+11
+698
+88
+9
+31
+14
+15
+12
+9
+9
+4
+7
+887
+12
+43
+67
+11
+12
+10
+20
+9
+17
+89
+722
+CIFAR-10, Symmetric 50%
+0
+200
+400
+600
+800
+100
+200
+300
+400
+500
+600
+700
+800
+Figure 11. Confusion matrices of true class and predicted class for our algorithm for CIFAR-10, MNIST and FMNIST datasets.
+15
+
+(Cifar-10. Symmetric 30%)
+90
+80
+M
+70
+accuracy
+60
+50
+test
+40
+30
+20
+10
+0
+25
+50
+75
+100
+125
+150
+epoch
+Ours
+Trace
+Co-teaching
+Co-teaching+
+CDR
+JoCoROriginal
+Thin
+Thick
+Image
+Image
+Image
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 1
+0.08
+0.10
+0.12
+0.14
+0.16
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 1
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 1
+0.2
+0.4
+0.6
+0.8
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 2
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 2
+0.07
+0.08
+0.09
+0.10
+0.11
+0.12
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 2
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 3
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 3
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator 3
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Figure 12. Learned Annotators’ confusion for different image styles using our approach with the regularizer (λ=0.01, m=2) on MNIST
+dataset.
+16
+
+山a6Image
+Original
+Our (λ=0.01, m=2)
+Our (λ=0)
+Original
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator1 Original
+0.00
+0.05
+0.10
+0.15
+0.20
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator1 Original
+0.00
+0.05
+0.10
+0.15
+0.20
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator1 Original
+0.00
+0.05
+0.10
+0.15
+0.20
+Thick
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator1 Thick
+0.0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator1 Thick
+0.0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator1 Thick
+0.0
+0.2
+0.4
+0.6
+0.8
+Thin
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator1 Thin
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator1 Thin
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator1 Thin
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Figure 13. Original and Predicted confusion for Annotator 1 using different models: our approach with regularizer (λ = 0.01, m=2) and
+without it (λ = 0) on MNIST dataset.
+17
+
+6Image
+Original
+Our (λ=0.01, m=2)
+Our (λ=0)
+Original
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator2 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator2 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator2 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Thick
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator2 Thick
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator2 Thick
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator2 Thick
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Thin
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator2 Thin
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator2 Thin
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator2 Thin
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+0.16
+Figure 14. Original and Predicted confusion for Annotator 2 using different models: our approach with regularizer (λ = 0.01, m=2) and
+without it (λ = 0) on MNIST dataset.
+18
+
+6Image
+Original
+Our (λ=0.01, m=2)
+Our (λ=0)
+Original
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator3 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator3 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator3 Original
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Thick
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator3 Thick
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator3 Thick
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator3 Thick
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Thin
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+Annotator3 Thin
+0.0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator3 Thin
+0.0
+0.2
+0.4
+0.6
+0.8
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Annotator3 Thin
+0.0
+0.2
+0.4
+0.6
+0.8
+Figure 15. Original and Predicted confusion for Annotator 3 using different models: our approach with regularizer (λ = 0.01, m=2) and
+without it (λ = 0) on MNIST dataset.
+19
+
+6Input
+Thin
+Thick
+Fractured
+Pred
+GT
+Figure 16. Visualisation of the predictions of the annotators’ segmentations (Thin, Thick and Fractured) together with the predictions of
+the estimated true labels using our algorithm in comparison with the test image and GT. Black is true positive, White is true negative, Red
+represents false positive, whilst Green is false negative.
+20
+
+44400006.

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+Exciton dissociation mediated by phonons in organic photovoltaics
+Stepan Fomichev,1, 2, ∗ Leonard Ruocco,1, 2, ∗ Alexandra Tully,1, 2 and Mona Berciu1, 2, 3
+1Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, V6T 1Z1 Canada
+2Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, V6T 1Z4 Canada
+3Leibniz Institute for Solid State and Materials Research (IFW) Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany
+(Dated: January 10, 2023)
+It is well known that phonons can overscreen the bare Coulomb electron-electron repulsion, turning
+it into the effective attraction that binds the Cooper pairs responsible for BCS superconductivity.
+Here, we use a simple lattice model to prove that the counterpart of this is also possible, whereby
+phonons overscreen the bare electron-hole attraction and may turn it repulsive at short distances,
+driving exciton dissociation in certain regions of the parameter space. We argue that this phonon-
+mediated short-range screening plays an important role in the physics of organic solar cell materials
+(and other materials with strong electron-phonon coupling) and could point the way to new strategies
+for optimizing their efficiencies.
+I.
+INTRODUCTION
+Organic solar cells (OSCs) have been heralded as a rev-
+olutionary technology in the renewable energy sector due
+to their flexible and light-weight nature and low produc-
+tion cost.1–4 While power conversion efficiencies of OSC
+devices have been improving,5 they have not yet reached
+levels high enough for OSCs to realize their promise; this
+is largely due to the challenge of efficiently extracting free
+charge carriers without detrimental losses.6,7
+All light-harvesting devices start by capturing a pho-
+ton to excite a bound electron-hole pair – an exciton.
+Voltage is ultimately produced through the generation
+of free charge carriers, requiring the dissociation of the
+exciton through some internal mechanism.
+Conventional (inorganic) solar cells, such as those
+based on Si or GaAs, have highly effective charge screen-
+ing. Because the screened Coulomb attraction is weak,
+the Wannier excitons it creates are highly extended and
+have small binding energies (few tens of meV). A combi-
+nation of thermal fluctuations and external electric fields
+is therefore sufficient to drive dissociation.
+By contrast, OSC materials have poor charge screen-
+ing, resulting in small Frenkel excitons with large binding
+energies of a hundred meV or more.8,9 These are stable
+against thermal fluctuations and fairly long-lived, lead-
+ing to high recombination losses and reduced efficiencies.
+This is why understanding and engineering exciton dis-
+sociation in OSCs remains a foundational challenge.
+To date, the most investigated approach to engi-
+neering dissociation is to use bulk-heterojunction inter-
+faces combining donor and acceptor materials, chosen so
+that the potential gradient at their interface helps over-
+come the high binding energies. This setup was shown
+to produce higher yields, which was attributed to en-
+hanced dissociation of so-called charge-transfer states at
+the donor/acceptor (D/A) interface.10,11 Charge-transfer
+states are believed to be relatively short-lived excitons
+∗ These authors contributed equally.
+FIG. 1.
+Lattice distortion from an exciton.
+Left panel:
+when the electron and the hole are far apart (red and blue
+circles, respectively) their excess charge induces local lattice
+distortions, giving rise to polarons.
+Right panel: A small
+Frenkel exciton produces a much weaker electric potential and
+thus a much smaller lattice distortion.
+composed of an electron and a hole that span neigh-
+bouring molecular sites. While such excitons are quite
+commonly generated in the bulk,12 they delocalize more
+easily when they span a D/A interface.13–15 However,
+more work is needed to understand both the nature of
+these states, and how they can be engineered to optimize
+exciton dissociation.
+Alongside charge screening, the vibrational character-
+istics (phonon modes) of OSCs are also relevant to disso-
+ciation – and even less well-understood. Phonons couple
+strongly to molecular orbitals, as evidenced by photoe-
+mission experiments,16 and thus may be playing a role
+in the exciton dynamics.17 Most of the studies to date
+have focused on the role of phonons in the formation of
+charge transfer states,18,19 and how electron-phonon cou-
+pling affects the yield across the D/A interface.20–25
+Here we present a fundamentally different way whereby
+electron-phonon coupling can influence exciton dissocia-
+tion, even in the absence of a D/A interface. We show
+that sufficiently strong electron-phonon coupling can be
+directly responsible for exciton dissociation, despite the
+presence of significant Coulomb attraction between the
+electron and the hole.
+The basic idea is sketched out in Fig.
+1, where we
+arXiv:2301.03530v1  [cond-mat.str-el]  9 Jan 2023
+
+2
+compare the e��ects of electron-phonon coupling when the
+hole and electron are far apart (left panel) versus when
+bound in a small exciton (right panel). The addition of
+an excess carrier results in a local lattice distortion that
+dresses that carrier into a polaron. Because the electron
+and the hole have opposite charges, in a polar material
+they create opposite lattice distortions in their vicinity.
+However, when they are bound into a small exciton, their
+clouds essentially cancel each other out, and locally there
+is no distortion. Another way to say this is that there is
+no excess local charge in the presence of a small exciton
+– hence no local lattice distortion is expected.
+In this picture, electron-phonon coupling is seen to
+lower the energy of the dissociated state through polaron
+formation, while having little effect on the exciton bind-
+ing energy. For large enough electron-phonon coupling
+this leads to outright dissociation, as we show next. Even
+when that is not the case, our work shows that one must
+take polaron formation into consideration when choosing
+the donor/acceptor materials, because the polaronic con-
+tribution to the energetic landscape can be considerable.
+It is important to acknowledge that the idea of exci-
+ton dissociation driven by electron-phonon coupling was
+proposed previously by Sumi in Ref. 26, where he used a
+variational approximation to study the effect of Fr¨ohlich
+coupling on an exciton. His prediction of a sharp transi-
+tion between bound (exciton) and unbound (free electron
+and hole polarons) states was later discredited by Ger-
+lach and L¨owen,27 who proved that sharp transitions are
+forbidden in this class of Hamiltonians and concluded
+that overscreening is impossible in this context. We find
+a smooth crossover between the two types of states, fully
+consistent with the mathematical proof of Ref. 27. Our
+work shows that the contradiction between Refs. 26 and
+27 is not because overscreening is impossible, but be-
+cause the predicted sharp transition was an artifact of
+the variational approximation28 used by Sumi.
+The article is organized as follows: Sec. II introduces
+the model we use to study this problem, and Sec. III
+explains our formalism and approach. Key results are
+shown in Sec. IV, while Sec. V contains an extended dis-
+cussion of the various approximations made in the model
+and the relevance of this phenomenology in the context
+of OSCs.
+II.
+THE MODEL
+We consider a single electron-hole pair in a one-
+dimensional (1D) ionic chain, where each site supports a
+single on-site orbital and a dispersionless Einstein phonon
+mode. The single electron-hole pair assumption is reason-
+able if, for example, the concentration of photo-generated
+electron-hole pairs in the material is very low. We focus
+on the 1D chain because here it is known that Coulomb
+attraction always results in the formation of strongly
+bound excitons, unlike in higher dimensions where ex-
+citons can be either exponentially weakly bound (in 2D)
+or unstable unless the attraction is sufficiently strong (in
+3D).29 Thus, demonstrating dissociation in 1D would im-
+ply similar behaviour in higher dimensions, given that the
+exciton is even more loosely bound there.
+Our Hamiltonian reads:
+ˆH = ˆTe + ˆVe−h + ˆHph + ˆVe−ph + ˆVh−ph.
+(1)
+Here, ˆTe = �
+kσ ϵkc†
+kσckσ is the kinetic energy of free
+electrons in the conduction band, described by a tight-
+binding model with a dispersion ϵk = −2t cos k defined
+by the hopping t and momentum k ∈ (−π, π] of the bare
+electron (the lattice constant is set to a = 1, also ℏ = 1).
+The creation operator c†
+kσ adds an electron with momen-
+tum k and spin σ in this band. Its real space counterpart
+is c†
+nσ, where n = 1 . . . N indexes the sites of the chain,
+with N → ∞. Hole creation operators in real space are
+denoted by h†
+nσ. For simplicity, we assume that holes are
+localized (we reflect on this assumption in Sec. V).
+The electron-hole interaction ˆVe−h is modeled as an
+on-site Coulomb attraction
+ˆVe−h = −U
+�
+n,σ,σ′
+h†
+nσhnσc†
+nσ′cnσ′,
+(2)
+characterized by U > 0. Longer (but finite) range attrac-
+tions can be treated similarly and lead to quantitative
+changes only, at the cost of adding more parameters.
+Optical phonons are described with an Einstein model:
+ˆHph = Ω
+�
+n
+b†
+nbn
+where b†
+n creates a phonon with energy Ω at site n.
+Finally, the Holstein carrier-lattice couplings are:
+ˆVe−ph =Me
+�
+nσ
+c†
+nσcnσ(bn + b†
+n)
+(3)
+ˆVh−ph =Mh
+�
+nσ
+h†
+nσhnσ(bn + b†
+n)
+(4)
+with electron/hole-phonon couplings Me and Mh, respec-
+tively.
+Even after all these simplifications, there are four di-
+mensionless parameters: U/t, Ω/t, Me/t, Mh/t. To avoid
+further complications, we set the temperature T = 0.
+This is justified because we are interested in cases where
+all energy scales (including the exciton binding energy)
+are much larger than the thermal energy, as is typically
+the case in organic photovoltaics.
+III.
+METHODS
+Finite Coulomb attraction in 1D always leads to a
+ground-state with a stable, bound exciton. Our aim is to
+investigate the influence of the carrier-phonon couplings
+on the stability of the exciton. To do this, we calculate
+the Green’s function
+Gij(z) ≡ ⟨0| cihi ˆG(z)h†
+ic†
+j |0⟩
+(5)
+
+3
+where we reserve the index i to label the site hosting
+the immobile hole (the spin degree of freedom is irrele-
+vant for this calculation and we ignore them from now).
+The electron can move and the propagator above is the
+Fourier transform (at energy z = ω + iη) of the am-
+plitude of probability that if the hole is at site i, the
+electron moves from site j to site i within a given time
+interval, with both the initial and the final states having
+no phonons bn|0⟩ = 0. The broadening η → 0 introduces
+an artificial lifetime ∝ 1/η for the pair to recombine, and
+ˆG(z) = (z − ˆH)−1 is the resolvent. The associated local
+density of states (LDOS), plotted in the figures, is de-
+fined as A(ω) = −ImGii(z)/π; invariance to translations
+ensures that the LDOS is the same at all sites i.
+The propagator of Eq.
+(5) for the full interacting
+Hamiltonian is calculated using a novel, generalized ver-
+sion of the Momentum Average approximation (MA) – a
+method well established and validated for studying sin-
+gle polarons30–33 and bipolarons.34–36 This generalization
+allows, for the first time, to include into the variational
+space configurations with two phonon clouds located ar-
+bitrarily far apart: a hole cloud at site i, and an electron
+cloud elsewhere in the chain.
+We now briefly describe this method, before moving to
+discuss the results.
+A.
+Non-interacting spectrum
+The first step is to obtain the Green’s function in the
+absence of carrier-phonon coupling (Me = Mh = 0). The
+Green’s function G(i,0)
+ij
+(z) corresponding to ˆH0 = ˆTe +
+ˆVe−h + ˆHph, i.e. for the system without carrier-phonon
+coupling, can be calculated analytically (see Appendix
+A for details). The spectrum extracted from the poles
+of this Green’s function has a discrete eigenstate at ω =
+−
+√
+4t2 + U 2 and a continuum for ω ∈ [−2t, 2t].
+The
+continuum describes the electron unbound to the hole,
+i.e. free to move throughout the system. The discrete
+eigenstate is the energy of the bound exciton, lying below
+this continuum for any value of U > 0. All these features
+would be shifted by nΩ if there were n phonons in the
+system, but for the propagator of interest to us n = 0.
+B.
+Turning on interactions: Lang-Firsov
+transformation
+In the presence of carrier-phonon couplings (finite
+Me, Mh), if the carriers are not bound then they each cre-
+ate phonon clouds in their vicinity, turning into polarons.
+In the bound state their clouds combine, resulting in an
+exciton-polaron.
+Because the hole cannot move in our simplified model,
+and because its coupling to the lattice is local, its phonon
+cloud is definitely located at hole site i.
+We then use
+the Lang-Firsov transformation Ui = exp[ Mf
+Ω (bi − b†
+i)] to
+integrate out the hole-phonon coupling:
+˜Hi = U†
+i ˆHUi = ˆTe −
+�
+U + 2MeMh
+Ω
+�
+c†
+ici − M 2
+h
+Ω +
++ Ω
+�
+l
+b†
+l bl + Me
+�
+l
+c†
+l cl(b†
+l + bl)
+(6)
+after noting that U†
+i blUi = bl − δi,l
+Mh
+Ω . This transforma-
+tion is exact and shows the hole-polaron formation en-
+ergy −M 2
+h/Ω but also a change of the effective Coulomb
+attraction experienced by the electron when at site i,
+U → ˜U = U+ 2MeMh
+Ω
+, arising from the electron’s coupling
+to the hole’s cloud in addition to the Coulomb interac-
+tion with the hole. The propagator for the electron-hole
+pair
+Gij(z) ≡ ⟨0| cihi ˆG(z)h†
+ic†
+j |0⟩
+(7)
+is then rewritten in terms of the transformed Hamilto-
+nian:
+Gij(z) = e−
+M2
+h
+2Ω2
+∞
+�
+n=0
+1
+n!
+�Mh
+Ω
+�n
+Hij(n, ˜z)
+(8)
+where the new propagators
+Hij(n, ˜z) = ⟨0| hiciUi ˜G(˜z)h†
+ic†
+jb†n
+i |0⟩
+(9)
+describe the propagation of the electron in the presence
+of phonons created by the hole. To obtain Eq. (8) we
+used the Baker–Campbell–Hausdorff formula to rewrite
+U†
+i |0⟩ = e−M 2
+h/2Ω2 �∞
+n=0
+1
+n!
+�
+b†
+i
+Mh
+Ω
+�n
+|0⟩, and we intro-
+duced ˜z = z + M 2
+h/Ω and the transformed resolvent
+U†
+i ˆG(z)Ui ≡ ˜G(˜z) = (˜z − ˆhi)−1
+where
+ˆhi = ˜Hi + M 2
+h
+Ω
+= ˆTe − ˜Uc†
+ici + ˆHph + ˆVe−ph
+describes the electron’s kinetic energy, effective interac-
+tion with the hole located at i, and coupling to the lattice.
+So far, everything is exact.
+C.
+Analogy to the disorder MA
+Note that ˆhi obtained above is formally equivalent to
+the Hamiltonian for an electron with Holstein coupling
+in the presence of an on-site ‘disorder’ at site i. In pre-
+vious work, we have already demonstrated that for such
+problems, even the simplest version of the variational mo-
+mentum average (MA) approximation, namely the one-
+site MA(0) version, is quantitatively accurate if t/Ω is
+not too large.37,38 We use the same approximation here,
+straightforwardly generalized to include the presence of
+phonons created by the hole at site i. Specifically, we im-
+plement an MA where the variational space allows for the
+
+4
+presence of two phonon clouds: one at site i due primar-
+ily to the hole, and one at any other site of the system,
+created by the electron. We note that the electron cloud
+can be allowed to spread over more sites,39 increasing the
+accuracy of the approximation: however, the resulting
+improvements are quantitatively small and do not affect
+the physics. For our purposes it suffices to proceed with
+the one-site cloud approximation, which predicts energies
+to within an accuracy of a few percent.30,37,38,40
+Proceeding by analogy with the disorder MA calcula-
+tion, the equations-of-motion (EOMs) for the propaga-
+tors in this two-cloud generalization of MA are obtained
+by repeated use of the Dyson identity ˆG = ˆG0 + ˆG ˆV ˆG0
+with ˆV = ˆVe−ph. The resulting system of equations (B1-
+B3) and its derivation are shown for completeness in Ap-
+pendix B. This linear system that emerges turns out to
+be amenable to further simplifications driven by the in-
+tuition that not all propagators contribute equally: in-
+deed, we find that about half the propagators may be
+set to zero (halving the size of the system) with no no-
+ticeable changes to the resulting spectrum. More details
+on this further approximation and the intuition behind it
+are given in C, and in Appendix D we show some results
+that justify the validity of this futher approximation.
+D.
+Exciton wavefunction and the phonon cloud
+Once the Green’s functions Gij are obtained by solv-
+ing the linear system, to further elucidate the nature
+of the ground-state properties of our model we char-
+acterize the spatial extent of the exciton wavefunction,
+as well as calculate the size of its phonon cloud.
+To
+obtain the former, we use the Lehmann decomposition
+Gij(z) = �
+n ⟨0|hici|ψn⟩ ⟨ψn|c†
+jh†
+i|0⟩/(z − En), where
+ˆH|ψn⟩ = En|ψn⟩ are the eigenstates with one electron
+and one hole.
+At the exciton energy E0, and if η is
+much smaller than the gap to the continuum, there is
+only one dominant contribution to the Lehmann sum:
+Gij(z = E0 + iη) ≈ ⟨0|hici|ψ0⟩ ⟨ψ0|c†
+jh†
+i|0⟩/iη. Therefore
+we can use
+ρij(E0) = |⟨0|hicj|ψ0⟩|2
+|⟨0|hici|ψ0⟩|2 ≈ |Gij(E0)|2
+|Gii(E0)|2
+(10)
+to characterize the probability that the electron is at a
+distance |j −i| from the hole in the exciton ground-state,
+scaled such that ρii(E0) = 1.
+To calculate the average number of phonons Nph
+in the exciton cloud, we use the Hellmann-Feynman
+theorem:41,42
+Nph = ⟨ψ0|
+�
+l
+b†
+l bl|ψ0⟩ = ∂E0
+∂Ω .
+(11)
+The derivative is computed numerically with the finite-
+difference approach.
+Both of these metrics give addi-
+tional glimpses at the impact of phonons on the dissoci-
+ation process.
+IV.
+RESULTS
+A.
+Exciton dissociation driven by electron-phonon
+coupling
+Armed with the methods from the previous section,
+we calculate the spectrum of a system with one electron
+and one hole, in the presence of short range (on-site)
+Coulomb attraction of magnitude U > 0, and of Hol-
+stein carrier-phonon couplings Me and Mh, respectively,
+to an optical dispersionless phonon mode of energy Ω.
+As stated previously, we focus on 1D chains, where the
+carriers’ tendency to bind into an exciton is enhanced.
+The electron’s nearest neighbor hopping is t = 1; mean-
+while the hole is localized, modeling either a valence band
+with a very large effective mass or a hole trapped by an
+acceptor impurity.
+Exciton dissociation driven by the electron-phonon
+coupling is demonstrated graphically in Fig.
+2.
+The
+panels show the contour plot of the LDOS A(ω) at the
+hole site versus energy and coupling Me, when U = 1,
+Ω = 0.5 and Me = −Mh (panel a); Me = −0.5Mh (panel
+b); Me = −2Mh (panel c); and Me = Mh (panel d).
+At Me = Mh = 0, the lowest energy feature in the
+electron+hole spectrum is a discrete peak marking the
+existence of the exciton, just as discussed in Sec. III A. If
+MeMh < 0, the discrete peak merges smoothly with the
+continuum at M (c)
+e
+and the exciton dissociates into un-
+bound electron- and hole-polarons for Me > M (c)
+e . There
+is no discontinuity in the LDOS at M (c)
+e : thus, there is
+no contradiction between our result and Ref.
+27.
+By
+contrast, if MeMh > 0 (panel d), the exciton is further
+stabilized by increasing coupling.29
+The carrier-phonon coupling M is set by the gradient
+of the carrier-lattice potential with respect to a small
+lattice displacement. Because the hole and the electron
+have opposite charge, their respective carrier-lattice po-
+tentials have opposite signs and thus Me and Mh have
+opposite signs. Physically, this is because a lattice dis-
+tortion that is energetically favorable for an electron is
+generically unfavorable for a hole (left panel of Fig. 1).
+Moreover, a very small Frenkel exciton, with the electron
+and hole at the same site, creates no local charge imbal-
+ance so no lattice distortion is expected (right panel of
+Fig. 1). In the atomic limit (t = 0), a vanishing exciton-
+polaron binding energy −(Me+Mh)2/Ω ≈ 0 implies that
+Me ≈ −Mh. Of course, one can envision more complex
+situations where |Me| ̸= |Mh|, however panels (b) and (c)
+of Fig. 2 show the same dissociation phenomenology for
+different ratios Mh/Me < 0, demonstrating that exciton
+dissociation does not require fine-tuning: it is guaranteed
+to happen at large enough couplings. On the other hand,
+the exciton is always stable if Mh/Me > 0 (see panel (d)
+of Fig. 2), because in this case the cloud created by the
+exciton is larger than the sum of the individual clouds
+created by the two unbound carriers, further stabilizing
+the exciton.27,29
+
+5
+FIG. 2. Contour plots of the LDOS A(ω) at the hole site when Ω = 0.5 and U = 1. The electron-phonon coupling Me is shown
+on the x axis (the corresponding Mh is indicated on the figure). The dashed red line shows where we expect the lower edge of the
+continuum of eigenstates describing unbound electron- and hole-polarons, based on their individually calculated MA energies.
+Its good agreement with the calculated spectral weight provides a validation of the generalized MA we developed. The fast
+oscillations in the continuum weight are finite size effects, due to the cutoff |l − i|m = 50 for the maximum distance between
+the two clouds; the maximum numbers of phonons in the two clouds are set to nm = km = 20, sufficient for convergence. The
+discrete peak appearing below the continuum at small Me is the exciton bound state, broadened into a Lorentzian by the finite
+η = 0.01. With increasing coupling, the exciton approaches the continuum and eventually merges smoothly with it, marking
+its dissociation into a pair of unbound electron and hole polarons. This behaviour is robust so long as the couplings are of
+opposite sign, so that MeMh < 0, see panels (a)-(c). In contrast, when MeMh > 0, the exciton is always stable, see panel (d).
+B.
+Exciton dissociation phase diagram
+Figure 3 traces the crossover (blue line) between the
+ground-states with an exciton-polaron and those with
+unbound electron- and hole-polarons. The dashed line
+shows the perturbation theory prediction (details in Ap-
+pendix E). The agreement is excellent at small U, as
+expected, while at larger U perturbation theory overes-
+timates the critical coupling needed for dissociation.
+C.
+Exciton-polaron characteristics
+Next, we calculate the average number of phonons Nph
+in the exciton cloud, and also the probability ρij that the
+electron is at a distance |j−i| from the hole in the exciton
+ground-state, scaled such that ρii = 1 (see Sec. III D for
+details).
+Representative results are shown in Fig. 4. For com-
+pleteness, panel (a) shows the LDOS versus ω and Me,
+with dissociation occurring slightly above Me = 0.6.
+Panel (b) shows Nph of the exciton-polaron (solid yel-
+low line), compared to the sum of the ground-state av-
+erage numbers of phonons in the electron-polaron and
+the hole-polaron clouds (red dashed line); the latter are
+calculated individually and then summed. As expected,
+when tightly bound by an attractive U, the electron and
+the hole largely cancel each other’s lattice distortions, re-
+sulting in many fewer phonons than for the free polarons.
+Panels (c)-(e) show ρij vs. j − i for Me = 0.4, 0.5, 0.6,
+respectively (see vertical dotted lines in panels (a) and
+(b)). At small couplings, ρij is sharply peaked at the hole
+
+-1.0
+1.0
+(a)
+Me= Mh
+0.8
+-1.5
+tanh(Aoo(w)
+0.6
+-2.0
+0.4
+-2.5
+0.2
+-3.0,
+0.0
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+Me-1.0
+1.0
+(b)
+Me = - 0.5Mh
+0.8
+-1.5
+tanh(Aoo(w)
+0.6
+-2.0
+0.4
+-2.5
+0.2
+-3.0,
+0.0
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+Me-1.0
+1.0
+(c)
+ 2Mh
+0.8
+-1.5
+tanh(Aoo(w)
+0.6
+1/m
+-2.0
+0.4
+-2.5
+0.2
+-3.0,
+0.0
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+Me-1.0
+1.0
+(d)
+Me
+Mr
+0.8
+-1.5
+tanh(Aoo(w)
+0.6
+-2.0
+0.4
+-2.5
+0.2
+-3.0,
+0.0
+0
+0.2
+0.4
+0.6
+0.8
+1.0
+Me6
+FIG. 3.
+Exciton dissociation phase diagram.
+The critical
+electron-phonon coupling for dissociation increases with the
+Coulomb attraction U: it is calculated with MA (blue solid)
+and with perturbation theory (orange dashed). The orange
+region above the critical line indicates the region where we ex-
+pect dissociated electron and hole polarons, whereas the blue
+region below the line represents the bound exciton-polaron
+region. Other parameters are Ω = 0.5, Me = −Mh.
+site i, as expected for a strongly bound, small Frenkel ex-
+citon. As the coupling increases, ρij acquires “fat tails”,
+that are consistent with a larger exciton.
+Just before
+dissociation, ρij spreads over very many sites, consis-
+tent with the smooth crossover to an unbound electron-
+polaron that is (nearly) equally likely to be at any dis-
+tance from the hole.
+V.
+CONCLUSIONS
+We have shown that strong carrier-phonon coupling
+favors the dissociation of excitons into free polarons,
+even on 1D chains where excitons should be stable for
+any electron-hole attraction.
+This phenomenology is
+the counterpart to what drives BCS superconductivity.43
+There, phonons overscreen the electron-electron repul-
+sion turning it into an effective attraction. Here, phonons
+screen the electron-hole attraction and can turn it repul-
+sive, at sufficiently strong coupling.
+This phenomenology is robust and should be consid-
+ered when analyzing exciton stability in materials with
+carrier-phonon coupling because the critical coupling for
+dissociation need not be very large.
+Figure 4 shows a
+critical value Me = −Mh ≈ 0.6, which corresponds to a
+weak effective Holstein coupling λc = M 2
+e /2tΩ ≈ 0.36 for
+the electron, even though the bare exciton binding energy
+is a considerable 0.5t for those parameters. Indeed, panel
+(b) of Fig. 4 confirms that the average phonon numbers
+are small. Of course, to some extent this is because of
+the rather large phonon frequency Ω = 0.5t used there,
+although such ratios are reasonable in some organic ma-
+terials.
+Regarding the main approximations in our model:
+FIG. 4. Characterization of the phonon cloud of the exciton-
+polaron. a) Contour plot of the LDOS at the hole site, as a
+function of the coupling Me and energy ω. The yellow solid
+line tracks the exciton energy while the dashed red line tracks
+the lower edge of the continuum; their intersection marks
+the dissociation point.
+We track the exciton energy up to
+Me = 0.6, where its binding energy becomes comparable to η.
+b) Average number of phonons Nph in the exciton cloud (solid
+yellow line) and in the combined electron- and hole-polaron
+clouds (red dashed line). c)-e) Probability ρij that the elec-
+tron is at a distance |j−i| from the hole in the exciton ground-
+state, scaled such that ρii = 1, for Me = 0.4, 0.5, 0.6, respec-
+tively. Other parameters are Ω = 0.5, U = 1.5, Me = −Mh,
+η = 0.01, nm = km = 20, |l − i|m = 50.
+(i) we do not expect different dimensionality to change
+this phenomenology.
+In 3D, a bare exciton is stable
+only if the Coulomb attraction is above a critical value.29
+Whether the critical value is 0 (like in 1D) or finite (like
+in 3D) is irrelevant: strong enough carrier-phonon cou-
+pling will lower the effective attraction below this critical
+value and make the exciton unstable. Our MA method
+can be straightforwardly used to study higher-D systems.
+(ii) the assumption that the hole is immobile is also
+not essential: ‘releasing’ the hole does not change this
+picture qualitatively, only quantitatively. Moreover, in
+
+0.6
+unbound electron
+0.5
++ hole polarons
+0.4
+M 0.3
+0.2
+0.1
+exciton polaron
+0.2
+0.0
+0.8
+1.0
+0.4
+0.6
+U1.0
+-2.0
+exciton
+-2.2
+polarons
+0.8
+-2.4
+tanh(Aoo(k, w))
+0.6
+-2.6
+3
+-2.8
+0.4
+-3.0
+-3.2
+0.2
+-3.4
+0.0
+0
+0.2
+0.4
+0.6
+0.8
+Mew
+..............................
+exciton
+polarons
+2
+................
+1
+..
+0
+0.2
+0.4
+0.6
+0.8
+Me7
+FIG. 5. Schematic of the effective potential for the exciton-
+polaron. Screened electron-hole interaction (black line), ob-
+tained by summing the bare long-range Coulomb attraction
+(dashed line) and the contribution from phonon screening
+(blue line). Top: when the coupling is weak, the combined
+polaron radius D is large and the screening is weak. Mid-
+dle: strong coupling leads to small polarons with a strong
+short-range repulsion. The total potential has a minimum at
+r ∼ D. Bottom: for a rapidly-decreasing bare attraction, a
+metastable exciton may be trapped at the r = 0 local mini-
+mum, before tunneling into a dissociated state.
+the context of OSC materials doped with either acceptor
+or donor molecules, it is possible to envision trapping one
+species of the carriers on such molecules.
+(iii) the assumptions that the coupling is to a single
+optical mode and that it is of Holstein type are also not
+essential. Regardless of such details, polaron formation
+associated with local excess charge leads to a lowering of
+the energy. That is the only ingredient necessary for the
+mechanism discussed here.
+(iv) The assumption of a short-range Coulomb attrac-
+tion is non-trivial, however, and relaxing it can lead to
+qualitative changes. This is because the phonon screen-
+ing discussed here acts only at electron-hole distances
+r < D, where D is the sum of the radii of the two po-
+larons. If the electron and hole are sufficiently far, so that
+each can create its polaron cloud (r > D), the phonon
+screening vanishes. This contribution looks roughly like
+the blue lines in Fig.
+5, where ∆EB ≈ −2MeMh/Ω
+is the difference between the exciton-polaron and the
+free polarons formation energies. While ∆EB increases
+with increasing coupling, D decreases as the polarons be-
+come smaller. If the Coulomb attraction decreases rather
+slowly with r (dashed line), it is possible that as the
+coupling goes from weak (top panel) to strong (middle
+panel), the total potential has a well whose minimum
+moves from r ∼ 0 to r ∼ D. The latter well can still trap
+a stable exciton in 1D, because both the lower dimension-
+ality and the increased effective mass of strongly-coupled
+Holstein polarons would favor a bound state.
+We be-
+lieve that this explains why exciton dissociation was not
+observed in Ref. 44. However, in higher dimensions rele-
+vant for OSCs and/or for lighter Peierls polarons,33 such
+a ‘donut’-shaped trap might not suffice to bind the po-
+larons and the ground-state at strong coupling would still
+exhibit dissociation.
+A new scenario can occur if the bare Coulomb attrac-
+tion decreases significantly from r = 0 to r = D.
+As
+sketched in Fig.
+5(c), r = 0 can be a local minimum
+of the total potential (black line) followed by a potential
+barrier and a very shallow potential well for r > D. A
+Frenkel exciton with radius smaller than D can then be
+metastable, with a lifetime inversely proportional to the
+probability of tunneling through the barrier.
+Even though the ground state is the dissociated state,
+small excitons loaded optically into the metastable state
+might live long enough to control the OSC’s behavior.
+This may explain the very puzzling fact that some OSC
+materials, like pure C60 films, have both very strongly
+bound excitons45,46 and finite, albeit small, charge sepa-
+ration efficiency.47 The latter would represent the small
+fraction of excitons that tunnel out and dissociate. This
+scenario is also qualitatively consistent with the ob-
+servation that a dilute (∼10%) concentration of donor
+molecules increases the charge separation efficiency. Such
+molecules boost light absorption, so the metastable exci-
+ton state is populated more efficiently. This will increase
+the concentration of charge-separated pairs accordingly
+if the donor molecules are dilute enough to allow charge
+separation to proceed, explaining why peak efficiency oc-
+curs at a very low donor molecules concentration.47 The
+above scenario cannot be verified with MA; however, a
+recent study found a weak potential barrier due to nonlo-
+cal phonon screening in lead halide perovskites.48 While
+their parameters are very different than ours, their find-
+ing supports the possible appearance of this new scenario
+in the right circumstances.
+The results presented in this work illustrate some of
+the interesting physics expected in the many OSCs that
+have strong carrier-phonon coupling, and point towards
+possible ways to exploit it. We plan to investigate some
+of these topics in more detail in future works.
+
+Vtot(R)
+△EB
+D
+R
+U
+Vtot(R)
+△EB
+D
+-U
+Vtot(R)
+AEB
+>R
+-U8
+ACKNOWLEDGMENTS
+We thank Sarah Burke for bringing this problem to
+our attention and for many useful discussions. We thank
+David Reichman, Holger Fehske and Krzysztof Bieniasz
+for insightful comments. We acknowledge support from
+the Max Planck-UBC-UTokyo Centre for Quantum Ma-
+terials and the Canada First Research Excellence Fund,
+Quantum Materials and Future Technologies Program of
+the Stewart Blusson Quantum Matter Institute, and from
+the Natural Sciences and Engineering Research Council
+of Canada (NSERC). We gratefully acknowledge the use
+of computing resources from the Stewart Blusson Quan-
+tum Matter Institute computing cluster LISA.
+Appendix A: Free carrier Green’s function
+The Green’s function G(i,0)
+ij
+(z) corresponding to ˆH0 =
+ˆTe + ˆVe−h + ˆHph, i.e.
+for the system without carrier-
+phonon coupling, can be calculated analytically. In the
+absence of electron-phonon coupling there is only an elec-
+tron hopping on a 1D tight-binding lattice, subject to an
+on-site attractive potential from the static hole located
+at i. The corresponding Hamiltonian is H0 = T − Uc†
+ici.
+Here we calculate its lattice Green’s function:
+G(i,0)
+lj
+(z) = ⟨0| cl[z − H0]−1c†
+j |0⟩
+(A1)
+Applying Dyson’s identity, we find the EOM:
+G(i,0)
+l,j (z) = gl−j(z) − Ugi−j(z)G(i,0)
+l,i
+(z)
+(A2)
+where the free lattice Green’s functions gl−j(z)
+=
+⟨0| cl[z − T]−1c†
+j |0⟩
+can
+be
+calculated
+analytically:
+gδ(z) = |ζ(z)||δ|/√z − 2t√z + 2t, with ζ(z) = z/2t −
+�
+z/2t − 1
+�
+z/2t + 1.
+Equation (A2) can be solved trivially to find:
+G(i,0)
+li
+(z) = G(i,0)
+l−i (z) =
+gl−i(z)
+1 + Ug0(z)
+(A3)
+and
+G(i,0)
+ll
+(z) = g0(z) − U [gi−l(z)]2
+1 + Ug0(z).
+(A4)
+The propagators ˜G(i,0)
+il
+(z) appearing in the main text
+and in other appendices have the same expressions but
+with U → ˜U, where ˜U is the overscreened Coulomb at-
+traction defined in Sec. III.
+Appendix B: Green’s function with carrier-lattice
+coupling
+Here the MA equations of motion are obtained by re-
+peated application of the Dyson identity ˜G(˜z) = ˆG(i)
+0 (˜z)+
+˜G(˜z) ˆVe−ph ˆG(i)
+0 (˜z) where ˆG(i)
+0 (z) = (z − ˆTe + ˜Uc†
+ici)−1
+is the resolvent in the absence of electron-phonon cou-
+pling. We note that its corresponding Green’s functions
+˜G(i,0)
+ij
+(z) = ⟨0|ci ˆG(i)
+0 (z)c†
+j|0⟩ equal those calculated in
+Sec. A upon replacing U → ˜U.
+Using Dyson���s identity once, we find:
+Hij(n, ˜z) = ˜G(i,0)
+ij
+(˜z − nΩ)
+�Mh
+Ω
+�n
+e−M 2
+h/2Ω2+
++ ˜G(i,0)
+ij
+(˜z − nΩ)Me [nHii(n − 1, ˜z) + Hii(n + 1, ˜z)] +
++
+�
+l̸=i
+˜G(i,0)
+lj
+(˜z − nΩ)MeFill(n, 1, ˜z).
+(B1)
+Here, the terms on the 2nd line arise when the electron
+travels to site i and adds to or removes from the phonons
+already present there, while the last line describes terms
+where the electron moves to some other site l and starts
+a new cloud there, with the corresponding generalized
+two-cloud propagator:
+Fijl(n, k, ˜z) ≡ ⟨0| cihiUi ˜G(˜z)h†
+ic†
+j(b†
+i)n(b†
+l )k |0⟩ .
+(B2)
+The equation of motion (B1) is exact. Solving it neces-
+sitates calculating the propagators Fill that appear in it.
+We generate their equations of motion using again the
+Dyson identity, but now also imposing the variational
+constraint consistent with the one-site MA(0) approxi-
+mation for the electron cloud, namely that additional
+phonons cannot be created away from the two existing
+clouds. The resulting EOMs are
+Fill(n, k, ˜z) =Me ˜G(i,0)
+ll
+(˜z − (n + k)Ω) [kFill(n, k − 1, ˜z) + Fill(n, k + 1, ˜z)]
++ Me ˜G(i,0)
+il
+(˜z − (n + k)Ω) [kFiil(n, k − 1, ˜z) + Fiil(n, k + 1, ˜z)]
+Fiil(n, k, ˜z) =Me ˜G(i,0)
+ii
+(˜z − (n + k)Ω) [kFiil(n, k − 1, ˜z) + Fiil(n, k + 1, ˜z)]
++ Me ˜G(i,0)
+il
+(˜z − (n + k)Ω) [kFiil(n, k − 1, ˜z) + Fiil(n, k + 1, ˜z)] .
+(B3)
+Eqs. (B1-B3) define a linear, inhomogeneous system
+of coupled equations that can be numerically solved for
+
+9
+each value of z, with the resulting Hij(n, ˜z) then used in
+Eq. (8) to construct Gij(z). However, this approach is
+computationally intensive because one needs large cut-
+offs for the maximum numbers km, nm of phonons in the
+two clouds, as well as for the maximum distance |l − i|m
+between the clouds, before convergence is reached. An
+improved approach is discussed in Appendix C.
+Appendix C: Simplifying the EOMs
+A much more efficient yet still accurate solution to Eqs.
+(B1-B3) can be obtained by taking advantage of the fact
+that for the energies of interest, which lie below the free
+electron continuum, the free propagators ˜G(i,0)
+il
+(z) de-
+crease exponentially with the distance |l − i|. If we keep
+only the largest term with l = i, then Eqs. (B3) split into
+two uncoupled recurrence relations, one for Fill and one
+for Fiil, with only the former needed in Eq. (B1). This
+former recurrence relation can be solved with the ansatz:
+Fill(n, k, ˜z) = A(i,l)
+k
+(˜z − nΩ)Fill(n, k − 1, ˜z)
+(C1)
+where we note that Fill(n, 0, ˜z) ≡ Hil(n, ˜z). The contin-
+ued fractions
+A(i,l)
+k
+(z) =
+kMe ˜G(i,0)
+ll
+(z − kΩ)
+1 − Me ˜G(i,0)
+ll
+(z − kΩ)A(i,l)
+k+1(z)
+(C2)
+are calculated starting from A(i,l)
+km+1(z) = 0 for a suffi-
+ciently large km to ensure the desired accuracy. In par-
+ticular, this means that we can replace Fill(n, 1, ˜z) =
+A(i,l)
+1
+(˜z − nΩ)Hil(n, ˜z) in Eq. (B1) to convert it into a
+linear system linking only the Hij propagators. This still
+requires a summation over all the sites in the system,
+which in practice means summing over sites l up to a
+distance large enough from i that the sum converges.
+An efficient solution of such a linear system was pro-
+posed in Refs. 37 and 38 and we adopt it here. It is
+based on the observation that for |l − i| ≫ 1, the local
+potential ˜U created by the hole becomes irrelevant and
+the impurity Green’s function reduces to the free electron
+propagator
+˜G(i,0)
+ll
+(˜z) → g0(˜z) = 1
+N
+�
+k
+1
+˜z − ϵk
+=
+1
+√˜z − 2t√˜z + 2t.
+(C3)
+As a result, for |l − i| ≫ 1, the continued fractions ap-
+proach an asymptotic value that becomes independent
+of i, l: A(i,l)
+1
+(˜z − nΩ) → ΣMA(˜z − nΩ)/Me . Physically,
+ΣMA(z) is the MA(0) self-energy of the electron-polaron
+in the absence of the ‘impurity’ potential created by the
+hole located at i (see Ref. 30 for a derivation)
+ΣMA(z) =
+M 2
+e g0(z − Ω)
+1 −
+2M 2
+e g0(z − Ω)g0(z − 2Ω)
+1 − 3M 2
+e g0(z − 2Ω)g0(z − 3Ω)
+1 − . . .
+(C4)
+Because this asymptotic value is independent of l, we
+can define a renormalized energy
+vil(˜z − nΩ) = MeA(i,l)
+1
+(˜z − nΩ) − ΣMA(˜z − nΩ)
+(C5)
+which vanishes fast with increasing |l − i|. The sum in
+Eq. (B1) can be recast in terms of it by renormalizing
+the energy argument of the free propagators:
+Hij(n, ˜z) = ˜G(i,0)
+ij
+(˜˜zn)
+�Mh
+Ω
+�n
+e−
+M2
+h
+2Ω2 + ˜G(i,0)
+ij
+(˜˜zn)Me [nHii(n − 1, ˜z) + Hii(n + 1, ˜z)]
++
+�
+l̸=i
+˜G(i,0)
+lj
+(˜˜zn)vil(˜z − nΩ)Hil(n, ˜z)
+(C6)
+where we defined ˜˜zn ≡ ˜z −nΩ−ΣMA(˜z −nΩ). Equations
+(C6) converge much more quickly with the summation
+over l and can be solved efficiently.
+The accuracy of the approximation of replacing the
+coupled Eqs. (B1-B3) with the much more compact and
+efficienct Eq. (C6) is validated in Appendix D.
+Appendix D: Full vs approximate variational
+solutions
+The full variational solution of the particle+hole prop-
+agator can be obtained by simultaneously solving Eqs.
+(8), (B1) and (B3). They can be solved numerically, but
+this is slow because exceedingly large truncation cutoffs
+(system sizes) are required for convergence.
+Above in
+Appendix C, we proposed a much more efficient approx-
+imation which replaces Eqs. (B1)-(B3) with Eqs. (C6).
+To validate this approximation, in Fig. 6 we show a
+typical comparison of the results of the two methods for
+the LDOS at the hole site, focusing on the lower-energy
+part of the spectrum, of interest for the dissociation issue.
+Evidently, the agreement is very good. Similar diagrams
+were produced in all parameter regimes explored in this
+paper, thus effectively validating our approximation.
+
+10
+FIG. 6. Comparison of the LDOS at the hole site from solving
+the full variational solution described by Eqs. (8),(B1),(B3),
+shown in the left panel, versus the simplified and much more
+efficient Eq. (C6), shown in the right panel. Visually, the
+two are nearly indistinguishable, with most differences coming
+in at higher energies.
+Model parameters are U = 1, Ω =
+0.5, η = 0.01, Me = −Mh and convergence parameters are
+nm = km = 12, |l − i|m = 50.
+Appendix E: Perturbation theory for exciton
+dissociation
+Here we summarize the perturbation theory (PT) cal-
+culation used to draw the dissociation line in Fig.
+3
+in the main text. We begin by estimating the ground-
+state energies for the individual polarons.
+The result
+for the (static) hole-polaron is Eh
+P = −M 2
+h/Ω. To find
+the electron-polaron’s PT counterpart, we use the sin-
+gle polaron Green’s function at the same one-site MA(0)
+level of approximation:30 G(k, z) = [z − ϵk − ΣMA(z)]−1
+where the full expression for ΣMA(z) is shown in Eq.
+(C4).
+To lowest non-trivial order in PT, it becomes
+ΣMA ≈ M 2
+e g0(ω − Ω). Using this expression to find the
+lowest k = 0 pole, we find the polaron ground-state en-
+ergy to be:
+Ee
+P (k = 0) = −2t −
+M 2
+e
+�
+Ω(Ω + 4t)
+(E1)
+The PT-predicted lower edge of the continuum is then at
+Emin = Ee
+P (k = 0) + Eh
+P .
+To find the bound exciton energy, we proceed similarly,
+essentially solving the EOMs to lowest order in the cou-
+plings, and then finding the location of the lowest peak
+for k = 0.
+For simplicity, we only list here the result
+when Me = −Mh. We find the exciton ground-state en-
+ergy to be given by Eexc = z0 + αg0(z0 − Ω)M 2
+e where
+z0 = −
+√
+U 2 + 4t2 is the bare exciton energy, and
+α =
+4G(i,0)
+ii
+(z0)[g0(z0 − Ω) − 2G(i,0)
+ii
+(z0 − Ω)]
+1 + 2G(i,0)
+ii
+(z0)[g0(z0 − Ω) − 2G(i,0)
+ii
+(z0 − Ω)]
+The dissociation occurs when Eexc = Emin.
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+

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+A-posteriori QMC-FEM error estimation
+for Bayesian inversion and optimal control
+with entropic risk measure
+Marcello Longo∗, Christoph Schwab∗, Andreas Stein∗†
+January 10, 2023
+Abstract
+We propose a novel a-posteriori error estimation technique where the
+target quantities of interest are ratios of high-dimensional integrals, as occur
+e.g. in PDE constrained Bayesian inversion and PDE constrained optimal
+control subject to an entropic risk measure. We consider in particular
+parametric, elliptic PDEs with affine-parametric diffusion coefficient, on
+high-dimensional parameter spaces. We combine our recent a-posteriori
+Quasi-Monte Carlo (QMC) error analysis, with Finite Element a-posteriori
+error estimation. The proposed approach yields a computable a-posteriori
+estimator which is reliable, up to higher order terms. The estimator’s
+reliability is uniform with respect to the PDE discretization, and robust
+with respect to the parametric dimension of the uncertain PDE input.
+1
+Introduction
+The efficient numerical approximation of high-dimensional, parametric partial
+differential equations (PDEs for short) received increasing attention during the
+past years. In this work, we address two classes of high-dimensional numerical
+integration problems which arise in connection with data assimilation and PDE
+constrained optimization. We illustrate the abstract concepts for a parametric,
+linear elliptic PDE. The first class is the so-called Bayesian inverse problem (BIP).
+There, we are interested in the posterior expectation of a (linear) functional of
+the solution u of a parametric PDE, conditional on observation data subject to
+additive, centered Gaussian observation noise [18,19]. See also [10]. A second
+problem class is PDE-constrained optimization. Specifically, the optimal control
+problem (OCP) of parametric PDEs under an entropic risk measure [7], where
+the state variable satisfies a parametric PDE constraint [9].
+When numerically approximating solutions of a BIP or an OCP, it is essential
+to quantify the error due to the numerical discretization in order to meet a
+prescribed numerical tolerance without wasting computational resources. As
+solving PDEs exactly is in general not possible, discretizations such as Finite
+Element Methods (FEM) must be used instead. Additionally, the parametric
+∗Seminar for Applied Mathematics, ETH Zürich
+†Corresponding author. Email: [email protected]
+1
+arXiv:2301.03327v1  [math.NA]  9 Jan 2023
+
+uncertainty in the forward PDE model is passed on to the solution, and this
+must be taken into account both in the computation and in the error estimation.
+This justifies the need for an a-posteriori error analysis.
+Assuming that the uncertain PDE coefficients can be described by means of a
+parameter vector y ∈ U :=
+�
+− 1
+2, 1
+2
+�s where s ∈ N, s ≫ 1, e.g. (6) below, we can
+employ suitable quasi-Monte Carlo (QMC) rules to approximate integrals over U.
+Here, we select extrapolated polynomial lattice (EPL) rules as first introduced
+in [3,5]. This choice is motivated by the deterministic nature or their quadrature
+nodes Pm, |Pm| = bm for some prime b [15], and good convergence properties
+under quantified parametric regularity of the integrand functions with respect
+to y ∈ U, uniformly in the dimension s [3]. Moreover, it was shown in [5,14]
+that under assumptions, EPL quadratures allow for computable a-posteriori
+quadrature error estimators that are asymptotically exact as m → ∞, with
+dimension robust ratio between estimated and actual quadrature error.
+Both, BIP and OCP problems for PDEs with parametric input take the form
+Z′
+Z ∈ Y,
+where Z =
+�
+U
+Θ(y) dy,
+Z′ =
+�
+U
+Θ′(y) dy,
+(1)
+for some suitable integrable functions Θ: U → R and Θ′ : U → Y, where Y
+is a separable Hilbert space and Z, Z′ are Bochner integrals with respect to a
+product measure dy on the possibly high-dimensional parameter space U. In
+particular, we have Y = R for the BIP case and Y ∈ L2(D) for the OCP case,
+with D being the physical domain of the considered PDE. Approximating the
+high-dimensional integrals with averages over polynomial lattices Pm ⊂ U yields
+a first approximation
+Z′
+m
+Zm
+∈ Y,
+where Zm = 1
+bm
+�
+y∈Pm
+Θ(y),
+Z′
+m = 1
+bm
+�
+y∈Pm
+Θ′(y).
+(2)
+Then, since the integrands Θ, Θ′ depend on the solution of a y-parametric
+PDE, we discretize the parametric PDEs for y ∈ Pm, resulting in computable,
+parametric integrand functions Θh(y), Θ′
+h(y) and in the computable estimates
+Z′
+m,h
+Zm,h
+∈ Y,
+where Zm,h = 1
+bm
+�
+y∈Pm
+Θh(y),
+Z′
+m,h = 1
+bm
+�
+y∈Pm
+Θ′
+h(y).
+(3)
+Here, the parameter h > 0 denotes the meshwidth of conforming Lagrangian
+Finite Element discretizations. We present a computable a-posteriori estimator
+for the combined Finite Element discretization and quadrature error
+err =
+����
+Z′
+Z −
+Z′
+m,h
+Zm,h
+����
+Y
+.
+(4)
+In the rest of this section we introduce the setting and we describe the two
+problems of interest, namely the BIP and the OCP with entropic risk measure.
+Section 2 and Section 3 are devoted to the QMC and the FEM a-posteriori
+error analysis, respectively, and these results will be combined in Section 4. We
+present numerical experiments in Section 5 and summary and conclusions in
+Section 6.
+2
+
+1.1
+Affine-Parametric Forward PDE
+For brevity of presentation, we consider a model, linear elliptic PDE with
+homogeneous Dirichlet boundary conditions. Given a bounded polygon D ⊆ R2
+and a parameter sequence y ∈ U, s ∈ N, consider the following parametric,
+linear, second order elliptic PDE in variational form: find u(·, y) ∈ X = H1
+0(D)
+such that
+�
+D
+a(x, y)∇xu(x, y) · ∇xv(x) dx =
+�
+D
+f(x)v(x) dx
+∀v ∈ X.
+(5)
+We assume that a is affine-parametric, namely that we are given a family of
+functions {ψj}j∈N0 ⊆ L∞(D) such that, with essinf ψ0 > κ > 0 and bj :=
+1
+κ ∥ψj∥L∞(D), we have
+a(x, y) = ψ0(x) +
+s
+�
+j=1
+yjψj,
+�
+j≥1
+bj < 2.
+(6)
+Then essinf a(·, y) > essinf ψ0 − κ > 0 for all y ∈ U. By the Lax-Milgram
+lemma, the parametric weak solution u(·, y) ∈ X is well defined for any f ∈
+X ∗ = H−1(D), where ∗ denotes the topological dual. To justify the a-posteriori
+QMC error analysis of Section 2, we will additionally require the summability
+b = (bj)j≥1 ∈ ℓp(N)
+for some p ∈ (0, 1/2].
+(7)
+For the FEM approximation, we consider conforming subspaces1 Xh ⊆ X h ∈
+H ⊆ (0, ∞), dim(Xh) < ∞, that are linked to shape-regular, simplicial partitions
+{Th}h∈H of D [6, Section 8]. Assume that the resulting spaces are nested and
+conforming, that is Xh ⊆ Xh′ for any h, h′ ∈ H, h > h′ and that H accumulates
+at 0. We construct the Galerkin discretizations uh(y) ∈ Xh of (5), by solving
+�
+D
+a(x, y)∇xuh(x, y) · ∇xv(x) dx =
+�
+D
+f(x)v(x) dx
+∀v ∈ Xh.
+(8)
+To simplify notation, we write a(y) = a(·, y) and u(y) = u(·, y) and we omit the
+variable x for ∇x = ∇ and divx = div.
+1.2
+Bayesian inverse problem (BIP)
+Let X = {a ∈ L∞(D) : essinf a > 0} and fix f ∈ X ∗. Then, we can define the
+data-to-solution map S : X → X for the forward problem (5). We also define the
+observation functional O ∈ (X ∗)K, with a finite number K ∈ N of observations
+(e.g. representing sensors), and a goal functional (also called quantity of interest)
+G ∈ X ∗. We define the prior measure π0 to be the uniform distribution on U.
+The observations O(S(a)) are assumed to be additionally subject to additive
+observation noise η, which we assume to be centered Gaussian, i.e., η ∼ N(0, Γ)
+for some known, nondegenerate covariance matrix Γ ∈ RK×K. In other words,
+we assume given noisy observation data δ ∈ RK modeled as
+δ = O(S(a)) + η ∈ L2
+Γ(RK).
+(9)
+1In practice, h either parametrizes the local mesh-size maxT ∈Th |T|1/2, for quasi-uniform
+collections of partitions, or it relates to the refinement level in case of adaptive refinement [16].
+3
+
+We consider the Bayesian inverse problem of recovering the expected value of
+G(u), given observation data δ, that is Eπ0[G(u)|δ]. By Bayes’ theorem [19], the
+posterior distribution πδ of y|δ is absolutely continuous with respect to π0 and
+its Radon-Nikodym derivative with respect to the prior π0 is
+dπδ
+dπ0
+(y) = Θ(y)
+Z
+,
+(10)
+where Θ(y) := exp(− 1
+2|δ − O(S(a(y)))|2
+Γ) = exp(− 1
+2|δ − O(u(y))|2
+Γ) denotes the
+likelihood with the observation noise covariance-weighted, data-to-observation
+misfit, where |x|2
+Γ := x⊤Γ−1x and Z is defined in (1). As Θ(y) > 0 for all y ∈ U,
+Z > 0. In the present setting, Bayesian inversion amounts to the numerical
+evaluation of the posterior mean
+Eπδ[G(u)] = 1
+Z
+�
+U
+G(u(y))Θ(y) dy.
+This is (1) upon setting Θ′(y) := G(u(y))Θ(y) and Y = R. Define the FE
+solution operator Sh : X → Xh as the mapping Sha(y) = uh(y) via (8). The FE
+approximations of Θ, Θ′ used in (3) are then Θh = exp(− 1
+2|δ − O(uh(y))|2
+Γ) and
+Θ′
+h = G(uh(y))Θh(y), respectively.
+1.3
+Optimal control with entropic risk measure (OCP)
+Let Y = L2(D), assume a parameter independent target state ˆu ∈ Y and a
+nonempty, closed and convex set X ⊆ Y of admissible controls. Throughout
+the rest of the paper, we identify Y with its dual via Riesz representation and
+write ⟨·, ·⟩ for the inner product on Y. Once the affine parametric diffusion
+coefficient a(y) is fixed, (5) defines a linear solution operator Ly : Y → Y by
+Lyf = ι ◦ u(y) for all f ∈ Y, where ι denotes the continuous embedding X ⊂ Y.
+In particular, we view u(y) as a function of the right-hand side f of (5). For
+a function Φ: U → R and some θ ∈ (0, ∞), the entropic risk measure [12] is
+defined by
+R(Φ) = 1
+θ log
+��
+U
+exp(θΦ(y)) dy
+�
+.
+(11)
+The entropic risk is especially relevant when favoring a risk averse behavior [7].
+We consider the following minimization problem, for fixed constants α1, α2 > 0
+f ∗ := argmin
+f∈X
+J(f),
+J(f) := R( α1
+2 ∥Lyf − ˆu∥2
+Y) + α2
+2 ∥f∥2
+Y .
+(12)
+Due to convexity of R and α2 > 0, the functional J is strongly convex so that
+(12) is a well-posed minimization problem [9,12].
+Define the shorthand notation Φf(y) = α1
+2 ∥Lyf − ˆu∥2
+Y and the adjoint state
+given by qf(y) = α1Ly(Lyf − ˆu) ∈ Y. Under the above conditions on X, (12) is
+equivalent to the inequality ⟨J′(f ∗), f − f ∗⟩ ≥ 0 for all f ∈ X, where in analogy
+with [9, Lemma 3.6] the Fréchet derivative J′(f) ∈ Y of J at f ∈ X is
+J′(f) =
+1
+�
+U exp(θΦf(y)) dy
+�
+U
+exp(θΦf(y))qf(y) dy + α2f.
+(13)
+4
+
+Next, we replace in (12) the integral over U by QMC rules and the exact solution
+operator Ly by the Galerkin solution Ly
+h : Y → Xh defined by Ly
+hf = uh(y).
+Then, we obtain the discrete formulation
+f ∗
+m,h := argmin
+f∈X
+Jm,h(f),
+Jm,h(f) := Rm( α1
+2 ∥Ly
+hf − ˆu∥2
+Y) + α2
+2 ∥f∥2
+Y , (14)
+where Rm(Φ) = 1
+θ log
+�
+1
+bm
+�
+y∈Pm exp(θΦ(y))
+�
+is again convex, due to positivity
+of the QMC quadrature weights. The derivative J′
+m,h(f) ∈ Y of Jm,h is analogous
+to (13), again replacing the integrals by sample averages over Pm, Φh,f(y) =
+α1
+2 ∥Ly
+hf − ˆu∥2
+Y and qh,f(y) = α1Ly
+h(Ly
+hf − ˆu) ∈ Y. The next proposition recasts
+the error in the approximation f ∗
+m,h ≈ f ∗ to the form (4). Whenever it does not
+cause confusion, we will write q(y) = qf ∗
+m,h(y) and qh(y) = qh,f ∗
+m,h(y).
+Proposition 1. For Y = L2(D), assume Z, Z′ in (1) are defined by Θ(y) =
+exp(θΦf ∗
+m,h(y)) and Θ′(y) = q(y)Θ(y).
+Similarly, let Zm,h, Z′
+m,h in (3) be
+defined by Θh(y) = exp(θΦh,f ∗
+m,h(y)) and Θ′
+h(y) = qh(y)Θh(y).
+Then
+��f ∗ − f ∗
+m,h
+��
+Y ≤ 1
+α2
+��J′(f ∗
+m,h) − J′
+m,h(f ∗
+m,h)
+��
+Y = 1
+α2
+����
+Z′
+Z −
+Z′
+m,h
+Zm,h
+����
+Y
+.
+(15)
+Proof. The inequalities ⟨J′(f ∗), f − f ∗⟩ ≥ 0 and ⟨J′
+m,h(f ∗
+m,h), f − f ∗
+m,h⟩ ≥ 0 for
+all f ∈ X imply that ⟨J′
+m,h(f ∗
+m,h) − J′(f ∗), f ∗ − f ∗
+m,h⟩ ≥ 0. Moreover, strong
+convexity of J yields the relation ⟨J′(f ∗)−J′(f ∗
+m,h)−α2(f ∗−f ∗
+m,h), f ∗−f ∗
+m,h⟩ ≥ 0.
+Thus, we get
+α2
+��f ∗ − f ∗
+m,h
+��2
+Y ≤ ⟨J′
+m,h(f ∗
+m,h) − J′(f ∗) + α2(f ∗ − f ∗
+m,h), f ∗ − f ∗
+m,h⟩
+≤ ⟨J′
+m,h(f ∗
+m,h) − J′(f ∗
+m,h), f ∗ − f ∗
+m,h⟩
+≤
+��J′
+m,h(f ∗
+m,h) − J′(f ∗
+m,h)
+��
+Y
+��f ∗ − f ∗
+m,h
+��
+Y ,
+which implies the inequality in (15).
+The equality follows by substituting
+J′
+m,h(f ∗
+m,h) =
+Z′
+m,h
+Zm,h + α2f ∗
+m,h and (13).
+Remark 1. Note that Θ, Θ′ as defined in Proposition 1, and hence Z, Z′, also
+implicitly depend on h via the discrete minimizer f ∗
+m,h. In particular, the exact
+minimizer f ∗ does not appear in the right hand side of (15). This fact will be
+crucial for the ensuing a-posteriori error estimation methodology.
+2
+A-posteriori QMC error estimation
+We develop computable a-posteriori error estimators for the PDE discretization
+error and for the QMC-quadrature error, the latter being reliable independent
+of the integration dimension s.
+2.1
+Parametric regularity
+To leverage the results from [5, 14] and overcome the curse of dimensionality,
+we need to quantify the regularity with respect to y ∈ U. To this end, we
+5
+
+write |ν| = �
+j νj, supp(ν) = {j : νj ̸= 0} and, given smooth F : U → Y and
+ν ∈ F := {ν ∈ NN
+0 : | supp(ν)| < ∞} we introduce the multi-index notation
+∂ν
+yF(y) = �
+j∈supp(ν) ∂νj
+yj F(y) for the derivatives with respect to y. Moreover,
+given β = (βj)j∈N ∈ ℓp(N) for some p ∈ (0, 1), n ∈ N and c > 0, we define the
+SPOD weights γ = (γu)u⊆{1:s} via
+γu =
+�
+ν∈{1:α}|u|
+(|ν| + n)!
+�
+j∈u
+cβνj
+j .
+(16)
+Definition 1. Let Y be a separable Hilbert space, α ∈ N, α ≥ 2. We define the
+weighted unanchored Sobolev space Ws,α,γ with dominating mixed smoothness
+as the completion of C∞(U, Y) with respect to the norm
+∥F∥s,α,γ := max
+u⊆{1:s} γ−1
+u
+�
+v⊆u
+�
+νu\v∈{1:α}|u\v|
+�
+[− 1
+2 , 1
+2 ]|v|
+�����
+�
+[− 1
+2 , 1
+2 ]s−|v| ∂
+(νu\v,αv)
+y
+F(y) dy{1:s}\v
+�����
+Y
+dyv,
+where µ = (νu\v, αv) ∈ F is such that µj = νj if j ∈ u \ v, µj = α if j ∈ v and
+µj = 0 otherwise. The inner integral is interpreted as a Bochner integral.
+The relevance of the space Ws,α,γ in our context is justified by the following
+result, which will be the starting point of our analysis. This is a consequence of
+the so-called component-by-component (CBC) construction as described in [5,14],
+which takes as input s, α, γ and m and returns a polynomial lattice Pm.
+Theorem 1. Let α ∈ N, α ≥ 2 and F : U → Y for some separable Hilbert space
+Y be such that F ∈ Ws,α,γ for some weights γ of the form (16) for β ∈ ℓp(N),
+p ∈ (0, 1/2]. Then, there exists a sequence (Pm)m∈N of polynomial lattice rules
+such that as m → ∞
+Em(F) :=
+�
+U
+F − 1
+bm
+�
+y∈Pm
+F(y) = ∥F∥s,α,γ O(b−m),
+as well as for all ε > 0
+Em(F) = 1
+bm
+�
+y∈Pm
+F(y) −
+1
+bm−1
+�
+y∈Pm−1
+F(y) + ∥F∥s,α,γ O(b−2m+ε).
+Here the constants hidden in O(·) are independent of s, m ∈ N and F, but depend
+on ε and γ. In addition, the point sets (Pm)m∈N can be constructed explicitly by
+a CBC construction in O(smbm + s2bm) operations.
+Proof. For the case Y = R, this is proved in [5, Theorem 4.1]. Otherwise, let
+v ∈ Y arbitrary such that ∥v∥Y = 1. Then, ∂ν
+y⟨F(y), v⟩ = ⟨∂ν
+yF(y), v⟩ for all
+ν ∈ F implies ∥⟨F, v⟩∥s,α,γ ≤ ∥F∥s,α,γ. Hence, we reduced to the case Y = R
+and by linearity of the integral and the quadrature we conclude.
+Corollary 1. Assume that the PDE (5) satisfies (6) and (7). Then we can
+construct polynomial lattice rules (Pm)m∈N (depending on b) in O(smbm + s2bm)
+operations such that for all ε > 0 it holds for the BIP
+Z − Zm = Zm − Zm−1 + O(b−2m+ε),
+Z′ − Z′
+m = Z′
+m − Z′
+m−1 + O(b−2m+ε),
+as m → ∞. The hidden constants in O(·) are independent of s, m ∈ N.
+6
+
+Proof. From [18, Section 4.1], [4, Theorem 3.1] and ν! := �
+j∈supp(ν) νj! ≤ |ν|!,
+we have that sups ∥Θ∥s,α,γ + sups ∥Θ′∥s,α,γ < ∞ for α = 1 + ⌊1/p⌋ and some
+SPOD weights (16) defined by a sequence β ∼ b and n = 0. Hence we can apply
+Theorem 1 and conclude.
+A similar result can be given for the OCP, based on the results in [8,9].
+Corollary 2. Assume that the PDE (5) satisfies (6) and (7). Then we can
+construct polynomial lattice rules (Pm)m∈N (depending on b) in O(smbm + s2bm)
+operations such that for all ε > 0 it holds for the OCP
+Z − Zm = Zm − Zm−1 + O(b−2m+ε),
+Z′ − Z′
+m = Z′
+m − Z′
+m−1 + O(b−2m+ε),
+as m → ∞. The hidden constants in O(·) are independent of s, m ∈ N.
+Proof. Applying the result of [8, Lemma 4.6] in [9, Theorems 5.4 and 5.6], we
+conclude that sups ∥Θ∥s,α,γ + sups ∥Θ′∥s,α,γ < ∞ for α = 1 + ⌊1/p⌋ and for
+some SPOD weights (16) defined by a sequence β ∼ b and n = 2.
+2.2
+Ratio error estimator
+Proposition 2. Assume that the PDE (5) satisfies (6) and (7). Then, for the
+BIP and the OCP, we can construct polynomial lattice rules (Pm)m∈N such that
+Z′
+Z − Z′
+m
+Zm
+=
+1
+bZm − Zm−1
+�Zm−1Z′
+m − ZmZ′
+m−1
+Zm
+�
++ O(b−2m+ε),
+(17)
+holds for all ε > 0 as m → ∞. The hidden constant in O(·) is independent of
+s, m.
+Proof. First, Z − Zm → 0 as m → ∞ implies that Zm > Z/2 > 0 for m
+sufficiently large. Next, due to either Corollary 1 or 2, for all ε > 0 we have
+Z′
+Z − Z′
+m
+Zm
+= (Z′ − Z′
+m)Zm − (Z − Zm)Z′
+m
+(Z − Zm)Zm + Z2m
+=
+1
+b−1(Z′
+m − Z′
+m−1 + O(b−2m+ε))Zm −
+1
+b−1(Zm − Zm−1 + O(b−2m+ε))Z′
+m
+1
+b−1(Zm − Zm−1 + O(b−2m+ε))Zm + Z2m
+=
+1
+b−1(Z′
+m − Z′
+m−1)
+1
+b−1(bZm − Zm−1) + O(b−2m+ε)
+−
+1
+b−1(Zm − Zm−1)Z′
+m
+(
+1
+b−1(bZm − Zm−1) + O(b−2m+ε))Zm
++ O(b−2m+ε).
+Clearly Am :=
+1
+b−1(bZm − Zm−1) → Z as m → ∞, so that Am > Z/2 > 0 for
+m sufficiently large. Hence, by a geometric sum argument, collecting in Bm all
+terms contained in O(b−2m+ε) in the denominator, we obtain for m → ∞ that
+1
+Am + Bm
+=
+1
+Am
+∞
+�
+k=0
+(−1)k
+�Bm
+Am
+�k
+=
+1
+Am
++ O(b−2m+ε).
+7
+
+Therefore, as m → ∞
+Z′
+Z − Z′
+m
+Zm
+=
+1
+b−1(Z′
+m − Z′
+m−1)
+Am
+−
+1
+b−1(Zm − Zm−1)Z′
+m
+AmZm
++ O(b−2m+ε),
+(18)
+which, upon rearranging the terms, is the claim.
+Since Z′
+Z − Z′
+m
+Zm
+= O(b−m) ≫ O(b−2m+ε), Proposition 2 states that
+Ebm(Θ′, Θ) :=
+1
+bZm − Zm−1
+�Zm−1Z′
+m − ZmZ′
+m−1
+Zm
+�
+(19)
+is a computable, asymptotically exact error estimator.
+3
+A-posteriori FEM error estimation
+In practice the parametric solution u(y) ∈ X is not exactly available, and hence
+Zm, Z′
+m are not computable. For any y ∈ U, u(y) will be approximated by the
+corresponding Galerkin discretizations uh(y) ∈ Xh.
+3.1
+Ratio error estimator
+Let Θh, Θ′
+h, Zh, Z′
+h be defined replacing u and q by uh, qh in the definitions of
+Θ, Θ′, Z, Z′, respectively. Similarly, let Zm,h, Z′
+m,h be defined replacing u and q
+by uh, qh in the definitions of Zm, Z′
+m, respectively.
+Proposition 3. Assume there exist ζm,h, ζ′
+m,h such that for some c > 0 inde-
+pendent of h ∈ H, |Zm − Zm,h| ≤ cζm,h and
+���Z′
+m − Z′
+m,h
+���
+Y ≤ cζ′
+m,h. Assume
+that ζm,h → 0 as h → 0. Then, there exists h0 ∈ H such that for all h ≤ h0
+����
+Z′
+m
+Zm
+−
+Z′
+m,h
+Zm,h
+����
+Y
+≤ c
+Zm,hζ′
+m,h +
+���Z′
+m,h
+���
+Y ζm,h
+Z2
+m,h − cζm,hZm,h
+.
+Proof. Due to the limit ζm,h → 0 there exists h1 ∈ H such that Zm,h ≥ Zm/2 > 0
+for all h ≤ h1, h ∈ H. Therefore, we can pick h0 ∈ H such that Zm,h > cζm,h
+for all h ≤ h0, h ∈ H. Thus
+����
+Z′
+m
+Zm
+−
+Z′
+m,h
+Zm,h
+����
+Y
+=
+�����
+(Z′
+m − Z′
+m,h)Zm,h − (Zm − Zm,h)Z′
+m,h
+(Zm − Zm,h)Zm,h + Z2
+m,h
+�����
+Y
+≤
+���Z′
+m − Z′
+m,h
+���
+Y Zm,h + |Zm − Zm,h|
+���Z′
+m,h
+���
+Y
+Z2
+m,h − |Zm − Zm,h| Zm,h
+≤ c
+Zm,hζ′
+m,h +
+���Z′
+m,h
+���
+Y ζm,h
+Z2
+m,h − cζm,hZm,h
+.
+8
+
+Due to this result, we are left with the task of finding computable ζm,h, ζ′
+m,h
+satisfying the conditions |Zm − Zm,h| ≤ cζm,h and
+���Z′
+m − Z′
+m,h
+���
+Y ≤ cζ′
+m,h for
+some c > 0 independent of m, h. Thus, in the following sections we will provide
+such error estimators for BIP and OCP.
+3.2
+FEM error estimators for BIP
+For a finite collection of observation functionals O = (O1, . . . , OK) ∈ (X ∗)K,
+define ∥O∥X ∗ =
+��K
+k=1 ∥Ok∥2
+X ∗. The starting point to estimate the FEM error
+will be the following well-known result, e.g. [16]. Let {Th}h∈H be a family of
+shape-regular, simplicial meshes of D and let Pk(Th) be the set of piecewise
+polynomial functions on Th of degree at most k ∈ N0 in each T ∈ Th. Let Eh be
+the set of interior edges of all elements T ∈ Th. We assume that Xh := P1(Th)∩X,
+that f ∈ L2(D) and that a(y) ∈ W 1,∞(D). Let hT = |T|1/2 for T ∈ Th and he
+the length of an edge e ∈ Eh. Then we define the a-posteriori error estimator
+η2
+y,h :=
+�
+T ∈Th
+�
+h2
+T ∥f + div(a(y)∇uh(y))∥2
+L2(T )
++ 1
+2
+�
+e⊆∂T,e∈Eh
+he ∥�a(y)∇uh(y)�∥2
+L2(e)
+�
+.
+(20)
+By [16, Theorem 6.3] there exists some c∗ > 0, only depending on D and the
+shape regularity constant of {Th}h∈H, and in particular independent of y ∈ U
+and h ∈ H, such that
+∥u(y) − uh(y)∥X ≤ c∗ηy,h.
+(21)
+For the important special case O ∈ (L2(D))K and G ∈ L2(D) we may derive
+sharper estimates. To simplify the presentation, we assume here that the physical
+domain D ⊆ R2 is a convex polygon (see also Remark 3 below), and introduce
+the L2(D)−residual estimator
+˜η2
+y,h :=
+�
+T ∈Th
+�
+h4
+T
+��f ∗
+m,h + div(a(y)∇uh(y))
+��2
+L2(T )
++ 1
+2
+�
+e⊆∂T,e∈Eh
+h3
+e ∥�a(y)∇uh(y)�∥2
+L2(e)
+�
+.
+(22)
+The additional factors hT , he are derived from a standard duality argument, see,
+e.g. [20, Section 1.11]. Then, there exists some c∗ > 0 depending only on D
+and the shape regularity constant of {Th}h∈H, and in particular independent of
+y ∈ U and h ∈ H, such that
+∥u(y) − uh(y)∥L2(D) ≤ c∗˜ηy,h.
+(23)
+Lemma 1. Fix a regular mesh Th of simplices in D and a parameter vector
+y ∈ U and assume (21). Then
+|Θ(y) − Θh(y)| ≤ Θh(y)(eχy,h − 1) =: ζy,h
+9
+
+holds for
+χy,h :=
+���Γ−1/2O
+���
+X ∗
+�
+|δ − O(uh(y))|Γ + 1
+2
+���Γ−1/2O
+���
+X ∗ c∗ηy,h
+�
+c∗ηy,h.
+Furthermore, if O ∈ (L2(D))K and G ∈ L2(D), then
+|Θ(y) − Θh(y)| ≤ Θh(y)(e˜χy,h − 1) := ˜ζy,h
+holds for
+˜χy,h :=
+���Γ−1/2O
+���
+L2(D)
+�
+|δ − O(uh(y))|Γ + 1
+2
+���Γ−1/2O
+���
+L2(D) c∗˜ηy,h
+�
+c∗˜ηy,h.
+Proof. We define ∆h(y) := − 1
+2 |δ − O(u(y))|2
+Γ + 1
+2 |δ − O(uh(y))|2
+Γ to obtain
+|Θ(y) − Θh(y)| = |Θh(y)(e∆h(y) − 1)| ≤ Θh(y)(e|∆h(y)| − 1).
+The first part of the claim now follows with (21) and
+|∆h(y)| ≤ 1
+2
+���Γ−1/2O
+���
+X ∗ |2δ − O(u(y) + uh(y))|Γ ∥u(y) − uh(y)∥X
+≤
+���Γ−1/2O
+���
+X ∗ |δ − O(uh(y))|Γ ∥u(y) − uh(y)∥X
++ 1
+2
+���Γ−1/2O
+���
+2
+X ∗ ∥u(y) − uh(y)∥2
+X .
+The second part of the proof follows analogously by replacing X by L2(D) and
+using (23) instead of (21).
+Lemma 2. Fix a regular mesh of simplices Th and y ∈ U and assume (21).
+Then there exists a constant c∗ > 0 such that for all y ∈ U and h ∈ H
+|Θ′(y) − Θ′
+h(y)| ≤ ∥G∥X ∗ (c∗ηy,hΘh(y)eχy,h + ζy,h ∥uh(y)∥X ) =: ζ′
+y,h.
+Furthermore, if O ∈ (L2(D))K and G ∈ L2(D), then
+|Θ′(y) − Θ′
+h(y)| ≤ ∥G∥L2(D)
+�
+c∗˜ηy,hΘh(y)e˜χy,h + ˜ζy,h ∥uh(y)∥L2(D)
+�
+=: ˜ζ′
+y,h.
+Proof. Since Θ′(y) = G(u(y))Θ(y) = G(u(y)Θ(y)), we get
+|Θ′(y) − Θ′
+h(y)| ≤ ∥G∥X ∗ ∥u(y)Θ(y) − uh(y)Θh(y)∥X
+≤ ∥G∥X ∗ (∥u(y) − uh(y)∥X Θ(y) + ∥uh(y)∥X |Θ(y) − Θh(y)|),
+and hence the claim follows with Lemma 1 since Θ(y) ≤ Θh(y)eχy,h. The second
+part of the claim follows analogously by the second part of Lemma 1.
+Remark 2. We remark that the estimates in the proof of Lemma 2 are conservative:
+we used that K > 1 in Lemma 1. For K = 1, i.e. for a single observation
+functional, goal-oriented AFEM results from [1] and the references there, can be
+used to obtain sharper a-posteriori error bounds.
+10
+
+We now can define ζm,h by averaging ζy,h for y ∈ Pm, that is ζm,h :=
+1
+bm
+�
+y∈Pm ζy,h. Then we get from Lemma 1
+|Zm − Zm,h| ≤ 1
+bm
+�
+y∈Pm
+|Θ(y) − Θh(y)| ≤ ζm,h.
+(24)
+Analogously, with ζ′
+m,h =
+1
+bm
+�
+y∈Pm ζ′
+y,h, Lemma 2 implies
+��Z′
+m − Z′
+m,h
+�� ≤ 1
+bm
+�
+y∈Pm
+|Θ′(y) − Θ′
+h(y)| ≤ ζ′
+m,h.
+(25)
+In particular, if we construct Th such that it also holds ηy,h → 0 as h → 0 for
+all y ∈ U, (24) and (25) verify the hypotheses of Proposition 3 with c = 1.
+3.3
+FEM error estimators for OCP with entropic risk
+In the case of OCP we require error estimates for the parametric state at the
+discrete optimal control f ∗
+m,h, i.e. u(y) = Lyf ∗
+m,h and for the corresponding
+adjoint state q(y) = α1Ly(Lyf ∗
+m,h − ˆu). The error will be measured in the
+L2(D)-norm. Again we assume that Xh := P1(Th) ∩ X, that f ∈ L2(D) and
+a(y) ∈ W 1,∞(D), and that D ⊆ R2 is a convex polygon.
+Lemma 3. Fix a mesh Th and y ∈ U and impose (23). With the notation of
+Proposition 1 and ˜ηy,h defined as in (22), we have
+|Θ(y) − Θh(y)| ≤ Θh(y)(eχy,h − 1) =: ζy,h,
+where
+χy,h := θc∗ �α1
+2 ˜η2
+y,h + α1
+��Ly
+hf ∗
+m,h − ˆu
+��
+L2(D) ˜ηy,h
+�
+.
+Proof. By twofold application of the triangle inequality we have
+���Φf ∗
+m,h(y) − Φh,f ∗
+m,h(y)
+��� ≤ α1
+2
+��(Ly − Ly
+h)f ∗
+m,h
+��2
+L2(D)
++ α1
+��Ly
+hf ∗
+m,h − ˆu
+��
+L2(D)
+��(Ly − Ly
+h)f ∗
+m,h
+��
+L2(D)
+≤ c∗ �α1
+2 ˜η2
+y,h + α1
+��Ly
+hf ∗
+m,h − ˆu
+��
+L2(D) ˜ηy,h
+�
+=: c∗ξy,h.
+Note that ξy,h is computable due to Remark 1. Then, it follows with ∆h(y) :=
+θ(Φf ∗
+m,h(y) − Φh,f ∗
+m,h(y)) that
+|Θ(y) − Θh(y)| =
+���Θh(y)(e∆h(y) − 1)
+��� ≤ Θh(y)(e|∆h(y)| − 1).
+Using a residual estimator in the form (22) for the adjoint problem yields
+∥q(y) − qh(y)∥L2(D) ≤ 2 max(c∗, 1)c∗˜˜ηy,h,
+(26)
+where
+˜˜η2
+y,h := α2
+1
+�
+T ∈Th
+�
+h4
+T ∥uh(y) − ˆu + div(a(y)∇qh(y))∥2
+L2(T )
+(27)
++ 1
+2
+�
+e⊆∂T,e∈Eh
+h3
+e ∥�a(y)∇qh(y)�∥2
+L2(e)
+�
++ (max
+T ∈Th h4
+T )˜η2
+y,h.
+11
+
+Lemma 4. Let Y = L2(D). Fix a mesh Th and y ∈ U and impose (23) and
+(26). With the notation of Proposition 1, we have
+∥Θ′(y) − Θ′
+h(y)∥Y ≤ ζy,h ∥qh(y)∥Y + 2c∗Θh(y)eχy,h ˜˜ηy,h =: ζ′
+y,h
+Proof. As in the proof of Lemma 2, we get by Θ(y) ≤ Θh(y)eχy,h that
+∥q(y)Θ(y) − qh(y)Θh(y)∥Y ≤ |Θ(y) − Θh(y)| ∥qh(y)∥Y
++ Θ(y) ∥q(y) − qh(y)∥Y
+≤ ζy,h ∥qh(y)∥Y + 2c∗Θh(y)eχy,h ˜˜ηy,h.
+Remark 3. If D ⊆ R2 is a non-convex polygon, the reliability assumption (23)
+and the corresponding definitions (22), (27) of ˜ηy,h, ˜˜ηy,h must be adapted by
+using weighted L2 norms, with weights near the re-entrant corners. We refer
+to [21, Theorem 3.1] for a precise result in the case of the Poisson equation.
+4
+Combined QMC-FEM estimator
+In view of Propositions 2 and 3 we employ the computable a-posteriori estimator
+ESTbm,h := ∥Ebm(Θ′
+h, Θh)∥Y +
+Zm,hζ′
+m,h +
+���Z′
+m,h
+���
+Y ζm,h
+Z2
+m,h − ζm,hZm,h
+.
+(28)
+Note that the QMC error estimator ∥Ebm(Θ′, Θ)∥Y derived from Proposition 2
+is itself approximated by the computable expression ∥Ebm(Θ′
+h, Θh)∥Y. In the
+next proposition we precise that the additional error committed due to this
+extra approximation is of higher asymptotic order, as m → ∞. We equip the set
+C0(U, Y) with the norm ∥F∥∞ = supy∈U ∥F(y)∥Y.
+Proposition 4. Fix a family of regular meshes {Th}h∈H such that for some
+˜C > 0 independent of s ∈ N, h ∈ H and some SPOD weights (16), it holds
+max(∥Θ∥s,α,γ , ∥Θ′∥s,α,γ , ∥Θh∥s,α,γ , ∥Θ′
+h∥s,α,γ) ≤ ˜C.
+(29)
+Assume that the spaces {Xh}h are contained in X and that they are selected so
+that ∥Θ − Θh∥∞ → 0 as h → 0. Then we can construct a sequence of polynomial
+lattices (Pm)m∈N in O(smbm + s2bm) operations such that, for some h0 ∈ H and
+some constant C > 0 (independent of m, h, s) we have for any h < h0 that
+∥Ebm(Θ′, Θ) − Ebm(Θ′
+h, Θh)∥Y
+≤ Cb−m(∥Θ − Θh∥∞ + ∥Θ′ − Θ′
+h∥∞ + ∥Θ − Θh∥s,α,γ + ∥Θ′ − Θ′
+h∥s,α,γ).
+Proof. Throughout the proof, C > 0 is a generic constant independent of m, h, s.
+We compute the difference of the numerators
+∆1 := Zm−1Z′
+m − ZmZ′
+m−1 − Zm−1,hZ′
+m,h + Zm,hZ′
+m−1,h
+= −(Zm − Zm−1)(Z′
+m − Z′
+m,h) − (Zm − Zm,h − Zm−1 + Zm−1,h)Z′
+m,h
++ (Zm − Zm,h)(Z′
+m − Z′
+m−1) + Zm,h(Z′
+m − Z′
+m,h − Z′
+m−1 + Z′
+m−1,h).
+12
+
+We have |Zm − Zm,h| ≤ ∥Θ − Θh∥∞ and
+���Z′
+m − Z′
+m,h
+���
+Y ≤ ∥Θ′ − Θ′
+h∥∞. From
+Theorem 1, we know that Zm = Zm−1 + ∥Θ∥s,α,γ O(b−m) and Z′
+m = Z′
+m−1 +
+∥Θ′∥s,α,γ O(b���m) as m → ∞, with hidden constants in O(·) independent of
+s, m, h. Furthermore Zm − Zm,h − Zm−1 + Zm−1,h = ∥Θ − Θh∥s,α,γ O(b−m),
+and Z′
+m − Z′
+m,h − Z′
+m−1 + Z′
+m−1,h = ∥Θ′ − Θ′
+h∥s,α,γ O(b−m) also follow by
+Theorem 1. Therefore, we have
+∥∆1∥Y ≤ Cb−m(∥Θ − Θh∥∞ + ∥Θ′ − Θ′
+h∥∞ + ∥Θ − Θh∥s,α,γ + ∥Θ′ − Θ′
+h∥s,α,γ).
+Next, we define T1 := Zm−1,hZ′
+m,h − Zm,hZ′
+m−1,h and obtain the estimate
+∥T1∥Y ≤ C(∥Θh∥s,α,γ + ∥Θ′
+h∥s,α,γ)b−m. Moreover,
+∆2 := (bZm − Zm−1)Zm − (bZm,h − Zm−1,h)Zm,h
+= (b(Zm − Zm,h) + (Zm−1,h − Zm−1))Zm + (bZm,h − Zm−1,h)(Zm − Zm,h)
+gives |∆2| ≤ C ∥Θ − Θh∥∞. Next, we observe that T2 = (bZm,h − Zm−1,h)Zm,h
+is bounded from below away from 0, for h sufficiently small. Therefore, for h
+sufficiently small we apply the elementary inequality
+����
+T1 + ∆1
+T2 + ∆2
+− T1
+T2
+����
+Y
+≤ max(∥∆1∥Y , ∥T1∆2∥Y)
+1 + |T2|
+|T2| (|T2| − |∆2|),
+valid for T1, T2, ∆1, ∆2 ∈ R with |∆2| < |T2|, which is satisfied since |∆2| → 0
+as h → 0. Combining all these observations we obtain the claim.
+Theorem 2. For either the BIP or the OCP, assume that D ⊆ R2 is a con-
+vex polygon and that the PDE (5) satisfies (6), (7), f ∈ L2(D) and b′ ∈
+ℓp(N), p ∈ (0, 1/2], with b′j = ∥ψj∥W 1,∞(D). Let Xh = P1(Th) ∩ X for a family
+of shape-regular meshes Th such that h = maxT ∈Th hT . Then, we can construct
+polynomial lattices (Pm)m∈N such that the estimator ESTbm,h in (28) satisfies
+����
+Z′
+Z −
+Z′
+m,h
+Zm,h
+����
+Y
+≤ ESTbm,h + O(b−2m+ε + b−mh),
+for any ε > 0 as m → ∞, h → 0. The constant in O(·) is independent of s, m
+and h, but depends on ε.
+Proof. Since bj ≤ b′
+j, from either Corollary 1 or 2, there exist SPOD weights γ′
+as in (16) with β′ ∼ b′, such that sups ∥Θ∥s,α,γ′ + ∥Θ′∥s,α,γ′ < ∞. Combining
+(20) with Lemma 1 and h → 0 we have ζm,h → 0 for the BIP case for any m ∈ N.
+Similarly, combining (22) with Lemma 3 yields the same observation for the
+OCP case. Therefore we can apply Propositions 2 and 3 to get that we can
+construct polynomial lattices so that as m → ∞
+����
+Z′
+Z −
+Z′
+m,h
+Zm,h
+����
+Y
+≤ ∥Ebm(Θ′, Θ)∥Y +
+Zm,hζ′
+m,h +
+���Z′
+m,h
+���
+Y ζm,h
+Z2
+m,h − ζm,hZm,h
++ O(b−2m+ε).
+Next, we say that ρ ∈ (1, ∞)N is (b′, ε)-admissible if �
+j≥1(ρj − 1)b′
+j ≤ ε, see [4].
+Then we define Tb′,ε = �
+ρ,(b′,ε)-adm.{y ∈ Cs : dist(yj, [− 1
+2, 1
+2]) ≤ ρj − 1}. Fol-
+lowing the computations in [2, Theorem 4.1], yield h0 ∈ H and ε > 0 sufficiently
+13
+
+small such that for all h ≤ h0, h ∈ H, we have supy∈Tb′,ε |Θ(y) − Θh(y)| +
+∥Θ′(y) − Θ′
+h(y)∥Y ≤ Ch. By [4, Theorem 3.1], this implies that for a constant
+C independent of h, s,
+∥Θ − Θh∥ ∞ + ∥Θ′ − Θ′
+h∥∞ + ∥Θ − Θh∥s,α,γ′ + ∥Θ′ − Θ′
+h∥s,α,γ′ ≤ Ch.
+Thus, (29) and ∥Θ − Θh∥∞ → 0 hold, and we apply Proposition 4 to conclude.
+5
+Numerical experiment
+We consider the PDE (5) on the physical domain D := (0, 1)2, with f ≡ 10, and
+parametric diffusion coefficient given by
+a(x, y) = 1
+2 +
+16
+�
+j=1
+yj
+(k2
+j,1 + k2
+j,2)2 sin(kj,1x1) sin(kj,2x2).
+The pairs (kj,1, kj,2) ∈ N2 are defined by the ordering of N2 such that for j ∈ N,
+k2
+j,1 + k2
+j,2 ≤ k2
+j+1,1 + k2
+j+1,2, and the ordering is arbitrary when equality holds.
+We investigate a BIP with observation functional O = (O1, . . . , O4) ∈ (L2(D))4,
+given by Ok(v) :=
+1
+0.01
+�
+Ik vdx for v ∈ L2(D) and k = 1, . . . , 4, where I1 :=
+[0.1, 0.2]2, I2 := [0.1, 0.2] × [0.8, 0.9], I3 := [0.8, 0.9] × [0.1, 0.2], I4 := [0.8, 0.9]2.
+We draw a (random) sample of a to compute the "ground truth" of observa-
+tions O(S(a)) on a sequence of regular FE meshes of triangles obtained by
+uniform refinement, with 525.313 degrees of freedom (dofs).
+We add ran-
+dom noise η ∼ N(0, σ2I4) to the observations, where σ is set as 10% of
+the average of O(S(a)) ∈ R4. The realized synthetic data is then given by
+δ = (0.5205, 0.5037, 0.5443, 0.4609)⊤.
+Our aim is to approximate Eπδ[G(u)] by the ratio estimator
+Z′
+m,h
+Zm,h , where
+G ∈ L2(D) is given by G(v) :=
+1
+0.5
+�
+[0.25,0.75]2 vdx for v ∈ L2(D). The FE
+mesh and the polynomial lattice rule, that eventually determine h and m, are
+refined successively based on the combined estimator in (28). For tolerances
+τFEM, τQMC > 0, we start from an initial FE mesh of D, that is uniformly refined
+until the stopping criterion
+Zm,hζ′
+m,h+∥Z′
+m,h∥Yζm,h
+Z2
+m,h−ζm,hZm,h
+≤ τFEM is met. Thereafter, we
+increase the number bm of lattice points until there holds |Ebm(Θ′
+h, Θh)| ≤ τQMC.
+We initialize by a FE mesh with 41 dofs and bm0 QMC points with base b = 2
+and m0 = 2, and set the tolerances to τFEM = τQMC = 2−6. To assess the total
+realized error, we compute a reference solution Z′
+ref
+Zref by a multilevel Monte Carlo
+ratio estimator, see [17], and report absolute error
+���
+Z′
+m,h
+Zm,h − Z′
+ref
+Zref
+���. The reference
+estimator uses 6 refinement levels with 545/525.313 dofs on the coarsest/finest
+level, respectively, and uniform (pseudo-) random numbers y. The number
+of samples is adjusted to balance statistical error and discretization bias on
+each level. The experiment has been implemented in MATLAB using the
+MooAFEM library [11] for the FE discretization. All arising linear systems are
+solved directly by the \-operator in MATLAB.
+The estimated and realized errors vs.
+the number of iterations (in the
+sense of refinement steps) are depicted Figure 1. Here, the FE a-posteriori
+14
+
+estimator gives negative values on rather coarse meshes, where c∗˜ηy,h > 1.
+Therefore, we discarded these "pre-asymptotic" values in the plot. We see
+that the FE a-posteriori estimator from Proposition 3 is rather conservative at
+first, but eventually approaches the actual error for finer meshes. The QMC
+estimator |Ebm(Θ′
+h, Θ′
+h)| is of the same magnitude as σ at first, and only two
+more refinement steps are needed once the FE mesh is sufficiently fine. The
+combined error estimate ESTbm,h aligns well with the realized error, as expected
+from our theoretical analysis.
+Figure 1: Results for the QMC-FEM ratio estimator with a-posterior ratio refinement.
+First the FE mesh is refined until the tolerance τFEM is achieved (dashed w. circles),
+then the QMC a-posterior refinement takes place (dashed w. triangles). The estimated
+error (solid w. stars) is conservative for coarse meshes, but eventually approaches the
+realized error (solid w. diamonds).
+6
+Conclusion
+In this paper we outlined the construction of an a-posteriori QMC-FEM estimator,
+that allows to quantify the approximation error to a) posterior expectation
+in Bayesian inverse problems and b) the optimal control under the entropic
+risk measure. The estimator is computable and viable for large number of
+parameters s and it is asymptotically an upper bound for the errors in a) and b).
+Furthermore, the particular ratio structure Z′
+Z of the sought quantities allows
+to tackle both the BIP and OCP, in a unified manner. In either case, we work
+under the assumption that the underlying model is a parametric elliptic PDE
+with affine-parametric diffusion. Nevertheless, the present QMC methodology to
+high-dimensional integration is applicable to non-affine parametric PDEs with
+quantified, holomorphic-parametric dependence, see [4] and the references there.
+Since the error estimators we consider ηy,h, ˜ηy,h, ˜˜ηy,h are expressed as sums of
+local error contributions for T ∈ Th, a possible direction of research is to employ
+the presently proposed estimators ζy,h, ζ′
+y,h to steer an adaptive QMC-FEM
+algorithm [13].
+15
+
+ total error
+Z
+Zm.h
+Total/estimated errors
+-QMC-error est.
+*-FEM+QMC-error est.
+... TFEM/TQMC
+LC
+7
+# IterationsReferences
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+17
+

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+Language-Informed Transfer Learning for Embodied Household Activities
+Yuqian Jiang 1*, Qiaozi Gao 2, Govind Thattai 2, Gaurav Sukhatme 2,3
+1 The University of Texas at Austin, 2 Amazon Alexa AI, 3 University of Southern California
+[email protected], {qzgao, thattg}@amazon.com, [email protected]
+Abstract
+For service robots to become general-purpose in everyday
+household environments, they need not only a large library
+of primitive skills, but also the ability to quickly learn novel
+tasks specified by users. Fine-tuning neural networks on a va-
+riety of downstream tasks has been successful in many vision
+and language domains, but research is still limited on trans-
+fer learning between diverse long-horizon tasks. We propose
+that, compared to reinforcement learning for a new household
+activity from scratch, home robots can benefit from transfer-
+ring the value and policy networks trained for similar tasks.
+We evaluate this idea in the BEHAVIOR simulation bench-
+mark which includes a large number of household activities
+and a set of action primitives. For easy mapping between state
+spaces of different tasks, we provide a text-based representa-
+tion and leverage language models to produce a common em-
+bedding space. The results show that the selection of similar
+source activities can be informed by the semantic similarity
+of state and goal descriptions with the target task. We further
+analyze the results and discuss ways to overcome the problem
+of catastrophic forgetting.
+Introduction
+Domestic service robots have been envisioned to help in a
+variety of household activities. Imagine a single robot that
+can be versatile enough from tidying up the rooms to play-
+ing with kids. Such a robot not only requires the sensing,
+navigation, and manipulation capabilities, but also needs to
+intelligently combine these skills to perform each activity as
+requested by the users.
+Since every home is different, a simple library of pre-
+programmed tasks will hardly serve the purpose. For ex-
+ample, when a user wants to clean the kitchen cupboard,
+the specific goal conditions they would like to achieve will
+depend on their personal preferences and constraints of the
+environment. Does the robot re-arrange the dishes in a cer-
+tain pattern? Does the robot dust the outside of the cup-
+board? The reality is that there could be an infinite number
+of combinations of goals, and a robot will most likely have
+to learn to solve new goals after it is deployed in the individ-
+ual homes.
+In this paper, we study the problem of learning novel
+user-specified household activities for a service robot that
+*Work completed during an internship with Amazon Alexa AI.
+is shipped with pre-trained policies for a set of standard ac-
+tivities. We propose to learn the new activity by transferring
+from the policy of a similar activity. Our hypothesis is that
+the transfer can be more efficient than learning the new activ-
+ity from scratch if their initial state and goal conditions are
+similar. Intuitively, a robot should be able to learn putting
+away cleaned dishes efficiently if it has a good policy for
+cleaning kitchen cupboard. Further, we can measure activity
+similarities by leveraging language models to embed their
+state and goal descriptions.
+We test our hypothesis using the BEHAVIOR bench-
+mark (Srivastava et al. 2021). BEHAVIOR simulates a large
+number of household activities for an embodied AI to learn.
+We first present a reinforcement learning (RL) approach to
+solve a subset of activities from scratch. The approach lever-
+ages text descriptions of the agent’s current state and goal to
+allow the policies to operate in a common state space. We
+then initialize the learner with each of the pretrained policies
+when training it on a new activity, and evaluate the hypothe-
+sis that the transfer performance corresponds to the semantic
+similarity between the activity text descriptions. We present
+some initial results to show the potential of this approach for
+enabling versatile and adaptive home robots.
+Related Work
+Transfer learning leverages the knowledge learned in a
+source domain to improve the performance of a learner on
+the target domain. Transfer learning in reinforcement learn-
+ing has been studied to transfer knowledge between different
+Markov Decision Processes (MDPs) (Zhu, Lin, and Zhou
+2021; Taylor and Stone 2009). While many approaches are
+evaluated in tasks with the same high-level goal and only
+different configurations in Mujoco, navigation, and Atari do-
+mains (Barreto et al. 2017; Schaul et al. 2015), a few re-
+cent transfer learning approaches have demonstrated posi-
+tive transfer between distinct Atari games (Rusu et al. 2016;
+Fernando et al. 2017). Soemers et al. introduces an approach
+that transfers policy and value networks between distinct
+board games that have different action spaces (Soemers et al.
+2021). Encouraged by these successes, we propose to trans-
+fer RL policies among distinct embodied household activi-
+ties which require high-level long-horizon reasoning about a
+large variety of goal conditions. Further, this work proposes
+to use language models on activity descriptions to inform the
+arXiv:2301.05318v1  [cs.RO]  12 Jan 2023
+
+selection of source domains.
+BEHAVIOR is a benchmark where embodied AI so-
+lutions are evaluated on household activities in a realis-
+tic physics simulation. The activities are selected from the
+American Time Use Survey to reflect the real distribution of
+household chores. There has been very little success using
+RL to solve BEHAVIOR in its original setting (Srivastava
+et al. 2021). In this paper, the method of providing the text-
+based, fully observable state representation is most similar
+to the work done by Shridhar et al. for the ALFRED bench-
+mark (Shridhar et al. 2021).
+Approach
+Our approach consists of two steps. In the first part, we in-
+troduce a text-based state representation for a RL agent to
+efficiently learn a set of diverse BEHAVIOR activities from
+scratch. The state representation is also in a common embed-
+ding space to allow easy knowledge transfer to other activi-
+ties. In the second part, we introduce how these pre-trained
+policies are re-used for learning new activities, and test our
+hypothesis that the semantic similarity between activity de-
+scriptions can be used to predict transfer performances.
+Learning Single Activities
+We introduce a different RL formulation from the original
+one in the BEHAVIOR benchmark, in order to speed up
+learning these activities using standard RL algorithms.
+Text-Based State and Goal Representation
+Given the
+low RL performance in the original setting of BEHAVIOR,
+we take a similar approach to ALFWORLD (Shridhar et al.
+2021) by providing full observability of the logical state
+in the form of language. The simulator backbone of BE-
+HAVIOR extracts logical predicates that describe the current
+states and relations of all objects in the world. We filter the
+logical predicates to the ones relevant to the activity, and use
+a template to generate text descriptions of the logical state.
+Similarly, the goal conditions are represented with text de-
+scriptions. Figure 1 shows the initial state for one instance
+of the cleaning kitchen cupboard activity. Figure 2 shows
+the goal definition of the cleaning kitchen cupboard activity.
+There are two goals: 1) dust every cabinet and 2) move all
+cups to one cabinet and all bowls to the other. For the exam-
+ple initial state, there are two ways to ground the second goal
+based on how the cups and bowls are assigned to cabinets,
+and each grounding leads to a distinct set of subgoals.
+Action Primitives
+The action space includes a set of dis-
+crete action primitives implemented in BEHAVIOR: GRASP,
+TOGGLE ON, TOGGLE OFF, OPEN, CLOSE, PLACE INSIDE,
+PLACE ON TOP. Each action primitive takes a parameter that
+refers to an object. For example, PLACE INSIDE(cabinet 0)
+means the robot will put the object currently in its gripper
+into the cabinet.
+Problem Formulation
+We formulate a BEHAVIOR ac-
+tivity as a Markov Decision Process denoted by the tuple
+M = (S, A, P, R). S is the space that consists of tok-
+enized state and goal descriptions. A is the space of action
+top cabinet 47 is dusty. top cabinet 47 is next to cup 1. bot-
+tom cabinet 41 is dusty. bottom cabinet 41 is on top cup 0.
+bottom cabinet 41 is next to cup 0. bottom cabinet 41 is
+next to bowl 1. countertop 26 is under bath towel 0. coun-
+tertop 26 is in reach of robot. countertop 26 is in same room
+as robot. bath towel 0 is on top countertop 26. bath towel 0
+is in reach of robot. soap 0 is on top countertop 26.
+soap 0 is in reach of robot. bowl 0 is on top counter-
+top 26. bowl 0 is in reach of robot. bowl 1 is inside
+bottom cabinet 41. bowl 1 is next to bottom cabinet 41.
+cup 0 is inside bottom cabinet 41. cup 0 is next to bot-
+tom cabinet 41. cup 1 is inside top cabinet 47. cup 1 is next
+to top cabinet 47. room floor kitchen 0 is in reach of robot.
+room floor kitchen 0 is in field of view of robot.
+Figure 1: An example initial state of cleaning kitchen cup-
+board
+For every cabinet, the following is NOT true:
+the cabinet is dusty.
+For at least one cabinet, for every bowl, the bowl is inside
+the cabinet, and the following is NOT true:
+cup1 is inside the cabinet.
+For at least one cabinet, for every cup, the cup is inside the
+cabinet, and the following is NOT true:
+bowl1 is inside the cabinet.
+Figure 2: An example goal definition of cleaning kitchen
+cupboard
+primitives, parameterized by the objects relevant to the ac-
+tivity. P(·|s, a) is the unknown stochastic transition prob-
+abilities. R : S × A × S → R is the reward function.
+Given the grounded subgoals of the activity, R is defined
+as follows: if a is not executable at s, R(s, a, s′) = −1; oth-
+erwise, let g(s) be the number of subgoals satisfied in the
+state s, R(s, a, s′) =
+g(s′)−g(s)
+total number of subgoals · c where c
+is a large constant. The reward function penalizes choosing
+action primitives that are not executable, such as TOGGLE
+OFF(cup 0), and generously rewards achieving new sub-
+goals. The objective is to learn a policy π : S → A that
+maximizes the expected total reward.
+Actor-Critic Policy
+The policy can be trained by pol-
+icy gradient methods such as PPO (Schulman et al. 2017).
+Figure 3 shows the actor-critic architecture. We use a pre-
+trained DistilBert model (Sanh et al. 2020) to tokenize and
+encode the input text. The actor network outputs a tuple of
+the action primitive index and the object index.
+Transfer Learning
+Since the aim of this work is not to achieve top performances
+on BEHAVIOR, but rather to explore the connection be-
+tween transfer performance and activity similarity, we adopt
+a straightforward method to re-use pre-trained policies and
+compare the learning curves.
+
+Figure 3: Actor-critic network architecture for learning one
+BEHAVIOR activity.
+State and Action Mappings
+Since S is a space of tok-
+enized state and goal descriptions, the state space is common
+for all activities. However, the action primitives are param-
+eterized by the objects in the scene, so the action space can
+have different sizes. To re-use a policy for a new activity, we
+copy all the weights in the network (Figure 3) except for the
+actor output layer. Then we resize the actor output layer to
+match the new action space and randomly initialize it before
+training.
+Semantic Similarity
+Given a new activity with an initial
+state and a set of goal conditions, the text-based state and
+goal representation constructed for the MDP formulation is
+also a unique description of this activity. We use the pre-
+trained SimCSE model (Gao, Yao, and Chen 2022) to embed
+activity descriptions, and compute the consine similarity be-
+tween the embeddings of any pair of activities.
+Transfer Metric
+We evaluate the transfer performance of
+each pair of activities by the transfer ratio (or transfer score)
+metric (Taylor and Stone 2009; Rusu et al. 2016). The trans-
+fer ratio measures the ratio of the total reward given to the
+transfer learner and the total reward given to the non-transfer
+learner after a certain number of training steps. It can be
+computed by the ratio of the area under the transfer learning
+curve over the area under the non-transfer learning curve.
+Experiments
+We choose to study 7 activities from BEHAVIOR: storing
+food, cleaning kitchen cupboard, putting away Halloween
+decorations, collect misplaced items, putting away cleaned
+dishes, locking every window, cleaning microwave oven.
+The policies are trained with the PPO algorithm as imple-
+mented in the stable-baselines3 library (Raffin et al. 2021).
+An episode terminates when all the subgoals are achieved
+or the maximum number of steps (64) has been taken. The
+hyperparameter c in the reward function is set to 200. As
+a result, the highest total reward of an episode is 200, i.e.
+achieving all subgoals without any penalty. The lowest total
+Figure 4: Semantic similarities between source and target
+activities.
+reward is -64, i.e. always executing invalid actions.
+Training from Scratch
+To obtain a policy for each activ-
+ity, we train for 512 episodes and take the top performing
+policy out of 3 runs. Table 1 shows the mean reward per
+episode achieved at the end of training by the top policy for
+each activity. Note that there is a wide gap between how
+well these activities are solved by our policies. The policies
+for locking every window and cleaning microwave oven are
+near optimal, whereas the policy for cleaning kitchen cup-
+board never manages to achieve all subgoals during training.
+This difference is due to the solution length and the stochas-
+ticity of executing the action primitives. Some activities re-
+quire executing more than 10 actions in the correct order,
+and some actions (e.g. grasp) have a low success rate in pro-
+ducing the desired effects. The uncertain action effects re-
+flect the challenge for real robots, since the task-level policy
+should know how to recover when there are failures during
+execution.
+Since it’s much faster to learn window and microwave
+than the other activities, they are only used as source tasks
+but not target tasks in the transfer experiments below.
+Semantic Similarity
+Figure 4 summarizes the semantic
+similarity in a matrix. Each row is a source activity and
+each column is a target activity. A high number (or warm
+color) means the descriptions of the two activities are close
+in the embedding space, whereas a low number (or cool
+color) indicates that the embeddings are distant. It may not
+be intuitive why some activities are more similar than oth-
+ers based on their abbreviated names. For example, stor-
+ing food, cleaning kitchen cupboard, putting away dishes,
+putting away Halloween decorations all involve moving ob-
+
+Grasp cup_o
+(0,
+5)
+123.456
+ActionOutput Layer
+Value Output Layer
+Actor Layer 3
+Critic Layer 3
+Actor Layer 2
+Critic Layer 2
+Actor Layer 1
+Critic Layer 1
+Features Extractor (DistilBert Encoder
+Tokenizedtextobservationstarget
+source
+food
+cupboard
+halloween
+ misplaced
+dishes
+0.5
+window
+0.19
+0.35
+0.18
+0.07
+0.24
+0.3
+0.37
+0.25
+0.1
+0.25
+ 0.4
+microwave
+X
+poor
+0.39
+0.37
+0.14
+0.39
+0.3
+X
+cupboard
+0.39
+0.3
+0.07
+0.43
+X
+0.2
+halloween
+0.37
+0.3
+0.2
+0.4
+X
+misplaced
+0.14
+0.07
+0.2
+0.19
+0.1
+X
+dishes
+0.39
+0.43
+0.4
+0.19food
+cupboard
+halloween
+misplaced
+dishes
+window
+microwave
+-8.5
+-34.5
+1.1
+4.0
+-7.0
+196.0
+189.0
+Table 1: Mean reward per episode achieved at the end of training.
+Figure 5: Transfer ratios of the first 80 episodes.
+jects into cabinets, so their similarity scores are high when
+taking into account the full descriptions.
+Transfer Ratios
+Figure 5 presents the transfer ratio ma-
+trix after 80 episodes (or about 5000 steps). A ratio above
+1 indicates positive transfer, i.e. the transfer learner receives
+higher total reward during training. Comparing with the sim-
+ilarity score matrix, we can make two observations. First, a
+high-quality source policy can lead to positive transfer, even
+if the activity is not similar. The activities storing food and
+putting away Halloween decorations (two difficult tasks) are
+not similar to locking every window or cleaning microwave
+oven (two easy tasks), but we see high transfer ratios in the
+first two rows of their columns. Second, for each target ac-
+tivity, higher semantic similarity has a higher chance of pos-
+itive transfer. Cleaning kitchen cupboard and putting away
+cleaned dishes have a high semantic similarity (0.43). The
+only positive transfer to cupboard was from dishes and vice
+versa. On the other hand, collecting misplaced items is se-
+mantically very different from all other activities, and gets
+some of the worst transfer ratios.
+Catastrophic Forgetting
+While there are clear signs that
+re-using policies can jump start learning a new activity, the
+benefits of transfer quickly disappear as catastrophic forget-
+ting takes place. Figure 6 shows the transfer ratios after 160
+episodes (or about 10,000 steps). The general observations
+in Figure 5 still hold, but the ratios are getting lower and
+Figure 6: Transfer ratios of the first 160 episodes.
+there are fewer cases of positive transfer.
+For future studies, one of the ideas to transfer knowl-
+edge without suffering from the conflicting goals is by de-
+coupling the task-independent knowledge from the task-
+dependent knowledge. In the case of household activities,
+there is a lot of shared knowledge across activities, espe-
+cially the preconditions and effects of actions. For example,
+TOGGLE OFF(cup 0) is an invalid action in any activity. To
+this end, successor features (Barreto et al. 2017) and uni-
+versal value function approximation (Schaul et al. 2015) are
+both methods to learn representations that decouple the dy-
+namics from the rewards so they will generalize over differ-
+ent goals. Meanwhile, there are neural representations de-
+signed to avoid catastrophic forgetting. Progressive neural
+nets (Rusu et al. 2016) add a new column of network while
+preserving the weights learned in previous tasks.
+Conclusion
+We propose that home robots can efficiently learn novel
+household tasks from similar but distinct activities, and
+present our analysis in the BEHAVIOR benchmark. Our ex-
+periments show encouraging results: activity similarity mea-
+sured by language embeddings can be used as a predictor for
+transfer performance, and a high-quality source policy of an
+easy but different activity can sometimes lead to a jump-
+start. We also observe the problem of catastrophic forgetting
+and suggest future research in this direction.
+
+target
+food
+cupboard halloween misplaced
+source
+dishes
+2.00
+window 
+2.24
+0.78
+1.52
+0.30
+0.64
+1.75
+microwave
+6.61
+0.52
+2.08
+1.15
+0.75
+1.50
+pooy
+X
+0.81
+1.01
+0.71
+0.54
+1.25
+0.70
+X
+cupboard
+1.00
+0.32
+1.09
+1.00
+2.39
+0.29
+X
+halloween
+0.28
+0.58
+0.75
+X
+0.50
+misplaced
+1.43
+0.33
+0.79
+0.74
+0.25
+1.78
+1.26
+1.30
+0.19
+X
+dishes
+0.00target
+source
+food
+cupboard halloween misplaced
+dishes
+2.00
+window
+1.08
+0.76
+0.92
+0.43
+0.48
+1.75
+microwave
+3.28
+0.32
+1.33
+0.47
+0.51
+1.50
+food
+X
+0.76
+0.67
+0.74
+0.62
+1.25
+X
+cupboard
+0.88
+0.72
+0.40
+1.09
+1.00
+X
+1.76
+0.46
+0.40
+0.62
+0.75
+halloween
+X
+0.50
+misplaced
+1.10
+0.23
+0.65
+0.84
+0.25
+1.40
+1.00
+0.21
+X
+dishes
+0.87
+0.00References
+Barreto, A.; Dabney, W.; Munos, R.; Hunt, J. J.; Schaul, T.;
+van Hasselt, H. P.; and Silver, D. 2017. Successor Features
+for Transfer in Reinforcement Learning.
+In Advances in
+Neural Information Processing Systems, volume 30. Curran
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+Fernando, C.; Banarse, D.; Blundell, C.; Zwols, Y.; Ha, D.;
+Rusu, A. A.; Pritzel, A.; and Wierstra, D. 2017. Pathnet:
+Evolution Channels Gradient Descent in Super Neural Net-
+works. arXiv preprint arXiv:1701.08734.
+Gao, T.; Yao, X.; and Chen, D. 2022.
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+ALFWorld: Aligning
+Text and Embodied Environments for Interactive Learning.
+arXiv:2010.03768.
+Soemers, D. J. N. J.; Mella, V.; Piette, E.; Stephenson, M.;
+Browne, C.; and Teytaud, O. 2021. Transfer of Fully Convo-
+lutional Policy-Value Networks Between Games and Game
+Variants. arXiv:2102.12375.
+Srivastava, S.; Li, C.; Lingelbach, M.; Mart´ın-Mart´ın, R.;
+Xia, F.; Vainio, K.; Lian, Z.; Gokmen, C.; Buch, S.; Liu,
+C. K.; Savarese, S.; Gweon, H.; Wu, J.; and Fei-Fei, L. 2021.
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+ities in Virtual, Interactive, and Ecological Environments.
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+Learning in Deep Reinforcement Learning: A Survey.
+arXiv:2009.07888 [cs, stat].
+

AtE4T4oBgHgl3EQf5A4w/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf,len=504
+page_content='Language-Informed Transfer Learning for Embodied Household Activities  Yuqian Jiang 1*, Qiaozi Gao 2, Govind Thattai 2, Gaurav Sukhatme 2,3  1 The University of Texas at Austin, 2 Amazon Alexa AI, 3 University of Southern California  jiangyuqian@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='edu, {qzgao, thattg}@amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='com, gaurav@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='edu  Abstract  For service robots to become general-purpose in everyday  household environments, they need not only a large library  of primitive skills, but also the ability to quickly learn novel  tasks specified by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Fine-tuning neural networks on a va-  riety of downstream tasks has been successful in many vision  and language domains, but research is still limited on trans-  fer learning between diverse long-horizon tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' We propose  that, compared to reinforcement learning for a new household  activity from scratch, home robots can benefit from transfer-  ring the value and policy networks trained for similar tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  We evaluate this idea in the BEHAVIOR simulation bench-  mark which includes a large number of household activities  and a set of action primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' For easy mapping between state  spaces of different tasks, we provide a text-based representa-  tion and leverage language models to produce a common em-  bedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The results show that the selection of similar  source activities can be informed by the semantic similarity  of state and goal descriptions with the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' We further  analyze the results and discuss ways to overcome the problem  of catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Introduction  Domestic service robots have been envisioned to help in a  variety of household activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Imagine a single robot that  can be versatile enough from tidying up the rooms to play-  ing with kids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Such a robot not only requires the sensing,  navigation, and manipulation capabilities, but also needs to  intelligently combine these skills to perform each activity as  requested by the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Since every home is different, a simple library of pre-  programmed tasks will hardly serve the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' For ex-  ample, when a user wants to clean the kitchen cupboard,  the specific goal conditions they would like to achieve will  depend on their personal preferences and constraints of the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Does the robot re-arrange the dishes in a cer-  tain pattern?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Does the robot dust the outside of the cup-  board?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The reality is that there could be an infinite number  of combinations of goals, and a robot will most likely have  to learn to solve new goals after it is deployed in the individ-  ual homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  In this paper, we study the problem of learning novel  user-specified household activities for a service robot that  Work completed during an internship with Amazon Alexa AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  is shipped with pre-trained policies for a set of standard ac-  tivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' We propose to learn the new activity by transferring  from the policy of a similar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Our hypothesis is that  the transfer can be more efficient than learning the new activ-  ity from scratch if their initial state and goal conditions are  similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Intuitively, a robot should be able to learn putting  away cleaned dishes efficiently if it has a good policy for  cleaning kitchen cupboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Further, we can measure activity  similarities by leveraging language models to embed their  state and goal descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  We test our hypothesis using the BEHAVIOR bench-  mark (Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' BEHAVIOR simulates a large  number of household activities for an embodied AI to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  We first present a reinforcement learning (RL) approach to  solve a subset of activities from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The approach lever-  ages text descriptions of the agent’s current state and goal to  allow the policies to operate in a common state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' We  then initialize the learner with each of the pretrained policies  when training it on a new activity, and evaluate the hypothe-  sis that the transfer performance corresponds to the semantic  similarity between the activity text descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' We present  some initial results to show the potential of this approach for  enabling versatile and adaptive home robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Related Work  Transfer learning leverages the knowledge learned in a  source domain to improve the performance of a learner on  the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Transfer learning in reinforcement learn-  ing has been studied to transfer knowledge between different  Markov Decision Processes (MDPs) (Zhu, Lin, and Zhou  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Taylor and Stone 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' While many approaches are  evaluated in tasks with the same high-level goal and only  different configurations in Mujoco, navigation, and Atari do-  mains (Barreto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Schaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2015), a few re-  cent transfer learning approaches have demonstrated posi-  tive transfer between distinct Atari games (Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Fernando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Soemers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' introduces an approach  that transfers policy and value networks between distinct  board games that have different action spaces (Soemers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Encouraged by these successes, we propose to trans-  fer RL policies among distinct embodied household activi-  ties which require high-level long-horizon reasoning about a  large variety of goal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Further, this work proposes  to use language models on activity descriptions to inform the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='05318v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='RO]  12 Jan 2023  selection of source domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  BEHAVIOR is a benchmark where embodied AI so-  lutions are evaluated on household activities in a realis-  tic physics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The activities are selected from the  American Time Use Survey to reflect the real distribution of  household chores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' There has been very little success using  RL to solve BEHAVIOR in its original setting (Srivastava  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' In this paper, the method of providing the text-  based, fully observable state representation is most similar  to the work done by Shridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' for the ALFRED bench-  mark (Shridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Approach  Our approach consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' In the first part, we in-  troduce a text-based state representation for a RL agent to  efficiently learn a set of diverse BEHAVIOR activities from  scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The state representation is also in a common embed-  ding space to allow easy knowledge transfer to other activi-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' In the second part, we introduce how these pre-trained  policies are re-used for learning new activities, and test our  hypothesis that the semantic similarity between activity de-  scriptions can be used to predict transfer performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Learning Single Activities  We introduce a different RL formulation from the original  one in the BEHAVIOR benchmark, in order to speed up  learning these activities using standard RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Text-Based State and Goal Representation  Given the  low RL performance in the original setting of BEHAVIOR,  we take a similar approach to ALFWORLD (Shridhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  2021) by providing full observability of the logical state  in the form of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The simulator backbone of BE-  HAVIOR extracts logical predicates that describe the current  states and relations of all objects in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' We filter the  logical predicates to the ones relevant to the activity, and use  a template to generate text descriptions of the logical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Similarly, the goal conditions are represented with text de-  scriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Figure 1 shows the initial state for one instance  of the cleaning kitchen cupboard activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Figure 2 shows  the goal definition of the cleaning kitchen cupboard activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  There are two goals: 1) dust every cabinet and 2) move all  cups to one cabinet and all bowls to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' For the exam-  ple initial state, there are two ways to ground the second goal  based on how the cups and bowls are assigned to cabinets,  and each grounding leads to a distinct set of subgoals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Action Primitives  The action space includes a set of dis-  crete action primitives implemented in BEHAVIOR: GRASP,  TOGGLE ON, TOGGLE OFF, OPEN, CLOSE, PLACE INSIDE,  PLACE ON TOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Each action primitive takes a parameter that  refers to an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' For example, PLACE INSIDE(cabinet 0)  means the robot will put the object currently in its gripper  into the cabinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Problem Formulation  We formulate a BEHAVIOR ac-  tivity as a Markov Decision Process denoted by the tuple  M = (S, A, P, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' S is the space that consists of tok-  enized state and goal descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' A is the space of action  top cabinet 47 is dusty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' top cabinet 47 is next to cup 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bot-  tom cabinet 41 is dusty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bottom cabinet 41 is on top cup 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  bottom cabinet 41 is next to cup 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bottom cabinet 41 is  next to bowl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' countertop 26 is under bath towel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' coun-  tertop 26 is in reach of robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' countertop 26 is in same room  as robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bath towel 0 is on top countertop 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bath towel 0  is in reach of robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' soap 0 is on top countertop 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  soap 0 is in reach of robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bowl 0 is on top counter-  top 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bowl 0 is in reach of robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bowl 1 is inside  bottom cabinet 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' bowl 1 is next to bottom cabinet 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  cup 0 is inside bottom cabinet 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' cup 0 is next to bot-  tom cabinet 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' cup 1 is inside top cabinet 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' cup 1 is next  to top cabinet 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' room floor kitchen 0 is in reach of robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  room floor kitchen 0 is in field of view of robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Figure 1: An example initial state of cleaning kitchen cup-  board  For every cabinet, the following is NOT true:  the cabinet is dusty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  For at least one cabinet, for every bowl, the bowl is inside  the cabinet, and the following is NOT true:  cup1 is inside the cabinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  For at least one cabinet, for every cup, the cup is inside the  cabinet, and the following is NOT true:  bowl1 is inside the cabinet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Figure 2: An example goal definition of cleaning kitchen  cupboard  primitives, parameterized by the objects relevant to the ac-  tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' P(·|s, a) is the unknown stochastic transition prob-  abilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' R : S × A × S → R is the reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Given the grounded subgoals of the activity, R is defined  as follows: if a is not executable at s, R(s, a, s′) = −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' oth-  erwise, let g(s) be the number of subgoals satisfied in the  state s, R(s, a, s′) =  g(s′)−g(s)  total number of subgoals · c where c  is a large constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The reward function penalizes choosing  action primitives that are not executable, such as TOGGLE  OFF(cup 0), and generously rewards achieving new sub-  goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The objective is to learn a policy π : S → A that  maximizes the expected total reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Actor-Critic Policy  The policy can be trained by pol-  icy gradient methods such as PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Figure 3 shows the actor-critic architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' We use a pre-  trained DistilBert model (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2020) to tokenize and  encode the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The actor network outputs a tuple of  the action primitive index and the object index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Transfer Learning  Since the aim of this work is not to achieve top performances  on BEHAVIOR, but rather to explore the connection be-  tween transfer performance and activity similarity, we adopt  a straightforward method to re-use pre-trained policies and  compare the learning curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Figure 3: Actor-critic network architecture for learning one  BEHAVIOR activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  State and Action Mappings  Since S is a space of tok-  enized state and goal descriptions, the state space is common  for all activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' However, the action primitives are param-  eterized by the objects in the scene, so the action space can  have different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' To re-use a policy for a new activity, we  copy all the weights in the network (Figure 3) except for the  actor output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Then we resize the actor output layer to  match the new action space and randomly initialize it before  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Semantic Similarity  Given a new activity with an initial  state and a set of goal conditions, the text-based state and  goal representation constructed for the MDP formulation is  also a unique description of this activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' We use the pre-  trained SimCSE model (Gao, Yao, and Chen 2022) to embed  activity descriptions, and compute the consine similarity be-  tween the embeddings of any pair of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Transfer Metric  We evaluate the transfer performance of  each pair of activities by the transfer ratio (or transfer score)  metric (Taylor and Stone 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The trans-  fer ratio measures the ratio of the total reward given to the  transfer learner and the total reward given to the non-transfer  learner after a certain number of training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' It can be  computed by the ratio of the area under the transfer learning  curve over the area under the non-transfer learning curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Experiments  We choose to study 7 activities from BEHAVIOR: storing  food, cleaning kitchen cupboard, putting away Halloween  decorations, collect misplaced items, putting away cleaned  dishes, locking every window, cleaning microwave oven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  The policies are trained with the PPO algorithm as imple-  mented in the stable-baselines3 library (Raffin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  An episode terminates when all the subgoals are achieved  or the maximum number of steps (64) has been taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The  hyperparameter c in the reward function is set to 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' As  a result, the highest total reward of an episode is 200, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  achieving all subgoals without any penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The lowest total  Figure 4: Semantic similarities between source and target  activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  reward is -64, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' always executing invalid actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Training from Scratch  To obtain a policy for each activ-  ity, we train for 512 episodes and take the top performing  policy out of 3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Table 1 shows the mean reward per  episode achieved at the end of training by the top policy for  each activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Note that there is a wide gap between how  well these activities are solved by our policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The policies  for locking every window and cleaning microwave oven are  near optimal, whereas the policy for cleaning kitchen cup-  board never manages to achieve all subgoals during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  This difference is due to the solution length and the stochas-  ticity of executing the action primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Some activities re-  quire executing more than 10 actions in the correct order,  and some actions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' grasp) have a low success rate in pro-  ducing the desired effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The uncertain action effects re-  flect the challenge for real robots, since the task-level policy  should know how to recover when there are failures during  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Since it’s much faster to learn window and microwave  than the other activities, they are only used as source tasks  but not target tasks in the transfer experiments below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Semantic Similarity  Figure 4 summarizes the semantic  similarity in a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Each row is a source activity and  each column is a target activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' A high number (or warm  color) means the descriptions of the two activities are close  in the embedding space, whereas a low number (or cool  color) indicates that the embeddings are distant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' It may not  be intuitive why some activities are more similar than oth-  ers based on their abbreviated names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' For example, stor-  ing food, cleaning kitchen cupboard, putting away dishes,  putting away Halloween decorations all involve moving ob-  Grasp cup_o  (0,  5)  123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='456  ActionOutput Layer  Value Output Layer  Actor Layer 3  Critic Layer 3  Actor Layer 2  Critic Layer 2  Actor Layer 1  Critic Layer 1  Features Extractor (DistilBert Encoder  Tokenizedtextobservationstarget  source  food  cupboard  halloween  misplaced  dishes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='5  window  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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+page_content='4  microwave  X  poor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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+page_content='3  X  cupboard  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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+page_content='43  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='2  halloween  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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+page_content='4  X  misplaced  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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+page_content='1  X  dishes  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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+page_content='19food  cupboard  halloween  misplaced  dishes  window  microwave  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='5  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='0  196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='0  189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='0  Table 1: Mean reward per episode achieved at the end of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Figure 5: Transfer ratios of the first 80 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  jects into cabinets, so their similarity scores are high when  taking into account the full descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Transfer Ratios  Figure 5 presents the transfer ratio ma-  trix after 80 episodes (or about 5000 steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' A ratio above  1 indicates positive transfer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' the transfer learner receives  higher total reward during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Comparing with the sim-  ilarity score matrix, we can make two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' First, a  high-quality source policy can lead to positive transfer, even  if the activity is not similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The activities storing food and  putting away Halloween decorations (two difficult tasks) are  not similar to locking every window or cleaning microwave  oven (two easy tasks), but we see high transfer ratios in the  first two rows of their columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Second, for each target ac-  tivity, higher semantic similarity has a higher chance of pos-  itive transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Cleaning kitchen cupboard and putting away  cleaned dishes have a high semantic similarity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The  only positive transfer to cupboard was from dishes and vice  versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' On the other hand, collecting misplaced items is se-  mantically very different from all other activities, and gets  some of the worst transfer ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Catastrophic Forgetting  While there are clear signs that  re-using policies can jump start learning a new activity, the  benefits of transfer quickly disappear as catastrophic forget-  ting takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Figure 6 shows the transfer ratios after 160  episodes (or about 10,000 steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' The general observations  in Figure 5 still hold, but the ratios are getting lower and  Figure 6: Transfer ratios of the first 160 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  there are fewer cases of positive transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  For future studies, one of the ideas to transfer knowl-  edge without suffering from the conflicting goals is by de-  coupling the task-independent knowledge from the task-  dependent knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' In the case of household activities,  there is a lot of shared knowledge across activities, espe-  cially the preconditions and effects of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' For example,  TOGGLE OFF(cup 0) is an invalid action in any activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' To  this end, successor features (Barreto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2017) and uni-  versal value function approximation (Schaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2015) are  both methods to learn representations that decouple the dy-  namics from the rewards so they will generalize over differ-  ent goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Meanwhile, there are neural representations de-  signed to avoid catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Progressive neural  nets (Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' 2016) add a new column of network while  preserving the weights learned in previous tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  Conclusion  We propose that home robots can efficiently learn novel  household tasks from similar but distinct activities, and  present our analysis in the BEHAVIOR benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=' Our ex-  periments show encouraging results: activity similarity mea-  sured by language embeddings can be used as a predictor for  transfer performance, and a high-quality source policy of an  easy but different activity can sometimes lead to a jump-  start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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+page_content=' Trischler,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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+page_content='  ALFWorld: Aligning  Text and Embodied Environments for Interactive Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='  arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
+page_content='03768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE4T4oBgHgl3EQf5A4w/content/2301.05318v1.pdf'}
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BdFQT4oBgHgl3EQf9zeq/content/tmp_files/2301.13452v1.pdf.txt

@@ -0,0 +1,2126 @@
+Distribution of the number of pivots needed using Gaussian elimination with
+partial pivoting on random matrices∗
+John Peca-Medlin†
+Abstract. Gaussian elimination with partial pivoting (GEPP) remains the most common method to solve dense
+linear systems. Each GEPP step uses a row transposition pivot movement if needed to ensure the
+leading pivot entry is maximal in magnitude for the leading column of the remaining untriangularized
+subsystem. We will use theoretical and numerical approaches to study how often this pivot movement
+is needed. We provide full distributional descriptions for the number of pivot movements needed
+using GEPP using particular Haar random ensembles, as well as compare these models to other
+common transformations from randomized numerical linear algebra. Additionally, we introduce new
+random ensembles with fixed pivot movement counts and fixed sparsity, α. Experiments estimating
+the empirical spectral density (ESD) of these random ensembles leads to a new conjecture on a
+universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure
+on the complex unit disk that depends on α and is an interpolation of the uniform measure on the
+unit disk and the Dirac measure at the origin.
+Key words. Gaussian elimination, partial pivoting, butterfly matrices, Stirling numbers of the first kind, nu-
+merical linear algebra, universality
+AMS subject classifications. 60B20, 15A23, 65F99
+1. Introduction and background. Gaussian elimination (GE) is the most used method
+to solve linear systems
+(1.1)
+Ax = b
+for A ∈ Rn×n, and remains a staple of introductory linear algebra courses. If no leading prin-
+cipal minors of A vanish, then GE iteratively transforms (1.1) into two equivalent triangular
+systems, resulting in the factorization A = LU for L unipotent lower triangular and U upper
+triangular matrices using 2
+3n2(1 +o(1)) FLOPs. If A is nonsingular but does have a vanishing
+principal minor, then GE would need to be combined with a selected pivoting strategy to
+ensure GE can be continued at each intermediate step without encountering a zero pivot. The
+row and column movements used to ensure nonzero pivots would then result in additional
+permutation matrices P, Q for a modified GE factorization PAQ = LU. Even when pivoting
+is not necessary, it remains desirable for added computational stability for certain pivoting
+strategies.
+This includes GE with partial pivoting (GEPP), the most prominent pivoting
+strategy for dense linear systems, which is the default strategy used by MATLAB with its
+built-in lu function. GEPP uses only row permutations and so results in a final PA = LU
+factorization. (See Subsection 1.3 for relevant description of GE, while Section 3 provides
+further background for GEPP.)
+With high performance computing, choosing a desired pivoting strategy with GE often
+becomes a balancing act that takes into account the total computation time (viz., total FLOP
+∗Submitted to the editors February 1, 2023.
+†Department of Mathematics, University of Arizona, Tucson, AZ ([email protected]).
+1
+arXiv:2301.13452v1  [math.NA]  31 Jan 2023
+
+2
+J. PECA-MEDLIN
+count) rather than just accuracy (viz., the numerical stability of computed solutions). The
+cost of moving large amounts of data can be very expensive on high performance machines.
+For example, numerical experiments on a hybrid CPU/GPU setup have shown GEPP used
+with moderately sized random matrices have pivoting account for over 20 percent of the total
+computation time [1]. Hence, limiting pivot movements using GEPP is desirable to save time.
+Parker introduced a preconditioning method through the use of random butterfly matrices
+to remove the need for pivoting overall for any nonsingular linear system [16].
+Butterfly
+matrices are a recursively defined class of orthogonal matrices (see Section 4 for a full definition
+of random butterfly matrices) for which matrix-vector multiplication Ax is computed using
+3n log2 n FLOPs rather than the O(n2) FLOPs needed using a general dense matrix. Parker
+established for U, V iid random butterfly matrices, then Ax = b can be transformed into the
+equivalent system
+(1.2)
+UAy = Ub
+and
+x = V ∗y
+for which GE without pivoting (GENP) can be carried out with probability near 1. The above
+computations shows then transforming (1.1) int to (1.2) can be performed using O(n2 log2 n)
+FLOPs, and hence does not impact the leading order complexity of GE.
+In [17], Peca-Medlin and Trogdon further explored the numerical stability of using GE with
+a variety of pivoting strategies in addition to using randomized preconditioning methods,
+which included random butterfly and Haar orthogonal transformations.
+One output was
+certain preconditioners had the impact that running the preconditioner followed then by GE
+with another pivoting strategy could “upgrade” the pivoting strategy in terms of the numerical
+accuracy for the computed solution. For instance, even though GENP often leads to accuracy
+far from that achieved using GEPP or GE with complete pivoting (GECP), a preconditioned
+matrix using GEPP would lead to accuracy on par with using GECP. Adding one step of
+iterative refinement would further seal this alignment in accuracy.
+A natural direction that arose out of this previous analysis in [17] was to better understand
+how many actual pivot movements are needed with these different pivoting strategies on the
+preconditioned linear systems. The goal of this paper is to provide a clearer answer to this
+question with respect to GEPP. GEPP can use a pivot movement at each GE step, for up to
+n − 1 total pivot movements. So applying GEPP to a random linear system will result in the
+number of pivot movements being a random variable with support in 0, 1, . . . , n − 1. We will
+study the question of how much movement one should expect if they choose to use GEPP in
+conjunction with randomized preconditioning methods1.
+Our results include both theoretical and numerical approaches focusing on applying GEPP
+to several random matrix ensembles. The theoretical results rely on input matrices from Haar
+orthogonal and butterfly ensembles (see Subsection 1.2 for a review of Haar measures). Our
+numerical studies use further study these models in relation to other common transformations
+from randomized numerical linear algebra transformations, which expand studies in [17].
+1.1. Outline of paper. This paper is structured to explore the question of how much
+actual pivot movement is necessary when using GEPP with a variety of randomized precon-
+1This is a simpler focus than the more general study of the distribution of the GEPP permutation matrix
+factor.
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+3
+ditioning methods on particular nonsingular linear systems. The remainder of Section 1 will
+establish notation and preliminary background on GE (Subsection 1.3) and the Stirling-1
+distribution (Subsection 1.4). This includes connections to previous statistical results of dis-
+tributions using Stirling numbers of the first kind, as well as formally establishing standard
+statistics for this distribution that will be used for comparison in later numerical results in
+Section 6.
+Section 2 provides a statement of the main theoretical result, Theorem 2.1, that provides
+the full distribution for the number of GEPP pivot movements needed for particular random
+ensembles. This includes the distribution of pivot movements when
+I. the resulting GEPP permutation matrix factor P is uniformly distributed among the
+permutation matrices, which uses the Stirling-1 distribution, Υn, that satisfies
+(1.3)
+P(Υn = k) = |s(n, k)|
+n!
+for k = 1, 2, . . . , n where s(n, k) is the Stirling number of the first kind; and
+II. when GEPP is applied to a Haar-butterfly matrix, which results in a scaled Bernoulli
+distribution.
+The remainder of Section 2 provides results and implications of (I) in connection with QR
+factorizations and other standard Haar unitary models. The proof of Theorem 2.1 is postponed
+under Sections 3 and 4.
+Section 3 provides the necessary background to establish a proof of part (I) of Theo-
+rem 2.1. This includes introducing explicit decompositions of permutations in Sn, the group
+of permutations of n objects, that connect explicitly to GEPP permutation matrix factors as
+well as uniform sampling of Sn. Section 4 provides more background for P(B)
+N , the butterfly
+permutation matrices, and yields a proof for part (II) of Theorem 2.1. Additionally, explicit
+structural configurations of exact pivot movement locations of Haar-butterfly matrices are
+established, that yield a distribution on the pivot location configurations.
+Section 5 builds on top of Theorem 2.1 to introduce a new ensemble of random matrices
+that align with uniformly sampling from GLn(R)/Un(R), the left cosets of the group of non-
+singular upper triangular matrices in the general linear group. This ensemble can be used
+to sample from random ensembles with fixed pivot movement distributions. General random
+ensembles are introduced with fixed sparsity conditions. A new conjecture is provided for the
+asymptotic empirical spectral distribution for this generalized random ensemble with fixed
+sparsity that connects to and subsumes the famous circular law in random matrix theory.
+Section 6 uses numerical experiments to further explore the distribution of pivot move-
+ments needed using other transformations from randomized numerical linear algebra. These
+experiments focus on three initial models, two that need minimal GEPP pivot movements and
+one that require the maximal number of GEPP pivot movements. These approaches build
+on top of the numerical experiments used in [17], as well as connect to other random models
+used in earlier sections.
+1.2. Notation and preliminaries. For convenience, N will be reserved for powers of 2, with
+N = 2n. For A ∈ Fn×m where F = R or C, Aij denotes the entry in the ith row and jth column
+of A, while Aα,β will denote the submatrix of A with row indices α ⊂ [n] := {1, 2, . . . , n} and
+
+4
+J. PECA-MEDLIN
+β ⊂ [m]. Let ei denote the standard basis elements of Fn and Eij = eieT
+j , the standard basis
+elements of Fn×m. I denotes the identity matrix and 0 the zero matrix or vector (with the
+dimensions implicit from context if not stated explicitly). If A ∈ Fn×n is nonsingular, then
+A0 := I. Let D = {z ∈ C : |z| < 1} denote the unit complex disk, with ∂D denoting the unit
+complex circle. We will write Sn−1 = {x ∈ Fn : ∥x∥2 = 1}, where ∥ · ∥2 denotes the standard
+ℓ2-norm.
+Let Sn denote the symmetric group on n elements. Recall every permutation σ ∈ Sn can
+be written in cycle notation, with σ = τ1τ2 · · · τj where τi = (ai1 ai2 · · · aik) is a k-cycle, such
+that τi(aim) = aim+1 for m < k and τi(aik) = ai1. Moreover, recall every permutation can be
+written as a product of disjoint cycles. For σ ∈ Sn, let Pσ denote the orthogonal permutation
+matrix such that Pσei = eσ(i). For example, for (1 2) ∈ S2, then
+(1.4)
+P(1 2) =
+�0
+1
+1
+0
+�
+.
+Let Pn denote the n×n permutation matrices, i.e., the left regular representation of the action
+of Sn on [n].
+Let ∥·∥max denote the element-wise max norm of a matrix defined by ∥A∥max = maxi,j |Aij|.
+Define A ⊕ B ∈ F(n1+m1)×(n2+m2) to be the block diagonal matrix with blocks A ∈ Fn1×m1
+and B ∈ Fn2×m2. Define A ⊗ B ∈ Fn1n2×m1m2 to be the Kronecker product of A ∈ Rn1×m1
+and B ∈ Rn2×m2, given by
+(1.5)
+A ⊗ B =
+�
+��
+A11B
+· · ·
+A1,m1B
+...
+...
+...
+An1,1B
+· · ·
+An1,m1B
+�
+�� .
+Recall Kronecker products satisfy the mixed-product property: if all matrix sizes are compat-
+ible for the necessary matrix multiplications, then
+(1.6)
+(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD),
+i.e., the product of Kronecker products is the Kronecker product of the products. As a result,
+Kronecker products inherit certain shared properties of their input matrices. For example, if
+A and B are both orthogonal or unitary matrices, then so is A ⊗ B. Similarly, if A ∈ Pn and
+B ∈ Pm then A ⊗ B ∈ Pnm.
+Let GLn(F) denote the group of nonsingular matrices with entries in F. Let Un(F) denote
+the subgroup of nonsingular upper triangular matrices and Ln(F) denote the subgroup of
+unipotent (i.e., with all diagonal entries equal to 1) lower triangular matrices. O(n) and U(n)
+denotes the orthogonal and unitary groups of n × n matrices and SO(n), SU(n) denote the
+respective special orthogonal and special unitary subgroups; note O(n) will be used for the
+orthogonal matrices while O(n) is the classical “big-oh” notation. Recall if H is a subgroup
+of G, then G/H = {xH : x ∈ G} will denote the set of left-cosets of H in G and G\H =
+{Hx : x ∈ G} the set of right-cosets of H in G.
+We write X ∼ Y if X and Y are equal in distribution.
+Standard distributions that
+will be used in this document include X ∼ N(0, 1) to denote a standard Gaussian random
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+5
+variable (with probability density (2π)−1/2e−x2/2); X ∼ NC(0, 1) to denote a standard complex
+Gaussian random variable (with X ∼ (Z1 + iZ2)/
+√
+2 for Z1, Z2 iid N(0, 1)); X ∼ Uniform(A)
+to denote a uniform random variable with support on a compact set A with probability
+density
+1
+|A|1A (for |A| either denoting the cardinality of A if A is finite or the corresponding
+appropriate Lebesgue-measure of A); ξ ∼ Bernoulli(p) to denote a Bernoulli random variable
+with parameter p ∈ [0, 1] where P(ξ = 1) = p = 1 − P(ξ = 0); and ξ ∼ Rademacher
+to denote a Rademacher random variable that takes only the values 1 and −1 with equal
+probability (i.e., ξ ∼ (−1)Bernoulli(1/2)). A random variable is called continuous if its associated
+probability density is a continuous function. Let Gin(n, m) denote the n×m Ginibre ensemble,
+consisting of random matrices with independent and identically distributed (iid) standard
+Gaussian entries; GinC(n, m) will denote the similarly defined complex Ginibre ensemble,
+whose entries are iid standard complex Gaussian random variables. Let GOE(n) and GUE(n)
+denote the Gaussian Orthogonal and Gaussian Unitary Ensembles, respectively; recall these
+can be sampled using the Ginibre ensembles as follows: if G ∼ Gin(n, n) and H ∼ GinC(n, n),
+then (G + GT )/
+√
+2 ∼ GOE(n) and (H + H∗)/
+√
+2 ∼ GUE(n).
+Let ϵmachine denote the machine-epsilon, which is the minimal positive number such that
+fl(1 + ϵmachine) ̸= 1 using floating-point arithmetic.2 If using t-bit mantissa precision, then
+ϵmachine = 2−t. Our later experiments in Section 6 will use double precision in MATLAB,
+which uses a 52-bit mantissa.
+Standard models from randomized numerical linear algebra will be used for comparison in
+Section 6. These will include the Walsh transformation and Discrete Cosine Transformations
+(DCT), which were previously used in [17]. Sampling for the following experiments will use
+native (deterministic) MATLAB functions (viz., the Fast Walsh-Hadamard transform fwht
+and the default Type II Discrete cosine transform dct) applied after an independent row
+sign transformation chosen uniformly from {±1}N. See [20, 24] for an overview of numerical
+properties of the Walsh and DCT transforms, and [13] for a thorough survey that provides
+proper context for use of these transforms and other tools from randomized numerical linear
+algebra.
+Additionally, we will utilize left and right invariance properties of the Haar measure on
+locally compact Hausdorff topological groups, first established by Weil [25]. For a compact
+group G, this measure can be normalized to yield a probability measure Haar(G), which in-
+herits the invariance and regularity properties of the original measure and yields a means
+to uniformly sample from compact groups, such as O(n) and SO(N). Recall every nonsin-
+gular matrix A ∈ Fn×n has a QR factorization, with A = QR for R upper triangular with
+positive diagonal entries and Q ∈ O(n) if F = R or Q ∈ U(n) if F = C. Stewart provided
+an outline to sample from Haar(O(n)) by using Gin(n, n) through the QR factorization: if
+A ∼ Gin(n, n) and A = QR is the QR decomposition of A where R has positive diagonal
+entries, then Q ∼ Haar(O(n)) [19]. Similarly, Haar(U(n)) can be sampled using GinC(n, n).
+Our experiments will employ efficient sampling methods for Haar(O(n)) that use Gaussian
+Householder reflectors, in line with the QR factorization of Gin(n, n) (see [15] for an outline
+of this method).
+2We will use the IEEE standard model for floating-point arithmetic.
+
+6
+J. PECA-MEDLIN
+1.3. Gaussian elimination and growth factors. GENP iteratively works through the bot-
+tom right untriangularized n − k + 1 dimensional submatrices of the GE transformed matrix
+A(k) to result in the factorization A = LU for L a unipotent lower triangular matrix and U
+an upper triangular matrix. A(k) represents the resulting transformed matrix of A at the kth
+GE step that is zero below the first k − 1 diagonals and
+(1.7)
+Lij =
+A(j)
+ij
+A(j)
+jj
+for i > j, with A(1) = A and A(n−1) = U. When GENP can be completed (viz., when all
+leading principal minors are nonzero), the final factorization A = LU can be reused with
+different input b to solve the computationally simpler triangular systems
+(1.8)
+Ly = b
+and
+Ux = y.
+Moreover, if A has nonvanishing principal minors, then the resulting LU factorization is
+unique. See standard references, such as [10], for an explicit outline of GE.
+If GENP cannot be completed, then a pivoting strategy can be applied so that GE can
+continue at each step, which can involve row or column movements that ensure the leading
+diagonal entry (i.e., the pivot) of the untriangularized subsystem is nonzero. Different pivot-
+ing strategies then result in the modified GE factorization PAQ = LU for P, Q permutation
+matrices. GEPP remains the most popular pivoting strategy, which uses only row permu-
+tations to ensure the leading pivot at the kth GE step is maximal in magnitude among the
+lower entries in its column. By construction, the L from the resulting GEPP factorization
+PA = LU satisfies ∥L∥max = 1. If there is ever a “tie” during an intermediate GEPP pivot
+search, which occurs when |Aj
+ij| = |Aj
+jj| and would result in |Lij| = 1 for some i > j, then
+the L and U factors are not unique with respect to row transformed linear systems, i.e., if A
+has the GEPP factorization PA = LU and B = QA for Q a permutation matrix, then we do
+not necessarily have the GEPP factorization (PQT )B = LU. When ties are avoided, GEPP
+results in unique L and U factors.
+Theorem 1.1 ([17]).
+Let A be a nonsingular square matrix. Then the L and U factors in
+the GEPP factorization PA = LU are invariant under row permutations on A iff |Lij| < 1
+for all i > j.
+Moreover, when no ties are encountered with a nonsingular A with B = QA defined as above,
+then GEPP does necessarily result in the factorization (PQT )B = LU.
+Even when pivoting is not necessary, pivoting can remain desirable for its numerical stabil-
+ity properties when using floating-point arithmetic. Wilkinson first established the backward
+stability of GEPP by showing the growth factor,
+(1.9)
+ρ(A) = maxk ∥A(k)∥max
+∥A∥max
+,
+satisfies the upper exponential bound 1 ≤ ρ(A) ≤ 2n−1 for all matrices A [26]. The growth
+factor controls the backwards relative error for computed solutions ˆx using GE, as Wilkinson
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+7
+further established through
+(1.10)
+∥ˆx − x∥∞
+∥x∥∞
+≤ 4n2κ∞(A)ρ(A)ϵmachine
+for κ∞(A) = ∥A∥∞∥A−1∥∞ the ℓ∞-condition number. Section 6 will consider particular linear
+models that maximize the GEPP growth factors.
+In practice, GEPP implementations result in computed solutions with higher accuracy far
+from the worst-case exponential behavior Wilkinson’s analysis first highlighted. Understand-
+ing this behavior remains an important question in numerical analysis. This was partially
+answered by Huang and Tikhomirov through the use of average-case analysis of GEPP using
+A ∼ Gin(n, n): they showed with probability near 1, both the number of bits of prevision
+needed to solve Ax = b to m bits of accuracy is m + O(log n) while also the computed and
+exact GEPP permutation matrix factors align [11].
+1.4. Stirling-1 distribution. This section will delve further into some properties of the
+Stirling-1 distribution, Υn, with probability mass function given by (1.3). Recall the Stirling
+numbers of the first kind, s(n, k), arise as the coefficients using the generating function
+(1.11)
+(x)n =
+n
+�
+k=0
+s(n, k)xk
+for (x)n = x(x − 1) · · · (x − n + 1), where s(n, k) = 0 if not 1 ≤ k ≤ n except s(0, 0) = 1.3 The
+absolute Stirling numbers of the first kind, |s(n, k)|, can similarly be generated using (1.11)
+along with |s(n, k)| = (−1)n+ks(n, k); alternatively, |s(n, k)| are determined by the generating
+function
+(1.12)
+⟨x⟩n =
+n
+�
+k=0
+|s(n, k)|xk
+for ⟨x⟩n = x(x + 1) · · · (x + n − 1). (1.12) further establishes the relation
+(1.13)
+|s(n, k)| = sn−k(1, 2, . . . , n − 1)
+where
+(1.14)
+sj(a1, a2, . . . , am) =
+�
+i1<···<ij
+j�
+ℓ=1
+aik
+denotes the elementary symmetric polynomials. This relationship can be used to establish the
+recurrence
+(1.15)
+|s(n, k)| = |s(n − 1, k − 1)| + (n − 1)|s(n − 1, k)|
+3The notation for Stirling numbers is inconsistent throughout the literature. We are adopting the convention
+used in [5].
+
+8
+J. PECA-MEDLIN
+for k > 0.
+Plugging x = 1 into (1.12) can be used to establish the identity
+(1.16)
+n! =
+n
+�
+k=1
+|s(n, k)|,
+which yields (1.3) denotes a valid probability density. An alternative justification for (1.16)
+follows from the standard interpretation that |s(n, k)| counts the number of permutations
+σ ∈ Sn that have exactly k cycles in their disjoint cycle decomposition, where fixed points
+are counted as 1-cycles4. This interpretation using Sn can be used to provide a combinatorial
+proof of (1.16): the left hand side of (1.16) is the number of elements of Sn, and the right
+hand side is the sum of each subset of permutations with a fixed number of k cycles for k
+ranging from 1 (viz., the n-cycles, of which there are |s(n, 1)| = (n−1)!) to n (viz., the identity
+permutation, in which each object comprises its own cycle of length 1, so that |s(n, n)| = 1).5
+Stirling numbers have appeared in connection with statistical problems dating back to
+their original formulation by Stirling in the 1730s (cf. [4]). Probabilistic tools have been
+used to establish and analyze properties of Stirling numbers in the mid- to late-20th century
+[3, 4, 9]. Υn has appeared as a variant of a more general ensemble of Stirling distributions
+but has not been studied extensively in the literature. For instance, the mean and variance
+have been computed for Υn (cf. [3]), but general higher moment computations have not been
+touched. Applying successive derivatives in x to (1.12) and then plugging in x = 16 yields
+EΥn = Hn
+and
+Var Υn = Hn − H(2)
+n ,
+(1.17)
+where
+(1.18)
+H(m)
+n
+=
+n
+�
+j=1
+1
+jm
+are the generalized Harmonic numbers and Hn = H(1)
+n
+the standard Harmonic numbers. Well
+known asymptotic results as n → ∞ using Harmonic numbers include Hn − log n → γ ≈
+4This correspondence is justified by noting
+�n
+k
+�
+, the number of permutations σ ∈ Sn with k disjoint cycles
+in their disjoint cycle decomposition, satisfies both the initial conditions along with the recurrence (1.15) (a
+combinatorial argument yields the analogous recurrence by separating whether 1 comprises its own cycle, which
+aligns with |s(n − 1, k − 1)|, or if 1 is contained in a larger cycle, which aligns with (n − 1)|s(n − 1, k)| since
+there are then n − 1 places to insert 1 into an existing cycle).
+5Similarly, the Stirling numbers of the second kind, S(n, k), can be defined as the number of partitions of
+n objects into k nonempty sets, which can be connected to Sn. S(n, k) and s(n, k) further are related as they
+yield the coordinates n, k for lower triangular n × n matrices that are mutual inverses (cf. pg. 144 in [5]).
+6For example, using
+d
+dx⟨x⟩n = ⟨x⟩n ·
+�n−1
+�
+j=0
+1
+x + j
+�
+=
+n
+�
+k=1
+|s(n, k)|kxk−1
+and then plugging in x = 1 yields n!Hn = �n
+k=1 k|s(n, k)| = n!EΥn.
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+9
+0.5772156649, the Euler–Mascheroni constant, and H(m)
+n
+→ ζ(m) for m > 1, where
+(1.19)
+ζ(m) =
+�
+n≥1
+1
+nm
+is the Riemann-zeta function, with ζ(2) = π2
+6 .
+Continuing with higher moment computations using (1.12) then yields EΥm
+n is a function
+of H(j)
+n
+for j = 1, 2, . . . , m. Moreover, after computing also the third and fourth moments
+of Υn, gathering all resulting terms that use only Hn = H(1)
+n
+results in a Bell polynomial of
+order m with inputs Hn for each component, i.e.,
+(1.20)
+EΥn mod (H(2)
+n , H(3)
+n , . . . , H(n)
+n ) = Bn(Hn, . . . , Hn).
+This aligns also with the above formulas for EΥn = Hn = B1(Hn) and EΥ2
+n = Var(Υn) +
+(EΥn)2 = H2
+n + Hn − H(2)
+n
+= B2(Hn, Hn) − H(2)
+n . A future research area can explore finding
+closed forms for the higher moments of Υn, including establishing (1.20) for higher orders, as
+well as other general Stirling distributions.
+1.5. Butterfly matrices. This section will define and introduce relevant background for
+butterfly matrices and random butterfly matrices, including Haar-butterfly matrices, Bs(N, ΣS).
+See [17, 23] for a fuller discussion of additional numerical and spectral properties of butterfly
+matrices and random butterfly matrices.
+The butterfly matrices of order N are an ensemble of special orthogonal matrices defined
+recursively as follows: let {1} as the order 1 butterfly matrices; for each order N > 1 butterfly
+matrix B, there exist order N/2 butterfly matrices A1, A2 and order N/2 symmetric matrices
+C, S such that CS = SC and C2 + S2 = IN/2 where
+(1.21)
+B =
+� C
+S
+−S
+C
+� �A1
+0
+0
+A2
+�
+=
+� CA1
+SA2
+−SA1
+CA2
+�
+.
+The simple butterfly matrices are formed such that A1 = A2 at each recursive step. The
+scalar butterfly matrices, B(N), are formed using (C, S) = (cos θI, sin θI) for some angle θ
+at each recursive step, where Bs(N) then denotes the simple scalar butterfly matrices. Note
+then each B ∈ B(N) is of the form
+(1.22)
+B = (B(θ) ⊗ I2)(A1 ⊕ A2)
+for
+(1.23)
+B(θ) =
+� cos θ
+sin θ
+− sin θ
+cos θ
+�
+the (counter-clockwise) rotation matrix, while each B ∈ Bs(N) can then be written of the
+form
+(1.24)
+B = B(θ) =
+n
+�
+j=1
+B(θn−j+1)
+
+10
+J. PECA-MEDLIN
+for θ ∈ [0, 2π)n. Note Bs(N) ⊂ B(N) ⊂ SO(N), with equality when N ≤ 2. While B(N) is
+not multiplicatively closed, Bs(N) forms a closed subgroup of SO(N) with
+(1.25)
+B(θ)B(ψ) = B(θ + ψ)
+and
+B(θ)−1 = B(−θ)
+for B(θ), B(ψ) ∈ Bs(N).
+Let Σ be a collection of dimension 2k pairs (Ck, Sk) of random symmetric matrices with
+CkSk = SkCk and C2
+k + S2
+k = I2k for k ≥ 1. We will write B(N, Σ) and Bs(N, Σ) to denote
+the ensembles of random butterfly matrices and random simple butterfly matrices formed by
+independently sampling (C, S) from Σ at each recursive step. Let
+ΣS = {(cos θ(k)I2k−1, sin θ(k)I2k−1) : θ(k) iid Uniform([0, 2π), k ≥ 1}
+and
+(1.26)
+ΣD = {
+2k−1
+�
+j=1
+(cos θ(k)
+j , sin θ(k)
+j ) : θ(k)
+j
+iid Uniform([0, 2π), k ≥ 1}.
+(1.27)
+A large focus for the remainder of this paper is on the Haar-butterfly matrices, Bs(N, ΣS),
+while numerical experiments in Section 6 will also use the other random scalar butterfly ensem-
+ble, B(N, ΣS), along with the random diagonal butterfly ensembles, B(N, ΣD) and Bs(N, ΣD).
+Since Bs(N) is a compact abelian group, it has a Haar measure that enables uniform sampling
+of its elements. The name of Haar-butterfly matrices for Bs(N, ΣS) is precisely because this
+construction aligns exactly with this Haar measure on Bs(N).
+Proposition 1.2 ([23]).
+Bs(N, ΣS) ∼ Haar(Bs(N))
+Using the mixed-product property, matrix factorizations of each Kronecker component
+lead to a matrix factorization of Kronecker products. In particular, this holds for the LU
+factorizations of Bs(N) using GENP and GEPP (see Proposition 4.1).
+In particular, the
+permutation matrix factors from the GEPP factorization of B ∈ Bs(N) are from the butterfly
+permutation matrices, P(B)
+N , which are defined recursively as P(B)
+2
+= {I2, P(1 2)} and
+(1.28)
+P(B)
+N
+=
+�
+�
+�
+n
+�
+j=1
+P ej
+(1 2) : ej ∈ {0, 1}
+�
+�
+� = P(B)
+2
+⊗ P (B)
+N/2
+for N > 2. These resulting permutations are explicit examples of perfect shuffles. See [6] for a
+thorough overview of perfect shuffles and some of their inherent applications and properties.
+The mixed-product property further yields P(B)
+N
+comprises a subgroup of permutation matrices
+that is isomorphic to (Z/2Z)n.
+Moreover, if B ∼ Bs(N, ΣS), then P ∼ Haar(P(B)
+N ) for
+PB = LU the GEPP factorization of B (see Corollary 4.2).
+2. Distribution of GEPP pivot movements for particular random matrices. We first will
+state the main theoretical result on the distribution of the number of GEPP pivot movements
+used that applies to particular input random matrix models. These will make use of Υn, the
+Stirling-1 distribution (see Subsection 1.4), and P(B)
+N , the butterfly permutation matrices (see
+Subsection 1.5).
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+11
+Theorem 2.1. (I) If A is a n × n random matrix with independent columns whose first
+n − 1 columns have continuous iid entries, then P ∼ Uniform(Pn) for PA = LU the GEPP
+factorization of A for n ≥ 2. Moreover, the number of GEPP pivot movements needed for A
+is equal in distribution to n − Υn.
+(II) If B ∼ Bs(N, ΣS), then P ∼ Uniform(P(B)
+N ) for PB = LU the GEPP factorization.
+Moreover, the number of GEPP pivot movements needed for B is equal in distribution to
+N
+2 Bernoulli
+�
+1 − 1
+N
+�
+.
+The proof of Theorem 2.1 will be postponed until Section 3 for (I) and Section 4 for (II).
+Part (I) gives the most general result that yields pivot movements following the Stirling-1
+distribution. This includes iid random matrices when the entry distribution is continuous.
+Corollary 2.2. If A is a n×n matrix with continuous iid entries, then the number of GEPP
+pivot movements needed for A is equal in distribution to n − Υn.
+If A is an iid matrix with entries taken from a distribution ξ with atoms (i.e., P(ξ = c) > 0
+for some c), then GEPP would yield a different distribution on the number of pivot movements.
+Example 2.3. If A is 2 × 2 where Aij are each iid Bernoulli(p), then GEPP yields the
+permutation matrix factor P = P ζ
+(1 2) where ζ ∼ Bernoulli(p(1 − p)). This follows since a
+pivot movement is needed only if A11 = 0 and A21 = 1. A pivot is needed with probability
+p(1 − p) ≤ 1
+4, so the number of GEPP pivot movements is equal in distribution to ζ.
+Other continuous random models that do not fit the conditions in (I) would yield different
+distributions on the resulting GEPP permutation matrix factors.
+Example 2.4. Consider G ∼ GOE(2), where G11, G22 ∼ N(0, 2) and G21 = G12 ∼ N(0, 1)
+are independent. The resulting number of GEPP pivot movements for G is equal in distribu-
+tion to Bernoulli(p) for
+p = P(|G21| > |G11|) = P(|Z1| >
+√
+2|Z2|) = P(Z2
+1/Z2
+2 > 2)
+= P(F1,1 > 2) = 1
+π
+� ∞
+2
+dx
+√x(1 + x) = 2
+π arctan
+� 1
+√
+2
+�
+≈ 0.391826552
+using Zi ∼ N(0, 1) iid and Fµ,ν denotes the F-distribution with µ > 0 numerator degrees of
+freedom (d.f.) and ν > 0 denominator d.f.
+Example 2.5. Consider now G ∼ GUE(2), where G11, G22 ∼ N(0, 1) while G12 = G21 ∼
+NC(0, 1), then the resulting number of GEPP pivot movements would similarly be equal in
+distribution to Bernoulli(q) for
+q = P(|G21| > |G11|) = P
+��
+(Z2
+1 + Z2
+2)/2 > |Z3|
+�
+= P((Z2
+1 + Z2
+2)/2 > Z2
+3)
+= P(F2,1 > 1) =
+� ∞
+1
+dx
+(1 + 2x)3/2 =
+1
+√
+3 ≈ 0.577350269
+Remark 2.6. In comparison to Examples 2.3 to 2.5, Corollary 2.2 yields if G is a continuous
+iid 2 × 2 matrix, then the number of GEPP pivots needed would be equal in distribution to
+2 − Υ2 ∼ Bernoulli(1
+2), where we note P(Υ2 = 1) = |s(2,1)|
+2!
+= 1
+2 = |s(2,2)|
+2!
+= P(Υ2 = 2).
+
+12
+J. PECA-MEDLIN
+A further result from Theorem 2.1 and Corollary 2.2 involves the relationship of the
+LU factorization of an iid matrix to its QR factorization. If A has the GEPP factorization
+PA = LU, then its QR factorization A = QR yields PQ = AR−1 = L(UR−1).7 In particular,
+the resulting permutation matrix and hence the pivot movements that would be needed using
+GEPP on Q and A are identical when no ties are encountered by Theorem 1.1. This obser-
+vation then can be combined with Stewart’s realization of Haar orthogonal and Haar unitary
+sampling using Ginibre ensembles (cf. Subsection 1.2) to yield:
+Corollary 2.7. If A ∼ Haar(O(n)) or A ∼ Haar(U(n)), then the number of GEPP pivot
+movements needed for A is equal in distribution to n − Υn.
+A similar approach can then yield information about the pivot movements needed on
+Haar(SO(n)) and Haar(SU(n)).
+Note Stewart’s sampling method for Haar(O(n)) can be
+indirectly rephrased in terms of the Subgroup algorithm, which was formally established by
+Diaconis and Shahshahani [7]. The Subgroup algorithm enables a uniform sampling method
+for a compact group by using a subgroup and its associated cosets:
+Theorem 2.8 (Subgroup algorithm,[7]). If G is a compact group, H is a closed subgroup of
+G, and H/G is the set of left-costs of H, then if x ∼ Uniform(G/H) and y ∼ Haar(H), then
+xy ∼ Haar(G).
+Theorem 2.8 can analogously be stated using right cosets.
+Stewart’s approach for sampling Haar orthogonal matrices can be realized in light of The-
+orem 2.8 by iteratively using Householder reflectors as coset representatives of the subgroup
+of orthogonal matrices whose first row and column have 1 in the leading diagonal and are
+zero elsewhere.8 More directly, one can then realize Haar(SO(n)) by using the coset repre-
+sentatives of O(n)/ SO(n) of D(x) = diag(x, 1, . . . , 1) for x = ±1: if A ∼ Haar(SO(n)) and
+x ∼ Uniform(±1), then D(x)A ∼ Haar(O(n)).9 Moreover, group homomorphisms can be used
+to yield uniform measures on cosets as push forward measures of Haar measures. In particular,
+since SO(n) is a normal subgroup of O(n) (since it is the kernel of the group homomorphism
+det : O(n) → R), then the natural quotient map C2 = {±1} ∼= O(n)/ SO(n) ∼= O(n)\ SO(n)
+yields C2 × SO(n) ∼= O(n). This yields uniform sampling from both C2 and SO(n) using
+push forwards of the Haar sampling on O(n) along with the natural projection maps to each
+component, which similarly holds using instead U(n) and SU(n):
+Corollary 2.9. x ∼ Uniform(±1) and A ∼ Haar(SO(n)) iff AD(x) ∼ Haar(O(n)), while
+y ∼ Uniform(T) and A ∼ Haar(SU(n)) iff AD(y) ∼ Haar(U(n)).
+Hence, mimicking the result aligning the GEPP permutation matrix factors between A and
+the Q from the A = QR factorization, we similarly have identical P factors for A ∼ Haar(O(n))
+and B ∼ Haar(SO(n)) where A = BD(x).10
+Corollary 2.10. If A ∼ Haar(SO(n)) or A ∼ Haar(SU(n)), then the number of GEPP pivot
+7Note UR−1 ∈ U(F, n) when U, R ∈ U(F, n) since U(F, n) is a group.
+8The Householder reflectors then can be uniformly sampled by choosing x ∈ Sn−1 uniformly.
+9Similarly Haar(U(n)) and Haar(SU(n)) can be related using the coset representatives D(x) for x ∈ T.
+10The argument from the paragraph before Corollary 2.7 is identical after replacing R ∈ U(F, n) with
+D(x) ∈ U(F, n).
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+13
+movements needed for A is equal in distribution to n − Υn.
+In [16, 17], the authors studied particular random preconditioning transformations of the
+form UAV ∗ for iid U, V so that the resulting output can almost surely (a.s.) have a LU
+factorization. In [17], the authors further studied the numerical properties for GE pivoting
+strategies, including GEPP, after applying these random preconditioning transformations.
+Relating this to our current topic, one could ask how many pivot movements are needed using
+GEPP after these random transformations?
+For the Haar orthogonal case (as well as the unitary, special orthogonal and special unitary
+cases), the result is immediate: Let A be a nonsingular real matrix. Suppose U, V are iid
+Haar(O(n)) and let B = AV ∗, which is independent of U. Let B = QR be the QR factorization
+of B. Then UAV ∗ = UB = U(QR) = (UQ)R. The permutation matrix factor resulting
+from GEPP for UAV ∗ is then identical (if no ties are encountered, which holds a.s.) to the
+permutation matrix factor needed for UQ. Since Haar(O(n)) is right invariant, then UQ ∼ U.
+This then yields the resulting number of pivots under this two-sided transformations is equal
+in distribution to the (I) case.
+Corollary 2.11. If A is a n×n nonsingular matrix, and U, V are iid from the Haar measure
+on O(n), U(n), SO(n) or SU(n), then the number of GEPP pivot movements needed for UAV ∗
+is equal in distribution to n − Υn.
+Remark 2.12. Extensions of (II) from Theorem 2.1 are less direct, since (II) relies heavily
+on explicit structural properties of Haar-butterfly matrices, Bs(N, ΣS). Analogous results can
+be established if the focus is restricted to matrices in
+n
+�
+F2×2.
+3. Permutations from GEPP and uniform sampling of Sn. Recall how the permuta-
+tion matrix factor is formed when applying GEPP to a matrix. First a search for a largest
+magnitude element is performed on the first column of A = A(1). After that value is found,
+say at index i1 (for i1 ≥ 1), then the corresponding row permutation for the transposition
+σ1 = (1 i1) is applied to A (using P (1) = Pσ1), after which the standard GE step follows to
+iteratively eliminate each element under the first diagonal value to form A(2) using the pivot
+element and the resulting lower triangular GE elimination factor ˜L(1,1), so that
+(3.1)
+A(2) = (L(1,1))−1P (1)A(1).
+Then GE continues with a search for the largest magnitude element on the leading column of
+A(2)
+2:n,2:n, which results in a transposition of the form σ2 = (2 i2) for i2 ≥ 2. The corresponding
+row permutation is performed (using P (2)) followed by the standard GE elimination step using
+the second pivot (using ˜L(2,2)), with which one groups the lower triangular and permutation
+matrix factors using the relation L(j,k) = P (j)L(j−1,k)P (j)11, so that
+(3.2)
+A(3) = (L(2,2))−1P (2)A(2) = (L(2,1)L(2,2))−1P (2)P (1)A(1).
+The process then continues, moving one column at a time, which results in the final GEPP
+factorization PA = LU for P = P (n−1) · · · P (2)P (1), L = L(n−1,n−1) · · · L(n−1,2)L(n−1,1) and
+U = A(n−1).
+11Note P (j) = (P (j))−1 since these permutation matrices correspond to transpositions that have order 2.
+
+14
+J. PECA-MEDLIN
+Hence, the resulting permutation matrix factor is built up step by step using σk = (k ik),
+resulting in
+(3.3)
+P = P (n−1) · · · P (2)P (1) = Pσn−1 · · · Pσ2Pσ1 = Pσn−1···σ2σ1 = Pσ
+for
+(3.4)
+σ = (n − 1 in−1) · · · (2 i2)(1 i1)
+where j ≤ ij ≤ n for each j.
+Remark 3.1. If ik = k, then (3.4) can abstain from including the trivial permutation (k ik).
+In particular, (3.4) can trivially be expanded to σ = (n in)σ where necessarily in = n.
+(3.4) is useful because every permutation can be put in this form.
+Lemma 3.2. Every permutation in Sn can be written in the form (3.4).
+In particular, every permutation is realizable as corresponding to the GEPP permutation
+matrix factor for some input nonsingular matrix
+Proof. By counting, it is clear n! inputs can be used to form σ (n choices for i1, n−1 choices
+for in−1, and so on). Moreover, we see this correspondence is one-to-one: suppose σ and σ′
+are formed using distinct inputs, and let k be the minimal index such that ik ̸= i′
+k. Without
+loss of generality, assume k = 1. Let ρ = σ(1 i1) and ρ′ = σ′(1 i1). Then ρ(1) = σ(i1) = 1
+while ρ′(1) = σ′(i1) > 1; it follows σ ̸= σ′, which yields this mapping is an injection and hence
+a bijection since |Sn| is finite.
+Alternatively, this can be realized through induction: this clearly holds for n = 2 since
+S2 = {1, (1 2)}. Now assume it holds for n − 1, then one can realize (using the inductive
+hypothesis) that Sn−1 ∼= {(n − 1 in−1) · · · (2 i2) : j ≤ ij ≤ n} as a subgroup of Sn, by
+recognizing the permutations that fix 1 in Sn is isomorphic to Sn−1. Moreover, adopting the
+crude labeling of Sn−1 for this subgroup of Sn, [Sn : Sn−1] = n!/(n−1)! = n, and every (right)
+coset representative for Sn−1 can be provided by Sn−1(1 i1) so that Sn = �n+1
+j=1 Sn−1(1 j).
+The coset decomposition structure for Sn used in the alternative proof of Lemma 3.2 was
+utilized by Diaconis and Shashahani through an application of the Subgroup algorithm for
+generating a Haar measure on Sn [7]. Their result yields a means to sample uniformly from
+Sn as the convolution of the Haar measure on Sn−1 and the uniform measure on {(1 j) : 1 ≤
+j ≤ n}. Explicitly, you can generate a uniform permutation σ ∈ Sn by first uniformly (and
+independently) sampling both ρ ∈ Sn−1 and σ1 ∈ {(1 j) : 1 ≤ j ≤ n}, and then forming
+σ = ρσ1. Iteratively applying this, one can uniformly sample a permutation by uniformly
+sampling each ik ∈ {k, k + 1, . . . , n}, which yields a permutation in the form (3.4).
+This
+establishes:
+Corollary 3.3. If ik ∼ Uniform{k, k + 1, . . . , n} for each k = 1, . . . , n − 1, then
+(3.5)
+(n − 1 in−1) · · · (2 i2)(1 i1) ∼ Uniform(Sn).
+Moreover, the steps where pivot movements occur can be read off directly from σ when
+it is written in the form (3.4). If Pσ is the resulting permutation matrix factor from applying
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+15
+GEPP to A, then we explicitly know at step k in GE, the pivot search on A(k) resulted in
+the transposition (k ik). This translates directly to how many pivot movements are needed
+through a particular implementation of GEPP by counting how many of these transpositions
+have ik > k. (So no pivot movement is needed if k = ik, since this translates to the leading
+pivot entry being maximal in its respective column.)
+It follows then, explicitly, if A is a random matrix such that the resulting GEPP permu-
+tation matrix factor P satisfies P ∼ Uniform(Pn) = Haar(Pn)12, then X, the number of pivot
+movements needed during this implementation of GEPP on A, necessarily satisfies
+(3.6)
+P(X = k) = #{σ ∈ Sn : j = ij for n − k indices j}
+|Sn|
+.
+Furthermore, for σ of the form (3.4), the number of times j > ij then corresponds precisely
+to the number of disjoint cycles in the representation of σ: iff ik = k, then k is contained
+in a cycle with elements no larger than k (each successive row transposition will fix k); so
+k is the maximal element in its cycle. (Recall if k is a fixed point of a permutation, then k
+comprises its own 1-cycle.) If ik > k, then k belongs to a cycle with a larger element. So
+counting the number of times ik = k is then equivalent to counting the number of disjoint
+cycles in the disjoint cycle decomposition of σ, since these can be enumerated by using their
+maximal elements.
+As is established in Subsection 1.4, these then align exactly with the
+absolute Stirling numbers of the first kind, |s(n, k)|.
+Example 3.4. Consider σ = (5 6)(3 4)(2 4)(1 3) ∈ S6. By default in = n, so i6 = 6 will
+be considered as not needing a pivot movement on the nth GE step (this is always vacuously
+true since GE completes after n − 1 steps). Here, we have only ik = k for k = 4 and k = 6, so
+we should expect σ consists of precisely 2 cycles in its disjoint cycle decomposition, and 4 and
+6 are the maximal elements in those two cycles. This is verified by finding the final disjoint
+cycle form of σ = (1 4 2 3)(5 6), for which 4 and 6 are the maximal elements of each of the
+disjoint cycles.
+Remark 3.5. Connecting the above conversation with the form (3.4), one can form a n-
+cycle in Sn by requiring ik > k for all k up to n − 1. Moreover, connecting this further to
+Corollary 3.3, one can uniformly sample n-cycles by sampling ik ∼ Uniform{k + 1, . . . , n} for
+k = 1, 2, . . . , n − 1. Let
+(3.7)
+Pn -cycles
+n
+= {Pσ : σ is a n-cycle in Sn}
+denote the subset of permutation matrices Pn that correspond to the n-cycles. If the cor-
+responding GEPP permutation matrix factor is in Pn -cycles
+n
+for a n × n matrix A, then ev-
+ery GE step required a pivot movement for A.
+Moreover, if ik ∼ Uniform{k + 1, . . . , n}
+for each k is used to generate the corresponding n-cycle σ = (n − 1 in−1) · · · (1 i1), then
+Pσ ∼ Uniform(Pn -cycles
+n
+).
+3.1. Proof of (I) of Theorem 2.1. Now we have the sufficient tools to establish:
+12This is equivalent to the statement P = Pσ for σ ∼ Uniform(Sn).
+
+16
+J. PECA-MEDLIN
+Theorem 2.1: (I) If A is a n × n random matrix with independent columns
+whose first n − 1 columns have continuous iid entries, then P ∼ Uniform(Pn)
+for PA = LU the GEPP factorization of A for n ≥ 2. Moreover, the number
+of GEPP pivot movements needed for A is equal in distribution to n − Υn.
+Proof. Suppose A satisfies the (I) hypothesis, and let P be the associated GEPP permu-
+tation matrix factor for A.
+We will prove P ∼ Uniform(Pn) using induction on n ≥ 2.
+Suppose n = 2, so the first column of A has continuous iid entries A11 and A21.
+Us-
+ing GEPP, a pivot will be needed only if |A21| > |A11|, but P(|A11| > |A21|) =
+1
+2 since
+A11 ∼ A21 (and P(|A11| = |A21|) = 0 since these are continuous random variables). Hence,
+P = P ζ
+(1 2) ∼ Haar(P2) for ζ ∼ Bernoulli(1
+2).
+Now assume the result holds for any random matrix of dimension n − 1 with independent
+columns whose first n − 2 columns have continuous iid entries.
+Let A be the dimension
+n matrix satisfying the statement. Using GEPP on A, for the first pivot search using the
+leading column of A = A(1), we have
+(3.8)
+P(max(|A11|, |A21|, . . . , |An1|) = |A11|) = P(max(|A11|, |A21|, . . . , |An1|) = |Ak1|)
+for each k = 1, 2, . . . , n since A11 ∼ Ak1 are continuous iid.
+It follows the first GEPP
+row transposition is of the form P (1) = Pσ1 for σ1 = (1 i1) for i1 ∼ Uniform{1, 2, . . . , n},
+where i1 = argmaxk |Ak1|. Now applying the GE elimination factor L(1,1) results in A(2) =
+(L(1,1))−1P (1)A(1). Moreover, ˜A(1) = P (1)A(1) still satisfies the (I) hypothesis since row per-
+mutations preserve each of the column independence and iid properties of A(1). A(2) then has
+entries of the form
+(3.9)
+A(2)
+ij = ˜Aij − ˜A1j · L(1)
+i1 = ˜Aij − ˜A1j ·
+˜Ai1
+˜A11
+.
+for i, j ≥ 2. In particular, since the columns are independent and have continuous iid entries,
+then L(1)
+i1
+is independent of ˜Aij for each i when j ≥ 2, while ˜A1j · L(1)
+i1
+is independent of
+˜Aij for each i ≥ 2, so that B = A(2)
+2:n,2:n also satisfies the (I) hypothesis.
+Now we can
+apply the inductive hypothesis to B to yield a GEPP factorization PB = LU where P ∼
+Uniform(Pn−1). Embedding Pn−1 into Pn using the map Q �→ 1 ⊕ Q yields then the resulting
+GEPP permutation matrix factor
+(3.10)
+Pσ = (1 ⊕ P)P (1)
+for A. Moreover, since P ∼ Uniform(Pn−1), then 1 ⊕ P = Pρ for ρ ∼ Uniform(Sn−1)13 while
+P (1) = Pσ1 for σ1 ∼ Uniform{(1 j) : j = 1, 2, . . . , n}, so that by the Subgroup algorithm
+we have σ = ρσ1 ∼ Uniform(Sn). It follows Pσ ∼ Uniform(Pn). This establishes the first
+statement part of (I) from Theorem 2.1.
+Now suppose X is the number of pivot movements needed using GEPP on A. The prior
+13Now associating Sn−1 with the isomorphic subgroup of Sn that fixes 1.
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+17
+conversation up to (3.6) establishes the correspondence
+P(X = n − k) = #{σ ∈ Sn : j = ij for k indices j}
+n!
+= #{σ ∈ Sn : σ has k cycles in its disjoint cycle decomposition}
+n!
+= |s(n, k)|
+n!
+= P(Υn = k)
+for k = 1, 2, . . . , n, which yields the desired result n − X ∼ Υn.
+4. Butterfly permutations. Using the Kronecker product factorization for B(θ) ∈ Bs(N)
+(where N = 2n) along with the mixed-product property, then matrix factorizations of each
+Kronecker component yield the matrix factorization of the resulting butterfly matrix. This was
+used in [17] to yield both the eigenvalue (Schur) decomposition as well as the LU factorizations
+of scalar simple butterfly matrices, Bs(N). For completeness, we will restate this latter result:
+Proposition 4.1 ([17]).
+Let B = B(θ) ∈ Bs(N).
+(I) If cos θi ̸= 0 for all i, then B has the GENP factorization B = LθUθ where Lθ =
+�n
+j=1 Lθn−j+1 and Uθ = �n
+j=1 Uθn−j+1 for
+(4.1)
+Lθ =
+�
+1
+0
+− tan θ
+1
+�
+and
+Uθ =
+�cos θ
+sin θ
+0
+sec θ
+�
+.
+(II) Using θ ∈ [0, 2π)n, let
+(4.2)
+ej =
+� 1
+if | tan θj| ≤ 1,
+0
+if | tan θj| > 1
+for each j. Let θ′ ∈ [0, 2π)n be such that θ′
+j = π
+2 ej + (−1)ejθj = θjej + ( π
+2 − θj)(1 − ej)
+for each j. If | tan θj| ̸= 1 for any j, then the GEPP factorization of B is PB = LU where
+P = Pθ = �n
+j=1 Pθn−j+1, L = Lθ′, and U = Uθ′Dθ for
+(4.3)
+Pθ = P ej
+(1 2)
+and
+Dθ = (−1)1−ej ⊕ 1.
+Moreover, (PB)(k) = B(θ′)(k)Dθ for all k where B(θ′) ∈ Bs(N).
+In particular, P ∈ P(B)
+N
+(cf. (1.28)) for PB = LU the GEPP factorization of B ∈ Bs(N).
+Note if θ ∼ Uniform([0, 2π)) then P(| tan θ| ≤ 1) = 1
+2, so the resulting GEPP permutation
+matrix factor Pθ for B(θ) ∼ Bs(N, ΣS), a Haar-butterfly matrix, then satisfies
+(4.4)
+P(Pθ = Q) = 2−n = 1
+N =
+1
+|P(B)
+N |
+for each Q ∈ P(B)
+N
+(using also Propositions 1.2 and 4.1). This establishes the first part of (II)
+from Theorem 2.1:
+
+18
+J. PECA-MEDLIN
+Corollary 4.2. If B ∼ Bs(N, ΣS), then P ∼ Uniform(P(B)
+N ) for the GEPP factorization
+PB = LU.
+Next, we can connect the resulting GEPP permutation matrix factors for Haar-butterfly
+matrices to the corresponding number of total pivot movements needed.
+This can be ac-
+complished by finding the associated permutation σ ∈ Sn written in the form (3.4) for the
+corresponding butterfly permutation.
+Proposition 4.3. If n = 1, P(B)
+2
+= P2. For n > 1, then Pσ ∈ P(B)
+2N where σ ∈ S2N written
+in the form (3.4) is one of four options: either σ = 1 if Pσ = I2 ⊗ IN or σ ̸= 1 is the product
+of N disjoint transpositions, where σ = (N 2N) · · · (2 N + 2)(1 N + 1) if Pσ = P(1 2) ⊗ IN, or
+(4.5)
+σ = (a1 + N a2 + N) · · · (aN−1 + N aN + N)(a1 a2) · · · (aN−1 aN)
+if Pσ = I2 ⊗ Pρ, or
+σ = (a2 a1 + N)(a1 a2 + N) · · · (aN aN−1 + N)(aN−1 aN + N)
+if Pσ = P(1 2) ⊗ Pρ,
+where Pρ ∈ P(B)
+N
+with ρ = (a1 a2) · · · (aN−1 aN) ∈ SN in the form (3.4) such that a2k−1 < a2k
+for each k unless ρ = 1 and a2k−1 > a2k+1 for each k when n > 2.
+Proof. We will use induction on n = log2 N along with the fact P(B)
+2N = P(B)
+2
+⊗ P(B)
+N . For
+n = 2, starting with P(B)
+2
+= P2 = {I2, P(1 2)} we have
+P(B)
+4
+= P2 ⊗ P2 = {I2 ⊗ I2, P(1 2) ⊗ I2, I2 ⊗ P(1 2), P(1 2) ⊗ P(1 2)}
+(4.6)
+= {I4, P(2 4)(1 3), P(3 4)(1 2), P(2 3)(1 4)}.
+(4.7)
+The corresponding permutations as written above then all satisfy the form (3.4), where we
+abstain from including the trivial permutations (ik k) when ik = k. Moreover, writing each
+associated non-trivial permutation can be written as the product of N = 2 disjoint transposi-
+tions of the form (a1 a2)(a3 a4), which then have a1 > a3 and a2k−1 < a2k for k = 1, 2. (Note
+the (distinct) transpositions used must necessarily be disjoint since necessarily P 2
+σ = Pσ2 = I.)
+Assume the result holds for n. The n+1 case follows by just reading off the corresponding
+permutations I2 ⊗ Pρ and P(1 2) ⊗ Pρ for ρ = (a1 a2) · · · (aN−1 aN) such that Pρ ∈ P(B)
+N
+(for
+ρ already in form (3.4)). For ρ = 1, then P1 = IN yields I2 ⊗ IN = I2N and P(1 2) ⊗ IN =
+P(N 2N)···(2 N+2)(1 N+1); if Pρ ∈ P(B)
+N
+for ρ = (a1 a2) · · · (aN−1 aN) ∈ SN where a2k+1 <
+a2k−1 < a2k for each k, then
+(4.8)
+Pσ = I2 ⊗ Pρ = P(a1+N a2+N)···(aN−1+N aN+N)(a1 a2)···(aN−1 aN)
+and
+Pσ = P(1 2) ⊗ Pρ = P(a2 a1+N)(a1 a2+N)···(aN aN−1+N)(aN−1 aN+N).
+Moreover, each associated permutation σ in the form (3.4) then is either the trivial per-
+mutation or is the product of N disjoint transpositions and can be written in the form
+(b1 b2) · · · (b2N−1 b2N) where b2k+1 < b2k−1 < b2k for each k. This holds directly by con-
+struction for the associated permutations for I2 ⊗ IN, P(1 2) ⊗ IN and I2 ⊗ Pρ, which follows
+since ak ≤ N along with a2k+1 < a2k−1 < a2k for all k. It remains to show σ can be written
+in this form when Pσ = P(1 2) ⊗ Pρ.
+By (4.8), σ = (a2 a1 + N)(a1 a2 + N) · · · (aN aN−1 + N)(aN−1 aN + N). Note this is the
+product of N disjoint transpositions again since ak ≤ N and ai ̸= aj for all i ̸= j. Moreover,
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+19
+since the aj are distinct, we can label b2k−1 = a(N−k+1) for k = 1, 2, . . . , N using the ordered
+statistics subscript (so a(1) < a(2) < · · · < a(N)), with then b2k = ρ(a(N−k+1)) + N. We can
+then further write σ = (b1 b2) · · · (b2N−1 b2N) since disjoint cycles commute, for which it then
+follows also b2k+1 < b2k−1 < b2k for each k.
+Example 4.4. The corresponding permutations are {1, (1 2)} = S2 for P(B)
+2
+, {1, (2 4)(1 3),
+(3 4)(1 2), (2 3)(1 4)} for P(B)
+4
+, and
+(4.9)
+�
+�
+�
+�
+�
+�
+�
+1, (4 8)(3 7)(2 6)(1 5),
+(6 8)(5 7)(2 4)(1 3), (4 6)(3 5)(2 8)(1 7),
+(7 8)(5 6)(3 4)(1 2), (4 7)(3 8)(2 5)(1 6),
+(6 7)(5 8)(2 3)(1 4), (4 5)(3 6)(2 7)(1 8)
+�
+�
+�
+�
+�
+�
+�
+for P(B)
+8
+.
+Remark 4.5. If no pivot movements are needed on the first GE step, then no pivot move-
+ments will be needed at any GE step (using (3.4) and Proposition 4.3).
+Furthermore, it
+follows inductively that half the time all of the pivot movements occur in the first N/2 steps;
+a quarter of the time all of the pivot movements occur as the first N/4 steps of each N/2
+partitioning; an eighth of the time all of the pivots occur at the first N/8 steps of each N/4
+partitioning; and so on, where 2−k of the time the pivots occur at each step in the first half
+of each N21−k partitioning of the GE steps, which stops with exactly one permutation (i.e.,
+2−n =
+1
+N of the time) does pivoting occur at precisely every other GE step. The remaining
+permutation accounts for no pivoting ever occurring when Pθ = I. This yields a Cantor set-
+like decomposition of [n] into the possible configuration of locations of where the pivots can
+occur using GEPP. To find out which configuration will be used on a particular simple scalar
+butterfly matrix (i.e., the exact pivoting locations one will encounter), this is determined by
+the first step where k = ik.
+For example, using the associated permutations from P(B)
+8
+from Example 4.4, we see
+4 permutations have pivot movements only at the first half of the total GE steps (i.e.,
+(4 8)(3 7)(2 6), (1 5), (4 6)(3 5)(2 8)(1 7), (4 7)(3 8)(2 5)(1 6), and (4 5)(3 6)(2 7)(1 8)
+each yield pivot movements at GE steps 1 through 4); 2 permutations have pivot movements
+only at the first half of each half partitioning of the total GE steps (i.e, (6 8)(5 7)(2 4)(1 3)
+and (6 7)(5 8)(2 3)(1 4) yield pivot movements only at steps 1,2 and 5,6); 1 partition has
+pivot movements at every other GE step (i.e., (7 8)(5 6)(3 4)(1 2) yields pivot movements
+only at steps 1,3,5,7); with only one permutation (the trivial permutation) having no GE pivot
+movements.
+Moreover, applying GEPP to Haar-butterfly matrices, where then each butterfly permuta-
+tion occurs with equal probability, then induces a probability measure on these configurations.
+Figure 1 shows the possible GEPP pivot movement locations associated with Haar-butterfly
+permutations of size N = 28, along with the probability for each particular configuration.
+4.1. Proof of (II) of Theorem 2.1. Now we have the sufficient tools to establish:
+Theorem 2.1: (II) If B ∼ Bs(N, ΣS), then P ∼ Uniform(P(B)
+N ) for PB = LU
+the GEPP factorization.
+Moreover, the number of GEPP pivot movements
+needed for B is equal in distribution to N
+2 Bernoulli
+�
+1 − 1
+N
+�
+.
+
+20
+J. PECA-MEDLIN
+0
+50
+100
+150
+200
+250
+2-8
+2-8
+2-7
+2-6
+2-5
+2-4
+2-3
+2-2
+2-1
+Figure 1: GEPP pivot movement configurations for Haar-butterfly permutations for N = 28
+and their associated probabilities, pk, with the exact pivot movement locations indicated by
+blue
+Proof. Corollary 4.2 yields directly that applying GEPP to a Haar-butterfly matrix results
+in a uniform butterfly permutation matrix factor, which is the first part of (II). Moreover,
+using the associated permutations from GEPP in the form (3.4) to then be able to read off
+explicitly which GE steps need pivot movements, then we have by Proposition 4.3 that ik > k
+precisely for N/2 indices k for each non-trivial case and for precisely 0 times in the trivial
+case. It follows then if Y is the number of pivot movements needed when using GEPP on
+B(θ) ∼ Bs(N, ΣS), then since Pθ ∼ Uniform(P(B)
+N ) we have
+P(Y = 0) = P(Pθ = IN) = 2−n = 1
+N , and
+(4.10)
+P
+�
+Y = N
+2
+�
+= P(Pθ ̸= IN) = 1 − P(Pθ = IN) = 1 − 1
+N .
+(4.11)
+Hence, Y ∼ N
+2 Bernoulli(1 − 1
+N ).
+5. Random ensembles PLmax
+n
+(ξ), PLn(ξ), PLmax
+n
+(ξ, α) and PLn(ξ, α). Using the
+prior discussions, we can build a random ensemble of matrices that require a maximal number
+of GEPP pivot movements. As mentioned in Remark 3.5, a maximal number of GEPP pivot
+movements occurs when the corresponding permutation matrix factor P ∈ Pn -cycles
+n
+. Using
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+21
+this, we can define
+(5.1)
+PLmax
+n
+(ξ) = {PL : P ∼ Uniform(Pn -cycles
+n
+), L ∈ Ln independent of P, Lij ∼ ξ iid for i > j}.
+Recall, by construction, the corresponding GEPP lower triangular matrix factor L satisfies
+∥L∥max = 1. In particular, |Lij| ≤ 1 for all i > j. Moreover, Theorem 1.1 yields the L factor is
+invariant under row permutations if |Lij| < 1 for all i > j. Hence, if A = PLU where U ∈ Un
+and PL ∼ PLn(ξ) for any distribution ξ with |ξ| < 1, then A will always require n − 1 GEPP
+pivot movements. Similar ensembles can be constructed where the P factor is restricted to
+particular pivot configurations.
+We will also study the more general model
+(5.2)
+PLn(ξ) = {PL : P ∼ Uniform(Pn), L ∈ Ln independent of P, Lij ∼ ξ iid for i > j}.
+When |ξ| < 1, then A = PLU for PL ∼ PLn(ξ) and U ∈ Un corresponds to the GEPP LU
+factorization of A.
+Remark 5.1. Two natural distributions to consider are ξ ∼ Uniform([−1, 1]) and ξ ∼
+Uniform(D).
+Using GEPP on GLn(F), the left-coset representatives of Un(F) in GLn(F),
+GLn(F)/Un(F), are precisely of the form PL for P ∈ Pn and L ∈ Ln where |Lij| ≤ 1 for all i >
+j. Hence, PLn(ξ) then corresponds to uniformly sampled representatives of GLn(R)/Un(R)
+when ξ ∼ Uniform([−1, 1]) and uniformly sampled representatives of GLn(C)/Un(C) when
+ξ ∼ Uniform(D).
+5.1. Eigenvalues of PLmax
+n
+(ξ) and PLn(ξ). Suppose PL ∼ PLmax
+n
+(ξ). Since L ∈ Ln,
+then its eigenvalues are exactly 1 with multiplicity n. Since P ∈ Pn -cycles
+n
+, then its eigenvalues
+are exactly the nth roots of unity, e2πi/n. The spectral pictures for each P and L separately
+fall exactly on ∂D, and these are deterministic despite each matrix being random.
+So a natural next question is what does the spectral picture look like for their product, PL?
+The eigenvalues no longer stay on ∂D, but they appear to remain asymptotically concentrated
+inside D when scaled by
+�
+nσ2/2 when Eξ = 0 (i.e., ξ is centered) and σ2 = E|ξ|2 is the variance
+of ξ. Figure 2 shows the (computed) eigenvalue locations for PLmax
+n
+(ξ) scaled by
+�
+nσ2/2
+using n = 214 = 16, 384 and ξ sampled from Uniform([−1, 1]), Uniform(D), Rademacher and
+N(0, 1). Noticeably, Figure 2 further suggests a universality result for this limiting behavior.
+Recall the empirical spectral distribution (ESD) of a n × n matrix A is the probability
+measure
+(5.3)
+µA = 1
+n
+n
+�
+k=1
+δλk(A),
+which gives equal weight to the location of each eigenvalue of A. Note if A is a random matrix,
+then µA is a random probability measure.
+Empirically, Figure 2 then suggests µA is converging to a probability measure on D that is
+an interpolation between the uniform measure on D and the Dirac measure at the origin when
+A = PL/
+�
+nσ2/2 for PL ∼ PLn(ξ) with ξ having 0 mean and finite variance. Although the
+eigenvalues of A have a higher density near the origin, they can never include the origin since
+PL is nonsingular.
+
+22
+J. PECA-MEDLIN
+(a) ξ ∼ Uniform([−1, 1])
+(b) ξ ∼ Uniform(D)
+(c) ξ ∼ Rademacher
+(d) ξ ∼ N(0, 1)
+Figure 2: Computed eigenvalues (in blue) for PL/
+�
+nσ2/2 where n = 214 = 16, 384 and
+PL ∼ PLmax
+n
+(ξ) for (a) ξ ∼ Uniform([−1, 1]) where σ2 =
+1
+3, (b) ξ ∼ Uniform(D) where
+σ2 = 1
+2, (c) ξ ∼ Rademacher where σ2 = 1, and (d) ξ ∼ N(0, 1) where σ2 = 1, mapped
+against the unit complex circle ∂D (in red)
+The pictures are indistinguishable to Figure 2 when instead using PLn(ξ), where P ∼
+Uniform(Pn) instead of P ∼ Uniform(Pn -cycles
+n
+). In both cases, the ESD of P limit to the
+uniform measure on ∂D in probability in the weak star sense [8].
+However, replacing P
+with another random matrix from an ensemble whose ESDs similarly limit to the uniform
+measure on ∂D (e.g., using Haar sampling for O(n), U(n), or Bs(n) [8, 23]) yield different
+spectral pictures that extend beyond D when still using the scaling
+�
+nσ2/2. This points to
+the significance of the divisor of 2 in the above scaling, which corresponds to the density of
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+23
+the L factor of 1
+2; for example, UL has full density when U ∼ Haar(O(n)) or U ∼ Bs(N, ΣS).
+Since multiplication by permutation matrices preserves density, one might expect the
+same scaling when uniformly sampling permutation matrices versus corresponding n-cycle
+permutation matrices. Both should adequately mix up the rows of L so that the eigenvalues
+move away from 1. However, preserving density is not sufficient on its own, as can be seen if
+using a diagonal matrix D, since then DL will only have eigenvalues located at the diagonal
+entries of D (since L is unipotent lower triangular). A future area of research can study a
+more general random ensemble that could replace P in PL ∼ PLn(ξ) with another random
+matrix that sufficiently randomizes the rows of L without changing the sparsity of L.
+5.2. Fixed sparsity random ensembles PLmax
+n
+(ξ, α) and PLn(ξ, α). We can similarly
+study random ensembles with (approximately) fixed sparsity. Let the sparsity of a matrix
+A be determined by α = α(A), the ratio of the number of nonzero entries of A to the total
+number of entries of A. If α = 1, then A has full density, and if α = 0 then A is the zero
+matrix. A full lower triangular matrix has density given by α =
+�n+1
+2
+�
+/n2 = 1
+2(1 + 1
+n) ≈ 1
+2 for
+large n. We can then construct a matrix with fixed (approximate) sparsity by zeroing out all
+entries above a specific diagonal. For a n × n matrix that has zero entries only above a set
+diagonal k ∈ [−n, n] with entries at indices (i, i + k) (where k = 0 is the main diagonal, k > 0
+is the super diagonal, and k < 0 are the sub diagonals), one can determine the sparsity by
+computing:
+(5.4)
+gn(k) = 1
+n2
+�1
+2(n + k)(n + k + 1) − k(k + 1) · 1(k > 0)
+�
+.
+This is the ratio of nonzero entries to the total number of matrix entries for a matrix that
+is zero above and full at or below the kth diagonal; the triangular numbers naturally show
+up for the numerator. Note gn(k) is a quadratic polynomial in k for fixed n, so gn(k) can be
+extended to include non-integral k. In particular, one could uniquely solve for
+(5.5)
+kα ∈ [−n, n]
+such that
+gn(kα) = α.
+Using this, we can introduce another random ensemble
+(5.6)
+PLn(ξ, α) =
+�
+PL : P ∼ Uniform(Pn), L independent of P, Lij = 0
+if i + ⌊kα⌋ ≤ j,
+Lij ∼ ξ iid
+if i + ⌊kα⌋ > j
+�
+We can similarly define PLmax
+n
+(ξ, α) by requiring P ∼ Uniform(Pn -cycles
+n
+) (as well as other
+ensembles with fixed pivot configurations). Note if PL ∼ PLn(ξ, 1
+2) then P(L+In) ∼ PLn(ξ).
+Known results for asymptotic limiting behavior of ESDs to probability measures on D
+include the famous Circular law introduced by Bai in [2] and proven with a fourth moment
+condition by Tao and Vu [21].
+Theorem 5.2 (Circular law [21]).
+Let A be a n × n matrix with iid entries sampled from
+ξ where Eξ = 0, σ2 = E|ξ|2 and E|ξ|4 < ∞. Then µA/
+√
+nσ2 converges weakly to the uniform
+measure on D in probability and almost surely.
+
+24
+J. PECA-MEDLIN
+Theorem 5.2 yields the asymptotic spectral picture for PLn(ξ, 1) when ξ is centered with
+finite fourth moment. Figure 3a shows the map of the computed eigenvalues PLn(N(0, 1), 1) ∼
+Gin(n, n) for n = 214, while Figure 3b plots n random sample points from Uniform(D).
+Comparing this to the Ginibre picture (i.e., Figure 3a) also exemplifies how the asymptotic
+result does not hold for fixed n. The Ginibre picture has repulsions between its eigenvalues
+that lead to points being similarly spaced, while the asymptotic endpoint (i.e., Figure 3b) has
+Poisson clumping.
+(a) ξ ∼ N(0, 1), α = 1
+(b) n = 214 iid samples from Uniform(D)
+(c) ξ ∼ N(0, 1), α = 3
+4
+(d) ξ ∼ N(0, 1), α = 1
+4
+Figure 3: Computed eigenvalues (in blue) for PL/
+√
+αnσ2 where n = 214 = 16, 384 and
+PL ∼ PLn(ξ, α) for ξ ∼ N(0, 1) (where σ2 = 1) and (a) α = 1, (c) α = 3/4, and (d) α = 1/4,
+along with (b) n iid samples from Uniform(D), mapped against the unit complex circle ∂D
+(in red)
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+25
+Using A ∼ PLn(ξ, α) for fixed α ∈ (0, 1] and ξ ∼ N(0, 1), empirical data suggest a similar
+asymptotic result for the ESD of A/
+√
+αnσ2. Note the scaling matches that of Theorem 5.2
+where α = 1 as well as that seen in Figure 2 where α = 1
+2. In particular, following the trajec-
+tory for α = 1 in Figure 3a, α = 3
+4 in Figure 3d, α = 1
+2 in Figure 2 (recall P ∼ Uniform(Pn)
+and P ∼ Uniform(Pn -cycles
+n
+) empirically result in indistinguishable spectral pictures), and
+α = 1
+4 in Figure 3c, suggests the scaling by
+√
+αnσ2 have the corresponding ESDs limit to
+να, a fixed probability measure with support on D that depends on α (and not on ξ), which
+further converge to ν, the uniform measure on D, as α → 1 and converge to δ0, the Dirac
+measure at the origin, as α → 0. So the limiting measure is an interpolation between ν and
+δ0.
+Together, these suggest different universality classes than those included by the Circular
+law. Previous studies of sparsity and the Circular law have studied iid ensembles that have
+sparsity α = αn converging to 0 slow enough whose ESDs still limit to ν [14, 18]. Other studies
+have similarly explored the impact of sparsity on the extreme eigenvalues in the Hermitian
+case, which has ESDs limiting to the semicircular law [12]. Results in the literature for fixed
+sparsity random ensembles remain sparse. The above discussion provides supporting evidence
+for the following conjecture:
+Conjecture 5.3. Fix α ∈ (0, 1]. Let A = PLQ be the n × n matrix, where P and Q are iid
+uniformly chosen permutation matrices and L is a n×n random matrix independent of P and
+Q whose nonzero entries are iid from ξ with Eξ = 0, E|ξ| = σ2 and E|ξ|4 < ∞, where Lij = 0
+if i + ⌊kα⌋ < j. Then there exists a probability measure, να, on D that is an interpolation
+between the uniform measure on D, ν, and the Dirac measure at the origin, δ0, such that
+µAn/
+√
+αnσ2 converges weakly to να in probability and almost surely. Furthermore, να → ν as
+α → 1 and να → δ0 as α → 0, with both convergences holding uniformly with respect to the
+total variation distance.
+Remark 5.4. For α = 1, this is the circular law.
+Remark 5.5. Note the right permutation matrix Q term is irrelevant, since A is similar
+to QPL, and QP ∼ P since the uniform measure on Sn is left- and right-invariant (it is the
+Haar measure on Pn). So the study of the above ensembles reduces to the ensembles of the
+form PLn(ξ, α) for centered ξ with finite fourth moments.
+Remark 5.6. Additional random ensembles that can be studied in light of Conjecture 5.3
+include perturbations of PLn(ξ, α) for deterministic matrices (similar to those studied in
+[14, 21]), as well as P(L + In) for PL ∼ PLn(ξ, α). This latter model is interesting when
+α ≤ 1
+2 since the corresponding L factor has eigenvalues of 0 with multiplicity n; in particular,
+L and hence PL does not have full rank. Using experiments for several fixed α < 1
+2, then
+the nullity of PL ∼ PLn(N(0, 1), α) appears to be approximately (1 −
+√
+2α)n, while 0 is an
+eigenvalue of multiplicity approximately (1−2α)n; when α ≥ 1
+2, both are 0 (a.s.). Conversely,
+now considering A = P(L + In), then 0 is never an eigenvalue for A and so A is always full
+rank for all α.
+6. Numerical experiments. The final section will focus on a set of experiments that will
+study the number of GEPP pivot movements needed on particular random linear systems.
+These experiments will expand on results first presented in [17], which studied the impact on
+
+26
+J. PECA-MEDLIN
+the growth factors and relative error computations when using common random transforma-
+tions from numerical linear algebra on particular fixed linear systems, Ax = b. Both initial
+models represent scenarios when no GEPP pivot movements are needed, so we will refer to
+both here as min-movement models. Carrying forward the naming scheme from [17], the two
+linear systems studied include:
+1. the min-movement (na¨ıve) model14, with A = In where ρ(In) = 1 is minimized, and
+2. the min-movement (worst-case) model15, with A = An a particular linear model that
+maximizes the growth factor ρ(An) = 2n−1.
+In the na¨ıve model, the authors studied the 1-sided random transformation ΩI = Ω, which
+provides a means to directly study the corresponding random matrix, Ω, itself. The worst-case
+model considered the 2-sided random transformations, UAnV ∗, where U, V were independently
+sampled random matrices; this 2-sided transformation follows the construction used by Parker
+that would remove the need for pivoting in GEPP (with high probability), so that UAnV ∗ =
+LU has a GE factorization [16]. The matrix An is of the form
+(6.1)
+An = In −
+�
+i>j
+Eij +
+n−1
+�
+j=1
+Ein.
+Wilkinson introduced An to establish the growth factor bound ρ(A) ≤ 2n−1 is sharp [26]. By
+construction, no GEPP pivoting would be needed at any intermediate GE step when using
+An, so the final GENP and GEPP factorizations of An both align, with An = LnUn for
+Ln = In − �
+i>j Eij and Un = In − Enn + �n
+k=1 2k−1Ekn. It follows ρ(An) = |Unn| = 2n−1.
+For example,
+(6.2)
+A4 =
+�
+���
+1
+0
+0
+1
+−1
+1
+0
+1
+−1
+−1
+1
+1
+−1
+−1
+−1
+1
+�
+��� =
+�
+���
+1
+0
+0
+0
+−1
+1
+0
+0
+−1
+−1
+1
+0
+−1
+−1
+−1
+1
+�
+���
+�
+���
+1
+0
+0
+1
+0
+1
+0
+2
+0
+0
+1
+4
+0
+0
+0
+8
+�
+��� = L4U4
+has ρ(A4) = 8 = 23.
+With respect to GEPP, both the na¨ıve and worst-case models use input matrices that
+need 0 total pivot movements. We will study both of these min-movement models in addition
+to a third model, that looks at the other extreme in terms of the number of GEPP pivot
+movements:
+3. the max-movement model, where A ∼ PLmax
+n
+(ξ) for ξ ∼ Uniform([−1, 1]).
+Note if A = PL ∼ PLmax
+n
+(ξ) when |ξ| < 1 a.s., then ρ(A) = ρ(L) = 1.
+We will consider only the 1-sided transformation case for the na¨ıve model (so that we
+can study the random matrices themselves) and only the 2-sided transformation cases for the
+worst-case and max-movement models. Together, these three models will allow us to study the
+14The na¨ıve model was named to reflect that using any method is unnecessary to solve the trivial linear
+system Ix = x = b.
+15The worst-case moniker was chosen to reflect the numerical stability of computed solutions using GEPP,
+which is controlled by the growth factors: solving Anx = b sees relative errors of order O(1) starting when
+n ≈ 60.
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+27
+impact of random transformations on two different systems where no GEPP pivot movements
+are needed as well as one (independently sampled random) system where a maximal number
+of GEPP pivot movements are needed.
+For each set of experiments, we will consider the following random transformations for
+fixed N = 2n:
+• Bs(N, ΣS)
+• B(N, ΣS)
+• Bs(N, ΣD)
+• B(N, ΣD)
+• Walsh transform
+• Haar(O(N))
+• Discrete Cosine Transform (DCT II)
+To ease the following discussion, we choose N = 24 = 16 and N = 28 = 256 to focus on as we
+feel they are representative of the behavior we saw for other choices of N. For the na¨ıve model,
+which will study the pivot movements for each of the associated random matrices themselves
+(using the 1-sided preconditioning with A = I), our experiments will additionally use:
+• GOE(N)
+• GUE(N)
+• Bernoulli(1
+2)
+These models were touched on for N = 2 in Examples 2.3 to 2.5.
+Each of the butterfly models is sampled using custom MATLAB recursive functions with
+iid uniformly chosen angles16 in line with methods outlined in [17, 23]. See Subsection 1.2 for
+more information and sampling techniques of the Walsh, Haar orthogonal, DCT, GOE(N)
+and GUE(N) transformations. The Bernoulli ensemble uses iid Bernoulli(1
+2) entries17. Each
+set of experiments (using N = 24 and N = 28 for all three models) will use 10,000 trials using
+MATLAB in double precision, where ϵ = 2−52 (ϵ ≈ 2.220446 · 10−16).
+6.1. Min-movement (na¨ıve) model. For the na¨ıve model, our goal is to study the number
+of GEPP pivot movements needed for 10 different random ensembles.
+These allow us to
+gauge the impact on (1-sided) random transformations on the simplest linear system, Ix =
+b, in terms of the number of GEPP pivot movements needed.
+Starting with I, no pivot
+movements are needed, so the transformed system ΩI = Ω then allows us to study how much
+this transformation introduces new pivot movements. This model also then enables us to
+directly study the number of pivot movements needed for each random matrix, Ω.
+Table 1 shows the sample medians, means (¯x), and standard deviations (s) for the 10,000
+trials each for N = 24 and N = 28, while Figure 4 summarizes the total number of GEPP
+pivot movements encountered for each sampled random matrix across each set of trials. Note
+the axes in Figure 4 show each possible step output for 0 to N − 1.
+16The number of angles used depends on if the cosine and sine matrices are scalar and if the butterfly matrix
+is simple. For Bs(N, ΣS), then one uniform angle is sampled at each recursive step, for n = log2 N total uniform
+angles needed, while similarly B(N, ΣS) and Bs(N, ΣD) both sample a total of N − 1 uniform angles, with
+B(N, ΣD) using 1
+2Nn total uniform angles. These compare to Haar(O(N)), which (using Givens rotations to
+find the QR factorization of Gin(N, N)) can be sampled using
+�N
+2
+�
+= 1
+2N(N − 1) uniform angles. The above
+
+28
+J. PECA-MEDLIN
+N = 16
+N = 256
+Median
+¯x
+s
+Median
+¯x
+s
+Bs(N, ΣS)
+8
+7.492
+1.951
+128
+127.501
+7.978
+B(N, ΣS)
+11
+10.934
+2.622
+232
+230.392
+14.418
+Bs(N, ΣD)
+12
+11.287
+2.459
+245
+241.915
+10.308
+B(N, ΣD)
+13
+12.535
+1.430
+250
+249.901
+2.112
+Walsh
+6
+6
+-
+120
+120
+-
+Haar(O(N))
+13
+12.624
+1.345
+250
+249.884
+2.123
+DCT II
+13
+13
+-
+249
+249
+-
+GOE(N)
+11
+10.954
+1.780
+249
+248.696
+2.359
+GUE(N)
+11
+11.132
+1.761
+249
+248.867
+2.348
+Bernoulli
+11
+10.509
+1.774
+248
+247.783
+2.467
+Table 1: Pivot counts for numerical experiments for GEPP with 10,000 trials, for random
+matrices of orders N = 24 and N = 28
+0
+5
+10
+15
+0
+50
+100
+150
+200
+250
+Figure 4: Histogram of 104 samples of pivot movement counts for random matrices of order
+N = 24 and N = 28.
+ordering reflects this ordering of complexity.
+17Bernoulli matrices are sampled using native MATLAB functions, round(rand(n)).
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+29
+6.1.1. Discussion. For each set of experiments, the Haar-butterfly and Walsh transforms
+introduce the least amount of additional movement from the initial minimal GEPP movement
+setup, each with total pivot movements at most N/2 while the remaining models had total
+pivot movements closer to the upper bound of N − 1.
+By construction, both the Walsh and DCT models have deterministic output. This follows
+since the transformations are of the form WD where W is a deterministic associated matrix
+(the default Fast-Walsh Hadamard matrix or the DCT matrix used by the native MATLAB
+functions for the associated multiplication operators) while D is a random row sign matrix
+sampled uniformly from {±1}N. Hence, if the GEPP factorization of W is PW = LU, then
+the GEPP factorization of WD is PWD = L(UD). So the permutation matrix factor is
+independent of D for these models.
+This is reflected in Figure 4 and Table 1 (e.g., both
+sample standard deviations are 0).
+The two Haar random ensembles studied (viz., Bs(N, ΣS) and Haar(O(N))) have the full
+distribution on the number of GEPP pivot movements determined by Theorem 2.1, using also
+Corollary 2.7. From Figure 4, these two models also appear to represent relative extreme
+models among the random ensembles considered in these trials, with the resulting uniform
+GEPP permutation matrix factor yielding the most pivot movements.
+For Haar-butterfly matrices, we can directly compare the sample statistics against the
+exact distribution statistics for YN ∼ N
+2 Bernoulli(1 − 1
+N ). We can compute exactly
+EYN = N
+2
+�
+1 − 1
+N
+�
+and
+(6.3)
+σYN = N
+2
+��
+1 − 1
+N
+�
+· 1
+N ,
+(6.4)
+This yields that EY16 = 7.5 and σY16 ≈ 1.93649167, which align (as expected) with the
+associated sample mean of 7.492 and sample standard deviation of 1.951 from Table 1 for
+N = 16. Similarly, the exact values EY256 = 127.5 and σY256 ≈ 7.98435971 align with the
+sample statistics ¯x = 127.501 and s = 7.978 for N = 256.
+Moreover, as can be seen in
+Figure 4, the trials resulted only in total GEPP movements of 0 or N
+2 , as should be expected
+for a scaled Bernoulli distribution. This agrees with the result from [11] that the computed
+GEPP permutation matrix factors using floating-point arithmetic and exact arithmetic align
+with very high probability for Gin(n, n). Other standard sample statistic comparisons for the
+Haar-butterfly matrices similarly align, as expected18.
+Similarly, we can compare the output for the Haar(O(N)) trials, which have XN, the total
+GEPP pivot movements needed, equal in distribution to N − ΥN. We can compute exactly
+EXN = N − EΥN = N − HN
+and
+(6.5)
+σXN = σΥN =
+�
+HN − H(2)
+N .
+(6.6)
+18For example, the sample medians exactly match the exact medians of N/2. Also, we can compare the
+sample proportion ˆpN to the population success parameter pN = 1 − 1
+N . These again compare very favorably,
+where 1 − ˆp16 = 0.0635 aligns with 1 − p16 =
+1
+16 = 0.0625 and 1 − ˆp256 = 0.0039 aligns with 1 − p256 =
+1
+256 =
+0.00390625.
+
+30
+J. PECA-MEDLIN
+This yields that EX16 ≈ 12.619271006771006 and σX16 ≈ 1.340291930806123, which align
+with the associated sample mean of 12.624 and sample standard deviation of 1.345 from
+Table 1 for N = 16. Similarly, the exact values EX256 ≈ 249.8756550371827 and σX256 ≈
+2.117382706670809 align with the sample statistics ¯x = 249.696 and s = 2.123 for N = 256.
+Figure 4 shows the butterfly models pivot movements lie strictly between the pivot move-
+ments for Haar-butterfly matrices and Haar(O(N)), with the increase in associated number
+of uniform angles needed for the butterfly models leading to the sample distributions progres-
+sively moving toward the Haar(O(N)) model for both N = 16 and N = 256. While B(N, ΣD)
+results in pivot movements very close to those modeled by the uniform GEPP permutation ma-
+trix factors, B(N, ΣS) and Bs(N, ΣD) lie strictly in between both the Haar-butterfly and Haar
+orthogonal pivot movements. Moreover, the remaining random models for GOE(N), GUE(N)
+and Bernoulli have pivot movement distributions staying to the left of the Haar orthogonal
+model, which move closer to the Haar orthogonal model distribution as N increases. This
+suggests as N increases for these remaining models, the resulting random GEPP permutation
+matrix moves closer to a uniform permutation matrix.
+Remark 6.1. For both the Haar-butterfly and Haar orthogonal models, the 1-sided na¨ıve
+model is equivalent to the 2-sided na¨ıve model since UINV ∗ = UV ∗ ∼ U by the right-
+invariance of the Haar measure. This does not hold, however, for any other random matrices
+in the na¨ıve experiments.
+Remark 6.2. For the Bernoulli model, it is possible to generate a singular random matrix,
+which occurs with probability 2−N(1 + o(1)) [22]. Of the 10,000 trials, this occurred 48 times
+for N = 16 and 0 times for N = 256. Table 1 and Figure 4 show the summary statistics
+and overall GEPP pivot counts for the remaining 9,952 nonsingular Bernoulli matrices when
+N = 16.
+6.2. Min-movement (worst-case) model. For the worst-case model, we want to study
+the number of GEPP pivot movements needed when starting with a fixed linear system, AN,
+that again requires no pivot movements. This provides a means to measure how much new
+GEPP pivot movements are generated by these random transformations. For this model, we
+will consider only the 2-sided transformation, UANV ∗, where U and V are iid samples from
+each random model used in the experiments.
+Analogously to the na¨ıve model, Table 2 shows the sample medians, means (¯x), and stan-
+dard deviations (s) for the 10,000 trials again for N = 24 and N = 28, while Figure 5 sum-
+marizes the total number of GEPP pivot movements encountered for each sampled UANV ∗
+across each set of trials.
+6.2.1. Discussion. Only the Haar(O(N)) model for UANV ∗ is sampled from a distri-
+bution determined in Theorem 2.1, since Corollary 2.11 yields resulting GEPP permutation
+matrix factor is a uniform permutation matrix, so that XN, the number of GEPP pivot move-
+ments, is equal in distribution to N −ΥN. Since AN does not preserve the Kronecker product
+structure, the Haar-butterfly pivot movement distribution is not preserved. Hence, the num-
+ber of GEPP pivot movements is no longer a scaled Bernoulli distribution and now has full
+support on 0, 1, . . . , N − 1.
+Again, both the Haar-butterfly and Haar orthogonal models provide representatives for
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+31
+N = 16
+N = 256
+Median
+¯x
+s
+Median
+¯x
+s
+Bs(N, ΣS)
+11
+10.563
+2.380
+179
+181.784
+27.493
+B(N, ΣS)
+12
+12.217
+1.760
+249
+247.936
+4.322
+Bs(N, ΣD)
+12
+11.967
+1.995
+249
+247.493
+5.521
+B(N, ΣD)
+13
+12.558
+1.406
+250
+249.876
+2.090
+Walsh
+12
+12.245
+1.617
+250
+249.654
+2.253
+Haar(O(N))
+13
+12.636
+1.335
+250
+249.900
+2.129
+DCT II
+12
+11.820
+1.719
+250
+249.447
+2.313
+Table 2: Pivot counts for numerical experiments for GEPP with 10,000 trials, for 2-sided
+transformation of Worst-case model of orders N = 24 and N = 28
+0
+5
+10
+15
+0
+50
+100
+150
+200
+250
+Figure 5: Histogram of 104 samples of pivot movement counts for 2-sided random transfor-
+mations of order N = 24 and N = 28 worst-case model, UANV ∗.
+the extremes in the number of pivot movements introduced by these random transformations.
+As in the na¨ıve model, the Haar-butterfly transformation introduced the least amount of new
+GEPP pivot movements for the initial minimal GEPP pivot movement model AN.
+Of the remaining models, only B(N, ΣS) and Bs(N, ΣD) have resulting distributions that
+do not appear to align with the Haar orthogonal model, although they both appear much
+closer to the Haar orthogonal than Haar-butterfly models.
+The remaining models’ align-
+
+32
+J. PECA-MEDLIN
+ment with the Haar orthogonal models manifests even for the small N = 16 experiments for
+B(N, ΣD) and the Walsh transform: the exact values EX16 ≈ 12.619271006771006 and σX16 ≈
+1.340291930806123 compare to the remaining respective samples means of 12.558 and 12.245
+and sample standard deviations of 1.406 and 1.617 for the B(N, ΣD) and Walsh models. This
+alignment is even more pronounced for N = 256: the exact values EX256 ≈ 249.8756550371827
+and σX256 ≈ 2.117382706670809 line up very well for B(N, ΣD), Walsh, and DCT II models,
+whose sample means range from 249.447 to 249.876 and whose sample standard deviations
+range from 2.090 to 2.313. Moreover, the remaining models have sample medians each of
+250 that exactly match that for the Haar orthogonal model for N = 256, while the sample
+medians match or are smaller by one than the true Haar orthogonal sample median of 13 for
+N = 16. Again, these suggest performance for the non-butterfly models moving toward the
+uniform row permutations as N increases.
+6.3. Max-movement model. While the min-movement models studied the impact of
+random transformations on the number of pivot movements introduced to initial models that
+require no GEPP pivot movements, the max-movement model will instead study the impact
+of the random transformations on a model that has maximal GEPP pivot movements, PL ∼
+PLmax
+N
+(ξ) for ξ ∼ Uniform([−1, 1]). (Unlike the min-movements models, the input matrix
+PL is random.) This provides a means to measure how much GEPP pivot movement can
+be removed by these random transformations. As in the worst-case model, we will consider
+only the 2-sided transformation, UPLV ∗, where U and V are iid samples from each random
+model.
+Table 3 shows the sample medians, means (¯x), and standard deviations (s) for the 10,000
+trials each for N = 24 and N = 28, while Figure 6 summarizes the total number of GEPP
+pivot movements encountered for each sampled matrix UPLV ∗.
+N = 16
+N = 256
+Median
+¯x
+s
+Median
+¯x
+s
+Bs(N, ΣS)
+13
+12.580
+1.348
+250
+249.864
+2.126
+B(N, ΣS)
+13
+12.594
+1.369
+250
+249.899
+2.090
+Bs(N, ΣD)
+13
+12.613
+1.357
+250
+249.901
+2.120
+B(N, ΣD)
+13
+12.626
+1.322
+250
+249.887
+2.121
+Walsh
+13
+12.630
+1.332
+250
+249.879
+2.123
+Haar(O(N))
+13
+12.625
+1.339
+250
+249.833
+2.130
+DCT II
+13
+12.573
+1.344
+250
+249.923
+2.116
+Table 3: Pivot counts for numerical experiments for GEPP with 10,000 trials, for 2-sided
+transformation of max-movement model of orders N = 24 and N = 28
+6.3.1. Discussion. As in the worst-case model, only the Haar orthogonal transformed
+model UPLV ∗ has its distribution determined by Theorem 2.1, where Corollary 2.11 again
+yields XN, the number of GEPP pivot movements, correspond to uniform row permutations,
+so XN ∼ N − ΥN. Unlike both min-movement models, all of the resulting experiments align
+strongly with this uniform row permutation model. All of the sample means are within 0.05 of
+
+DISTRIBUTION OF THE NUMBER OF PIVOTS NEEDED USING GEPP
+33
+0
+5
+10
+15
+0
+50
+100
+150
+200
+250
+Figure 6: Histogram of 104 samples of pivot movement counts for 2-sided random transfor-
+mations of order N = 24 and N = 28 maximal movement real model, UPLV ∗.
+the exact means EXN and all of the sample standard deviations are within 0.02 of the exact
+standard deviations σXN for both N = 16 and N = 256 (see Table 3). Moreover, every set
+of experiments exactly matched the true medians for the uniform row permutation models of
+13 for N = 16 and 250 for N = 256. Hence, this suggests every random transformation had
+essentially equivalent dampening impacts on the total GEPP pivot movements when starting
+with a maximal pivot movement model.
+6.4. Conclusions. The Haar orthogonal model, which had GEPP pivot movements Xn ∼
+n − Υn, remains a strong comparison point for each random transformation across each min-
+and max-movement model. In each case, Xn represents both an upper bound for overall per-
+formance in terms of numbers of pivot movements, as well as a limiting class for most random
+transformations, which further suggest a universality result in terms of GEPP pivot move-
+ments. Since the Haar orthogonal model results in uniform GEPP permutation matrix factors,
+this suggests most random transformation classes have sufficient random mixing properties
+using both minimal and maximal GEPP movement input models. Undesirably, however, this
+model asymptotically is concentrated near the upper bound of n − 1 in terms of total pivot
+movements, with average n − Hn = (n − 1)(1 + o(1)).
+The Haar-butterfly model introduced the least amount of additional pivot movements
+among the min-movement models, while the remaining butterfly models introduced increasing
+pivot movements as they increased in randomness (i.e., from B(N, ΣS) and BS(N, ΣD) to
+
+34
+J. PECA-MEDLIN
+B(N, ΣD)). However, only the Haar-butterfly models remained far from the upper bound
+for the min-movement models. In [17], a future direction wanted to explore the impact of
+combining random transformations (that remove the need of GEPP pivoting) with GEPP on
+the total number of pivot movements. To address this concern, these experiments suggest
+the butterfly models do the least amount of damage in terms of introducing new GEPP pivot
+movements when starting with a linear system with little GEPP pivot movements necessary.
+However, no models had strong dampening performance when starting with a max-movement
+input system.
+7. Acknowledgements. The author would like to thank a referee on a previous paper who
+had asked about the number of movements still needed after using a random transformation
+on a linear system, which led to the particular direction pursued here. Additionally, the author
+thanks Tom Trogdon and Nick Ercolani for many helpful thoughts and insights during the
+project.
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+

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+arXiv:2301.00560v1  [quant-ph]  2 Jan 2023
+PauliComposer: Compute Tensor Products of Pauli Matrices Efficiently
+Sebasti´an V. Romero
+1∗ and Juan Santos-Su´arez
+2†
+1TECNALIA, Basque Research and Technology Alliance (BRTA), 48160 Derio, Spain
+2Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE),
+Universidade de Santiago de Compostela, 15705 Santiago de Compostela, Spain
+(Dated: January 3, 2023)
+We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices. This
+is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calcu-
+lations. The strength of this strategy is benchmarked against state-of-the-art techniques, showing
+a remarkable acceleration. As a side product, we provide an optimized method for one key calculus
+in quantum simulations: the Pauli basis decomposition of Hamiltonians.
+I.
+INTRODUCTION
+Pauli matrices [1] are one of the most important and
+well-known set of matrices within the field of quantum
+physics. They are particularly important both in physics
+and chemistry when used to describe Hamiltonians of
+many-body spin glasses [2–7] or for quantum simula-
+tions [8–13].
+The vast majority of these systems are
+out of analytic control so that non-equilibrium states
+are usually studied through exact diagonalization which
+requires their Hamiltonians to be written in its matrix
+form. While this task may be regarded as a trivial mat-
+ter in a mathematical sense, it involves the calculation of
+an exponentially growing number of operations.
+In this work, we present the PauliComposer (PC) al-
+gorithm which significantly expedites this calculation. It
+exploits the fact that any Pauli word only has one ele-
+ment different from zero per row and column, so a num-
+ber of calculations can be avoided.
+Additionally, each
+matrix entry can be computed without performing any
+multiplications. This algorithm can be used to boost in-
+ner calculations where several tensor products involving
+Pauli matrices appear. In particular, those that appear
+while building Hamiltonians as weighted sums of Pauli
+strings or decomposing an operator in the Pauli basis.
+The PC algorithm could be implemented in compu-
+tational frameworks in which this sort of operations are
+crucial, such as the Python modules Qiskit [14], Penny-
+Lane [15], OpenFermion [16] and Cirq [17]. It can also
+potentially be used in many other applications, such as
+the Pauli basis decomposition of the Fock space [18] and
+conventional computation of Ising model Hamiltonians
+to solve optimization problems [19–22], among others.
+The rest of the article is organized as follows: in Sec-
+tion II we describe the algorithm formulation in depth,
+showing a pseudocode-written routine for its computa-
+tion. In Section III, a set of benchmark tests is performed
+to show that a remarkable speed-up can be achieved
+∗ [email protected]
+† [email protected]
+when compared to state-of-the-art techniques.
+In Sec-
+tion IV, we show how this Pauli Composer algorithm
+can be used to solve relevant problems. Finally, the con-
+clusions drawn from the presented results are given in
+Section V. We provide proofs for several statements and
+details of the algorithm in the appendices.
+II.
+ALGORITHM FORMULATION
+In this section we discuss the PC algorithm formulation
+in detail. Pauli matrices are hermitian, involutory and
+unitary matrices that together with the identity form the
+set σ{0,1,2,3} = {I, X, Y, Z}. Given an input string x =
+xn−1 . . . x0 ∈ {0, 1, 2, 3}n, the PC algorithm constructs
+P(x) := σxn−1 ⊗ σxn−2 ⊗ · · · ⊗ σx0.
+(1)
+Let us denote its matrix elements as Pj,k(x) with
+j, k = 0, . . . , 2n − 1. It is important to remark that for
+each row j, there will be a single column k(j) such that
+Pj,k(j) ̸= 0 (see Appendix A). The solution amounts to a
+map from the initial Pauli string to the positions and val-
+ues of the 2n nonzero elements. This calculation will be
+done sequentially, hence the complexity of the algorithm
+will be bounded from below by this number.
+As a first step, it is worth noting that Pauli string
+matrices are either real (all elements are ±1) or purely
+imaginary (all are ±i). This depends on nY , the number
+of Y operators in P(x). We can redefine ˜Y := iY , so that
+˜σ{0,1,2,3} = {I, X, ˜Y , Z} and
+˜P(x) := ˜σxn−1 ⊗ · · · ⊗ ˜σx0.
+As a result, every entry in ˜P(x) will be ±1. This implies
+that there is no need to compute any multiplication: the
+problem reduces to locating the nonzero entries in ˜P(x)
+and tracking sign changes. The original P(x) can be re-
+covered as P(x) = (−i)nY mod 4 ˜P(x).
+We will now present an iterative procedure to compute
+˜P by finding for each row j the nonzero column number
+k(j) and its corresponding value ˜Pj,k(j). For the first row,
+j = 0, the nonzero element ˜P0,k(0), can be found at
+k(0) = [y(xn−1) . . . y(x0)]10,
+(2)
+where [an−1 . . . a0]10 is the decimal representation of the
+bit string a = an−12n−1 + · · ·+ a020 and y(xi) tracks the
+
+2
+diagonality of σxi, being equal to 0 if xi ∈ {0, 3} and 1
+otherwise. The value of this entry is
+˜P0,k(0) = +1 =⇒ P0,k(0) = (−i)nY mod 4.
+(3)
+The following entries can be computed iteratively. At
+the end of stage l, with l = 0, · · · , n − 1, all nonzero
+elements in the first 2l+1 rows of Pj,k(j) will have been
+computed using the information given by the substring
+xl . . . x0. At the next step, l + 1, the following 2l rows
+are filled using the ones that had already been computed,
+where the row-column relation k(j) is given by
+k(j + 2l) = k(j) + (−1)y(xl)2l,
+j = 0, . . . , 2l − 1.
+(4)
+The second term of the RHS of this relation takes into ac-
+count the way that the blocks of zeros returned at stage
+l affect the new relative location of the nonzero blocks
+within the new 2l+1 × 2l+1 subcomposition. Its corre-
+sponding values are obtained from the previous ones, up
+to a possible change of sign given by
+Pj+2l,k(j+2l) = ǫlPj,k(j),
+(5)
+with ǫl equal to 1 if xl ∈ {0, 1} and −1 otherwise. This ǫl
+is nothing but a parameter that takes into account if σxl
+introduces a sign flip. In Alg. 1 a pseudocode that sum-
+marises the presented algorithm using (2)-(5), is shown.
+For the particular case of diagonal Pauli strings (only
+I and Z matrices), there is no need to compute the row-
+column relation k(j), just the sign assignment is enough.
+Even if this is also the case for anti-diagonal matrices, we
+focus on the diagonal case due to its relevance in combi-
+natorial problems [19–22]. See Alg. 2 for the pseudocode
+of this case (PDC stands for Pauli Diagonal Composer).
+The PC algorithm is able to circumvent the calculation
+of a significant amount of operations. When generic Kro-
+necker product routines (see Appendix B) are used for
+the same task, the amount of multiplications needed for
+computing a Pauli string is O[n22n] and O[n2n] for dense
+and sparse matrices, respectively. In contrast, the PC al-
+gorithm, considering the worst-case scenarios, needs
+• {I, Z}⊗n: O[2n] changes of sign.
+• Otherwise: O[2n] sums and O[2n] changes of sign.
+In all cases this novel algorithm can significantly out-
+perform those that are not specifically designed for Pauli
+matrices.
+On top of that, this method is also advantageous
+for computing weighted Pauli strings.
+Following (3),
+W := ωP, with arbitrary ω, can be computed by defining
+W0,k(0) = ω(−i)nY mod 4 which avoids having to do any
+extra multiplication. This change is reflected in Alg. 1 by
+changing line 6 to m(0) ← ω(−i)nY mod 4 and line 4 to
+m(0) ← ω in Alg. 2. This is specially important as it can
+be used to compute Hamiltonians written as a weighted
+sum of Pauli strings, where H = �
+x ωxP(x).
+Algorithm 1: PC: compose n Pauli matrices
+input : xn−1xn−2 . . . x0 ← string with xi ∈ {0, 1, 2, 3}
+1 n ← len(x)
+2 nY ← number of Y matrices in x
+3 j ← range(0, 2n − 1)
+// rows
+4 k, m ← empty 2n-array
+// columns/entries
+5 k(0) ← y(xn−1) . . . y(x0) in base 10
+6 m(0) ← (−i)nY mod 4
+7 for l ∈ range(0, n − 1) do
+8
+k(2l : 2l+1 − 1) ← k(0 : 2l − 1) + (−1)y(xl)2l
+9
+if xl ∈ {0, 1} then
+// ǫl = 1
+10
+m(2l : 2l+1 − 1) ← m(0 : 2l − 1)
+11
+else
+// ǫl = −1
+12
+m(2l : 2l+1 − 1) ← −m(0 : 2l − 1)
+output: P(x) as a sparse matrix stacking (j, k, m)
+Algorithm 2: PDC: compose n diagonal Pauli matrices
+input : xn−1xn−2 . . . x0 ← string with xi ∈ {0, 3}
+1 n ← len(x)
+2 j, k ← range(0, 2n − 1)
+// rows/columns
+3 m ← empty 2n-array
+// entries
+4 m(0) ← 1
+5 for l ∈ range(0, n − 1) do
+6
+if xl = 0 then
+// ǫl = 1
+7
+m(2l : 2l+1 − 1) ← m(0 : 2l − 1)
+8
+else
+// ǫl = −1
+9
+m(2l : 2l+1 − 1) ← −m(0 : 2l − 1)
+output: P(x) as a sparse matrix stacking (j, k, m)
+III.
+BENCHMARKING
+In this section we analyse the improvement that the
+PC strategy introduces against the methods presented in
+Appendix B in two figures of merit: memory storage and
+execution times. For this purpose, we use MATLAB [23]
+(which incorporates optimized routines of the well-known
+BLAS and LAPACK libraries [24–28]) and, only for the
+PC, also Python [29] since many quantum computing
+libraries are written in this language [14–17]. See Tab. I
+for a full description of the computational resources used.
+Concerning memory needs, with this algorithm only 2n
+nonzero elements out of 22n are stored. This is exactly
+the same as using sparse matrices, thus, no major im-
+provement is to be expected. As for the computational
+time, we compare how different algorithms behave as the
+length n of the Pauli string increases. In Fig. 1 execu-
+Table I. Computer and software specifications.
+Processor
+Intel® Core™ i7-11850H (16×2.50 GHz)
+RAM
+32.0 GB (DDR4)
+OS
+Ubuntu 22.04.1 LTS (×64)
+MATLAB [23]
+9.12.0.1884302 (R2022a)
+Python [29]
+3.9.12
+NumPy [30]
+1.23.2
+SciPy [31]
+1.9.0
+Qiskit [14]
+0.38.0
+PennyLane [15]
+0.23.1
+
+3
+2
+4
+6
+8 10 12 14 16 18 20 22 24 26 28 30
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+n
+Execution times (s)
+Naive
+Mixed
+Alg993 [32]
+Tree
+PC/PDC (M)
+PC/PDC (P)
+Figure 1. Execution times for computing general (solid line)
+and diagonal n-Pauli strings (dashed line) using different
+methods. Here, M stands for MATLAB and P for Python.
+tion times for general and diagonal Pauli strings (solid
+and dashed lines, respectively) are shown. For the Pauli
+Composer methods, we use the PC routine (Alg. 1) for
+the general case and the PDC routine (Alg. 2) for the di-
+agonal one. In accordance to our theoretical analysis, the
+PC algorithm proves to be the best performing routine.
+On a more technical note, when using the PC rou-
+tine, matrices with complex values (nY odd) take twice as
+much time as real valued ones (nY even). Consequently,
+we compute their execution times separately and then
+average them. Moreover, it is convenient to choose when
+to use PC or PDC as the latter can be up to 10 times faster.
+IV.
+REAL USE CASES OF THE PAULI
+COMPOSER ALGORITHM
+The PC algorithm can be used to perform useful cal-
+culations in physics. In this section, the Pauli basis de-
+composition of a Hamiltonian and the construction of a
+Hamiltonian as a sum of weighted Pauli strings are dis-
+cussed in detail. Another worth mentioning scenario is
+the digital implementation of the complex exponential of
+a Pauli string, i.e. e−iθP (x) = cos(θ)I − i sin(θ)P(x).
+A.
+Pauli basis decomposition of a Hamiltonian
+The decomposition of a Hamiltonian written as a 2n ×
+2n matrix into the Pauli basis is a common problem in
+quantum computing. Given a general Hamiltonian H,
+this decomposition can be written as
+H =
+�
+x ωx
+�
+σxn−1 ⊗ · · · ⊗ σx0
+�
+=
+�
+x ωxP(x),
+(6)
+with x = xn−1 . . . x0 and P(x) as in (1). The coefficients
+ωx are obtained from the orthogonal projection as
+ωx = 1
+2n tr[P(x)H] = 1
+2n
+2n−1
+�
+j=0
+Pj,k(j)(x)Hk(j),j.
+(7)
+Following the discussion in Section II, the double sum
+collapses to a single one in (7) since there is only one
+nonzero element per row and column.
+Additionally, in some special cases, it can be known in
+advance if some set of ωx will vanish:
+• If H is symmetric, strings with an odd number of
+Y matrices can be avoided (2n−1(2n + 1) terms).
+• If H is diagonal, only strings composed by I and Z
+will contribute (2n terms).
+The amount of operations made by this Pauli Decom-
+poser (PD) is given by the following list
+• If H is diagonal (O[2n] strings): O[22n] operations.
+• Otherwise (O[22n] strings): O[23n] operations.
+This PD algorithm checks if the input matrix satisfies
+one of the special cases defined above, discards all van-
+ishing Pauli strings and computes the coefficients of the
+remaining ones using the PC routine and (7). This work-
+flow considerably enhances our results, especially for di-
+agonal matrices.
+In Tab. II, we tested the most extended methods for de-
+composing matrices into weighted sums of Pauli strings
+against PD using Python [29] to compare their perfor-
+mance. In particular, we used the SparsePauliOp class
+from Qiskit [14] and the decompose hamiltonian func-
+tion from PennyLane [15] (only works with hermitian
+Hamiltonians). Four types of random 2n × 2n matrices
+were generated, namely non-hermitian HNH, hermitian
+HH, symmetric HS and diagonal HD matrices. The PD
+vastly outperforms Qiskit and PennyLane routines, spe-
+cially for the symmetric and diagonal cases.
+B.
+Building of a Hamiltonian as a sum of weighted
+Pauli strings
+Many Hamiltonians are written in terms of weighted
+Pauli strings. As mentioned, our method can compute
+weighted Pauli strings directly without performing extra
+computations.
+In Fig. 2 we show a performance com-
+parison of the presented methods for computing Hamil-
+tonians written as sums of weighted Pauli strings. The
+Hamiltonian used is similar to the one proposed in [21],
+H =
+n−1
+�
+i=0
+αiσi
+3 +
+n−1
+�
+i<j
+βi,jσi
+3σj
+3,
+(8)
+
+4
+Table II. Execution times (in seconds) for decomposing an arbitrary 2n × 2n matrix. In brackets we see the number of threads
+used by each routine. Here, PC and PDC run under Python code as well as Qiskit [14] and PennyLane [15].
+n
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Non-hermitian matrix HNH
+PC (×1)
+0.0005
+0.0021
+0.012
+0.078
+0.55
+4.06
+31.2
+254
+2008
+Qiskit (×16)
+0.0015
+0.0050
+0.020
+0.14
+1.16
+8.78
+92.38
+1398
+26938
+Hermitian matrix HH
+PC (×1)
+0.0004
+0.0021
+0.012
+0.078
+0.56
+4.24
+32.86
+261
+2007
+Qiskit (×16)
+0.0010
+0.0035
+0.018
+0.10
+1.47
+12.02
+108.3
+1295
+26848
+PennyLane (×16)
+0.0013
+0.0060
+0.030
+0.15
+2.23
+10.66
+97.6
+2019
+35014
+Symmetric matrix HS
+PC (×1)
+0.0003
+0.0010
+0.0058
+0.036
+0.24
+1.78
+14.05
+108
+794
+Qiskit (×16)
+0.0010
+0.0036
+0.018
+0.10
+1.45
+11.07
+104.6
+1320
+26399
+PennyLane (×16)
+0.0011
+0.0054
+0.027
+0.13
+1.36
+9.22
+91.52
+1477
+31583
+Diagonal matrix HD
+PDC (×1)
+0.0001
+0.0002
+0.0006
+0.0018
+0.0068
+0.025
+0.094
+0.37
+1.49
+Qiskit (×16)
+0.0010
+0.0035
+0.018
+0.10
+1.46
+11.0
+103.3
+1270
+25977
+PennyLane (×16)
+0.0010
+0.0047
+0.023
+0.11
+1.20
+8.29
+86.17
+1370
+30941
+being the corresponding weigths ⃗α = [α0, . . . , αn−1] and
+⃗β = [β0,1, . . . , β0,n−1, β1,2, . . . , βn−2,n−1] arbitrary and σi
+3
+as defined in (B1) ∀i, j. This Hamiltonian is computed
+using Alg. 3, which uses the PDC routine (see Alg. 2)
+with two inputs: the string x ∈ {0, 3}n to compute and
+the weights to consider. In the PDC case, we use two
+strategies: compute each weighted term of (8) directly
+and compute each Pauli string and then multiply it by
+its corresponding weight (solid and dashed lines in Fig. 2,
+respectively). This is done by changing lines 6 to H ←
+H + αiPDC(str1) and 10 to H ← H + βi,jPDC(str2)
+in Alg. 3 for the second one.
+There is no remarkable
+difference between both methods.
+2
+4
+6
+8 10 12 14 16 18 20 22 24 26 28 30
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+103
+104
+n
+Execution times (s)
+Naive
+Tree
+PDC (M)
+PDC (P)
+Figure 2.
+Execution times for computing (8) using Alg. 3
+(solid line) and computing previously the Pauli string for then
+multiply it by its corresponding weight (dashed line).
+Algorithm 3: Ising model Hamiltonian computation
+input : ⃗α, ⃗β ← lists of weights
+1 n ← len(⃗α)
+2 H ← 2n × 2n sparse matrix of zeros
+3 for i ∈ range(0, n − 1) do
+4
+str1 ← string of n zeros
+// n identities
+5
+str1(i) ← 3
+// Z in the i-th position
+6
+H ← H + PDC(str1, αi)
+7
+for j ∈ range(i + 1, n − 1) do
+8
+str2 ← copy(str1)
+9
+str2(j) ← 3
+// Z in the j-th position
+10
+H ← H + PDC(str2, βi,j)
+output: Hamiltonian H as a sparse matrix
+V.
+CONCLUSIONS
+The fast and reliable computation of tensor products
+of Pauli matrices is crucial in the field of quantum me-
+chanics and, in particular, of quantum computing.
+In
+this article we propose a novel algorithm with proven
+theoretical and experimental enhancements over similar
+methods of this key yet computationally tedious task.
+This is achieved by taking advantage of the properties of
+Pauli matrices and the tensor product definition, which
+implies that one can avoid trivial operations such as mul-
+tiplying constants by one and waste time computing ele-
+ments with value zero that could be known in advance.
+Concerning memory resources, it is convenient to store
+the obtained results as sparse matrices since only 2n out
+of 22n entries will not be zero for a Pauli string of length
+n, i.e. the density of the resultant matrix will be 2−n
+(see Appendix A).
+
+5
+Our benchmark tests suggest that the Pauli Composer
+algorithm and its variants can achieve a remarkable accel-
+eration when compared to the most well-known methods
+for the same purpose both for single Pauli strings and
+real use cases. In particular, the most considerable out-
+performance can be seen in Tab. II for the symmetric and
+diagonal matrix decomposition over the Pauli basis.
+Finally, its simple implementation (Alg. 1-2) can po-
+tentially allow to integrate the PC routines into quantum
+simulation packages to enhance inner calculations.
+ACKNOWLEDGMENTS
+We would like to thank Javier Mas Sol´e, Yue Ban
+and Mikel Garc´ıa de Andoin for the helpful discus-
+sions that led to the present article.
+This research is
+funded by the QUANTEK project (ELKARTEK pro-
+gram from the Basque Government, expedient no. KK-
+2021/00070) and the project “BRTA QUANTUM: Hacia
+una especializaci´on armonizada en tecnolog´ıas cu´anticas
+en BRTA” (expedient no. KK-2022/00041). The work
+of JSS has received support from Xunta de Galicia
+(Centro singular de investigaci´on de Galicia acceditation
+2019-2022) by European Union ERDF, from the Span-
+ish Research State Agency (grant PID2020-114157GB-
+100) and from MICIN with funding from the European
+Union NextGenerationEU (PRTR-C17.I1) and the Gali-
+cian Regional Government with own funding through
+the “Planes Complementarios de I+D+I con las Comu-
+nidades Aut´onomas” in Quantum Communication.
+Data and code availability statement.
+The data
+and code used in the current study are available upon
+reasonable request from the corresponding authors.
+Appendix A: Some proofs regarding Pauli strings
+In this section we prove two key properties of Pauli
+strings on which our algorithm is based.
+Theorem A.1. A Pauli string P(x) of length n given
+by (1) has only 2n nonzero entries.
+Proof. With the help of Fig. 3, we can compute the num-
+ber of zeros in the resulting matrix as
+n0(n) = 2
+�
+2n−1 × 2n−1�
++ 4
+�
+2n−2 × 2n−2�
++ 8
+�
+2n−3 × 2n−3�
++ · · · + 2n(1 × 1)
+=
+2n−1
+�
+k=n
+2k = 2n (2n − 1) .
+(A1)
+In other words, P(x) will have only 2n nonzero terms.
+We can prove (A1) by induction easily: n0(n = 1) is true
+n−1
+�
+i=0
+σxn−i−1 =
+
+
+0
+· · ·
+0
+0
+· · ·
+· · ·
+0
+0
+· · ·
+0
+0
+0
+0
+· · ·
+0
+0
+· · ·
+· · ·
+0
+0
+· · ·
+0
+
+
+Figure 3. Scheme for computing the number of zeros of an
+arbitrary composition of n Pauli matrices.
+since n0(1) = 21(21 − 1) = 2 and if we assume that n0(n)
+holds, we can see that
+n0(n + 1) =
+2(n+1)−1
+�
+k=n+1
+2k = 2n+1 �
+2n+1 − 1
+�
+also holds true.
+From this result and the unitarity of P(x), we can infer
+another important aspect.
+Corollary A.1.1. A Pauli string P(x) of length n given
+by (1) has only one nonzero entry per row and column.
+Proof. Since the tensor product of unitary matrices is
+also unitary, then |det P(x)| = 1. From Th. A.1, only 2n
+entries of the resulting 2n × 2n matrix are nonzero. So
+the logical conclusion to be drawn is that the unique way
+to locate them without having a row and a column full
+of zeros, thus returning a zero determinant, is that each
+row and column must have only one nonzero entry.
+Appendix B: Standard methods for computing
+tensor products
+For the sake of completeness, in this appendix, we will
+briefly review the well established algorithms that were
+used in the benchmark [32–34]. First, one can consider
+what we call the Naive algorithm, which consists on per-
+forming the calculations directly. It is clearly highly inef-
+ficient as it scales in the number of operations as O[n2n]
+for sparse Pauli matrices. Second, the Mixed algorithm
+uses the mixed-product property
+n−1
+�
+i=0
+σxn−i−1 =
+n−1
+�
+i=0
+σi
+xn−i−1,
+with
+σi
+xi :=
+
+
+
+
+
+I⊗n−1 ⊗ σx0
+if i = 0
+I⊗n−i−1 ⊗ σxi ⊗ I⊗i
+if 0 < i < n − 1
+σxn−1 ⊗ I⊗n−1
+if i = n − 1
+, (B1)
+
+6
+to simplify the calculation into a simple product of block
+diagonal matrices. Based on this procedure, Algorithm
+993 is presented in [32]. It can be shown that this method
+performs over O[n2n] operations. Besides that, as Fig. 1
+suggests, the fact that it requires to transpose and re-
+shape several matrices has a non-negligible effect that
+fatally increases its computation time. Finally, the Tree
+routine starts storing pairs of tensor products as
+�
+σxn−2i−1 ⊗ σxn−2i−2
+�n/2−1
+i=0
+if n is even
+�
+σxn−1
+�
+∪
+�
+σxn−2i−1 ⊗ σxn−2i−2
+�⌊n/2⌋
+i=0
+otherwise
+,
+and proceeds with the resultant matrices following the
+same logic, which allows to compute (1) by iteratively
+grouping its terms by pairs.
+For better results, this
+method can be parallelized.
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+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='00560v1  [quant-ph]  2 Jan 2023  PauliComposer: Compute Tensor Products of Pauli Matrices Efficiently  Sebasti´an V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Romero  1∗ and Juan Santos-Su´arez  2†  1TECNALIA, Basque Research and Technology Alliance (BRTA), 48160 Derio, Spain  2Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE),  Universidade de Santiago de Compostela, 15705 Santiago de Compostela, Spain  (Dated: January 3, 2023)  We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This  is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calcu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' The strength of this strategy is benchmarked against state-of-the-art techniques, showing  a remarkable acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' As a side product, we provide an optimized method for one key calculus  in quantum simulations: the Pauli basis decomposition of Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  INTRODUCTION  Pauli matrices [1] are one of the most important and  well-known set of matrices within the field of quantum  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' They are particularly important both in physics  and chemistry when used to describe Hamiltonians of  many-body spin glasses [2–7] or for quantum simula-  tions [8–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  The vast majority of these systems are  out of analytic control so that non-equilibrium states  are usually studied through exact diagonalization which  requires their Hamiltonians to be written in its matrix  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' While this task may be regarded as a trivial mat-  ter in a mathematical sense, it involves the calculation of  an exponentially growing number of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  In this work, we present the PauliComposer (PC) al-  gorithm which significantly expedites this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' It  exploits the fact that any Pauli word only has one ele-  ment different from zero per row and column, so a num-  ber of calculations can be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Additionally, each  matrix entry can be computed without performing any  multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This algorithm can be used to boost in-  ner calculations where several tensor products involving  Pauli matrices appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In particular, those that appear  while building Hamiltonians as weighted sums of Pauli  strings or decomposing an operator in the Pauli basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  The PC algorithm could be implemented in compu-  tational frameworks in which this sort of operations are  crucial, such as the Python modules Qiskit [14], Penny-  Lane [15], OpenFermion [16] and Cirq [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' It can also  potentially be used in many other applications, such as  the Pauli basis decomposition of the Fock space [18] and  conventional computation of Ising model Hamiltonians  to solve optimization problems [19–22], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  The rest of the article is organized as follows: in Sec-  tion II we describe the algorithm formulation in depth,  showing a pseudocode-written routine for its computa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In Section III, a set of benchmark tests is performed  to show that a remarkable speed-up can be achieved  ∗ sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='vidal@tecnalia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='com  † juansantos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='suarez@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='es  when compared to state-of-the-art techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  In Sec-  tion IV, we show how this Pauli Composer algorithm  can be used to solve relevant problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Finally, the con-  clusions drawn from the presented results are given in  Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' We provide proofs for several statements and  details of the algorithm in the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  ALGORITHM FORMULATION  In this section we discuss the PC algorithm formulation  in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Pauli matrices are hermitian, involutory and  unitary matrices that together with the identity form the  set σ{0,1,2,3} = {I, X, Y, Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Given an input string x =  xn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' x0 ∈ {0, 1, 2, 3}n, the PC algorithm constructs  P(x) := σxn−1 ⊗ σxn−2 ⊗ · · · ⊗ σx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  (1)  Let us denote its matrix elements as Pj,k(x) with  j, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' , 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' It is important to remark that for  each row j, there will be a single column k(j) such that  Pj,k(j) ̸= 0 (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' The solution amounts to a  map from the initial Pauli string to the positions and val-  ues of the 2n nonzero elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This calculation will be  done sequentially, hence the complexity of the algorithm  will be bounded from below by this number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  As a first step, it is worth noting that Pauli string  matrices are either real (all elements are ±1) or purely  imaginary (all are ±i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This depends on nY , the number  of Y operators in P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' We can redefine ˜Y := iY , so that  ˜σ{0,1,2,3} = {I, X, ˜Y , Z} and  ˜P(x) := ˜σxn−1 ⊗ · · · ⊗ ˜σx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  As a result, every entry in ˜P(x) will be ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This implies  that there is no need to compute any multiplication: the  problem reduces to locating the nonzero entries in ˜P(x)  and tracking sign changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' The original P(x) can be re-  covered as P(x) = (−i)nY mod 4 ˜P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  We will now present an iterative procedure to compute  ˜P by finding for each row j the nonzero column number  k(j) and its corresponding value ˜Pj,k(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' For the first row,  j = 0, the nonzero element ˜P0,k(0), can be found at  k(0) = [y(xn−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' y(x0)]10,  (2)  where [an−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' a0]10 is the decimal representation of the  bit string a = an−12n−1 + · · ·+ a020 and y(xi) tracks the  2  diagonality of σxi, being equal to 0 if xi ∈ {0, 3} and 1  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' The value of this entry is  ˜P0,k(0) = +1 =⇒ P0,k(0) = (−i)nY mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  (3)  The following entries can be computed iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' At  the end of stage l, with l = 0, · · · , n − 1, all nonzero  elements in the first 2l+1 rows of Pj,k(j) will have been  computed using the information given by the substring  xl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' At the next step, l + 1, the following 2l rows  are filled using the ones that had already been computed,  where the row-column relation k(j) is given by  k(j + 2l) = k(j) + (−1)y(xl)2l,  j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' , 2l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  (4)  The second term of the RHS of this relation takes into ac-  count the way that the blocks of zeros returned at stage  l affect the new relative location of the nonzero blocks  within the new 2l+1 × 2l+1 subcomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Its corre-  sponding values are obtained from the previous ones, up  to a possible change of sign given by  Pj+2l,k(j+2l) = ǫlPj,k(j),  (5)  with ǫl equal to 1 if xl ∈ {0, 1} and −1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This ǫl  is nothing but a parameter that takes into account if σxl  introduces a sign flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 1 a pseudocode that sum-  marises the presented algorithm using (2)-(5), is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  For the particular case of diagonal Pauli strings (only  I and Z matrices), there is no need to compute the row-  column relation k(j), just the sign assignment is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Even if this is also the case for anti-diagonal matrices, we  focus on the diagonal case due to its relevance in combi-  natorial problems [19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' See Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 2 for the pseudocode  of this case (PDC stands for Pauli Diagonal Composer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  The PC algorithm is able to circumvent the calculation  of a significant amount of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' When generic Kro-  necker product routines (see Appendix B) are used for  the same task, the amount of multiplications needed for  computing a Pauli string is O[n22n] and O[n2n] for dense  and sparse matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In contrast, the PC al-  gorithm, considering the worst-case scenarios, needs  {I, Z}⊗n: O[2n] changes of sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Otherwise: O[2n] sums and O[2n] changes of sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  In all cases this novel algorithm can significantly out-  perform those that are not specifically designed for Pauli  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  On top of that, this method is also advantageous  for computing weighted Pauli strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Following (3),  W := ωP, with arbitrary ω, can be computed by defining  W0,k(0) = ω(−i)nY mod 4 which avoids having to do any  extra multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This change is reflected in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 1 by  changing line 6 to m(0) ← ω(−i)nY mod 4 and line 4 to  m(0) ← ω in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This is specially important as it can  be used to compute Hamiltonians written as a weighted  sum of Pauli strings, where H = �  x ωxP(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Algorithm 1: PC: compose n Pauli matrices  input : xn−1xn−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' x0 ← string with xi ∈ {0, 1, 2, 3}  1 n ← len(x)  2 nY ← number of Y matrices in x  3 j ← range(0, 2n − 1)  // rows  4 k, m ← empty 2n-array  // columns/entries  5 k(0) ← y(xn−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' y(x0) in base 10  6 m(0) ← (−i)nY mod 4  7 for l ∈ range(0, n − 1) do  8  k(2l : 2l+1 − 1) ← k(0 : 2l − 1) + (−1)y(xl)2l  9  if xl ∈ {0, 1} then  // ǫl = 1  10  m(2l : 2l+1 − 1) ← m(0 : 2l − 1)  11  else  // ǫl = −1  12  m(2l : 2l+1 − 1) ← −m(0 : 2l − 1)  output: P(x) as a sparse matrix stacking (j, k, m)  Algorithm 2: PDC: compose n diagonal Pauli matrices  input : xn−1xn−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' x0 ← string with xi ∈ {0, 3}  1 n ← len(x)  2 j, k ← range(0, 2n − 1)  // rows/columns  3 m ← empty 2n-array  // entries  4 m(0) ← 1  5 for l ∈ range(0, n − 1) do  6  if xl = 0 then  // ǫl = 1  7  m(2l : 2l+1 − 1) ← m(0 : 2l − 1)  8  else  // ǫl = −1  9  m(2l : 2l+1 − 1) ← −m(0 : 2l − 1)  output: P(x) as a sparse matrix stacking (j, k, m)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  BENCHMARKING  In this section we analyse the improvement that the  PC strategy introduces against the methods presented in  Appendix B in two figures of merit: memory storage and  execution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' For this purpose, we use MATLAB [23]  (which incorporates optimized routines of the well-known  BLAS and LAPACK libraries [24–28]) and, only for the  PC, also Python [29] since many quantum computing  libraries are written in this language [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' I  for a full description of the computational resources used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Concerning memory needs, with this algorithm only 2n  nonzero elements out of 22n are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This is exactly  the same as using sparse matrices, thus, no major im-  provement is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' As for the computational  time, we compare how different algorithms behave as the  length n of the Pauli string increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 1 execu-  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Computer and software specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Processor  Intel® Core™ i7-11850H (16×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='50 GHz)  RAM  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0 GB (DDR4)  OS  Ubuntu 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='1 LTS (×64)  MATLAB [23]  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='1884302 (R2022a)  Python [29]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='12  NumPy [30]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='2  SciPy [31]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0  Qiskit [14]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0  PennyLane [15]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='1  3  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30  10−5  10−4  10−3  10−2  10−1  100  101  n  Execution times (s)  Naive  Mixed  Alg993 [32]  Tree  PC/PDC (M)  PC/PDC (P)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Execution times for computing general (solid line)  and diagonal n-Pauli strings (dashed line) using different  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Here, M stands for MATLAB and P for Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  tion times for general and diagonal Pauli strings (solid  and dashed lines, respectively) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' For the Pauli  Composer methods, we use the PC routine (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 1) for  the general case and the PDC routine (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 2) for the di-  agonal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In accordance to our theoretical analysis, the  PC algorithm proves to be the best performing routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  On a more technical note, when using the PC rou-  tine, matrices with complex values (nY odd) take twice as  much time as real valued ones (nY even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Consequently,  we compute their execution times separately and then  average them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Moreover, it is convenient to choose when  to use PC or PDC as the latter can be up to 10 times faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  REAL USE CASES OF THE PAULI  COMPOSER ALGORITHM  The PC algorithm can be used to perform useful cal-  culations in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In this section, the Pauli basis de-  composition of a Hamiltonian and the construction of a  Hamiltonian as a sum of weighted Pauli strings are dis-  cussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Another worth mentioning scenario is  the digital implementation of the complex exponential of  a Pauli string, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' e−iθP (x) = cos(θ)I − i sin(θ)P(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Pauli basis decomposition of a Hamiltonian  The decomposition of a Hamiltonian written as a 2n ×  2n matrix into the Pauli basis is a common problem in  quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Given a general Hamiltonian H,  this decomposition can be written as  H =  �  x ωx  �  σxn−1 ⊗ · · · ⊗ σx0  �  =  �  x ωxP(x),  (6)  with x = xn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' x0 and P(x) as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' The coefficients  ωx are obtained from the orthogonal projection as  ωx = 1  2n tr[P(x)H] = 1  2n  2n−1  �  j=0  Pj,k(j)(x)Hk(j),j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  (7)  Following the discussion in Section II, the double sum  collapses to a single one in (7) since there is only one  nonzero element per row and column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Additionally, in some special cases, it can be known in  advance if some set of ωx will vanish:  If H is symmetric, strings with an odd number of  Y matrices can be avoided (2n−1(2n + 1) terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  If H is diagonal, only strings composed by I and Z  will contribute (2n terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  The amount of operations made by this Pauli Decom-  poser (PD) is given by the following list  If H is diagonal (O[2n] strings): O[22n] operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Otherwise (O[22n] strings): O[23n] operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  This PD algorithm checks if the input matrix satisfies  one of the special cases defined above, discards all van-  ishing Pauli strings and computes the coefficients of the  remaining ones using the PC routine and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This work-  flow considerably enhances our results, especially for di-  agonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' II, we tested the most extended methods for de-  composing matrices into weighted sums of Pauli strings  against PD using Python [29] to compare their perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In particular, we used the SparsePauliOp class  from Qiskit [14] and the decompose hamiltonian func-  tion from PennyLane [15] (only works with hermitian  Hamiltonians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Four types of random 2n × 2n matrices  were generated, namely non-hermitian HNH, hermitian  HH, symmetric HS and diagonal HD matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' The PD  vastly outperforms Qiskit and PennyLane routines, spe-  cially for the symmetric and diagonal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Building of a Hamiltonian as a sum of weighted  Pauli strings  Many Hamiltonians are written in terms of weighted  Pauli strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' As mentioned, our method can compute  weighted Pauli strings directly without performing extra  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 2 we show a performance com-  parison of the presented methods for computing Hamil-  tonians written as sums of weighted Pauli strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' The  Hamiltonian used is similar to the one proposed in [21],  H =  n−1  �  i=0  αiσi  3 +  n−1  �  i<j  βi,jσi  3σj  3,  (8)  4  Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Execution times (in seconds) for decomposing an arbitrary 2n × 2n matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In brackets we see the number of threads  used by each routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Here, PC and PDC run under Python code as well as Qiskit [14] and PennyLane [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  n  2  3  4  5  6  7  8  9  10  Non-hermitian matrix HNH  PC (×1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='078  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='55  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='06  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='2  254  2008  Qiskit (×16)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='16  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='78  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='38  1398  26938  Hermitian matrix HH  PC (×1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='0021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='078  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='56  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='24  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='86  261  2007  Qiskit (×16)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
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+page_content='17  1370  30941  being the corresponding weigths ⃗α = [α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
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+page_content=' , αn−1] and  ⃗β = [β0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
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+page_content=' , β0,n−1, β1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' , βn−2,n−1] arbitrary and σi  3  as defined in (B1) ∀i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This Hamiltonian is computed  using Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 3, which uses the PDC routine (see Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 2)  with two inputs: the string x ∈ {0, 3}n to compute and  the weights to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In the PDC case, we use two  strategies: compute each weighted term of (8) directly  and compute each Pauli string and then multiply it by  its corresponding weight (solid and dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 2,  respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' This is done by changing lines 6 to H ←  H + αiPDC(str1) and 10 to H ← H + βi,jPDC(str2)  in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 3 for the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  There is no remarkable  difference between both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30  10−5  10−4  10−3  10−2  10−1  100  101  102  103  104  n  Execution times (s)  Naive  Tree  PDC (M)  PDC (P)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Execution times for computing (8) using Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 3  (solid line) and computing previously the Pauli string for then  multiply it by its corresponding weight (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Algorithm 3: Ising model Hamiltonian computation  input : ⃗α, ⃗β ← lists of weights  1 n ← len(⃗α)  2 H ← 2n × 2n sparse matrix of zeros  3 for i ∈ range(0, n − 1) do  4  str1 ← string of n zeros  // n identities  5  str1(i) ← 3  // Z in the i-th position  6  H ← H + PDC(str1, αi)  7  for j ∈ range(i + 1, n − 1) do  8  str2 ← copy(str1)  9  str2(j) ← 3  // Z in the j-th position  10  H ← H + PDC(str2, βi,j)  output: Hamiltonian H as a sparse matrix  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  CONCLUSIONS  The fast and reliable computation of tensor products  of Pauli matrices is crucial in the field of quantum me-  chanics and, in particular, of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  In  this article we propose a novel algorithm with proven  theoretical and experimental enhancements over similar  methods of this key yet computationally tedious task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  This is achieved by taking advantage of the properties of  Pauli matrices and the tensor product definition, which  implies that one can avoid trivial operations such as mul-  tiplying constants by one and waste time computing ele-  ments with value zero that could be known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Concerning memory resources, it is convenient to store  the obtained results as sparse matrices since only 2n out  of 22n entries will not be zero for a Pauli string of length  n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' the density of the resultant matrix will be 2−n  (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  5  Our benchmark tests suggest that the Pauli Composer  algorithm and its variants can achieve a remarkable accel-  eration when compared to the most well-known methods  for the same purpose both for single Pauli strings and  real use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' In particular, the most considerable out-  performance can be seen in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' II for the symmetric and  diagonal matrix decomposition over the Pauli basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Finally, its simple implementation (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 1-2) can po-  tentially allow to integrate the PC routines into quantum  simulation packages to enhance inner calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We would like to thank Javier Mas Sol´e, Yue Ban  and Mikel Garc´ıa de Andoin for the helpful discus-  sions that led to the present article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  This research is  funded by the QUANTEK project (ELKARTEK pro-  gram from the Basque Government, expedient no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' KK-  2021/00070) and the project “BRTA QUANTUM: Hacia  una especializaci´on armonizada en tecnolog´ıas cu´anticas  en BRTA” (expedient no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' KK-2022/00041).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' The work  of JSS has received support from Xunta de Galicia  (Centro singular de investigaci´on de Galicia acceditation  2019-2022) by European Union ERDF, from the Span-  ish Research State Agency (grant PID2020-114157GB-  100) and from MICIN with funding from the European  Union NextGenerationEU (PRTR-C17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='I1) and the Gali-  cian Regional Government with own funding through  the “Planes Complementarios de I+D+I con las Comu-  nidades Aut´onomas” in Quantum Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Data and code availability statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  The data  and code used in the current study are available upon  reasonable request from the corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Appendix A: Some proofs regarding Pauli strings  In this section we prove two key properties of Pauli  strings on which our algorithm is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' A Pauli string P(x) of length n given  by (1) has only 2n nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' With the help of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 3, we can compute the num-  ber of zeros in the resulting matrix as  n0(n) = 2  �  2n−1 × 2n−1�  + 4  �  2n−2 × 2n−2�  + 8  �  2n−3 × 2n−3�  + · · · + 2n(1 × 1)  =  2n−1  �  k=n  2k = 2n (2n − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  (A1)  In other words, P(x) will have only 2n nonzero terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  We can prove (A1) by induction easily: n0(n = 1) is true  n−1  �  i=0  σxn−i−1 =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  · ·  0  0  · ·  · ·  0  0  · ·  0  0  0  0  · ·  0  0  · ·  · ·  0  0  · ·  0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Scheme for computing the number of zeros of an  arbitrary composition of n Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  since n0(1) = 21(21 − 1) = 2 and if we assume that n0(n)  holds, we can see that  n0(n + 1) =  2(n+1)−1  �  k=n+1  2k = 2n+1 �  2n+1 − 1  �  also holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  From this result and the unitarity of P(x), we can infer  another important aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' A Pauli string P(x) of length n given  by (1) has only one nonzero entry per row and column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Since the tensor product of unitary matrices is  also unitary, then |det P(x)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' From Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='1, only 2n  entries of the resulting 2n × 2n matrix are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' So  the logical conclusion to be drawn is that the unique way  to locate them without having a row and a column full  of zeros, thus returning a zero determinant, is that each  row and column must have only one nonzero entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content='  Appendix B: Standard methods for computing  tensor products  For the sake of completeness, in this appendix, we will  briefly review the well established algorithms that were  used in the benchmark [32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' First, one can consider  what we call the Naive algorithm, which consists on per-  forming the calculations directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' It is clearly highly inef-  ficient as it scales in the number of operations as O[n2n]  for sparse Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Second, the Mixed algorithm  uses the mixed-product property  n−1  �  i=0  σxn−i−1 =  n−1  �  i=0  σi  xn−i−1,  with  σi  xi :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  I⊗n−1 ⊗ σx0  if i = 0  I⊗n−i−1 ⊗ σxi ⊗ I⊗i  if 0 < i < n − 1  σxn−1 ⊗ I⊗n−1  if i = n − 1  , (B1)  6  to simplify the calculation into a simple product of block  diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Based on this procedure, Algorithm  993 is presented in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' It can be shown that this method  performs over O[n2n] operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Besides that, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' 1  suggests, the fact that it requires to transpose and re-  shape several matrices has a non-negligible effect that  fatally increases its computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
+page_content=' Finally, the Tree  routine starts storing pairs of tensor products as  �  σxn−2i−1 ⊗ σxn−2i−2  �n/2−1  i=0  if n is even  �  σxn−1  �  ∪  �  σxn−2i−1 ⊗ σxn−2i−2  �⌊n/2⌋  i=0  otherwise  ,  and proceeds with the resultant matrices following the  same logic, which allows to compute (1) by iteratively  grouping its terms by pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfrfkX/content/2301.00560v1.pdf'}
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GdAyT4oBgHgl3EQfrflY/content/tmp_files/2301.00561v1.pdf.txt

@@ -0,0 +1,1425 @@
+arXiv:2301.00561v1  [cs.LG]  2 Jan 2023
+Local Differential Privacy for Sequential Decision Making in a Changing
+Environment
+Pratik Gajane
+Eindhoven University of Technology
+[email protected]
+Abstract
+We study the problem of preserving privacy while still pro-
+viding high utility in sequential decision making scenarios
+in a changing environment. We consider abruptly changing
+environment: the environment remains constant during peri-
+ods and it changes at unknown time instants. To formulate
+this problem, we propose a variant of multi-armed bandits
+called non-stationary stochastic corrupt bandits. We construct
+an algorithm called SW-KLUCB-CF and prove an upper
+bound on its utility using the performance measure of regret.
+The proven regret upper bound for SW-KLUCB-CF is near-
+optimal in the number of time steps and matches the best
+known bound for analogous problems in terms of the num-
+ber of time steps and the number of changes. Moreover, we
+present a provably optimal mechanism which can guarantee
+the desired level of local differential privacy while providing
+high utility.
+Introduction
+Several practically relevant applications including recom-
+mender systems, Internet advertising have been formulated
+as sequential decision making problems using the frame-
+work of multi-armed bandits. The importance of privacy in
+such sequential decision making problems has been exten-
+sively discussed in the literature (see for example, Thakurta
+and Smith (2013); Mishra and Thakurta (2015); Tossou and
+Dimitrakakis (2016)).
+Differential privacy, introduced by Dwork et al. (2006),
+is one of the popular approaches to address such privacy
+concerns. In sequential decision making problems, algo-
+rithms providing differential privacy preserve data privacy
+by adding appropriate statistical noise to the data. Duchi,
+Jordan, and Wainwright (2014) extend this notion to local
+differential privacy in which data remains private even from
+the algorithm. The main difference between global and local
+differential privacy is whether privacy is to be maintained
+from the algorithm or the (possibly unintended) recipient of
+the output of the algorithm. In global differential privacy,
+noise is added by the algorithm so the output does not re-
+veal private information about the input. In local differential
+privacy, noise is added to the input of the algorithm so that
+privacy is maintained even from the algorithm.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+To understand the motivation for local differential privacy,
+let us consider the practical application of Internet adver-
+tising 1. An advertising system receives, as input, feedback
+from the users which may reveal private information about
+them. The advertising system employs a suitable learning
+algorithm and selects ads for the users tailored to the feed-
+back given by them. These selected ads are then given to
+the advertisers as output. While using global differential pri-
+vacy, privacy is maintained from the advertisers by ensuring
+that the output of the learning algorithms does not reveal in-
+formation about the input (i.e., user information). Typically,
+advertising systems are established by leading social me-
+dia networks, web browsers and other popular websites. Ko-
+rolova (2010); Kosinski, Stillwell, and Graepel (2013) show
+that it is possible to accurately predict a range of highly sen-
+sitive personal attributes including age, sexual orientation,
+relationship status, political and religious affiliation using
+the feedback available to the advertising systems. Such pos-
+sible breach of privacy necessitates us to protect personal
+user information not only from the advertisers but also from
+the advertising systems. Local differential privacy is able to
+achieve this objective unlike global differential privacy.
+In this article, we propose to use low privacy regime using
+local differential privacy. In low privacy regime, the noise
+added to the data is small and the aim of the privacy mecha-
+nism is to send as much information about data as allowed,
+but no more (Kairouz, Oh, and Viswanath 2014). This is in
+alignment with our dual goal of using privacy in recommen-
+dation systems or Internet advertising, and other similar ap-
+plications: provide useful recommendations/ads to the users
+while respecting their privacy as much as possible.
+We measure the utility of our proposed algorithm using
+regret which is a measure of the total mistake cost (precise
+definitions will follow in the next Section). When rewards
+are bounded (as assumed in most works in the literature),
+the regret of any algorithm is trivially bounded linearly in the
+number of time steps T . An algorithm is said to be learning
+if its regret is bounded sub-linearly in T .
+Main Contributions
+1. We propose non-stationary stochastic corrupt bandits, a
+novel formulation which aims to preserve local differen-
+1We consider a simplistic scenario for illustrative purposes.
+
+tial privacy while still providing high utility for sequen-
+tial decision making in a non-stationary environment.
+2. We construct an algorithm called SW-KLUCB-CF for
+the considered problem.
+3. We prove an upper bound on the utility of SW-KLUCB-
+CF in terms of its regret. This upper bound is near-
+optimal in terms of the number of time steps and matches
+the best known bound for analogous problems in terms of
+the number of time steps and the number of changes.
+4. We provide an optimal mechanism to achieve a desired
+level of local differential privacy while achieving high
+utility.
+This work is an extension of Gajane, Urvoy, and Kauf-
+mann (2018) to non-stationary environments and reuses
+some of the concepts used there. However, it should be noted
+that the algorithms proposed in Gajane, Urvoy, and Kauf-
+mann (2018) will not be able to solve the problem con-
+sidered in this article. In fact, it is easy to construct non-
+stationary environments for which the algorithms proposed
+in Gajane, Urvoy, and Kaufmann (2018) (and all other dif-
+ferentially private algorithms designed for stationary envi-
+ronment) will suffer regret linear in the number of time steps
+T . On the other hand, the algorithm proposed in this article
+can guarantee regret sub-linear in T in such scenarios. Fur-
+thermore, due to the changing environment and the use of
+a sliding window, the regret analysis in our article presents
+challenges not encountered in stationary settings.
+Our extension to non-stationary environments is practi-
+cally relevant as the assumption of stationarity is some-
+times unrealistic in real-world applications. Such an exten-
+sion providing local differential privacy in non-stationary
+environments for the problem of data collection is given by
+Joseph et al. (2018). Our problem is different than Joseph
+et al. (2018) as we study learning to make optimal sequen-
+tial decisions in a non-stationary environment while provid-
+ing local differential privacy. Note that a naive strategy of
+restarting an algorithm (designed for a stationary environ-
+ment) after each change is not possible in the problem con-
+sidered here as the time instants at which the changes occur
+are unknown.
+Related Work
+In the context of sequential decision-
+making, global differential privacy has been studied in
+various settings including stochastic bandits (Mishra and
+Thakurta 2015; Tossou and Dimitrakakis 2016), adversar-
+ial bandits (Thakurta and Smith 2013; Tossou and Dimi-
+trakakis 2017) and collaborative bandits (Wang et al. 2020).
+In the context of sequential decision-making, local differ-
+ential privacy has been considered in stochastic bandit set-
+ting (Gajane, Urvoy, and Kaufmann 2018; Tao et al. 2022),
+contextual bandits (Zheng et al. 2020), collaborative bandits
+(Wang et al. 2020) and Markov decision processes (Chowd-
+hury and Zhou 2022; Garcelon et al. 2020). For a compre-
+hensive overview of differential privacy and its application
+to other problems, see Dwork and Roth (2014).
+The notion of using a sliding window mechanism (as we
+do in our proposed algorithm) to deal with a non-stationary
+environment has been employed in classical bandits (Gariv-
+ier and Moulines 2011) as well as Markov decision pro-
+cesses (Gajane, Ortner, and Auer 2018).
+Non-Stationary Stochastic Corrupt Bandits
+A non-stationary stochastic corrupt bandits problem is for-
+mally characterized by a set of arms A = {1, . . ., K}
+on which are indexed a list of unknown sub-Gaussian
+reward distributions {νa(1)}a∈A, . . . , {νa(LT )}a∈A,
+a
+list
+of
+unknown
+sub-Gaussian
+feedback
+distributions
+{ςa(1)}a∈A, . . . , {ςa(LT )}a∈A, and a list of known mean-
+corruption functions {ga}a∈A. Here, the total number of
+time steps (i.e., the horizon) is indicated as T . The environ-
+ment undergoes LT abrupt changes at unknown time steps
+called as breakpoints and it remains constant in the intervals
+between two successive breakpoints.
+For notational convenience, we assume that the first
+breakpoint occurs at t = 1. From ith breakpoint till the
+subsequent breakpoint (or the horizon, in case of the last
+breakpoint), if the learner pulls an arm a ∈ A at time t,
+they receive a (hidden) reward Rt drawn from the distri-
+bution νa(i) with mean µa(i) and observe a feedback Ft
+drawn from the distribution ςa(i) with mean λa(i). We as-
+sume that, for each arm, there exists a loose link between the
+reward and the feedback through a known corruption func-
+tion ga which maps the mean of the reward distribution to the
+mean of the feedback distribution : ga(µa(i)) = λa(i), ∀a ∈
+A and 1 ≤ i ≤ LT . Our proposed algorithm and the proven
+regret bound also work if the corruption function for an arm
+changes across time as long as the current corruption func-
+tion is known.
+Note that these ga functions may be completely different
+from one arm to another. For Bernoulli distributions, the re-
+ward distributions and the feedback distributions are in [0, 1]
+for all a ∈ A and we assume all the corruption functions
+{ga}a∈A to be continuous in this interval. We also assume
+the corruption functions {ga}a∈A to be strictly monotonic
+and denote the corresponding inverse functions by g−1
+a . The
+assumption of monotonicity is required for efficient learning
+as proved in Gajane, Urvoy, and Kaufmann (2018).
+Another way to define the link between the reward and
+the feedback is to provide a corruption scheme operator ˜ga
+which maps the rewards into feedback distributions.
+Randomized Response
+Randomized response (a privacy
+protection technique introduced by (Warner 1965)) can also
+be simulated by a Bernoulli corrupt bandits problem and the
+corresponding corruption scheme ˜ga is encoded as:
+Ma :=
+�
+0
+1
+0
+p00(a)
+1 − p11(a)
+1
+1 − p00(a)
+p11(a)
+�
+(1)
+Each item in Ma denotes the probability of observing a par-
+ticular feedback for a particular reward i.e., Ma(y, x) :=
+P
+�
+Feedback from arm a = y | Reward from arm a = x
+�
+.
+The corresponding corruption function is ga(x) = 1 −
+p00(a) + [p00(a) + p11(a) − 1] · x.
+To measure the utility of an algorithm for this problem,
+we define the notion of regret in the following. Let us de-
+note the mean reward of arm a at time step t as µa,t.
+
+The objective of an algorithm, which chooses the arm ˆat
+at time t based only on the previously observed feedback,
+F1, . . . , Ft−1, is to maximize the expected sum of rewards
+i.e., to achieve high utility. This is equivalent to minimiz-
+ing the regret, Regret(T ) := �T
+t=1 µ∗,t − E
+��T
+t=1 µˆat,t
+�
+,
+where µ∗,t := maxa∈A µa,t. Regret measures the perfor-
+mance of the algorithm against an omniscient policy that at
+each time step chooses the arm with the maximal mean re-
+ward. Thus, low regret translates to achieving high utility.
+The Proposed Algorithm
+To solve the problem at hand, we propose SW-KLUCB-
+CF, an adaptation of the kl-UCB algorithm of Capp´e et al.
+(2013). The algorithm takes as input: the window size w,
+a non-decreasing function f, the horizon T and the corrup-
+tions functions g1, . . . , gK. We assume that the horizon T
+is known; an unknown T can be handled using the doubling
+trick (Besson and Kaufmann 2018). We use d(x, y) to denote
+the Kullback–Leibler divergence between two Bernoulli dis-
+tributions with mean x and y. We also use a shorthand of
+x ∧ y to denote min(x, y).
+At each time time step t, the algorithm computes an
+Indexa(t), which is an upper-confidence bound on µa,t
+built from a confidence interval on λa,t based on the KL-
+divergence. The quantity Na(t, w) denotes the number of
+times arm a was chosen in the last w time steps until time t.
+Correspondingly, ˆλa(t, w) denotes the empirical mean of the
+feedback observed from arm a in the last w time steps until
+time t: ˆλa(t, w) :=
+1
+Na(t,w)
+�t
+s=min{1,t−w+1} Fs · 1(ˆas=a).
+Theorem 1 gives an upper bound on the regret of SW-
+KLUCB-CF. A more explicit bound is proved in the Ap-
+pendix.
+Theorem 1 The regret of SW-KLUCB-CF using f(x) :=
+log(x) + 3 log(log(x)) and w =
+�
+4eT
+LT +4 on a Bernoulli
+non-stationary stochastic corrupt bandits problem with
+strictly monotonic and continuous corruption functions
+{ga}a∈A at time T is upper-bounded by 2
+˜O
+
+�
+a∈A
+�
+LT T +
+LT
+�
+i=1
+�
+a̸=a∗(i)
+log
+��
+T
+LT
+�
+d(λa(i), ga(µ∗(i))
+
+ ,
+where a∗(i) and µ∗(i) are the optimum arm and the cor-
+responding optimal mean respectively after ith change and
+before the subsequent change.
+The lower bound on regret in terms T for classical
+non-stationary stochastic bandits is Ω(
+√
+T) (Garivier and
+Moulines 2011). Theorem 1 matches the lower bound up to
+logarithmic factors, so SW-KLUCB-CF has near-optimal
+regret guarantees in terms of the time horizon T . The
+best known regret upper bounds for classical non-stationary
+stochastic bandits (e.g., Auer, Gajane, and Ortner (2019))
+also feature logarithmic terms besides the lower bound,
+hence our regret bound is in line with the best known results
+2 ˜O ignores logarithmic factors and constants.
+Algorithm 1: Sliding Window KLUCB for Non-Stationary
+Stochastic Corrupt Bandits (SW-KLUCB-CF)
+Input: Window size w, a non-decreasing function
+f : N → R, T , monotonic and continuous corruption
+functions g1, . . . , gK and d(x, y) := KL(B(x), B(y)),
+1. Initialization: Pull each arm once.
+2. for time t = K, . . . , T − 1 do
+(a) Compute for each arm a ∈ A the quantity
+Indexa(t)
+:= max
+�
+q : Na(t, w) · d(ˆλa(t, w), ga(q)) ≤ f (t ∧ w)
+�
+(b) Pull arm ˆat+1 := argmax
+a∈A
+Indexa(t) and observe the
+feedback Ft+1.
+end for
+for analogous problems. Moreover, the bound in Theorem 1
+also matches the best known regret bound in terms of LT for
+classical non-stationary stochastic bandits which is O√LT .
+We can use SW-KLUCB-CF on non-stationary stochas-
+tic corrupts bandits where the corruption is done via random-
+ized response. The following corollary bounds the resulting
+regret.
+Corollary 1 The regret of SW-KLUCB-CF on a Bernoulli
+non-stationary stochastic corrupt bandits problem with ran-
+domized response using corruption matrices {M}a∈A at
+time T is upper-bounded by
+˜O
+
+�
+a∈A
+�
+LT T +
+LT
+�
+i=1
+�
+a̸=a∗(i)
+log
+��
+T
+LT
+�
+(p00(a) + p11(a) − 1)2
+
+ .
+This corollary follows from Theorem 1 and Pinsker’s in-
+equality: d(x, y) > 2(x−y)2. The term (p00(a)+p11(a)−1)
+can be understood as the slope of the corruption function ga.
+Corruption Mechanism to Preserve Local
+Privacy in Non-Stationary Environment
+First, let us formally define local differential privacy.
+Definition 1 (Locally differentially private mechanism) Any
+randomized mechanism M is ǫ-locally differentially private
+for ǫ ≥ 0, if for all d1, d2 ∈ Domain(M) and for all S ⊂
+Range(M),
+P[M(d1) ∈ S] ≤ eǫ · P[M(d2) ∈ S].
+As done in Gajane, Urvoy, and Kaufmann (2018), a straight-
+forward approach to achieve local differential privacy us-
+ing corrupt bandits is to employ a corruption scheme on the
+user feedback. This is similar to how randomized response
+is used in data collection by Wang, Wu, and Hu (2016).
+Definition 2 (ǫ-locally differentially private bandit feed-
+back corruption scheme) A bandit feedback corruption
+scheme ˜g is ǫ-locally differentially private for ǫ ≥ 0, if for
+
+all reward sequences Rt1, . . . , Rt2 and R′
+t1 . . . , R′
+t2, and for
+all S ⊂ Range(˜g),
+P[˜g(Rt1, . . . , Rt2) ∈ S] ≤ eǫ · P[˜g(R′
+t1, . . . , R′
+t2) ∈ S].
+When
+corruption
+is
+done
+by
+randomized
+re-
+sponse,
+local
+differential
+privacy
+requires
+that
+max1≤a≤K
+�
+p00(a)
+1−p11(a),
+p11(a)
+1−p00(a)
+�
+≤ eǫ. From Corollary 1,
+we can see that to achieve lower regret, p00(a) + p11(a) is
+to be maximized for all a ∈ A. Using Wang, Wu, and Hu
+(2016, Result 1), we can state that, in order to achieve ǫ-local
+differential privacy while maximizing p00(a) + p11(a),
+Ma =
+�
+0
+1
+0
+eǫ
+1+eǫ
+1
+1+eǫ
+1
+1
+1+eǫ
+eǫ
+1+eǫ
+�
+.
+(2)
+As it turns out, this is equivalent to the staircase mechanism
+for local privacy which is the optimal local differential pri-
+vacy mechanism for low privacy regime (Kairouz, Oh, and
+Viswanath 2016, Theorem 14). The trade-off between utility
+and privacy is controlled by ǫ.
+Using the corruption parameters from Eq. (2) with Corol-
+lary 1, we arrive at the following upper bound.
+Corollary 2 At time T , the regret of
+SW-KLUCB-
+CF
+with
+ǫ-locally
+differentially
+private
+bandit
+feedback
+corruption
+scheme
+given
+by
+Eq.
+(2)
+is
+˜O
+�
+�
+a∈A
+√LTT + �LT
+i=1
+�
+a̸=a∗(i)
+log
+��
+T
+LT
+�
+( eǫ−1
+eǫ+1)
+2
+�
+.
+The term
+� eǫ−1
+eǫ+1
+�2 in the above expression conveys the rela-
+tionship of the regret with the level of local differential pri-
+vacy symbolized by ǫ. For low values of ǫ,
+� eǫ−1
+eǫ+1
+�
+≈ ǫ/2.
+This is in line with other bandit algorithms providing differ-
+ential privacy (e.g., Mishra and Thakurta (2015)).
+Elements of Mathematical Analysis
+Here, we provide a proof outline for Theorem 1. Please refer
+to the Appendix for the complete proof.
+We start by bounding the expected number of times a sub-
+optimal arm (i.e., an arm other than the optimal arm at the
+time of selection) is pulled by the algorithm till horizon T .
+Recall that, at any time step t, SW-KLUCB-CF pulls an
+arm maximizing an index defined as
+Indexa(t)
+:= max
+�
+q : Na(t, w) · d
+�
+ˆλa(t, w), ga(q)
+�
+≤ f (t ∧ w)
+�
+= max g−1
+a
+��
+q : Na(t, w) · d
+�
+ˆλa(t, w), q
+�
+≤ f (t ∧ w)
+��
+.
+We further decompose the computation of index as follows,
+Indexa(t) :=
+�g−1
+a (ℓa(t))
+if ga is decreasing,
+g−1
+a (ua(t))
+if ga is increasing
+where,
+ℓa(t) := min
+�
+q : Na(t, w) · d
+�
+ˆλa(t, w), q
+�
+≤ f (t ∧ w)
+�
+,
+ua(t) := max
+�
+q : Na(t, w) · d
+�
+ˆλa(t, w), q
+�
+≤ f (t ∧ w)
+�
+.
+The interval [ℓa(t), ua(t)] is a KL-based confidence in-
+terval on the mean feedback λa,t of arm a at time t. This
+is in contrast to kl-UCB (Capp´e et al. 2013) where a con-
+fidence interval is placed on the mean reward. Furthermore,
+This differs from kl-UCB-CF (Gajane, Urvoy, and Kauf-
+mann 2018) where the mean feedback of an arm remains the
+same for all the time steps and f does not feature w.
+In our analysis, we use the fact that when an arm a is
+picked at time t+1 by SW-KLUCB-CF, one of the follow-
+ing is true: Either the mean feedback of the optimal arm a∗,t
+with mean reward µ∗,t is outside its confidence interval (i.e.,
+ga∗,t(µ∗,t) < ℓa∗,t(t) or ga∗,t(µ∗,t) > ua∗,t(t)) which is
+unlikely. Or, the mean feedback of the optimal arm is where
+it should be, and then the fact that arm a is selected indicates
+that the confidence interval on λa cannot be too small as ei-
+ther (ua(t) ≥ ga(µ∗,t)) or (ℓa(t) ≤ ga(µ∗,t)). The previous
+statement follows from considering various cases depending
+on whether the corruption functions ga and ga∗,t are increas-
+ing or decreasing. We then need to control the two terms in
+the decomposition of the expected number of draws of arm
+a. The term regarding the “unlikely” event, is bounded using
+the same technique as in the kl-UCB analysis, however with
+some added challenges due to the use of a sliding window. In
+particular, the analysis of a typical upper confidence bound
+algorithm for bandits relies on the fact that the confidence
+interval for any arm is always non-increasing, however this
+is not true while using a sliding window. To control the sec-
+ond term, depending on the monotonicity of the corruption
+functions ga and ga∗,t, we need to meticulously adapt the
+arguments in Capp´e et al. (2013) to control the number of
+draws of a suboptimal arm, as can be seen in the Appendix.
+Concluding Remarks
+In this work, we proposed the setting of non-stationary
+stochastic corrupt bandits for preserving privacy while still
+maintaining high utility in sequential decision making in a
+changing environment. We devised an algorithm called SW-
+KLUCB-CF and proved its regret upper bound which is
+near-optimal in the number of time steps and matches the
+best known bound for analogous problems in terms of the
+number of time steps and the number of changes. Moreover,
+we provided an optimal corruption scheme to be used with
+our algorithm in order to attain the dual goal of achieving
+high utility while maintaining the desired level of privacy.
+Interesting directions for future work include:
+1. Complete an empirical evaluation of the proposed algo-
+rithm on simulated as well as real-life data.
+2. Characterize the changes in the environment by a varia-
+tion budget (as done in Besbes, Gur, and Zeevi (2014) for
+classical bandits) instead of the number of changes.
+3. Incorporate contextual information in the learning pro-
+cess.
+4. Propose a Bayesian algorithm for non-stationary stochas-
+tic corrupt bandits.
+5. Propose a (near-)optimal differentially private algorithm
+which does not need to know the number of changes.
+
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+Zheng, K.; Cai, T.; Huang, W.; Li, Z.; and Wang, L. 2020.
+Locally Differentially Private (Contextual) Bandits Learn-
+ing. In Larochelle, H.; Ranzato, M.; Hadsell, R.; Balcan, M.;
+and Lin, H., eds., Advances in Neural Information Process-
+ing Systems, volume 33, 12300–12310. Curran Associates,
+Inc.
+
+Proof of Theorem 1
+Proof. The proof follows along the lines of the proof for Theorem 2 from Gajane, Urvoy, and Kaufmann (2018).
+The index used by SW-KLUCB-CFis defined by
+Indexa(t) := max
+�
+q : Na(t, w) · d
+�
+ˆλa(t, w), ga(q)
+�
+≤ f (t ∧ w)
+�
+= max g−1
+a
+��
+q : Na(t, w) · d
+�
+ˆλa(t, w), q
+�
+≤ f (t ∧ w)
+��
+.
+For the purpose of this proof, we further decompose the computation of index as follows,
+Indexa(t) :=
+�g−1
+a (ℓa(t))
+if ga is decreasing,
+g−1
+a (ua(t))
+if ga is increasing
+where,
+ℓa(t) := min
+�
+q : Na(t, w) · d
+�
+ˆλa(t, w), q
+�
+≤ f (t ∧ w)
+�
+and
+ua(t) := max
+�
+q : Na(t, w) · d
+�
+ˆλa(t, w), q
+�
+≤ f (t ∧ w)
+�
+.
+Note that, the optimal arm at time t is denoted as a∗,t and µ∗,t is the corresponding optimal mean. Along the same lines, let
+ℓ∗(t) := ℓa∗,t(t) and u∗(t) := ua∗,t(t).
+Let Na(t) be the number of times arm a has been pulled till time t. To get an upper bound on the regret of our algorithm,
+we first bound E[Na(t)] for all the non-optimal arms a (i.e., a ̸= a∗,t at time t). Recall that µi,t is the mean reward of arm i
+at time step t. Let us define T (w) as the set of indices t ∈ {K + 1, . . . , T } such that µi,s = µi,t for all i ∈ {1, . . . , K} and
+all t − w < s ≤ t. That is to say T (w) is the set of all time steps t ∈ {K + 1, . . . , T } for which there was no change in the
+previous w time steps. Recall that ˆat is the arm chosen by the algorithm at time step t. Then,
+E(Na(T )) = 1 +
+T −1
+�
+t=K
+P(ˆat+1 = a)
+≤ 1 + LT · w +
+�
+K≤t≤T −1, t∈T (w)
+P(ˆat+1 = a).
+Depending upon if ga and ga∗,t are increasing or decreasing there are four possible sub-cases:
+• Both ga∗,t and ga are increasing.
+(ˆat+1 = a)
+⊆
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, u∗(t) ≥ ga∗,t(µ∗,t)
+�
+=
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, g−1
+a∗,t(u∗(t)) ≥ µ∗,t
+�
+since ga∗,t is increasing
+=
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, g−1
+a (ua(t)) ≥ µ∗,t
+�
+since Indexa ≥ Indexa∗,t
+=
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+∪ (ˆat+1 = a, ua(t) ≥ ga(µ∗,t))
+since ga is increasing.
+∴ E(Na(T )) ≤1 + LT · w +
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
++
+�
+K≤t≤T −1, t∈T (w)
+P (ˆat+1 = a, ua(t) ≥ ga(µ∗,t)) .
+(3)
+• ga∗,t is decreasing and ga is increasing.
+(ˆat+1 = a)
+⊆
+�
+ℓ∗(t) > ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, ℓ∗(t) ≤ ga∗,t(µ∗,t)
+�
+=
+�
+ℓ∗(t) > ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, g−1
+a∗,t(ℓ∗(t)) ≥ µ∗,t
+�
+since ga∗,t is decreasing
+=
+�
+ℓ∗(t) > ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, g−1
+a (ua(t)) ≥ µ∗,t
+�
+since Indexa ≥ Indexa∗,t
+=
+�
+ℓ∗(t) > ga∗,t(µ∗,t)
+�
+∪ (ˆat+1 = a, ua(t) ≥ ga(µ∗,t))
+since ga is increasing.
+
+∴ E(Na(T )) ≤1 + LT · w +
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+ℓ∗(t) > ga∗,t(µ∗,t)
+�
++
+�
+K≤t≤T −1, t∈T (w)
+P (ˆat+1 = a, ua(t) ≥ ga(µ∗,t)) .
+(4)
+• ga∗,t is increasing and ga is decreasing.
+(ˆat+1 = a)
+⊆
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, u∗(t) ≥ ga∗,t(µ∗,t)
+�
+=
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, g−1
+a∗,t(u∗(t)) ≥ µ∗,t
+�
+since ga∗,t is increasing
+=
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+∪
+�
+ˆat+1 = a, g−1
+a (ℓa(t)) ≥ µ∗,t
+�
+since Indexa > Indexa∗,t
+=
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+∪ (ˆat+1 = a, ℓa(t) ≤ ga(µ∗,t))
+since ga is decreasing.
+∴ E(Na(T )) ≤1 + LT · w +
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
++
+�
+K≤t≤T −1, t∈T (w)
+P (ˆat+1 = a, ℓa(t) ≤ ga(µ∗,t)) .
+(5)
+• ga∗,t is decreasing and ga is decreasing.
+(ˆat+1 = a)
+⊆
+�
+ℓ∗(t) > ga∗,t(µa∗,t)
+�
+∪
+�
+ˆat+1 = a, ℓ∗(t) ≤ ga∗,t(µa∗,t
+�
+=
+�
+ℓ∗(t) > ga∗,t(µa∗,t)
+�
+∪
+�
+ˆat+1 = a, g−1
+a∗,t(ℓ∗(t)) ≥ µa∗,t
+�
+since ga∗,t is decreasing
+=
+�
+ℓ∗(t) > ga∗,t(µa∗,t)
+�
+∪
+�
+ˆat+1 = a, g−1
+a (ℓa(t)) ≥ µa∗,t
+�
+since Indexa > Indexa∗,t
+=
+�
+ℓ∗(t) > ga∗,t(µa∗,t)
+�
+∪
+�
+ˆat+1 = a, ℓa(t) ≤ ga(µa∗,t)
+�
+since ga is decreasing.
+∴ E(Na(T )) ≤1 + LT · w +
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+ℓ∗(t) > ga∗,t(µa∗,t)
+�
++
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+ˆat+1 = a, ℓa(t) ≤ ga(µa∗,t)
+�
+.
+(6)
+We first upper bound the two sums
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+and
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+ℓ∗(t) > ga∗,t(µa∗,t)
+�
+(7)
+using that ℓ∗(t) and u∗(t) are respectively lower and upper confidence bound on ga∗,t(µ∗,t). Recall that min {t, w} is denoted
+as t ∧ w.
+P
+�
+ua∗,t < ga∗,t(µ∗,t)
+�
+≤ P
+�
+ga∗,t(µ∗,t) > ˆλa∗,t(t, w) and Na∗,t(t, w) · d
+�
+ˆλa∗,t(t, w), ga∗,t(µ∗,t)
+�
+≥ f (t ∧ w)
+�
+≤ P
+�
+∃s ∈ {1, . . . , (t ∧ w)} : ga∗,t(µ∗,t) > ˆλa∗,t,s and s · d(ˆλa∗,t,s, ga∗,t(µ∗,t)) ≥ f (t ∧ w)
+�
+≤ min
+�
+1, e ⌈f (t ∧ w) log t⌉ e−f(t∧w)�
+,
+(8)
+where the upper bound follows from Lemma 2 in Capp´e et al. (2013), and the fact that ˆλa∗,t,s is the empirical mean of s
+Bernoulli samples with mean ga∗,t(µ∗,t). Similarly, one has
+P
+�
+ℓ∗(t) > ga∗,t(µa∗,t)
+�
+≤ min
+�
+1, e ⌈f (t ∧ w) log t⌉ e−f(t∧w)�
+.
+(9)
+As f(x) := log x + 3(log log x), for x ≥ 3,
+e⌈f(x) log x⌉ ≤ 4e log2 x.
+
+Then, using Eq. (8) and Eq. (9), the two quantities in Eq. (7) can be upper bounded by
+1 +
+T −1
+�
+t=3
+e ⌈f (t ∧ w) log t⌉ e−f(t∧w) ≤ 1 +
+T −1
+�
+t=3
+4e · log2 (t ∧ w) · e−f(t∧w)
+= 1 + 4e
+T −1
+�
+t=3
+1
+(t ∧ w) · log (t ∧ w)
+= 1 + 4e
+w
+�
+t=3
+1
+(t ∧ w) · log (t ∧ w) + 4e
+T
+�
+t=w+1
+1
+(t ∧ w) · log (t ∧ w)
+≤ 1 + 4e
+w
+�
+t=3
+1
+3 log 3 + 4e
+T
+�
+t=w+1
+1
+w log w
+≤ 1 +
+4ew
+3 log 3 +
+4eT
+w log w.
+This proves that
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+u∗(t) < ga∗,t(µ∗,t)
+�
+≤ 1 +
+4ew
+3 log 3 +
+4eT
+w log w
+and,
+(10)
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+ℓ∗(t) > ga∗,t(µa∗,t)
+�
+≤ 1 +
+4ew
+3 log 3 +
+4eT
+w log w .
+(11)
+We now turn our attention to the other two sums involved in the upper bound we gave for E(Na(T )). Let the unknown time-
+step at which ith change occurs be denoted as ti. For notational convenience, we assume that the first change occurs at t = 1 so
+t1 = 1 and change L+1 takes place at t = T +1 where T is the horizon. We introduce the notation d+(x, y) = d(x, y)·1(x<y)
+and d−(x, y) = d(x, y) · 1(x>y). So we can write, when ga is increasing,
+�
+K≤t≤T −1, t∈T (w)
+P (ˆat+1 = a, ua(t) ≥ ga(µ∗,t))
+≤
+L
+�
+i=1
+�
+ti≤t<ti+1−1, t∈T (w)
+P (ˆat+1 = a, ua(t) ≥ ga(µ∗,t))
+= E
+
+
+L
+�
+i=1
+�
+ti≤t<ti+1−1, t∈T (w)
+1ˆat+1=a · 1Na(t,w)·d+(ˆλa,Na(t,w),ga(µ∗,t))≤f(t∧w)
+
+
+≤ E
+
+
+L
+�
+i=1
+�
+ti≤t<ti+1−1, t∈T (w)
+t∧w
+�
+s=1
+1ˆat+1=a · 1Na(t,w)=s · 1s·d+(ˆλa,s,ga(µ∗,t))≤f(t∧w)
+
+
+≤ E
+
+
+L
+�
+i=1
+�
+ti≤t<ti+1−1, t∈T (w)
+t∧w
+�
+s=1
+1ˆat+1=a · 1Na(t)=s · 1s·d+(ˆλa,s,ga(µ∗,t))≤f(t∧w)
+
+
+≤ E
+�
+L
+�
+i=1
+t∧w
+�
+s=1
+1s·d+(ˆλa,s,ga(µ∗,t))≤f(t∧w)
+�
+ti≤t<ti+1−1, t∈T (w)
+1ˆat+1=a · 1Na(t)=s
+�
+��
+�
+≤1
+�
+.
+In the above, the penultimate steps follows from the fact that the event Na(t, w) = s is subsumed by the event Na(t) = s. So,
+one obtains, when ga is increasing,
+�
+K≤t≤T −1, t∈T (w)
+P (ˆat+1 = a, ua(t) ≥ ga(µ∗,t)) ≤ P
+� L
+�
+l=1
+t∧w
+�
+s=1
+s · d+(ˆλa,s, ga(µ∗,t)) ≤ f(t ∧ w)
+�
+.
+(12)
+Using similar arguments, one can show that when ga is decreasing,
+�
+K≤t≤T −1, t∈T (w)
+P
+�
+ˆat+1 = a, ℓa(t) ≤ ga(µa∗,t)
+�
+≤ P
+� L
+�
+l=1
+t∧w
+�
+s=1
+s · d−(ˆλa,s, ga(µ∗,t)) ≤ f(t ∧ w)
+�
+.
+(13)
+
+Recall that µa(i) is the mean reward of arm a after ith change and before the subsequent change. Correspondingly, let λa(i)
+be the mean feedback of arm a after ith change and and before the subsequent change. Furthermore, let µ∗(i) be the optimum
+mean after ith change and and before the subsequent change.
+Using Appendix A.2. of (Capp´e et al. 2013), the quantity in the right-hand side of (12) can be upper-bounded by
+L
+�
+i=1
+f(w)
+d(λa(i), ga(µ∗(i)) +
+L
+�
+i=1
+√
+2π
+�
+d′(λa(i), ga(µ∗(i))2
+(d(λa(i), ga(µ∗(i))3
+�
+f(w) +
+L
+�
+i=1
+2
+�d′(λa(i), ga(µ∗(i))
+d(λa(i), ga(µ∗(i))
+�2
++ 1.
+(14)
+For (13), noting that d−(x, y) = d+(1 − x, 1 − y), one has
+P
+�
+s · d−(ˆλa,s, ga(µ∗,t)) ≤ f(t ∧ w)
+�
+=P
+�
+s · d+(1 − ˆλa,s, 1 − ga(µ∗,t)) ≤ f(t ∧ w)
+�
+=P
+�
+s · d+(ˆµa,s, 1 − ga(µ∗,t)) ≤ f(t ∧ w)
+�
+,
+where ˆµa,s := 1−ˆλa,s, is the empirical mean of s observations of a Bernoulli random variable with mean 1−λa < 1−ga(µ∗,t).
+Hence, the analysis of (Capp´e et al. 2013) can be applied, and using that d(1 − x, 1 − y) = d(x, y) and d′(1 − x, 1 − y) =
+−d′(x, y), the right hand side of (13) can also be upper bound by (14).
+Combining inequalities (10), (11) and (12),(13), (14) with the initial decomposition of E[Na(T )], and substituting f(x) :=
+log(x) + 3 log log(x) yields in all cases,
+E[Na(T )] ≤ LT · w +
+4ew
+3 log 3 +
+4eT
+w log w +
+LT
+�
+i=1
+f(w)
+d(λa(i), ga(µ∗(i))
++
+LT
+�
+i=1
+√
+2π
+�
+d′(λa(i), ga(µ∗(i))2
+(d(λa(i), ga(µ∗(i))3
+�
+f(w)
++
+LT
+�
+i=1
+2
+�d′(λa(i), ga(µ∗(i))
+d(λa(i), ga(µ∗(i))
+�2
++ 5
+≤ (LT + 4) · w +
+4eT
+w log w +
+LT
+�
+i=1
+log(w) + 3 log log(w)
+d(λa(i), ga(µ∗(i))
++
+LT
+�
+i=1
+√
+2π
+�
+d′(λa(i), ga(µ∗(i))2
+(d(λa(i), ga(µ∗(i))3
+�
+log(w) + 3 log log(w)
++
+LT
+�
+i=1
+2
+�d′(λa(i), ga(µ∗(i))
+d(λa(i), ga(µ∗(i))
+�2
++ 5.
+(15)
+Minimizing the leading terms in the RHS from eq. (15) via taking the first derivative with respect to w and equating it to 0,
+leads to solving for w in
+w2 �
+log2 w
+�
+log w + 1
+=
+4eT
+LT + 4
+≃ w2 log (w2) =
+8eT
+LT + 4
+Here, w must be positive for the log to exist, so we can write w2 = eu for some u, and the equation becomes
+ueu =
+8eT
+LT + 4.
+This equation has no solution in an elementary expression, although it can be expressed in terms of the Lambert W function
+(Corless et al. 1996). Opting for an elementary expression for w, we can choose w =
+�
+4eT
+LT +4, which leads to the following
+
+bound,
+E[Na(T )] ≤
+�
+4e(LT + 4)T +
+�
+4e(LT + 4)T
+log
+��
+4eT
+LT +4
+� +
+LT
+�
+i=1
+log
+��
+4eT
+LT +4
+�
++ 3 log log
+��
+4eT
+LT +4
+�
+d(λa(i), ga(µ∗(i))
++
+LT
+�
+i=1
+√
+2π
+�
+d′(λa(i), ga(µ∗(i))2
+(d(λa(i), ga(µ∗(i))3
+�
+�
+�
+�log
+��
+4eT
+LT + 4
+�
++ 3 log log
+��
+4eT
+LT + 4
+�
++
+LT
+�
+i=1
+2
+�d′(λa(i), ga(µ∗(i))
+d(λa(i), ga(µ∗(i))
+�2
++ 5.
+Since the rewards are bounded in [0, 1] for Bernoulli non-stationary stochastic bandits, the regret is upper-bounded by,
+˜O
+
+�
+a∈A
+�
+LTT +
+�
+a̸=a∗(i)
+LT
+�
+i=1
+log
+��
+T
+LT
+�
+d(λa(i), ga(µ∗(i))
+
+ .
+Assuming that LT =
+�
+T β�
+for some β ∈ [0, 1), the expected regret is upper bounded as ˜O
+�
+T (1+β)/2�
+. In particular, if β = 0,
+the number of breakpoints is upper-bounded by L independently of T , then with w =
+�
+4eT
+L+4, the upper bound is ˜O
+�√
+LT
+�
+.
+

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+1
+DRL-GAN: A Hybrid Approach for Binary and
+Multiclass Network Intrusion Detection
+Caroline Strickland∗, Chandrika Saha†, Muhammad Zakar‡,
+Sareh Soltani Nejad§, Noshin Tasnim¶, Daniel Lizotte∥, Anwar Haque∗∗
+Department of Computer Science, The University of Western Ontario, London, Canada
+{∗cstrick4, †csaha, ‡mzakar, §ssolta7, ¶ntasnim3, ∥dlizotte, ∗∗ahaque32}@uwo.ca
+Abstract—Our increasingly connected world continues to face
+an ever-growing amount of network-based attacks. Intrusion
+detection systems (IDS) are an essential security technology for
+detecting these attacks. Although numerous machine learning-
+based IDS have been proposed for the detection of malicious
+network traffic, the majority have difficulty properly detecting
+and classifying the more uncommon attack types. In this paper,
+we implement a novel hybrid technique using synthetic data
+produced by a Generative Adversarial Network (GAN) to use as
+input for training a Deep Reinforcement Learning (DRL) model.
+Our GAN model is trained with the NSL-KDD dataset for four
+attack categories as well as normal network flow. Ultimately, our
+findings demonstrate that training the DRL on specific synthetic
+datasets can result in better performance in correctly classifying
+minority classes over training on the true imbalanced dataset.
+Index Terms—Network Security, Network Intrusion Detection
+System, Deep Reinforcement Learning, Generative Adversarial
+Networks, NSL-KDD, Machine Learning, Artificial Intelligence.
+I. INTRODUCTION
+T
+HE increasing volume and sophistication of network-
+based attacks motivate the development of effective tech-
+niques and tools to prevent service disruption, unauthorized
+access, and the disclosure of sensitive information [1]. An
+Intrusion Detection System (IDS) is an important defence tool
+against sophisticated and increasing network attacks, but these
+systems, especially Machine Learning (ML) based systems,
+require large, reliable, and valid network traffic datasets to be
+effective. Although the majority of recently available datasets
+cover a range of network attack types and traffic patterns
+and include information about the attacking infrastructure,
+modern networks are increasingly diversified such that existing
+datasets are often not enough to develop effective classification
+mechanisms. These datasets often suffer from a lack of traffic
+diversity and volume or fail to cover the full scope of known
+attack types. To cope up with these new changes, we require
+a more dynamic dataset that will improve the ability of
+an IDS to detect intrusions. Using deep learning techniques
+such as Generative Adversarial Networks (GANs), we can
+fabricate additional data using existing datasets to increase the
+classification accuracy of an IDS, especially for rare attack
+categories.
+Two methods of IDS are Signature-based Intrusion Detec-
+tion Systems (SNIDS) and Anomaly-based Intrusion Detec-
+tion Systems (ANIDS). The SNIDS approach is effective for
+known threats, as it looks for specific patterns (or ‘signatures’)
+such as byte sequences in network traffic, or known malicious
+instructions sequences used by malware [1]. Conversely, the
+ANIDS approach uses ML algorithms to analyze and monitor
+the network traffic in order to detect any suspicious activ-
+ity, thus being an effective method for catching unknown
+attacks [2].
+The emergence of deep learning and its integration with
+Reinforcement Learning (RL) has created a class of Deep
+Reinforcement Learning (DRL) methods that are able to detect
+the most recent and sophisticated types of network attacks.
+DRL combines artificial neural networks with a framework of
+RL that helps software agents (or ‘learning entities’) learn how
+to reach their goals. DRL combines function approximation
+and target optimization, mapping states and actions to the
+rewards they lead to [3]. This results in a ‘policy’ that our
+learning agents can follow to make the best decisions given the
+current state. To detect network attacks, DRL is used to train
+an agent such that, given a ‘state’ represented as a collection of
+feature values, will take the best ‘action’ (which, in our case,
+acts as a classification of attack type), in order to recognize
+an attack.
+Each network is different in that its behaviours and patterns
+evolve gradually. Naturally, vulnerabilities also evolve. The
+performance of IDS classification accuracy suffers as existing
+datasets gradually become out of date, invalid, and unreli-
+able. Moreover, reliable data cannot often be shared due to
+privacy concerns. Existing publicly available datasets do not
+include all of the existing network attack types, let alone the
+unknown vulnerabilities and attacks. To resolve this, we need
+more diverse and up-to-date datasets that properly reflect the
+characteristics of network intrusions in order to increase the
+performance of the IDS. Knowing this, we propose a SNIDS
+using DRL techniques. We use a collection of GAN models
+to generate varied datasets, then use DRL to implement an
+IDS and train the model on the GAN-generated datasets and
+compare our results.
+We use the open-source dataset NSL-KDD [4]. NSL-KDD
+is imbalanced with significantly less attack samples than
+normal traffic (especially for Probe, U2R, and R2L attacks).
+Thus, we used GAN to generate synthetic data so that there
+is a more even class balance. We then trained the DRL model
+on both the untouched NSL-KDD dataset as well as the GAN-
+generated data from each of our unique models for both binary
+and multiclass classification. Finally, we assess how training
+the DRL models using synthetic datasets compares in terms
+of IDS performance as well as individual class F1-scores.
+arXiv:2301.03368v1  [cs.CR]  5 Jan 2023
+
+2
+Overall, the primary contributions of this paper include:
+1) Using both conditional and unconditional CTGAN and
+copulaGAN models to generate tabular data. This is
+useful for increasing the minority class samples in im-
+balanced datasets, as well as providing large datasets for
+training ML models.
+2) Combining GAN and DRL techniques for the purpose of
+network intrusion detection and increasing the precision
+and recall for classifying underrepresented class data. We
+propose a framework that trains a GAN model to produce
+synthetic data, and then passes that data to a DRL model
+that acts as an IDS and either alerts a user to an attack
+or classifies the network traffic as benign.
+The remainder of this paper is organized as follows: Sec-
+tion II surveys related work for the purpose of network
+intrusion detection and presents the motivation and novelty
+behind this work. Section III discusses methodology and
+details necessary for implementation of our models. Section IV
+provides a comprehensive evaluation of the obtained results.
+Section V presents an interpretation of our findings. Finally,
+Section VI discusses directions for future work.
+II. RELATED WORK
+Hsu and Matsuoka [1] propose a DRL model for anomaly-
+based network intrusion detection. This approach treats the
+network traffic data as the RL environment state variables
+and the outcome of intrusion detection as the action. The
+correctness of the intrusion recognition result is used to
+determine the reward. The novelty of this work is that the
+DRL agent dynamically alternates between ‘detection mode’
+and ‘learning mode’ based on whether the current performance
+of the system is below a predefined threshold. In learning
+mode, the performance is evaluated through the reward and
+the model is updated with the new traffic data to improve the
+detection performance. In detection mode, a dummy reward is
+used to maintain operation and the true reward of the label is
+not calculated. The system was evaluated on pre-established
+benchmark datasets, NSL-KDD [4] and UNSW-NB15 [5],
+and consistently achieved over 90% in accuracy, recall, and
+precision performance metrics.
+Alavizadeh et al. [6] also propose a DRL-based contin-
+uously updating, self-learning NIDS. Their proposed Deep
+Q-Learning (DQL) model combines Q-learning based RL
+with a deep feed forward neural network to detect network
+intrusions. The model uses an ongoing trial and error auto-
+learning approach to improve its detection capabilities for
+different types of network intrusions. The model was evaluated
+on the NSL-KDD [4] dataset and outperformed some other ML
+techniques with 78% classification accuracy for the intrusion
+classes. This work is promising similarly to work in [1] due
+to its adaptive-learning capabilities, making it better suited
+for securing networks from the inevitably more sophisticated
+attacks cyber-attacks seen today.
+Benaddi et al. [7] developed a DRL-based IDS (DRL-IDS)
+for Wireless Sensor Networks (WSNs) [8] and Internet of
+Things (IoTs) [9]. Networking architectures like WSNs and
+IoTs are receiving increasingly more adoption in many areas
+such as healthcare, business, and smart cities and cyber-
+threats are the primary challenge for these networks [10]. They
+highlight that these networks are vulnerable to intrusions due
+to security flaws commonly found in IoT and WSN devices,
+zero-day vulnerabilities, and the openness of these networks
+to a large number of users. The DRL-IDS model improves
+intrusion detection performance while monitoring real-time
+network traffic. The model was evaluated against standard
+RL and K-Nearest Neighbours (KNN) based approaches using
+the NSL-KDD [4] dataset and performed better in terms of
+accuracy and detection rate with fewer false negatives.
+Lin et al. [11] propose a IDSGAN, a framework that uses
+GANs to generate adversarial malicious network traffic to
+deceive IDS. Their goal was to leverage GANs to improve
+IDS by exposing them to new, more combative and adversarial
+attack methods and types. This system modeled the black-box
+analogy of IDS from the perspective of an attacker that would
+generally not know about the internal details of the detec-
+tion system. A generator transformed known malicious traffic
+records into adversarial ones and a discriminator classified
+the records to learn about the originally unknown detection
+system. The authors demonstrated the validity of their system
+by only modifying the nonfunctional features of the records
+such that the modified records would still classify as an
+intrusion and not junk traffic. They evaluated their system
+using the standard NSL-KDD [4] dataset on multiple different
+detection models including Naive Bayes, Random Forest, and
+multilayer perceptron classifiers. IDSGAN achieved excellent
+results. The detection rate of the DoS attack type dropped from
+approximately 80% with normal records to less than 1% with
+modified, adversarial records.
+Ring et al. [12] used GANs to generate realistic flow-based
+network traffic data. They highlight that the ability of GANs
+to only process continuous attributes is a key challenge in
+using GANs to generate network traffic since network traffic
+data ultimately contains categorical features like IP addresses
+and ports. They propose three preprocessing techniques for
+converting categorical values in flow-based network traffic data
+into continuous values: (1) Simply treat features such as IP
+addresses and ports as numerical values (2) Create binary
+features from the categorical features (3) Use IP2Vec [13]
+to represent the categorical features as vectors. The authors
+evaluated these techniques on the CIDDS-001 [14] dataset and
+found that techniques (2) and (3) are effective at generating
+high-quality flow-based network traffic data. Finally, technique
+(1) is not well suited for this task, meaning that straightfor-
+ward numeric interpretation of categorical features should be
+avoided with GANs.
+Overall, there have been a handful of studies focused on
+using DRL to classify network traffic as normal or intrusion,
+as well as several that have used GANs to generate network
+traffic data. However, no study has combined these two ML
+approaches and evaluated the viability and effectiveness of this
+combination both in detecting and classifying network traffic
+as well as increasing the precision and recall performance for
+classifying previously underrepresented classes. Our proposed
+solution bridges this gap and improves the current state of
+knowledge in this field.
+
+3
+TABLE I
+NSL-KDD DATASET FEATURES
+F#
+Feature
+F#
+Feature
+F1
+Duration
+F22
+Is guest login
+F2
+Protocol type
+F23
+Count
+F3
+Service
+F24
+Srv count
+F4
+Flag
+F25
+Serror rate
+F5
+Src bytes
+F26
+Srv serror rate
+F6
+Dst bytes
+F27
+Rerror rate
+F7
+Land
+F28
+Srv rerror rate
+F8
+Wrong fragment
+F29
+Same srv rate
+F9
+Urgent
+F30
+Diff srv rate
+F10
+Hot
+F31
+Srv diff host rate
+F11
+Num failed logins
+F32
+Dst host count
+F12
+Logged in
+F33
+Dst host srv count
+F13
+Num compromised
+F34
+Dst host same srv rate
+F14
+Root shell
+F35
+Dst host diff srv rate
+F15
+Su attempted
+F36
+Dst host same src port rate
+F16
+Num root
+F37
+Dst host srv diff host rate
+F17
+Num file creations
+F38
+Dst host serror rate
+F18
+Num shells
+F39
+Dst host srv serror rate
+F19
+Num access files
+F40
+Dst host rerror rate
+F20*
+Num outbound cmds
+F41
+Dst host srv rerror rate
+F21
+Is host login
+F42
+Class label
+* Removed during data preprocessing.
+III. METHODOLOGY
+A. NSL-KDD Dataset
+NSL-KDD is an updated version of the KDD’99 dataset [4].
+Basic processing has been done, such as the removal of
+redundant records preventing classifiers from becoming biased
+towards more frequent records. The use of the NSL-KDD
+dataset has been very popular in studies on IDS, in a sense,
+becoming the de facto standard. It contains information which
+can help to build a host-based and network-based intrusion
+detection model to ensure network security in a variety of
+systems.
+The training and test set contains 125 973 and 22 544
+records, respectively. This includes 42 features, however we
+remove ‘Num outbound cmds’ as all records contain 0, so we
+are left with 41 features: 9 basic features, 12 content features
+for the connection, 9 temporal features calculated at two-
+second time windows, 10 statistical network traffic features,
+and the class label. Table I lists the features present in the
+dataset. The training set contains 22 attack types and the
+test set contains 37 attacks types. The 15 attack types not
+included in the training set make this dataset excellent for
+modelling unknown attacks. We opt to use the common 5-
+class classification of network traffic records: normal, DoS,
+Probe, R2L, and U2R. Table II describes these 5 classes in
+further detail. The class ID refers to the numerical mapping
+used by DRL and GAN.
+B. Machine Learning Performance Evaluation
+We used the accuracy and F1-score (which combines preci-
+sion and recall) metrics to evaluate the performance our DRL
+model and other ML algorithms. While the accuracy score only
+measures the percentage of correctly classified samples, this
+selection of performance metrics allows us to also evaluate the
+percentage of samples that were incorrectly classified. This is
+TABLE II
+NSL-KDD DATASET RECORD CLASSES
+ID
+Class
+Symbol
+# of Records
+Definition
+0
+Normal
+N
+77 054
+Normal network traffic
+record
+1
+DoS
+D
+53 387
+Denial of Service attack to
+prevent requests from
+intended users from being
+fulfilled
+2
+Probe
+P
+14 077
+Probing attack to gather
+information such as
+vulnerabilities about the
+target machine or network
+3
+R2L
+R
+3880
+An attacker tries to gain
+local access by sending
+packets to a remote
+machine
+4
+U2R
+U
+119
+An attacker with normal
+access tries to gain access
+to the root by exploiting
+system vulnerabilities
+Fig. 1. Confusion matrix for NIDS performance evaluation.
+especially important for NIDS as the accuracy performance
+metric is not enough to evaluate imbalanced datasets such as
+network traffic data which generally include significantly more
+normal traffic. These performance metrics are derived from the
+True Positive (TP), True Negative (TN), False Positive (FP),
+and False Negative (FN) values. Fig. 1 presents this confusion
+matrix used by our evaluation method.
+1) Accuracy: Accuracy measures the number of correct
+predictions out of the total predictions made by the model.
+In this case, accuracy measures the model’s ability to cor-
+rectly identify normal and attack traffic records. Equation 1
+formalizes the accuracy performance metric.
+Accuracy =
+TP + TN
+TP + FP + TN + FN
+(1)
+2) Precision: Precision measures the number of correct
+positive predictions out of the total number of positive predic-
+tions. In this case, precision measures the model’s degree of
+correctness in predicting attack records over the total number
+of attacks predicted [1], [15], [16]. Equation 2 formalizes the
+precision performance metric.
+Precision =
+TP
+TP + FP
+(2)
+3) Recall: Recall measures the number of correct positive
+predictions out of the total number of positive instances in
+the dataset. In this case, recall measures the model’s ability to
+
+Predicted Class
+Positive = A (Attack or Class)
+A
+N
+Negative = N (Normal
+A
+TP
+FP
+Actual
+Class
+N
+FN
+TN4
+correctly identify attack traffic records. From this definition,
+recall is also referred to as the true positive rate, detection rate,
+or sensitivity. Equation 3 formalizes the recall performance
+metric.
+Recall =
+TP
+TP + FN
+(3)
+4) F1-Score: F1-score is the harmonic mean of the pre-
+cision and recall values, essentially a combined measure of
+the two performance metrics. F1-score quantifies how dis-
+criminative the model is [17] and acts as a good indicator
+of performance since a decrease in either precision or recall
+results in a significant decrease in the F1-score. In addition,
+for multiclass classification we present both the unweighted
+and weighted F1-scores. The weighted F1-score accounts for
+label imbalance by considering the number of instances of
+each label when calculating the average F1-score. Equation 4
+shows how the F1-score is calculated.
+F1 Score = 2 · 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 · 𝑅𝑒𝑐𝑎𝑙𝑙
+𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + 𝑅𝑒𝑐𝑎𝑙𝑙
+=
+𝑇𝑃
+𝑇𝑃 + 1
+2 (𝐹𝑃 + 𝐹𝑁)
+(4)
+C. Statistical Evaluation of Synthetic Data
+To evaluate the synthetic data generated by the GAN models
+against the real data they were trained on, we used statistical
+metrics that compare the columns of the synthetic tables
+against those in the real tables. These statistical metrics are
+as follows:
+1) CSTest: The CSTest compares columns with discrete
+values using the Chi-squared test to compare their distribu-
+tions. The output of the test is an average of the CSTest 𝑝-
+values for each of the columns, which ultimately quantifies the
+probability that the compared columns were sampled from the
+same distribution.
+2) KSTest: The KSTest compares columns with continuous
+values using the two-sample Kolmogorov–Smirnov test and
+empirical Cumulative Distributed Function (CDF) to compare
+their distributions. The output of the test is an average of
+1 minus the KSTest D statistic for each of the columns.
+This result quantifies the maximum distance between the CDF
+expected and observed values.
+3) KSTestExtended: The KSTestExtended is an extension
+of the KSTest that converts all columns to numerical values
+using a hyper transformer and then applies the regular KSTest.
+D. Detection-based Evaluation of Synthetic Data
+Detection metrics use ML models to determine how distin-
+guishable the synthetic data is from the real data. To achieve
+this, both the synthetic and real tables are shuffled and a flag
+indicating whether the record is synthetic or not is added.
+Next, cross-validation is used with a selected ML model
+that predicts the flag, outputting 1 minus the average ROC
+AUC of all the cross-validation splits. Because the ROC AUC
+measures the separability of the classes from the model, a high
+detection metric score means that the model is unable to easily
+distinguish the synthetic records from the real ones.
+Fig. 2. Architecture of Generative Adversarial Networks.
+E. Generative Adversarial Network Models
+Goodfellow et al.
+[18] first proposed the idea of a GAN
+in 2014 as an unsupervised learning method that generates
+synthetic data using an input of real data. GANs are used
+to generate realistic synthetic data using real data, usually
+because obtaining more data can be difficult, time consuming,
+and costly. GANs use two independent models, a generator and
+a discriminator. By detecting patterns or similarity from given
+input data, the generator processes input data and produces
+more data. The discriminator is a classifier which determines
+the difference between the real data and the generated data.
+It produces a probability between 0 and 1 to define whether
+an instance belongs to the real data (closer to 1) or to the
+generated data (closer to 0). Fig. 2 highlights the overall
+workflow of GANs.
+F. Deep Reinforcement Learning Model
+DRL is a subfield of ML that combines both RL and deep
+learning. RL considers the problem of an agent learning to
+make decisions through trial and error, while DRL incorpo-
+rates deep learning, allowing agents to make decisions from
+unstructured input data without manual engineering of the state
+space.
+RL problems involve an agent learning how to map situa-
+tions to actions in order a maximize a numerical reward signal.
+It employs five key concepts:
+• Environment: The physical world that the agent operates
+within.
+• State: The agent’s belief of a configuration of the envi-
+ronment.
+• Reward: Numerical feedback from the environment.
+• Policy: A mapping from the agent’s state to actions.
+• Value: Expected future reward an agent would receive by
+taking an action in a certain state.
+Simply put, RL is the process of running an agent through
+sequences of state-actions pairs, observing the rewards that
+result, and using those rewards to formulate an optimal policy
+over time.
+For RL problems with small discrete state-actions spaces,
+the state-action mapping can be stored in a table to approxi-
+mate the mapping within a reasonable error value. However,
+for problems with large state-actions spaces, it is difficult to
+store such large amounts of data and, therefore, traditional RL
+methods suffer in terms of memory and performance. To over-
+come this, we can incorporate DRL, which is a combination of
+RL and deep neural networks. A neural network can be used
+
+Real Data
+Real
+Random
+Generated
+Generator
+Discriminator
+Noise
+Data
+Fake
+Fine Tune Training5
+Fig. 3. Architecture of Deep Reinforcement Learning.
+to approximate a value or policy function. Essentially, neural
+nets learn to map states to values rather than using a lookup
+table. Thus, a DRL model can independently learn to establish
+a successful function for gaining maximum long-term rewards
+in RL problems with large state-actions spaces.
+We have defined some characteristics within our DRL model
+in order for it to act as both a binary and multiclass classifier.
+For binary classification, we have defined our action space as
+follows:
+• 0 : No Alert (benign)
+• 1 : Alert (attack)
+And the rewards for this model are defined by:
+• +1 if agent correctly alerts to the correct type of attack.
+• 0 if agent does not raise an alert when it is not needed.
+• -1 if agent does not raise an alert when there is an attack.
+• -1 if agent raises alert when there is not one needed.
+For multiclass classification, we have defined our action
+space, also seen in Fig. 3, as follows:
+• 0 : No Alert (benign)
+• 1 : DoS
+• 2 : Probe
+• 3 : R2L
+• 4 : U2R
+And the rewards for this model are defined by:
+• +1 if agent correctly alerts to the correct type of attack.
+• 0 if agent does not raise an alert when it is not needed.
+• -1 if agent does not raise an alert when there is an attack.
+• -1 if agent raises alert when there is not one needed.
+• -1 if agent raises an alert to the incorrect type of attack.
+In terms of network security, alerting benign network traffic
+is typically safer than not alerting to an actual attack. Thus, we
+might consider that the reward for the latter two cases in the
+above enumeration should be greater than −1. However, this
+reward function was selected because identifying the wrong
+type of attack would lead to misdirection of resources, which
+we want to avoid.
+Finally, the state space for both binary and multiclass
+classification is a collection of 41 features, both numerical
+Normal
+DoS
+Probe
+R2L
+U2R
+Record Class
+0
+10000
+20000
+30000
+40000
+50000
+60000
+70000
+Number of Records
+67343
+45927
+11656
+995
+52
+9711
+7460
+2885
+2421
+67
+Train
+Test
+Fig. 4. Distribution of NSL-KDD dataset by record classes.
+Fig. 5.
+The action and state spaces of the proposed deep reinforcement
+learning model.
+and nominal, existing within the NSL-KDD dataset. Thus, we
+have an fairly complex and detailed state space. A visual of
+this environment can be seen in Fig. 3.
+In addition, we have assigned two distinct conditions for
+terminating an episode. An episode will be terminated if 1) it
+reaches a set timestep threshold, or 2) if an attack is issued
+and no alert has been made.
+IV. RESULTS
+The following subsections describe the experimental results
+from our proposed GAN and DRL models, followed by a
+comparative analysis of our proposed model with other state-
+of-the-art ML methods.
+A. GAN Models
+For our experiments, we trained two GAN models, CT-
+GAN [19] and CopulaGAN [20], using the implementations
+provided by the SDV open-source library [21], following work
+by [22] which showed promising results for generating net-
+work traffic data using models from this library. These models
+were trained on the NSL-KDD training data for 100 epochs
+with a batch size of 500. Both models used discriminator steps
+of 5, matching WGAN [23], an extended version of vanilla
+
+Agent
+State
+Action
+Reward
+State
+Action
+EnvironmentAgent
+Action
+Space
+NO ALERT
+Dos
+Probe
+U2R
+R2L
+Environment
+State
+Space
+flag
+protocol_type
+service
+logged_in6
+TABLE III
+STATISTICAL METRICS FOR SYNTHETIC DATA
+Synthetic Data
+CSTest
+KSTest
+KSTest
+Extended
+CTGAN
+0.9971
+0.9156
+0.9181
+CTGAN (Conditional)
+0.7468
+0.8655
+0.8571
+CopulaGAN
+0.9988
+0.9550
+0.9574
+CopulaGAN (Conditional)
+0.6982
+0.9000
+0.8864
+TABLE IV
+DISCRENMENT RESULTS FOR SYNTHETIC DATA USING
+LOGISTIC REGRESSION
+Synthetic Data
+Discernment Metric
+CTGAN
+0.7579
+CTGAN (Conditional)
+0.4139
+CopulaGAN
+0.6862
+CopulaGAN (Conditional)
+0.3948
+GAN. For the other hyperparameters, we opted to use the
+defaults provided by the SDV library.
+The trained CTGAN and CopulaGAN models were each
+used to generate two synthetic datasets:
+1) A dataset containing 200 000 records generated through
+regular sampling without conditions. As expected, these
+datasets contained records that closely matched the imbal-
+anced class distribution of the original NSL-KDD dataset.
+2) A dataset containing 20 000 records for each class gener-
+ated using conditional sampling through rejection. These
+datasets were used to explore the efficacy of using GANs
+to generate a balanced distribution from an highly imbal-
+anced training distribution.
+The statistical metric results showcased in Table III indicate
+that both CTGAN and CopulaGAN model the discrete and
+continuous features of the NSL-KDD dataset effectively. As
+dictated by the KSTest and KSTestExtended, CopulaGAN
+models continuous features better than CTGAN and main-
+tains parity for discrete features as indicated by the CSTest.
+Table IV highlights the results for a linear regression classifier
+used to evaluate detection performance of the synthetic data.
+Altogether, the classifier found it challenging to distinguish the
+synthetic records from the real ones, which indicates that the
+GANs are able to capture aspects of the true dataset. Table V
+and Table VI showcase the performance of ML models when
+trained to distinguish between various real and synthetic
+datasets. Across the board, there is comparable performance
+between the original real NSL-KDD dataset and the CTGAN
+and CopulaGAN synthetic datasets. Thus, there is promise in
+using synthetic data in place of real data.
+B. DRL Model
+The DRL models were implemented using both OpenAI
+Gym [24] and Tensorflow [25]. Training of the model was
+done in two distinct stages to investigate the variation in per-
+formance – binary classification and multiclass classification.
+0
+20000
+40000
+60000
+80000
+100000
+Timesteps
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Accuracy
+Original NSL-KDD
+CTGAN
+CTGAN (Conditional)
+CopulaGAN
+CopulaGAN (Conditional)
+Fig. 6. Results measuring the accuracy of binary classification after training
+the DRL model on both the original NSL-KDD dataset and each of the
+synthetic GAN datasets.
+1) Binary Classification: We begin with binary classifica-
+tion, using an action space of two (‘alert’ or ‘no alert’). While
+binary classification offers the user less knowledge on attack
+type specifications, it should perform the basic task of an IDS
+– alerting the user to an attack with a high accuracy.
+Initially, we trained the DRL model on the NSL-KDD
+training set, described in detail above. We did this to create
+a baseline to see how well our synthetic GAN-generated
+data performed in comparison. Prior to training our model,
+we converted all class labels using a binary mapping. If the
+class was originally ‘normal’, we assigned it a value of ‘0’,
+otherwise it was assigned a value of ‘1’, implying that the data
+point was an attack of some sort.
+For each model, proximal policy optimization (PPO2), a
+policy-gradient algorithm that directly optimizes the expected
+reward by estimating the gradient of the policy from the
+trajectories taken by the agent, is executed. We applied a
+custom multi-layer perceptron, a class of feedforward neural
+network [26], of three layers with size 128, 64, and 32.
+In addition, each model used a rectified linear unit (ReLU)
+activation function.
+Training on the NSL-KDD training dataset for 100,000
+timesteps resulted in an average accuracy of 89.5% and an F1-
+score of 0.906 on the test dataset. We then proceeded to train
+the DRL model on each of the GAN-generated datasets one-
+by-one and evaluate them individually on the NSL-KDD test
+dataset. The detailed results of these experiments can be seen
+in Table V, and viewed in terms of progressive performance
+for average accuracy in Fig. 6.
+Training on CTGAN synthetic data performs the best after
+the NSL-KDD trained model, with 85.7% accuracy and 0.869
+F1-score. Training using CopulaGAN synthetic data trails
+close behind with 82.9% accuracy and 0.838 F1-score. The
+conditional variations of both CopulaGAN and CTGAN per-
+form significantly worse than the three other datasets, reaching
+their peak of 70% and 66% respectively almost immediately
+and then dropping to just below 50%.
+2) Multiclass Classification: We then trained the DRL
+models to perform multiclass classification. Similar to binary
+classification, we are still detecting whether there is an attack
+
+7
+TABLE V
+MACHINE LEARNING PERFORMANCE FOR BINARY CLASSIFICATION
+Training Data
+Decision Tree
+AdaBoost
+Classifier
+Logistic Regression
+Classifier
+MLP Classifier
+Proposed DRL
+Accuracy
+F1
+Accuracy
+F1
+Accuracy
+F1
+Accuracy
+F1
+Accuracy
+F1
+NSL-KDD
+0.8407
+0.8414
+0.8221
+0.8270
+0.8700
+0.8802
+0.8054
+0.8080
+0.8951
+0.9064
+CTGAN
+0.8074
+0.8112
+0.8404
+0.8486
+0.8610
+0.8710
+0.8461
+0.8545
+0.8572
+0.8687
+CTGAN (Conditional)
+0.8801
+0.8927
+0.9086
+0.9226
+0.8740
+0.8853
+0.9077
+0.9220
+0.4662
+0.1172
+CopulaGAN
+0.7735
+0.7607
+0.8259
+0.8246
+0.8163
+0.8201
+0.7918
+0.7831
+0.8294
+0.8375
+CopulaGAN (Conditional)
+0.8287
+0.8333
+0.8743
+0.8881
+0.8256
+0.8311
+0.8947
+0.9074
+0.4901
+0.1893
+TABLE VI
+MACHINE LEARNING PERFORMANCE FOR MULTI-LABEL CLASSIFICATION
+Training Data
+Decision Tree
+MLP Classifier
+Proposed DRL
+Accuracy
+F1
+F1 (weighted)
+Accuracy
+F1
+F1 (weighted)
+Accuracy
+F1
+F1 (weighted)
+NSL-KDD
+0.7685
+0.5585
+0.7338
+0.7856
+0.6302
+0.7556
+0.7300
+0.4880
+0.6891
+CTGAN
+0.7475
+0.5297
+0.7336
+0.7765
+0.6467
+0.7572
+0.4247
+0.3033
+0.4503
+CTGAN (Conditional)
+0.6200
+0.4475
+0.6525
+0.7442
+0.5643
+0.7791
+0.5520
+0.3938
+0.4533
+CopulaGAN
+0.7031
+0.4165
+0.6618
+0.7374
+0.4606
+0.6863
+0.7023
+0.3967
+0.6345
+CopulaGAN (Conditional)
+0.6116
+0.3810
+0.6215
+0.7088
+0.4401
+0.6883
+0.4839
+0.2716
+0.4049
+TABLE VII
+CLASS-BASED F1 SCORES FOR MULTI-LABEL CLASSIFICATION
+Dataset
+Normal
+DoS
+Probe
+R2L
+U2R
+NSL-KDD
+0.7785
+0.8072
+0.4752
+0.1490
+0.0
+CTGAN
+0.5670
+0.4618
+0.3858
+0.0831
+0.0192
+CTGAN
+(Conditional)
+0.7662
+0.0
+0.4589
+0.5725
+0.1716
+CopulaGAN
+0.8139
+0.7101
+0.4593
+0.0
+0.0
+CopulaGAN
+(Conditional)
+0.8039
+0.0
+0.2201
+0.2097
+0.0512
+or not, however we now attempt to classify which type of
+attack is taking place. Instead of ‘0’ or ‘1’, our action space
+consists of 0, 1, 2, 3, and 4. As stated previously, 0 maps
+to ‘benign’, whereas 1, 2, 3, and 4 map to DoS, Probe,
+R2L, and U2R respectively. As our action space has increased
+in comparison to binary classification, our problem becomes
+significantly larger and more challenging.
+Like binary classification, we used a ReLU activation func-
+tion, however for the conditional versions of both CTGAN
+and copulaGAN we used a Sigmoid activation function, as we
+found that this results in a significant increase in performance
+on test data. For each model, we again used a custom multi-
+layer perceptron of three layers with size 128, 64, and 32.
+Again, we first analyzed the performance of our model after
+being trained on the real NSL-KDD dataset in order to create
+a benchmark. As seen in Table VI, our DRL model achieved
+73% accuracy and a 68.9% weighted F1-score.
+We then trained the DRL model on the four GAN-generated
+synthetic datasets discussed previously. The most promising
+results were seen in training the model on CopulaGAN. The
+model reaches an accuracy of 70.2% and a weighted F1-
+score of 63%. This is just a 2.7% drop in accuracy from
+training on the true NSL-KDD data. Training the DRL model
+on the remaining three synthetic datasets underperforms when
+compared to both the decision tree and MLP classifier.
+As discussed previously, an F1-score refers to both precision
+and recall being high. When we train on imbalanced datasets,
+the F1-scores in minority classes are typically quite low, as
+the ML model does a poor job of recognizing and properly
+classifying that test data. Looking at Table VII, we can see the
+F1-scores for each individual class for each of our training sets.
+Since NSL-KDD had extremely low records for both R2L and
+U2R, we can see that the F1-scores for these classes are also
+quite low at 0.1490 and 0.0, respectively.
+One of the major goals of our work was to determine if,
+by generating synthetic GAN data, we could inflate the F1-
+scores (more specifically, precision and recall) of the minority
+classes from our imbalanced dataset. In Table VII, we can
+see that training our DRL model with data generated from
+conditional CTGAN and conditional CopulaGAN improved
+upon the F1-scores for both R2L and U2R in the same way that
+we would expect to see if the true dataset naturally contained
+more records of these two class types. Training the DRL model
+on synthetic data from conditional CTGAN increased the F1-
+scores for R2L and U2R by 0.573 and 0.172 respectively.
+Training on synthetic data from conditional CopulaGAN im-
+proved the F1-scores for R2L and U2R by 0.210 and 0.051
+respectively. This demonstrates that the concept of using GAN
+models to generate synthetic data for a minority class and
+artificially inflating the training set in order to have better
+performance in classifying underrepresented classes is a viable
+option.
+
+8
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Timesteps
+1e6
+0.0
+0.2
+0.4
+0.6
+0.8
+F1-Score
+Normal
+DoS
+Probe
+R2L
+U2R
+(a) NSL-KDD
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Timesteps
+1e6
+0.0
+0.2
+0.4
+0.6
+0.8
+F1-Score
+Normal
+DoS
+Probe
+R2L
+U2R
+(b) CTGAN
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Timesteps
+1e6
+0.0
+0.2
+0.4
+0.6
+0.8
+F1-Score
+Normal
+DoS
+Probe
+R2L
+U2R
+(c) CTGAN (Conditional)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Timesteps
+1e6
+0.0
+0.2
+0.4
+0.6
+0.8
+F1-Score
+Normal
+DoS
+Probe
+R2L
+U2R
+(d) CopulaGAN
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Timesteps
+1e6
+0.0
+0.2
+0.4
+0.6
+0.8
+F1-Score
+Normal
+DoS
+Probe
+R2L
+U2R
+(e) CopulaGAN (Conditional)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Timesteps
+1e6
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+F1-Score
+Original NSL-KDD
+CTGAN
+CTGAN (Conditional)
+CopulaGAN
+CopulaGAN (Conditional)
+(f) Average Accuracy
+Fig. 7. Results measuring the F1-scores of multiclass classification (fig a-e) as well as the averages (fig f) after training the DRL model for 2 million timesteps
+on both NSL-KDD as well as each synthetic dataset.
+V. CONCLUSION
+In this paper, we have proposed a SNIDS which is able to
+perform binary and multiclass classification on network traffic
+data. We used DRL to implement this IDS. The model was
+trained using the NSL-KDD dataset, allowing it to detect a
+range of attack types on a network. To enhance the learning
+capabilities of our proposed model, GANs were used to
+fabricate training data. Our results demonstrate that this system
+is able to interact with the network and identify attack classes
+with competitive accuracy. As well, we show that generating
+synthetic data for underrepresented classes can improve the
+precision and recall within these classes, thus acting as a
+solution for imbalanced datasets.
+For binary classification, we obtained an 89.5% accuracy
+after training on the NLS-KDD dataset. We consider this our
+baseline model. When trained on the four synthetic datasets,
+data generated from unconditional CTGAN produced an accu-
+racy of 85.7%, the closest competition to the baseline model.
+For multiclass classification, we obtained a 73.0% accuracy
+after training on the NSL-KDD dataset. When trained on
+the four synthetic datasets, data generated from CopulaGAN
+produced an accuracy of 70.2%, the closest competition to the
+baseline model. Thus, clearly our GAN models generate data
+realistic enough to create competitive IDS.
+Further, both Table VII and Fig. 7 demonstrate an increase
+in F1-scores for minority classes on the IDS trained using
+GAN-generated data. Thus, while our overall accuracy de-
+creased, we are getting better precision and recall performance
+for the classes without sufficient data in the NSL-KDD dataset.
+This points to a solution for other ML models trying to learn
+from imbalanced datasets.
+VI. FUTURE WORK
+While our work demonstrated competitive classifiers and an
+increase in individual F1-scores for minority classes, there is
+still room for improvement.
+When training our GAN models, we have discussed that we
+used 41 features from the NSL-KDD dataset as input to our
+model. There are two major changes that we aim to implement
+in our future work. First, passing our input dataset through
+a pipeline of feature analysis methods, including (but not
+limited to) Pearson correlation, recursive feature elimination,
+and Lasso, with the aim to reduce our feature space. This
+has the potential to increase the quality of our generated
+dataset, thus increasing the evaluation metric scores for our
+DRL model. Secondly, supplementing the NSL-KDD dataset
+with data from under-represented classes in order to balance
+the dataset. Our work demonstrates that there is a notable
+increase in F1-score when the class has a significant amount
+of data being given as input to the GAN model. We plan to
+explore this idea, and see the limitations of our performance
+when the GAN is trained on significant sample sizes from each
+class, rather than just a small subset.
+We also plan to explore GAN models that are trained only
+on the minority classes of our true dataset classes. Thus,
+this generated data could potentially be merged with the true
+dataset to allow for heightened overall performance of the IDS,
+as we are synthetically creating balance.
+Finally, we plan to explore the performance of training both
+DQN, a value-iteration based method [27], and A3C [28], a
+value-iteration and policy-gradient method, on GAN-generated
+data to see how it compares with our PPO2 model. Both DQN
+and A3C are common DRL approaches, and have the potential
+to surpass the performance of our current model.
+
+9
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+2016.
+Caroline Strickland received a B.Sc in 2017 and M.Sc in 2019 from
+Memorial University of Newfoundland. She is currently in the third year of
+pursuing a PhD in Computer Science at the University of Western Ontario.
+Her past research has focused on using reinforcement learning for pattern
+formation within swarm systems, and her current research interests involve
+the intersection of hierarchical reinforcement learning with healthcare.
+Chandrika Saha received a B.Sc in Computer Science and Engineering from
+the University of Barishal, in 2019. Currently, she is pursuing her M.Sc. in
+Computer Science at the Western University, London, Ontario, Canada. Her
+research interest is Machine Learning, more specifically, deep learning and
+its application to network security.
+Muhammad Zakar received a B.Sc. in Computer Science from Western
+University, London, Canada, in 2021. Currently, he is pursuing a M.Sc. in
+Computer Science at Western University. His current research interests are
+in the areas of autonomous drones and vehicles, distributed systems, next-
+generation networks, and machine learning.
+Sareh Soltani Nejad received a B.Sc. in Computer Engineering from
+Amirkabir University of Technology (AUT), Tehran, Iran, in 2019. She is
+currently pursuing a M.Sc. in Computer Science at the University of Western
+Ontario, Canada. Her research interests broadly focus on Machine learning and
+Internet of Things applications in Smart Cities, Smart homes and Healthcare.
+Noshin Tasnim received a B.Sc. in Computer Science and Engineering from
+BRAC University, Bangladesh, in 2019. She is currently pursuing a M.Sc.
+in Computer Science with the Department of Computer Science, Western
+University, London, Canada. Her current research interests are in the areas of
+network security, and machine learning.
+Daniel Lizotte is currently an Associate Professor in the Department of
+Computer Science and in the Department of Epidemiology and Biostatistics,
+University of Western Ontario, London, ON, Canada. His research group in
+collaboration with community partners investigates different aspects of data-
+driven decision support in public health and health care. This work aligns
+with methodological research in the areas of machine learning, epidemiology,
+and biostatistics. He has received funding from the Natural Sciences and
+Engineering Research Council of Canada, the Canadian Institutes of Health
+Research, and the Social Sciences and Humanities Research Council of
+Canada, and he has served in various capacities on committees for the Machine
+Learning for Health Care, International Conference on Machine Learning, and
+NeurIPS conferences.
+Anwar Haque is an Assistant Professor in the Deptartment of Computer
+Science at the University of Western Ontario, Canada. Before joining Western,
+he was an Associate Director at Bell Canada. He is a leading international
+expert on next-generation communication network resources and performance
+management, cyber security, and smart city applications. Dr. Haque has
+authored/co-authored over 80 peer-reviewed research publications in leading
+journals and conferences, authored many industry technical papers, and held
+a number of patent/licenses. He has been awarded several national/provincial-
+level research grants, including NSERC, MITACS, OCE, and SOSCIP. Dr.
+Haque’s collaborative research grants are valued at more than $15 million.
+Dr. Haque is serving on the inaugural advisory committee for the newly
+established Bell-Western 5G research centre, and he established an industry
+consortium to promote and support smart systems and digital services research
+at Western. Dr. Haque is the director of the Western Information & Networking
+Group (WING) Lab at Western.
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