Semi-Supervised Learning by Local Behavioral Searching...

Appl. Math. Inf. Sci.8, No. 4, 1781-1787 (2014) 1781

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/080435

Semi-Supervised Learning by Local BehavioralSearching Strategy

Chun Zhang1, Junan Yang1,∗, Jiyang Zhang2, Dongsheng Li1 and Aixia Yong1

1 Electronic Engineering Institute, Key Laboratory of Electronic Restriction,230037 Hefei, China2 Jiangnan Institute of Computing Technology, 214083 Wuxi, China

Received: 5 Aug. 2013, Revised: 7 Nov. 2013, Accepted: 8 Nov. 2013Published online: 1 Jul. 2014

Abstract: Semi-supervised learning has attracted a significant amount of attention inpattern recognition and machine learning. Amongthese methods, a very popular type is semi-supervised support vectormachines. However, parameter selection in heat kernel functionduring the learning process is troublesome and harms the performanceimprovement of the hypothesis. To solve this problem, a novellocal behavioral searching strategy is proposed for semi-supervised learning in this paper. In detail, based on human behavioral learningtheory, the support vector machine is regularized with the un-normalizedgraph Laplacian. After building local distribution of featurespace, local behavioral paradigm considers the form of the underlying probability distribution in the neighborhood of a point. Validationof the proposed method is performed with extensive experiments. Results demonstrate that compared with traditional method, ourmethod can more effectively and stably enhance the learning performance.

Keywords: semi-supervised learning, behavioral learning, regularization, manifold learning, support vector machines

1 Introduction

Semi-supervised learning has received a significantinterest in pattern recognition and machine learning. Itexploits unlabeled data in addition to the limited labeledones to improve the learning performance [1]. Manysemi-supervised learning algorithms have been proposedduring the past decade, among which a very popular typeof algorithms is the semi-supervised support vectormachines (S3VMs).

Examples of this type include the semi-supervisedSVM [2], the transductive SVM (TSVM) [3], and theLaplacian SVM [4]. S3VM and the TSVM are built uponthe cluster assumption and use the unlabeled data toregularize the decision boundary. Specifically, thesemethods prefer the decision boundary to pass throughlow-density regions [5]. The Laplacian SVM is a S3VMthat exploits the datas manifold structure via the graphLaplacian. It encodes both the labeled and unlabeled databy a connected graph, where each instance is representedas a vertex and two vertices are connected by an edge ifthey have large similarity. The goal is to find class labelsfor the unlabeled data such that their inconsistencies with

both the supervised data and the underlying graphstructure are minimized.

However, while many efficient SVM methods havebeen developed for supervised learning, S3VMs stillsuffer from inefficiency issues. In particular, theoptimization problem of Bennett and Demirizs S3VM isformulated as a mixed-integer programming problem andso is computationally intractable in general. TSVM, onthe other hand, iteratively solves standard supervisedSVM problems. However, the number of iterationsrequired may be large since the TSVM is based on a localcombinatorial search that is guided by a label switchingprocedure. Unlike the TSVM, the Laplacian SVM focuseson regularization in reproducing Kernel Hilbert spacesand only needs to solve one small SVM with the labeleddata. But Laplacian SVM utilizes heat kernel weights toform edge weights when constructing data adjacencygraph. The performance of heat kernel weights highlydepends on parameter selection and how to exactly fixparameter in different applications may be troublesome.

Currently, Defense Advanced Research ProjectsAgency (DARPA) is soliciting innovative researchproposals in the area of machine learning for electronicwarfare applications and sets up the Behavioral Learning

∗ Corresponding author e-mail:[email protected].

c© 2014 NSPNatural Sciences Publishing Cor.

http://dx.doi.org/10.12785/amis/080435

1782 C. Zhang et. al. : Semi-Supervised Learning by Local Behavioral...

for Adaptive Electronic Warfare (BLADE) program [6] in2010. At the same time, more and more research fruitshold the viewpoint that human behavioral learning caneffectively improve the performance of machine learning[7,8,9]. Inspired by these booming trends, we propose anovel approach called Local Behavioral based LaplacianSVM (LB-LapSVM) to overcome the problem ofparameter selection in Laplacian SVM.

The rest of this paper is organized as follows. Section2 briefly describes the semi-supervised learningframework and its extension in Laplacian SVM. Section 3presents the local behavioral searching method and itsutility in LB-LapSVM. Section 4 performs extensiveexperiments. Finally, we provide some concludingremarks in Section 5.

2 Related work

Inspired by the success of large margin principle, S3VMsare extensions of supervised SVMs to semi-supervisedlearning by simultaneously learning the optimalhyperplane and the labels for unlabeled instances. It wasdisclosed that S3VMs realize the low-density assumptionby favoring the decision boundary going acrosslow-density regions.

2.1 semi-supervised learning framework

Formally, considering binary classification insemi-supervised learning, we are given a set ofl labeledsamples{xi,yi}

li=1, and a set ofu unlabeled samples

{xi}l+ui=l+1, wherexi ∈ R

N andyi ∈ {−1,+1}. Let us nowassume a general-purpose decision functionf . Theregularized functional to be minimized is defined as:

f ∗ = argminf∈HK

1l

l

∑i=1

V (xi,yi, f )+ γA ‖ f‖2K + γI ‖ f‖2

I (1)

where V represents a generic cost function of thecommitted errors on the labeled samples,HK is areproducing kernel Hilbert space (RKHS) induced by thekernel.γA controls the complexity off in the associatedHilbert spaceHK , and γI controls its complexity in theintrinsic geometry of the marginal data distribution. Forexample, if the probability distribution is supported on alow-dimensional manifold,‖ f‖2

I penalizes f along thatmanifold I. Note that this functional constitutes a generalregularization framework that takes into account all theavailable knowledge.

2.2 Laplacian SVM

The previous semi-supervised learning framework allowsus to develop many different algorithms just by playing

around with the loss function,V , and the regularizes,‖ f‖2.In this paper, we focus on the Laplacian SVM formulation,which basically uses a SVM as the learner core and thegraph Laplacian for manifold regularization.

A. Cost function of the errors

The Laplacian SVM uses the same hinge loss function asthe traditional SVM:

V (xi,yi, f )=max{0,1− yi f (xi)} (2)

where f represents the decision function implemented bythe selected classifier.

B. Decision function

We use as the decision functionf (x∗) = 〈w,φ(x∗)〉+ b,whereφ(·) is a nonlinear mapping to a higher (possiblyinfinite) dimensional Hilbert spaceH , and w and bdefine a linear regression in that space. By means of theRepresenter Theorem [10], weightsw can be expressed inthe dual problem as the expansion over labeled andunlabeled samplesw = ∑l+u

i=1 αiφ(xi) = ΦΦΦααα, whereΦΦΦ = [φ(x1), ...,φ(xl+u)]

T and ααα = [α1, ...,αl+u]. Then,the decision function is given by:

f (x∗) =l+u

∑i=1

αiK(xi,x∗)+b (3)

and K is the kernel matrix formed by kernel functions,K(xi,x j) =

⟨

φ(xi),φ(x j)⟩

. The key point here is that,without considering the mappingφ explicitly, anon-linear classifier can be constructed by selecting theproper kernel. Also, the regularization term can be fullyexpressed in terms of the kernel matrix and the expansioncoefficients:

‖ f‖2K = ‖w‖2 = (ΦΦΦααα)T (ΦΦΦααα) = αααT KKKααα (4)

C. Manifold regularization

The geometry of the data is modeled with a graph inwhich nodes represent both labeled and unlabeledsamples connected by weightsWi, j, regularizing thegraph follows from the smoothness (or manifold)assumption and intuitively is equivalent to penalize the”rapid changes” of the classification function evaluatedbetween close samples in the graph:

‖ f‖2I =

1

(l +u)2

l+u

∑i, j=1

Wi j( f (xi)− f (x j))2 = fT Lf (5)

where L = D − W is the graph Laplacian,D is thediagonal degree matrix ofW given by Dii = ∑l+u

j=1 Wi j,


Appl. Math. Inf. Sci.8, No. 4, 1781-1787 (2014) /www.naturalspublishing.com/Journals.asp 1783

Algorithm 1 A Summary of LapSVM algorithm [4].

Input: l labeled examples{xi,yi}li=1, u unlabeled examples

{xi}l+ui=l+1

Output: Estimated functionf : RN → R

1: Construct data adjacency graph with(l + u) nodes usinggraph kernel. Choose edge weights through heat kernel

weightsWi j = e−‖xi−x j‖2/4t .

2: Choose a kernel functionK(x,y). Compute the Gram matrixKi j = K(xi,x j).

3: Compute graph Laplacian matrix:L = D−W whereD is adiagonal matrix given byDii = ∑l+u

j=1 Wi j.4: ChooseγA andγI .5: Computeα∗ with the SVM QP solver for soft margin loss.6: Output functionf ∗(x) = ∑l+u

i=1 α∗i K(xi,x).

and f = [ f (x1), ..., f (xl+u)]T = Kααα, where we have

deliberately dropped the bias termb.The LapSVM algorithm is summarized in the

Algorithm 1. From Algorithm1 we can clearly see thatwhen LapSVM computes edge weights, heat kernel

weights Wi j = e−‖xi−x j‖2/4t are used. The value oft

varies in different applications. The performance ofLapSVM will be badly hurt if the value oft is falsely set.Whats more, when computing edge weights, heat kernelweights only focus on samplexi andx j. There is plentyadditional information to improve affinity measuring.From the above analysis, we proposed a novel methodcalled LB-LapSVM.

3 Local behavioral searching

In many real world situations, human are exposed to acombination of labeled data and far more unlabeled datawhen they need to make a classification decision.Understanding how human combine information fromlabeled and unlabeled data to draw inferences aboutconceptual boundaries can have significant social impact.In the realistic setting where labeled and unlabeled data isavailable, semi-supervised learning offers very explicitcomputational hypotheses that can be empirically testedin the laboratory. To help understand description in thissection, we start by providing a translation of relevantterms from semi-supervised learning to human behaviorallearning.

That is, when stimulus arrives, human use theirsupervised experiences from teachers and passiveexperiences from nature to complete concept learningtask. During the learning task, human take in to accountof some mechanism in their mind. When conceptcategory is obtained, the learning task is done.

Inspired by above analysis, we applied humanbehavioral learning strategy to LapSVM. In detail, whencomputing affinity between samplexi and x j, LapSVMonly focuses on samplexi and x j. However, its not thecase in human behavior. Given a pair of samples, in

Stimulus to human subjectInstance

Concept category

for human to learnClass

Concept learning

task for humanClassification

Supervised experience

from teacher

Labeled

Data

Passive experience

for human

Unlabeled

Data

Mechanism in the mind

of the human subject

Learning

Algorithm

Fig. 1: Translation from semi-supervised learning tohuman behavioral learning.

human empirical cognition, local neighborhood of thispair samples plays an important role in affinity measuring.Based on local distribution of feature space, humanbehavioral paradigm considers the form of the underlyingprobability distribution in the neighborhood of a point.

Instead of selecting a single parametert in heat kernelweights, we propose to calculate local behavioralparametersσi and σ j for data pointsxi and x j. Thedistance fromxi to x j as ”seen” byxi is d(xi,x j)/σi whilethe converse isd(x j,xi)/σ j. Therefore the square distanced2 may be generalized asd(xi,x j)d(x j,xi)/σiσ j = d2(xi,x j)/σiσ j. The affinitybetween a pair of points can thus be written as:

Wi, j = exp

(

−d2(xi,x j)

σiσ j

)

(6)

Using a specific parameter for each point allows localbehavioral searching of the point-to-point distancesaccording to the local statistics of the neighborhoodssurrounding pointsi and j. The selection of localbehavioral parameterσi can be done by studying the localstatistics of the neighborhood of pointxi. In order to catchhuman thought in neighborhood searching, we choose:

σi=d(xi,xK) (7)

wherexK is theK-th neighbor of pointxi. The selection ofK is independent and is a function of the data dimensionof the embedding space. Distance metricd(xi,x j) is setthe same to LapSVM, e.g.

∥

∥xi − x j∥

∥. So based on localbehavioral searching, edge weight betweenxi andx j canbe defined as:

Wi, j = exp

(

−∥

∥xi − x j∥

∥

2

σiσ j

)

(8)

We provide a visualization of the effect of thesuggested local behavioral searching strategy in Fig. 2.

Without loss of generality, binary classification istaken into account. In Fig. 2(a), we show input data


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(a) (b) (c)

Fig. 2: The effect of local behavioral searching strategy in LapSVM. (a) Input data points. (b) Heat kernel weights basedaffinity measuring. (c) Local behavioral searching based affinity measuring.

points. Squares and triangles each represent one class.They appear as the shape of a tight cluster resides withina background cluster. Fig. 2(b) shows the affinitymeasuring results based on heat kernel weights. Theaffinity between each point and its surrounding neighborsis indicated by the thickness of the line connecting them.Generally, the affinities across tight clusters are largerthan the affinities within the sparse background cluster.We can clearly discover that its not the case in Fig. 2(b).Thats to say, the standard approach to estimating affinitiesfails to capture the data structure. Fig. 2(c) represents thecorresponding visualization of affinities after localbehavioral searching. The affinities across clusters arenow significantly lower than the affinities within anysingle cluster. Local behavioral searching approachautomatically finds the two scales and results in highaffinities within clusters and low affinities across clusters.This is the information required for separation.

The local behavioral based LapSVM (LB-LapSVM)algorithm is summarized in the Algorithm2.

Algorithm 2 A Summary of LB-LapSVM algorithm.

Input: l labeled examples{xi,yi}li=1, u unlabeled examples

{xi}l+ui=l+1

Output: Estimated functionf : RN → R

1: Construct data adjacency graph with(l + u) nodes usinggraph kernel.

2: Compute local behavioral parametersσi and σ j throughformula (7) and build the affinity matrixW defined byformula (8).

3: Compute graph Laplacian matrix:L = D−W whereD is adiagonal matrix given byDii = ∑l+u

j=1 Wi j.4: ChooseγA andγI .5: Computeα∗ with the SVM QP solver for soft margin loss.6: Output functionf ∗(x) = ∑l+u

i=1 α∗i K(xi,x).

4 Experiments

We performed experiments on two moons data set andnine UCI data sets. Based on a survey of relatedapproaches, we chose to compare the proposedLB-LapSVM strategy with SVM (using labeled dataonly), TSVM, LapSVM. Other approaches lackout-of-sample extension, use different base-classifiers orparadigms, or are implementationally not preferable.

We use classification average error rate as theevaluation measure. The results are reported over 20 runsof each experiment, with different subsets of training andtesting data. To measure the inductive performance, werandomly split the data set into two halves and call themthe training and test sets. For each data set, 25% data arekept aside as test set to evaluate the performance oflearned hypothesis, while the remaining 75% data aretraining set, which is partitioned into original labeled andunlabeled set.

4.1 Two moons data set

The two moons data set contains 200 points belonging totwo non-linearly separable classes with only 1 labeledexample for each class. The RBF kernels are chosen foreach SVM based algorithms. In TSVM, the parameterNwhich controls the number of positive points in unlabeledset is set equal to that in labeled set. In LapSVM,parametersγA and γI are fixed as 0.03125 and 1. Thevalue of local searching range in LB-LapSVM is fixed as7. The best decision surfaces are shown in Fig. 3.

In Fig. 3, the solid rhombus and circle pointsrepresent labeled samples in each class, hollow squaresare unlabeled samples and black lines represent the bestdecision surfaces. Fig. 3 demonstrates how SVM andTSVM fail to find the optimal solution. The LapSVMdecision boundary seems to be acceptable, but theboundary is also helpless to those most complex areas.The proposed LB-LapSVM approach can effectively



SVM

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discover local intrinsic shape and cause the decisionsurface to appropriately adjust according to the geometryof the two classes. So the decision surface is intuitivelymost satisfying.

4.2 UCI data sets

In this section, we evaluate SVM, TSVM, LapSVM withthe proposed algorithm on nine UCI data sets shown inTable 1. Since these approaches are applicable fortwo-class problems, we chose the two-class data sets fromUCI data sets. The multiclass data sets in UCI areconverted into two-class data sets by choosing the twomost populated classes. The data sets are furtherprocessed by removing the samples with missing features,if any, binary encoding the categorical features, if any,and reducing the dimensionality using PCA to retain 95percent of the variance.

Table 1: Experimental UCI data sets.

Data set Size Attribute Class Pos/Neghouse 232 16 2 46.6%/53.4%heart 270 9 2 55.6%/44.4%vehicle 435 26 2 49.9%/50.1%wdbc 569 14 2 37.3%/62.7%isolet 600 51 2 50.0%/50.0%australian 690 14 2 55.5%/44.5%breastw 699 9 2 65.5%/34.5%diabetes 768 8 2 34.9%/65.1%sat 3041 36 2 50.4%/49.6%

The RBF kernels are chosen for each SVM basedalgorithms. As for these UCI data sets, we randomlyselect 10, 50 and 100 examples to be used as labeledexamples, and use the remaining data as unlabeled data.The experiments are repeated for 20 times and theaverage error rates are recorded in tables2 to 4.

Tables 2 to 4 summarize the average classificationerror rates of these algorithms under different numbers of

Table 2: Average error rate on UCI data sets with 10labeled examples.

Data setAverage error rate

SVM TSVM LapSVM LB-LapSVMhouse 0.123 0.112 0.114 0.106heart 0.288 0.265 0.253 0.242vehicle 0.257 0.212 0.241 0.227wdbc 0.209 0.195 0.188 0.164isolet 0.194 0.181 0.176 0.179australian 0.332 0.316 0.311 0.301breastw 0.043 0.041 0.041 0.038diabetes 0.358 0.334 0.317 0.314sat 0.024 0.022 0.018 0.019ave. 0.203 0.186 0.184 0.177







labeled examples. The rowave. in each table shows theaverage results over all the data sets. For each data setrow, the biggest improvement percentage has beenboldfaced. As can be seen from tables2 to 4, theproposed LB-LapSVM algorithm achieves highlycompetitive performance on most data sets under differentnumbers of labeled examples.

Moreover, tables 2 to 4 also show that if thealgorithms are compared through counting the number ofwinning data sets, i.e., the number of data sets on which




one algorithm achieves the best improvement amongcompared algorithms, LB-LapSVM is always the winner.In detail, under 10 labeled examples, LB-LapSVM has 6winning sets, while LapSVM and TSVM only have 2winning sets and 1 wining set respectively; under 50labeled examples, LB-LapSVM has 7 winning sets, whileLapSVM and TSVM only both have 1 winning setrespectively; under 100 labeled examples, LB-LapSVMhas 7 winning sets, while LapSVM only have 3 winningsets and TSVM with no wining set. So when comparedwith traditional methods, LB-LapSVM can moreeffectively and stably exploits unlabeled data to enhancethe learning performance.

4.3 Sensitivity to parameter setting

For further studying the performance of the comparedsemi-supervised learning algorithms, the averaged errorrates change of the compared algorithms along with theparameter setting are depicted in Fig. 4. For diversity, twomoons data set and three representative UCI data setswith 50 labeled examples are chosen. RegularizationparametersγA and γI are varied in steps in the range[10−6,1], and local behavioral parameterK is tuned in therange [3,5,7,9,11,13,15]. The results are done bycross-validation.

Fig. 4(a) clearly shows that the performance ofLaplacian based method strongly depends on setting ofregularization parametersγA andγI . To obtain outstandingresults, it is important to search these parameters on theright range. SinceγA controls the complexity off in theassociated Hilbert spaceHK and γI controls itscomplexity in the intrinsic geometry of the marginal datadistribution, we limit the regularization range at[10−6,1]based on empirical observations.

To perform fully validation with the sensitivity oflocal behavioral parameterK, three representative UCIdata sets are chosen. Results are shown in Fig. 4(b). It isobserved that performance will be badly hurt if the valueof K is too high or too low. Fixing the value ofK at [7,11]is recommended. The reason can be concluded that localbehavioral searching strategy in LapSVM can exactlydiscover the inner geometric structure between labeledand unlabeled examples. If the value ofK is set too low,the local behavioral searching scope can not cover all theaffinitive examples and affinity matrix can not fullyrepresent the inner structure between affinitive examples.Contrarily, if the value is fixed beyond normal scope,affinity measuring may suffer interfere from falsedistribution of adjacency graph and lead to descendenttrend of final classification precision.

5 Conclusion

We have proposed an algorithm for semi-supervisedlearning using human behavioral learning framework to

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Fig. 4: Illustrative examples of parameter selection forLB-LapSVM. (a) Sensitivity to regularization parametersγA and γI in two moons data set. (b) Sensitivity to localbehavioral parameterK in UCI data sets.

solve the confusion of parameter selection in LapSVM.This method brings together the ideas of human learningtheory, manifold learning, and kernel-based algorithms ina coherent and natural way to incorporate geometricstructure in regularization framework.

A series of experiments has been carried out in orderto evaluate performance of the proposed method andcompare it with other traditional techniques. From theseexperiments, it is to be noted that the LB-LapSVMalgorithm is effective and stable under different cases.

The main problem encountered is related to thecomputational cost, since a huge matrix consisting oflabeled and unlabeled samples must be inverted. We feelthat smart sampling strategies to select the mostinformative unlabeled samples could yield improveperformance, something to be considered in the future.

Acknowledgement

Both authors would like to acknowledge the support ofKey Laboratory of Electronic Restriction and NationalNatural Science Foundation (No. 61179036, No.60872113).



References

[1] X. J. Zhu, Semi-supervised learning literature survey, Univ.of Wisconsin-Madison, Wisconsin, Tech. Rep. ComputerSciences TR1530, Dec. (2007).

[2] K. Bennett and A. Demiriz, Semi-supervised support vectormachines, In Advances in Neural Information ProcessingSystems,11, 368-374 (1999).

[3] T. Joachims, Transductive inference for text classificationusing support vector machines, Proceedings 16thInternational Conference on Machine Learning, Bled,Slovenia, 200-209 (1999) .

[4] M. Belkin, P. Niyogi, V. Sindhwani, Manifold regularization:A geometric framework for learning from labeled andunlabeled examples, Journal of Machine Learning Research,7, 2399-2434 (2006).

[5] O. Chapelle, A. Zien, Semi-supervised learning by lowdensity separation, Proceedings of the 8th InternationalConference on Artificial Intelligence and Statistics, 57-64(2005) .

[6] DARPA, Behavioral Learning for Adaptive ElectronicWarfare, DARPA-BAA-10-79, (2010).

[7] T. Mitchell, The discipline of machine learning, CarnegieMellon University, Tech. Rep. CMU-ML-06-108, (2006).

[8] X.J Zhu, B.R. Gibson, K.S. Jun, Cognitive Models of Test-Item Effects in Human Category Learning, ICML, (2010).

[9] F. Khan, X.J Zhu, B. Mutlu, How Do Humans Teach:On Curriculum Learning and Teaching Dimension, NIPS,(2011).

[10] B. Scholkopf, A. Smola, Learning with Kernels-SupportVector Machines, Regularization, Optimization and Beyond,MIT Press Series, (2002).

Chun Zhang received theM. S. degree from ElectronicEngineering Institute,Hefei, China. Currently,he is a Ph.D. candidate inthe Department of ComputerEngineering at ElectronicEngineering Institute, Hefei,China. His research interestshave focused on Intelligent

Computing, Data Mining and Machine Learning.

Junan Yang receivedthe Ph. D. degree fromUniversity of Scienceand Technology of China.He is currently a professor inthe Department of ComputerEngineering at ElectronicEngineering Institute, Hefei,China. His research interestsinclude Neural Computing,

Large-scale machine and Computer Vision.

Jiyang Zhang receivedthe M. S. degree fromElectronic EngineeringInstitute, Hefei, China.He is currently a researchengineer at the JiangnanInstitute of ComputingTechnology, Wuxi, China.His research interests includeComputational Intelligence,

Signal Processing and Information Security.

Dongsheng Li receivedthe Ph. D. degree from HefeiUniversity of Technology,China. Currently, he isa professor in the Departmentof Computer Engineeringat Electronic EngineeringInstitute, Hefei, China.His research interestshave focused on Intelligent

Computing, Multichip Design, Parallel Signal Processing.

Aixia Yong receivedthe Ph. D. degree from HefeiUniversity of Technology,China. She is currentlyan associate professor inthe Department of ComputerEngineering at ElectronicEngineering Institute, Hefei,China. Her research interestsfocus on Image Processing,

Pattern Recognition and Machine Learning.



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