A Probability Distribution Kernel based on Whitening. Transformation

Transcription

1 AMSE JOURNALS-AMSE IIETA publcaton-2017-seres: Advances B; Vol. 60; N 1; pp Submtted Jan. 2017; Revsed March 15, 2017, Accepted Aprl 15, 2017 A Probablty Dstrbuton Kernel based on htenng Transformaton Jangsheng Gu, Yuanfeng Ch, Qng Zhang, Xaoan Bao The college of nformaton, Zhejang Sc-Tech Unversty Hangzhou , Chna (dewgjs@126.com) Abstract A kernel functon can change lnearly nseparable vectors nto lnearly separable vectors by mappng the orgnal feature space nto another feature space. But the dmenson of the new feature space s usually several tmes of the orgnal feature space, whch results n a more computatonal complexty. Ths paper ams at ntroducng a Drchlet probablty dstrbuton kernel based on whtenng transformaton whch s called DPT kernel functon for short. The DPT kernel functon frst mappng feature vectors of the orgnal feature space nto new vectors of another same dmenson feature space, and then classfes new vectors n the new feature space so as to acheve the purpose of classfyng orgnal feature vectors. The DPT kernel doesn t augment the dmenson of the new feature space. hat s more, the DPT kernel can effectvely elmnate the correlaton between vectors, and reduce the redundancy of data whch can further mprove the accuracy of classfcaton. In ths paper, we use the DPT kernel and other fve commonly used kernels on the three benchmark datasets (VOC2007, UIUCsport and Caltech101) for mage classfcaton experments. Experments show that the DPT kernel exhbts superor performances compared to the other state of the art kernels. Key words htenng transformaton, Drchlet probablty dstrbuton, Kernel functon, Image classfcaton 93

2 1. Introducton The study of kernel functons has caused a boom n mage classfcaton. People usually use a hgh effcency and low cost lnear classfer for mage classfcaton. But feature vectors are lnear nseparable n low dmensonal space. Accordng to recognton theory, the low dmensonal lnear nseparable vectors can become lnear separable n the hgh dmensonal feature space by kernel functons. Radal bass kernel (RBF) functon s a common kernel functon, and many scholars have made a deep research on the RBF kernel functon. Rahm uses a postve defnte shft-nvarant kernel to approxmate the RBF kernel functon, whch further mproves the performance of the RBF kernel functon [1]. Vempat uses the dsplay features to map the RBF kernel functon, whch makes a trade-off between the hgh computaton complexty and hgh classfcaton accuracy [2]. Addtve kernels are also commonly used kernel functons, and the addtve kernels nclude an Intersecton kernel [3], a Homker kernel and a Hellnger kernel [4]. Perronnn proposes that smply obtan the square-root of the feature vectors can lead to large mprovements, and the step perfect nterprets the basc prncples of the Hellnger kernel functon [5]. In order to compare the effectveness of addtve kernels, Vedald proposes a unfed framework for all addtve kernels [6]. That framework can use a closed form express all common kernels and experments show that the performance of the Homker kernel s the best. In order to adapt to a wde range of data samples, Gullaumn proposes a multple kernel learnng (MKL) for mage classfcaton [7]. The method can effectvely elmnate the nose samples and mproves the classfcaton accuracy of samples. In order to ncrease the dscrmnatve power of an object categorzaton method, t s effectve to model the ntra category dversty and the nter category correlaton among mages. Accordng to that dea, Yang ntroduces a group-senstve multple kernel learnng method to accommodate the dversty and the correlaton, the experments show that the group-senstve multple kernel learnng method has acheved qute good performance n three challengng datasets [8]. In order not to be lmted by the performance of the exstng kernel functon, Tom proposes a gene kernel functon [9], the gene kernel functon shows better performance compared wth the polynomal kernel functon, the gauss kernel functon and the sgmod kernel functon. Chen proposes a concept of an nner double kernel and the kernel n that nner double kernel functon can be arbtrary kernel functon [10]. The handwrtten recognton 94

3 rate of the nner double kernel functon s hgher than the Gaussan kernel functon and polynomal kernel functon. Ionescu proposes a new kernel functon that s called PQ kernel [11]. The PQ kernel defnes the number of concordant pars between two vectors s P. In the same way, the number of dscordant pars s Q. Experment shows that the PQ kernel whch combned wth the JS (Jensen-Shannon) kernel can further mprove the mage classfcaton accuracy. ang proposes a fast stochastc ntersecton kernel machne tranng algorthm (SIKMA), the method s more effectve than the lnear kernel n mage matchng, retreval and classfcaton [12]. Consderng the computatonal complexty of the kernel functons wll be very hgh, Chang proposes a probablty ntegral kernel as the smlarty measure of vectors and experments show that the probablty ntegral kernel reduced the computng tme dramatcally [13]. Although many people use kernel functon to measure the smlarty between feature vectors [14][15][16], but compared wth the orgnal features, the kernel functon ncreased feature space s dmenson after mappng features, and add the computatonal complexty. In order to solve that problem, we ntroduce a drchlet probablty dstrbuton kernel based on whtenng transformaton whch s called DPT kernel functon for short. The man purpose of the DPT kernel functon s to map the feature vectors of the orgnal feature space nto new vectors of another feature space whch has a same dmenson, and then classfy new vectors so as to acheve the purpose of classfyng orgnal feature vectors. The DPT kernel functon not only keeps the same dmenson between the new feature space and the orgnal feature space, but also effectvely elmnates the correlaton between the dmensons and reduces the data redundancy. In ths paper, we ntroduced the basc concept of whtenng transformaton at frst, and then the Drchlet probablty dstrbuton kernel (DPT) based on whtenng transformaton s ntroduced n detal. Fnally, the valdty of DPT kernel functon on mage classfcaton s verfed on VOC2007, UIUCsport and Caltech101 datasets. 2. Foundatons of the DPT kernel The whtenng transformaton s a knd of decorrelaton lnear transformaton, whch can transform a d dmenson vector J(x) whose mean s and covarance matrx s G nto a new 95

4 random vector z(x). Z x AJ x (1) where the dmenson of vector z(x) s also d, and the covarance matrx of vector z(x) s the unt matrx, A represents a whtenng vector that satsfes the condton A T A=G -1. The covarance matrx of vector z(x) s a unt matrx, so the random varables of the vector are rrelevant, and the varance of all random varables s 1, so the whtenng transform can effectvely elmnate the correlaton between vectors, and reduce the redundancy of data. There are many knds of whtenng transformatons, n ths paper, we choose the smplest Mahalanobs whtenng transform [17]. The mean of gradent vector H(x) of the ln-lkelhoo d functon for any probablty dstrbuton s: ln, E H x E p x 0 1 p x, p x, p x,, p x, dx E p x, dx p x (2) T 1, 2,..., s the dstrbuton parameter when the Drchlet probablty dstrbuton px (, ) have the max ln-lkelhood functon. The covarance matrx of vector Hxs: T T G E H x E H x H x E H x E H x H x Accordng to the defnton of the Mahalanobs whtenng transformaton, for any vector H(x) whose mean value s 0 and the covarance matrx s G, the vector (x) after whtenng transformaton s defned as the followng form: (3) 96

5 12 x G H x (4) In order to verfy the applcablty of Mahalanobs whtenng transformaton to ths paper, we calculate the expectaton and covarance matrx of vector (x) accordng to formula (5) and formula (6), E x E G H x G E H x 0 (5) Cov x E x E x x E x T E x x T E G H x G H x G E H x H x G G 1 2 T 1 2 GG By the defnton of covarance matrx, we can know that G s a sem postve defnte 12 symmetrc matrx, so there s a postve defnte symmetrc square root G G 1/2 G 1/2 =G. T (6) can make sure Cov x I (7) where I represents the unt vector. By formula (5) and formula (7), we can know that the expectaton of the vector (x) s 0, and the covarance matrx of the vector (x) s the unt matrx. Therefore, n order to effectvely elmnate the correlaton between vectors, and reduce the redundancy of data, we can carry out the whtenng transformaton of the gradent vector H(x). 3. DPT kernel The man dea of the DPT kernel functon s to map the feature vector x, y of the orgnal feature space nto vector(x), (y)of the new feature space, and then classfy new vectors so as to acheve the purpose of classfyng orgnal feature vectors. e use Drchlet probablty dstrbuton to get better gradent vector H(x).Gven a feature vector x=[x 1, x 2,, x w ] T, the formula of Drchlet probablty dstrbuton s: ( ) 1 D px, x ( ) (8)

6 here w s a feature dmenson whch s equal to the number of parameters of the model. t1 t denotes model parameter. e dt 1, 2,..., gamma functon. The DPT kernel functon s defned as: DPT ,,, T 0 1!, ndcates a K x y x y G H x G H y H x G H y (9) where vector (x) and vector (y) are obtaned by whtenng transformaton. H(x) s the gradent vector of the ln-lkelhood functon of Drchlet probablty dstrbuton, where expresses the dervaton of. ln p( x, ) H x 1 ln x B ln x 1 1 ln ln ln x ' 1 ln 1 ln x ' ' ' ' ,1,...,1,,.., ln x1,ln x2,...,ln x (10) snce ' d ln d (11) so 1 1 (12) ln x H x where s a dgamma functon, 1, ln x and are all nput functons, and 1 R s a vector whch all elements are 1. The covarance matrx of the formula (9) can be obtaned by formula 98

7 (13): ' (, )[ ln (, )][ ln (, )] { ( )} ( ) 1 (13) G p x p x p x dx dag B 2 d ln 2 d (14) dag(.) s the dagonal matrx whch s generated from the nput vectors. s a trgamma functon. B s a square matrx whch elements are all 1, and the dmenson of B s ww. G s a constant vector and the value of G has nothng to do wth x and y. The DPT kernel functon can be further wrtten as: T 1, ln ln 1 1 KDPT x y x 1 G y 1 (15) e set 1 1 (16) Therefore, the DPT kernel functon can be rewrtten as: T 1 K x, y [ln x ] G [ln y ] (17) DPT Accordng to the the formula (17), we can see that the value of the DPT kernel functon s closely related to the value of ln(x) and ln(y). 4. Parameter Optmzaton The Ln functon s very mportant for the DPT kernel functon. It should be noted that ln(x) when x0, at ths pont, the kernel functon s obvously unreasonable, so we modfy ln(x) to ln(x+f) usng small parameter f<<1. The optmzed DPT kernel functon s: 1 T ' 1 (18) KDPT [ln x f ] dag{ ( )} ( ) B [ln y f ] The basc problem of support vector learnng (SVL) s the selecton of kernel functon and kernel parameters, now we further study the modfed parameters f n the formula (18). Assume that the parameters are not affected by the change of parameters, we gve the margnal dstrbuton of Drchlet dstrbuton by Beta dstrbuton, as shown n formula (19): 0 1 m n1 px m n Bm n (19) 1 1 x 1x x 1 x p x,,, B,, 0 99

8 m=, n= 0 -. Frst, the varable v= ln(x) s used to nstead of the varable x to optmze the formula (19) nto a formula (20): p v; m, n m exp v 1 exp v Bm, n 1 n1 (20) And then accordng to the varable transformaton formula U=ln{exp(v)+f}, the ln(x) s optmzed to be wrtten n the form of ln(x+f), as shown n the formula (21): p U, m, n m exp U f 1 f exp U Bm, n 1 n1 (21) Functon ln(x+f) acts on the followng two aspects: on the one hand, t s roughly removed the ln(x) n the x f and only keep ln(f).the tny hstogram value s treated as 0, whch enhances the dfference between the approprate small x, lke a local contrast enhancement, whch reflects the negatve example theory [17]. At the same tme, by determnng the value of f, U= ln(x+f) forms a smooth dstrbuton on the top of ln(f), and t can keep a smooth dstrbuton state around the dstrbuton mode. Durng the process of determnng f, too small f pushes the lower boundary ln(f) away from the dstrbuton pattern, whch exaggerates the negatve example theory. On the other hand, The larger the f wll merge the low boundary ln(f) wth the dstrbuton pattern, whch s contrary to the theory of negatve examples. So, the sutable f can show the smooth dstrbuton between the dstrbuton pattern and the low boundary ln(f). In order to get the approprate f, we defne f based on the cumulatve percentage of the emprcal dstrbuton, we usually set the cumulatve percentage of 25%, the optmzaton parameter f=p -1 f (0.25)0.001, whch P f pxdx. 0 Assumng that the dmenson of the orgnal feature vector x s w1, then the feature vector (x) after the DPT kernel functon mappng has the followng form: 12 x G H x dag 1 B x f 1 2 ' { ( )} ( ) ln (22) Accordng to the formula (22), we can see that the dmenson of vector (x) s stll w1, so the dmenson of the feature space after the DPT kernel functon mappng s consstent wth the orgnal feature space. 5. Experments 100

9 In ths paper, we use three datasets (VOC2007 dataset, UIUCsport dataset and Caltech101 dataset) to carry out the mage classfcaton experments. Several state of the art kernels (Homker[6], JS+PQ[11], SIKMA[12], MKL[7], Lnear) were compared wth the DPT kernel on the three datasets. Our mage classfcaton s performed usng a PC wth an Intel Core CPU, 3.4 GHz, 8 GB RAM, and the algorthm s developed usng MATLAB R2014a and GCC4.6 compler under an Ubuntu12.04 (64-bt) operatng system Dataset In order to make experments more dversty, we choose VOC2007 dataset, UIUCsport dataset and Caltech101 dataset for mage classfcaton. Our frst set of experments s on VOC2007 dataset and VOC2007 dataset s composed of color mages wth mult labels. VOC2007 dataset contans 9963 mages and ncludes annotated objects. The VOC2007 dataset s dvded nto 20 categores (Aeroplane, Bcycle, Brd, Boat, Bottle and so on), each of whch contans 193~4015 mages. hat s more, a large number of mages are medum resoluton (500*333 pxels). e use an dentcal expermental setup as Chatfeld [18], and about 50% mages of the dataset are used for tranng/valdaton samples and others for testng samples. UIUCsport dataset s selected as the second dataset n our experments and UIUCsport dataset s composed of color mages wth a sngle label. UIUCsport dataset contans 8 categores (rowng, badmnton, polo, bocce, snowboardng, croquet, salng, rock clmbng) whch are all sports event. There are 1579 mages n the dataset and there are about 200 mages n each category. For each category, we randomly selected 70 mages for tranng and 60 for testng. The thrd dataset s Caltech101 and Caltech101 s composed of color and gray mages. There are 9144 mages n Caltech101 dataset and these mages are dstrbuted nto 102 categores (101 object categores and a background category), each of whch contans 40~800 mages. Furthermore, each mage of the dataset s about 350 x 200 pxels. For each category, we select 15 random mages for tranng and test on the rest Procedure In ths paper, the tranng samples and test samples of each data set need four steps: feature extracton, feature clusterng, feature codng, and hstogram formaton, then the tranng samples 101

10 are nput nto the SVM model to determne the decson functon of the model, the test samples are nput nto the SVM model to get the type of each mage n the test sample. The parameterbetween tranng samples and test samples are same n the four steps, for the sake of smplcty, we use the tranng sample to llustrate the parameters of the four steps. The frst step, usng SIFT dense samplng methods on three data sets of tranng mages for feature extracton, Then, feature ponts are randomly selected from the feature ponts extracted from each data set, whch s used for feature clusterng. The second step, the K mean algorthm s used to cluster the feature ponts of the three data sets nto 3000 feature types, and the number of teratons s set to 150.The thrd step, we use KCB codng to encode the features, n order to mprove the effcency of the computaton, each feature s encoded by consderng fve vsual words whch are closest to ther own [20].The fourth step, we usng spatal Pyramd model to form vsual word hstogram for each mage, the mage s dvded nto 3 layers, the three layer ncludes 1*1, 3*1 (three horzontal strps), and 2*2 (four quadrants) bns, for a total of 8 bns [19].The ffth step, The hstogram vector of the tranng mage s nput nto the SVM model to complete the tranng of the SVM model, and then test the mage nput to the SVM model to get the category of each mage, complete the SVM model test. e choose the DPT kernel (optmzaton parameter set to 0.001) and other fve kernels (Homker, JS+PQ, SIKMA, MKL, Lnear) for SVM model n mage classfcaton. e set the penalty parameter to 7.6, whch expresses the tolerance of the msclassfed data tems. hat s more, we use the one-versus-rest prncple to put the feature vectors of the testng mages nto SVM to get categores of the testng mages [19]. Specfc steps are as follows: frst set the frst class of the data set as a postve sample, all other categores are negatve samples, then each test mage s nput nto the SVM model to calculate the value of the decson functon, f the value of the decson functon s greater than 0, ndcates that the test mage s a postve sample, otherwse t s a negatve sample, then take the next category as a postve sample, and all the other categores as negatve samples, contnue to run the decson functon untl all categores as a postve sample. In ths paper, there are two expermental evaluaton ndex, accordng to PASCAL VOC2007's offcal statement: accuracy / recall curve s more ntutve and more senstve to the performance evaluaton than the recever operatng characterstc curve. In ths paper, we use the 102

11 map value of a method on a gven set of data ndcates the average accuracy of the method n all categores, f a map value s hgh, then the classfcaton performance s better. The PR curve s another ndcator of the experments n ths paper, we utlze the PR curve to analyze the classfcaton performance of dfferent methods. If the PR curve of a certan method s closer to the upper rght corner, the classfcaton performance of the method s better. To obtan relable results, we repeated the expermental process 5 tmes for each method Experments Results Table 1 and table 2 show the average precson for each category of sx methods (DPT, Homker, JS+PQ, SIKMA, MKL, Lnear) respectvely on VOC2007 and UIUCsport datasets, furthermore, the hghest average precson for each category s n bold. There are 102 categores n Caltech101 dataset, so we do not lst the average precson for each category one by one. e just gve the mean average precsons (maps) of sx methods (DPT, Homker, JS+PQ, SIKMA, MKL, Lnear) on Caltech101 dataset n table 3. e select four categores of the hghest classfcaton accuracy on Caltech101 dataset n fgure 1. hat s more, we select four categores of the lowest classfcaton accuracy on Caltech101 dataset n fgure 2. From table 1, table 2, and table 3, we can clearly see that the maps of fve kernels were well above the map of the baselne lnear kernel. hat s more, n table1 and table 2, the map of the DPT kernel n most of the categores s better than other kernels. In VOC2007 dataset, the DPT kernel has obtaned the best results for 19 out of 20 categores. In UIUCsport dataset, the DPT kernel has obtaned the best results for 7 out of 8 categores. To be concrete, n Tvmontor category of VOC2007 dataset, the map of the DPT kernel s above the map of the JS+PQ kernel, above the map of the SIKMA, above the map of the MKL method and above the map of the Homker kernel. In badmnton category of UIUCsport dataset, the map of the DPT kernel s above the map of the JS+PQ kernel, above the map of the SIKMA, above the map of the MKL method and above the map of the Homker kernel. Table 1. Performance on VOC2007 Category Lnear JS+PQ SIKMA MKL Homker DPT Aeroplane

12 Bcycle Brd Boat Bottle Bus Car Cat Char Cow Dnngtable Dog Horse Motorbke Person Pottedplant Sheep Sofa Tran Tvmontor map Table 2. Performance on UIUCsport Category Lnear JS+PQ SIKMA MKL Homker DPT rowng badmnton polo bocce snowboardng croquet salng rock clmbng map Table 3. Performance on Caltech101 Method Lnear JS+PQ SIKMA MKL Homker DPT map car_sde (100%) Motorbkes (99.96%) pagoda (100%) trlobte (100%) Fg. 1 Four categores of the hghest classfcaton accuracy on Caltech101 dataset 104

13 Beaver (17.91%) crocodle (20.76%) platypus (20.20%) wld_cat (12.00%) Fg. 2 Four categores of the lowest classfcaton accuracy on Caltech101 dataset hat s more, Precson/Recall (PR) analyss gves more ntutve and senstve evaluaton than the ROC analyss. So we choose PR analyss to evaluate the performance of dfferent kernels. e draw the PR curves of the sx kernels n fgure 3, fgure 4, and fgure 5. DPT kernel depcted n red lne, Homker kernel depcted n cyan lne, JS+PQ kernel depcted n blue lne, SIKMA method depcted n magenta lne, MKL method depcted n green lne and Lnear kernel depcted n black lne. Fg. 3 The PR curves of sx kernels on VOC2007 dataset Fg. 4 The PR curves of sx kernels on UIUCsport dataset 105

14 Fg. 5 The PR curves of sx kernels on Caltech101 dataset From the PR curves of VOC2007 dataset, t can be seen clearly that DPT kernel s better than other fve methods. On Caltech101 and UIUCsport datasets, the DPT kernel has an excellent PR curve agan. In terms of the PR curves, fve kernels are obvous better than the baselne lnear kernel. The MKL kernel combnes an RBF kernel and a lnear kernel for mage classfcaton, so t obtans a pretty good PR curve. However, the RBF kernel functon s very senstve to parameters, so the PR curve s lower than the Homker kernel functon whch s not affected by the parameters. The SIKMA kernel functon s not only senstve to the parameters, but also only takes nto account the smlarty between the orgnal vectors, so the PR curve s slghtly lower than the MKL kernel functon. JS+PQ kernel functon s related to the number of concordant and dscordant pars between two vectors, so t has a lmted descrpton and a poor PR curve. As can be seen n table 4, the tranng tme of sx kernels were measured n seconds. The classfcaton accuracy of lnear kernel functon s the lowest, and the tranng tme s the shortest. SIKMA kernel combnes a stochastc learnng method wth an ntersecton kernel, so the complexty of calculaton s lower than a tradton ntersecton kernel. Accordng to the formulaton (22), the DPT kernel has not mproved the feature dmenson, so t has a low computatonal complexty. The Homker kernel ncreases the feature dmenson from 1 to 7, so t ncreases the computaton tme. The JS+PQ kernel need to calculate the number of concordant and dscordant pars between two vectors, so the computaton complexty s very 106

15 hgh. Because the MKL kernel requres multple kernel matrces n the operaton, the computatonal complexty s hgh. From table 4, the computatonal complexty of the DPT kernel s smaller than Homker s, JS+PQ s and MKL s. Despte tranng tme of the DPT kernel s hgher than the Lnear and SIKMA kernel, the DPT kernel has made great progress n precson. Table 4. The tranng tme for sx kernels on three dataset Lnear SIKMA DPT Homker JS+PQ MKL VOC UIUCsport Caltech Concluson and Further ork Ths paper presents a new kernel, a drchlet probablty dstrbuton kernel based on whtenng transformaton, whch we called t as the DPT kernel. Far experments usng three benchmark datasets (VOC2007, UIUCsport and Caltech101) were carred out on sx methods (DPT, Homker, JS+PQ, SIKMA, MKL, Lnear). From the experments we conclude the followng results. The map of the DPT kernel s hgher than the maps of other fve kernels on three datasets. In terms of the computatonal tme, the DPT kernel s shorter than kernels (Homker, JS+PQ and MKL) and a lttle longer than the SIKMA kernel. On the one hand, the DPT kernel mproves the classfcaton precson wthout ncreasng the feature dmenson. On the other hand, the DPT kernel can effectvely elmnate the correlaton between vectors, and reduce the redundancy of data. In future work, we wll optmze the DPT kernel algorthm and thoroughly study the optmzaton parameter of the DPT kernel functon. Furthermore, we wll use a TF-IDF measure for features and elmnate the features whch have a low TF-IDF score. References 1. A. Rahm, B. Recht. Random Features for Large-Scale Kernel Machnes. Advances n Neural Informaton Processng Systems, 2007, vol. 20, pp S. Vempat, A. Vedald, A. Zsserman, et al. Generalzed RBF Feature Maps for Effcent Detecton// Brtsh Machne Vson Conference, BMVC 2010, Aberystwyth, UK, August 31 - September 3, Proceedngs. DBLP, 2010:

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