Efficient Sparsity Estimation via Marginal-Lasso Coding

Transcription

1 Effcent Sparsty Estmaton va Margnal-Lasso Codng Tzu-Y Hung 1,JwenLu 2, Yap-Peng Tan 1, and Shenghua Gao 3 1 School of Electrcal and Electronc Engneerng, Nanyang Technologcal Unversty, Sngapore 2 Advanced Dgtal Scences Center, Sngapore 3 ShanghaTech Unversty, Shangha, Chna Abstract. Ths paper presents a generc optmzaton framework for effcent feature quantzaton usng sparse codng whch can be appled to many computer vson tasks. Whle there are many works workng on sparse codng and dctonary learnng, none of them has exploted the advantages of the margnal regresson and the lasso smultaneously to provde more effcent and effectve solutons. In our work, we provde such an approach wth a theoretcal support. Therefore, the computatonal complexty of the proposed method can be two orders faster than that of the lasso wth sacrfcng the nevtable quantzaton error. On the other hand, the proposed method s more robust than the conventonal margnal regresson based methods. We also provde an adaptve regularzaton parameter selecton scheme and a dctonary learnng method ncorporated wth the proposed sparsty estmaton algorthm. Expermental results and detaled model analyss are presented to demonstrate the effcacy of our proposed methods. Keywords: Sparsty estmaton, margnal regresson, sparse codng, lasso, dctonary learnng, adaptve regularzaton parameter. 1 Introducton Sparse codng has been successfully appled to many machne learnng and computer vson tasks, ncludng mage classfcaton [30,31,8], face recognton [28,5], actvtybased person dentfcaton [22,15], etc. Sparse codng refers to a feature quantzaton mechansm whch obtans a codebook to encode each nput sgnal as a sparse hstogram feature based on the lnear combnaton of a few vsual words. Due to ts soft assgnment prncpal, t can make the quantzaton error much smaller [9,22]. Moreover, sparse codng s more robust to the nose and easly ncorporatng nto the bag-of-feature (BoF) framework. The least absoluton shrnkage and selecton operator whch s also known as lasso [27] n short s a popular sparse codng model n statstcs sgnal processng whch has wldly used n sparsty estmaton. It smultaneously mzes the quantzaton error, and mpose an L 1 penalty wth a regularzaton parameter λ whch controls the sparsty of the coeffcents. Whle there exsts many effcent algorthms [20,7,29], however, fndng the lasso solutons reman a computatonal task wth the complexty O(p 3 +dp 2 ) where d refers to the nput feature dmenson, and p refers to the amount of vsual D. Fleet et al. (Eds.): ECCV 2014, Part IV, LNCS 8692, pp , c Sprnger Internatonal Publshng Swtzerland 2014

2 Effcent Sparsty Estmaton va Margnal-Lasso Codng 579 words n the dctonary. Thus, t hardly supports large scale data analyss effcently and effectvely [6,23,11,1]. Recently, margnal regresson, has been revsted and shown ts effcent performance for vsual word selecton and sparsty estmaton [6,11,1]. For each feature, t calculates the correlaton wth each vsual word and mposes a tunng parameter to acheve the sparsty. Whle margnal regresson s smple and fast wth the complexty O(dp), t usually consders and utlzes a fxed tunng parameter whch s not always the good cut-off pont for each feature and could result n a large quantzaton error snce some coeffcents may be shrunk too much, but some may be ncluded too much nose. Hence, how to detere a sutable cut-off pont becomes an typcal problem n featurng codng. To address ths, the sure ndependent screenng (SIS) [6] method mposes an L 1 - norm penalty and an energy constrant E to each margnal regresson coeffcent vector. It sorts coeffcents n terms of ther absolute values and selects the top k coeffcents whose L 1 norm s bounded by E so that each sparse code can be represented by smlar sparsty level. However, ths method may also cause a large quantzaton error, because margnal regresson works well when vsual words have low correlaton n the dctonary [1,23]; however, n general, they are hghly correlated, and thus may fal to clarfy the relatonshp between vsual words and the nput feature. Whle there are many works workng on sparse codng and dctonary learnng, none of them has exploted the advantages of the margnal regresson and the lasso smultaneously to provde more effcent and effectve solutons. To ths end, we provde n ths paper such an approach wth a theoretcal support. We propose an effcent sparsty estmaton approach usng Margnal-Lasso Codng (MLC) whch quantzes each feature nto a small set of vsual words usng margnal regresson combned wth the lasso framework. Our model represents each sparse code wth smlar sparsty level bounded by a global sparsty energy and smultaneously shrnks ndvdual coeffcents to allevate the bas of sparse codes caused by the hghly correlated vsual words. Moreover, our approach automatcally deteres the shrnkng regularzaton parameter for each feature and further desgns a self-rato energy constrant of sparsty level whch s dfferent from the tradtonal constrant wth a case-dependent chosen value. Contrbuton: 1) We propose an effcent Margnal-Lasso Codng (MLC) framework for feature quantzaton wth sparsty estmaton, regularzatonparameterselecton and dctonary learnng. 2) We explot the advantages of the margnal regresson and the lasso smultaneously to provde more effcent and effectve solutons. 3) We detere the regularzaton parameter automatcally and adaptvely for each ndvdual feature and provde a self-rato energy bound. 4) We provde a theoretcal support for the proposed model. 5) We successfully apply the proposed method to varous recognton tasks. 2 Related Work Our approach s related to the general sparse codng model whch has been wdely used for sparse representaton [28,27,30,22], and there exst many algorthms to estmate sparse codes, such as least angle regresson [4], gradentdescent [7,29], and feature-sgn

3 580 T.-Y. Hung et al. search [20]. The feature-sgn search method [20,8,31] s a popular and effcent soluton for sparsty estmaton. It contnuously selects and updates potental canddates n an actve set to mantan these nonzero coeffcents and ther correspondng sgns untl reachng the constrant. However, wth a large dctonary sze, the actve set would become largeand the feature-sgn searchng process may be hard to terate [1,23]. Whle these methods am to obtan a small quantzaton error, they have a large computatonal cost [6,11]. Therefore, dfferent from tradtonal sparsty estmaton technques whch estmate sparse codes by contnuously selectng varables to the actve set based on the searchng crtera, we ncorporate margnal regresson coeffcents nto the sparse codng model whch s more effcent for sparsty estmaton. Recently, Fan and Lv [6] proposed a sure ndependent screenng (SIS) method usng margnal regresson to obtan sparse codes effcently and can easly deal wth largescale problems. Genovese et al. [11] provded a statstcal comparson of the lasso and margnal regresson. Due to the promsng performance n terms of accuracy and speed, Krshnakumar et al. [1] appled margnal regresson to learn sparse representaton for vsual tasks. Whle these solutons use hard-thresholdng-type methods, we propose n ths work a soft-thresholdng-type method wth a theoretcal support to use margnal regresson combned wth the lasso model for sparsty estmaton. 3 Effcent Sparsty Estmaton va Margnal-Lasso Codng 3.1 Sparsty Estmaton va Margnal Regresson Consder a regresson model x = Us + z wth an nput feature x R d, a coeffcent vector s R p, a column-wsely normalzed dctonary U = [u 1,...,u p ] R d p (p d) and a nose vector z R d. The feature quantzaton task s to quantze an nput feature x nto a feature hstogram s wth a few non-zero elements so that the lnear combnaton of U and s can gan a mzed quantzaton error. The mathematcal expresson can be shown as follows: 1 s 2 x Us 2 2 (1) To estmate the sparse code s, we frst compute the component-wse margnal regresson coeffcents â: â U T x (2) where â (k) = u T k x s the kth element of â and u k s the kth column of the dctonary U. Then, we threshold the coeffcents n terms of ther absolute values usng a tunng parameter t>0 [3,11] so that the coeffcents whose absolute values are larger than t are kept and the rest are set to zero. The mathematcal expresson can be wrtten as follows: ŝ (k) = { â(k) f â (k) >t 0 otherwse (3) Accordng to Eq. 3, however, utlzng a fxed tunng parameter may cause a large quantzaton error snce t s not always the good cut-off pont for each feature. Instead of

4 Effcent Sparsty Estmaton va Margnal-Lasso Codng 581 cut-off pont selecton, we can select the top k large coeffcents n terms of ther absolute values whose L 1 -norm s bounded by a constrant E [6,1]. The equaton can be reformulated as follows: ŝ (k) = { â(k) f k B 0 otherwse (4) where r B = {J 1,...,J r : r p : â (Jk) E} (5) k=1 We sort the coeffcents n terms of ther absolute values n descendng order whch denotes by ndexes J 1,...,J p where ŝ (J1) > ŝ (J2) > > ŝ (Jp). The two thresholdng approaches above are hard-thresholdng-type methods. 3.2 Margnal-Lasso Codng Whle margnal regresson has shown ts effectveness, there stll exsts three lmtatons: () Margnal regresson works well when the columns of dctonary have low correlaton [1,23]. However, n general, the columns of over-complete dctonares are usually hghly correlated, and thus, n a lnear regresson model, ther coeffcents can become poorly detered and exhbt hgh varance. () In addton, the fxed regularzaton parameter may cause a large quantzaton error. Some coeffcents may be shuck too much and some may be ncluded too much nose. () Moreover, the exstng margnal regresson solutons are hard-thresholdng-type methods, and there are no theoretcal supports. To address ths, we consder the followng condtons: (1) The regularzaton parameter should be chosen adaptvely and automatcally for each feature to mze ndvdual estmate of expected quantzaton error. (2) The L 1 -norm n the lasso framework should be consdered locally for vsual word selecton and coeffcent shrnkage to allevate the bas of sparse codes caused by the hghly correlated vsual words. (3) The L 1 -norm n the constrant should be bounded locally and globally for sparse representaton. To ths end, we ntroduce a Margnal-Lasso Codng (MLC) method to better characterze nput features and provde a theoretcally soft-thresholdng-type soluton. Smlar to the exstng works, we frst compute the margnal regresson coeffcents of each feature va Eq. 2. Then, we consder the followng optmzaton problem: s,λ 1 2 U T x s λ s 1 subject to s 1 E (6) The frst term s the quantzaton error, and the second term s a local L 1 penalty wth an adaptve regularzaton parameter λ whch controls the cut-off pont selecton and the shrnkage of coeffcents. The global sparsty constrant E bounds each sparse code to the smlar sparsty level. In ths model, we am to estmate a sparse code and a

5 582 T.-Y. Hung et al. parameter λ so that the quantzaton error can be mzed and the coeffcents can be shrunk smultaneously for sparse representaton. When consderng multple features, the optmzaton problems can be refned as follows: 1 s 1,...,s N 2 U T X S 2 F + N λ s 1 λ 1,...,λ N =1 (7) subject to s 1 E where X =[x 1,...,x N ] R d N be a set of features, and S =[s 1,...,s N ] R p N be a set of correspondng sparse codes. Snce the sparsty energy E s case-dependent varable, we provde here another self-rato energy e such that each feature can obtan ts sparse code by proportonally shrnkng ther margnal coeffcents to the desred level. The constrant can be reformulated as follows: 1 N s 1,...,s N 2 U T X S 2 F + λ s 1 λ 1,...,λ N =1 (8) subject to f λ(u T x) 1 U T x 1 e where f λ (A) refers to the soluton of Eq. 3 wth â = A and t = λ. Whle Eq. 7 consders the smlar L 1 -norm between sparse codes, Eq. 8 consders the smlar selfrato between each other. 3.3 Sparsty Estmaton When λ s fxed, we can solve the optmzaton problem of Eq. 6 by rewrtng the objectve as follows: 1 s 2 â s λ s 1 (9) Mnmzng the Eq. 9 s equvalent to mze ndvdual element errors wth ther correspondng absolute values. Thus, we can rewrte the objectve as follows: 1 p (â (k) s (k) ) 2 + λ s (k) (10) s 2 k=1 Snce the margnal regresson coeffcents are computed ndependently for each element, hence, we can decompose the problem nto p separate optmzaton tasks. For each task k, the optmzaton problem can be defned as follows: (â (k) s (k) ) 2 + λ s (k) (11) s (k) Then, by solvng the above optmzaton problem, we can obtan a soft-thresholdngtype optmal soluton of each element s (k) : â (k) λ f â (k) >λ s (k) = â (k) + λ f â (k) < λ (12) 0 otherwse = sgn(â (k) )( â (k) λ) +

6 Effcent Sparsty Estmaton va Margnal-Lasso Codng 583 where sgn(j) refers to the sgn of j,and(l) + means to keep the value when l s postve and set t to zero when l s negatve. The L 1 -norm can be defned as the sum of absolute elements s 1 = p k=1 sgn(â(k) )( â (k) λ) +. In Eq. 12, the regularzaton parameter λ controls the element selecton and the shrnkage of the coeffcents. If λ equals to zero, then s 1 = â 1. Instead, f λ>0, then some coeffcents wll be drectly set to zero and others wll be shrunk towards zero automatcally so that s 1 < â 1.By maxmzng s 1 bounded by the global sparsty constrant E, the ndvdual λ can be calculated as follows: λ =argmax λ s 1 =argmax λ subject to s 1 E p sgn(â (k) )( â (k) λ) + By dong so, we can estmate each element of the sparse code s (k) = sgn(â (k) )( â (k) λ ) + whch s a soft-thresholdng-type soluton. When consderng a set of features, the optmzaton problem of Eq. 7 or 8 can be solved va Algorthm 1. k=1 (13) 4 Margnal-Lasso Codng for Dctonary Learnng The dctonary U s a set of normalzed vsual words denotng by each column u k.to learn the dctonary, we teratvely optmze U and S by obtanng S wth a fxed U and updatng U based on a gven S. The sparsty estmaton S va the margnal-lasso model has been ntroduced n the prevous subsecton. In ths subsecton, we ntroduce the dctonary learnng method. Snce margnal regresson works well when the columns n the dctonary have low correlaton [6,11,1,23], we formulate the optmzaton problem of the dctonary learnng as follows: U N x Us γ U T U I 2 F =1 subject to u T k u k 1 k (18) We smultaneously mze the quantzaton errors denotng by the frst term and the correlatons between columns n the second term wth a regularzaton parameter γ. Ths optmzaton problem can be solved va the frst-order gradent descent method [23,1]: U q+1 =Π U {U q β F (U q )} (19) where N F (U) = x Us γ U T U I 2 F (20) and =1 F (U q )=2(U q SS T XS T )+4γ(U q U T q U q U q ) (21) F (U q ) s the gradent of F (U) wth respect to U at each teraton q, parameter β s the step sze, and Π U s the projecton functon whch maps each column u k to the

7 584 T.-Y. Hung et al. Algorthm 1. Effcent Sparsty Estmaton va Margnal-Lasso Codng Requre: Data X =[x 1,...,x N] R d N, a learned dctonary U =[u 1,...,u p] R d p, and a specfc value of the sparsty constrant E or a self-rato constrant e Ensure: Sparse coeffcents S =[s 1,...,s N ] R p N 1: For x, compute the margnal regresson coeffcents â (k) = ut k x u k 2 2 k (14) 2: Calculate the regularzaton parameter λ for each x based on the sparsty constrant E of Eq. 6 and 13 λ =argmax λ s 1 =argmax λ subject to s 1 E p k=1 sgn(â (k) )( â (k) λ ) + (15) Or calculate the regularzaton parameter λ for each x based on the self-rato constrant e n Eq. 8 λ =argmax λ s 1 =argmax λ subject to f λ (U T x ) 1 U T x 1 p k=1 e sgn(â (k) )( â (k) λ ) + (16) 3: Obtan sparse codes s (k) = â (k) λ f â (k) >λ â (k) + λ f â (k) < λ 0 otherwse (17) L 2 -norm unt ball. More specfcally, the dctonary s learned teratvely by fndng the local mum along the gradent drecton based on the small step sze and normalzng each column vector wth length smaller or equal to one untl convergence. The sparsty estmaton and the dctonary learnng process s shown n Algorthm 2. 5 Expermental Results In ths secton, we descrbe the expermental settngs and analyze the proposed method on the mage classfcaton, acton recognton and actvty-based human dentfcaton tasks. We denote the proposed margnal-lasso codng approach by MLC wth Matlab mplementaton. We compare the proposed method wth 5 algorthms: (1) The lasso model wth feature-sgn search (LASSO-FS): the code has been mplemented by ts authors n Matlab [20]. (2) The lasso model wth least angle regresson (LASSO-LAR): we used the code from SPAMS software mplemented by C++ [23]. (3) The margnal regresson model wth the hard-thresholdng method (MR-Hard): we mplemented t n

8 Effcent Sparsty Estmaton va Margnal-Lasso Codng 585 Algorthm 2. Sparsty Estmaton and Dctonary Learnng Requre: Data X = [x 1,...,x N] R d N, an ntal normalzed dctonary U = [u 1,...,u p] R d p, and an energy bound wth value E or self-rato e Ensure: Learned dctonary U R d p, sparse coeffcents S =[s 1,...,s N ] R p N 1: repeat 2: Obtan sparse code va Algorthm 1 usng the latest dctonary. 1 s 1,...,s N 2 U T X S 2 F + λ 1,...,λ N subject to s 1 E N λ s 1 =1 (22) 3: Update the dctonary based on the sparse codes obtaned from the prevous step. U N x Us γ U T U I 2 F =1 subject to u T k u k 1 k (23) 4: untl convergence Matlab accordng to Eq. 3. (4) The margnal regresson model wth the soft-thresholdng method (MR-Soft): we mplemented t n Matlab accordng to Eq. 3 wth soft thresholdng. (5) The margnal regresson model wth the sure ndependence screenng method (MR-SIS): we mplemented t n Matlab accordng to Eq Image Classfcaton We evaluate the performance of the proposed method for the mage classfcaton task on USPS and Scene 15 datasets. USPS s a handwrtten dgts dataset [14,31] whch conssts of 7291 tranng mages and 2007 test mages of sze wth dgts 0 to 9. We represent each sample by a 256-dmensonal vector. The dctonary szes are selected from 32, 64, 128, 256, 512, 1024 under dfferent tranng szes. For the penalty constrant, we adopt self-rato n Eq. (8), and the parameters are set emprcally va cross-valdaton. To evaluate the performance, we tran a lnear SVM classfer and compare to orgnal regresson (OR) and sparse codng (SC) methods. Table 1 shows the results under dfferent szes of tranng samples from 100 to Asshown,the proposed MLC method outperforms the other two methods n most of the cases and obtans the best accuracy result 95.3 under 5000 tranng sample sze. Scene 15 [24,21,19] contans 4485 mages wth 15 categores. Each category conssts of rangng from 200 to 400 mages wth average sze of pxels. Ths dataset s very dverse from ndoor such as lvng room, ktchen, offce, store, etc., to outdoor ncludng street, nsde cty, buldng, mountan, forest, etc. Followng the same settng as the prevous works [19,30], we randomly select 100 mages per category as tranng samples and the rest as test samples and ncorporate the proposed method wth spatal pyramd matchng whch denotes by MLcSPM. Table 2 shows the results. As shown, our MLcSPM method outperforms other methods, especally for the

9 586 T.-Y. Hung et al. Table 1. Recognton accuracy (%) of dfferent methods on the USPS dataset Sze of tranng set Method OR SC MLC Table 2. Recognton accuracy (%) of dfferent methods on the Scene 15 dataset Sze of tranng set Method 100 KSPM [19] KC [10] ScSPM [30] HardSPM SoftSPM SISSPM MLcSPM popular lasso-based ScSPM scheme. Further, the proposed MLcSPM outperforms other margnal regresson based methods. 5.2 Acton Recognton We analyze the characterstc of the proposed method usng the KTH acton recognton dataset. The KTH dataset [26] contans 25 persons. Each person performed 6 dfferent actvtes, ncludng boxng, handclappng, handwavng, joggng, runnng, and walkng, respectvely. There are 4 scenaros under each actvty, ncludng outdoors, outdoors wth scale varaton, outdoors wth dfferent clothes and ndoors, respectvely, resultng 599 vdeo sequences n total. We follow the orgnal expermental setup whch dvdes clps nto 2391 subclps wth 863 clps for the test set (9 subjects) and 1528 clps for the tranng set (16 subjects). We use three space-tme local feature descrptors n the experments, HOG, HOF and HOGHOF [18] obtaned by a 3D-Harrs detector [17], and quantze each local feature nto a sparse code va Algorthm 1. After that, we represent each vdeo clp usng the maxmum poolng method and use a non-lnear support vector machne [2] wth a X 2 kernel [18]. We select the parameters usng cross valdaton. To speedup the dctonary learnng process, we frstly learn an ntal dctonary usng onlne dctonary learnng method [23] wth 100 runs as the warm start. Comparson between Hard-Thresholdng and Soft-Thresholdng: We encode each HOF local feature usng a hard-thresholdng technque, MR-Hard, and a softthresholdng technque, MR-Soft, respectvely, wth dctonary sze 4000 n the tranng set as the tunnng parameter t vares. Fg. 1 (a) shows the average L 1 -norm energy of sparse codes under dfferent t. As shown, usng the soft-thresholdng scheme s a more effectve way for varable selecton and coeffcent shrnkage. Further, the soft-thresholdng

10 Effcent Sparsty Estmaton va Margnal-Lasso Codng 587 Fg. 1. Comparson of the soft and hard-thresholdng schemes. (a) The average L 1-norm energy of sparse codes under dfferent t. As shown, usng the soft-thresholdng scheme s a more effectve way for varable selecton and coeffcent shrnkage. (b) The average standard devaton of the L 1- norm energy between sparse codes as t changes. Sparse codes obtaned va the soft-thresholdng scheme are much stable than those obtaned va the hard-thresholdng scheme under dfferent t. scheme acheves smaller standard devaton between sparse codes as t vares as shown n Fg. 1 (b). In the recognton pont of vew, Fg. 2 shows the recognton accuracy of MR-Soft and MR-Hard wth the HOG, HOF, and HOGHOF features respectvely under dfferent t. The accuracy of the MR-soft method s much better than that of the MR-Hard method. Therefore, we can conclude that soft-thresholdng technque such as MR-Soft s much stable and robust than hard-thresholdng technque such as MR-Hard. Energy Dscusson: The sparsty constrant E n Eq. 6 needs to be specfed a value, and the value may vary wth dfferent datasets. Instead, the self-rato constrant e n Eq. 8 ams to keep the certan rato of the margnal regresson coeffcents n terms of L 1 - norm energy. For example, the self-rato e =5%means that, for each feature, only 5% of ts margnal regresson coeffcents n terms of the L 1 norm can be kept and the rest wll be set to zero. Thus, we can obtan sparse codes more easly by assgnng a desred rato. Fg. 3 shows the recognton accuracy between dfferent values of the self-rato constrant e. The results show that we obtan around 90% recognton accuracy by settng the self-rato e from 5% to 35%, and obtan poor recognton accuracy (below 10%) by settng e from 75% to 100%. Therefore, we can conclude that sparse representaton helps sgnal nterpretaton. Speed Comparson: In ths subsecton, we exae the feature quantzaton speed usng dfferent sparsty estmaton methods under three dfferent feature descrptors, HOG, HOF and HOGHOF, respectvely. The dctonary szes are set from 1000 to It s worth mentonng that all the methods except for LASSO-LAR were mplemented n Matlab, and thus t would be unfar to compare them wth the LASSO-LAR method together snce C++ program of LASSO-LAR has a bult-n speed advantage. Nevertheless, as shown n Table 3, the margnal regresson related methods stll take sgnfcantly less tme than those of the lasso solutons. In addton, MR-SIS and MLC have

11 588 T.-Y. Hung et al. Fg. 2. Accuracy of soft and hard-thresholdng wth the HOG, HOF and HOGHOF features as t vares. The accuracy of MR-soft s much better than that of MR-Hard n terms of recognton accuracy under three dfferent features. comparable quantzaton speed but perform slower than MR-Hard and MR-soft snce the former two consder the L 1 -norm energy constrant, and the latter two do not. Feature Quantzaton Error: Except for the speed comparson, Table 3 also shows the quantzaton errors. As shown, LASSO-FS and LASSO-LAR acheve the lowest quantzaton error. For the margnal regresson related methods, the energy-based methods (MR-SIS and MLC) acheve lower quantzaton error than those wthout L 1 - norm energy (MR-Hard and MR-Soft) constrant. In addton, the soft-thresholdng-lke methods acheve lower quantzaton error than the hard-thresholdng-lke methods parwsely,.e. MR-Soft aganst MR-Hard, and MLC aganst MR-SIS. Recognton Accuracy: Table 4 shows the recognton accuracy under three dfferent features wth the dctonary sze In ths experment, we select the sparsty constrant E va cross valdaton, and compare the proposed MLC model to other margnal related methods: MR-Hard, MR-Soft and MR-SIS. As shown, the proposed MLC approach obtans the best results, and the MR-Hard method obtans the worst and performs poorly under the HOF feature. In addton, the energy-based methods (MLC and MR-SIS) performs better than those wthout the energy constrant (MR-Hard and MR-Soft), and the soft-thresholdng-type methods have better performance aganst the hard-thresholdng-type methods parwsely (MLC aganst MR-SIS and MR-Soft aganst MR-Hard).

12 Effcent Sparsty Estmaton va Margnal-Lasso Codng 589 Fg. 3. Recognton accuracy under dfferent self-ratos Table 3. Comparson wth the exstng methods for the KTH dataset. The results are shown n terms of speed and quantzaton error. Notce that here we random sampled 1, 258 features from the tranng set. The actual numbers of the local features n the KTH dataset are 261, 946 for the tranng set and 136, 219 for the test set. tme (sec) Reconstructon error Dctonary sze Method HOG-LASSO-FS [20] HOG-LASSO-LAR [4] HOG-MR-Hard HOG-MR-Soft HOG-MR-SIS [6] HOG-MLC HOF-LASSO-FS [20] HOF-LASSO-LAR [4] HOF-MR-Hard HOF-MR-Soft HOF-MR-SIS [6] HOF-MLC HOGHOF-LASSO-FS [20] HOGHOF-LASSO-LAR [4] HOGHOF-MR-Hard HOGHOF-MR-Soft HOGHOF-MR-SIS [6] HOGHOF-MLC

13 590 T.-Y. Hung et al. Table 4. Comparson wth the exstng methods for the KTH dataset Feature-Method KTH HOG-MR-Hard HOG-MR-Soft HOG-MR-SIS [6] HOG-MLC HOF-MR-Hard HOF-MR-Soft HOF-MR-SIS [6] HOF-MLC HOGHOF-MR-Hard HOGHOF-MR-Soft HOGHOF-MR-SIS [6] HOGHOF-MLC Summary: Whle the lasso-lke models (LASSO-FS and LASSO-LAR) acheve the lowest quantzaton error, they take more tme to obtan sparse codes. Instead, the margnal regresson models wthout energy constrant (MR-Hard and MR-Soft) can reach the fastest speed to generate sparse codes; however, they also gan the hghest quantzaton errors. Whle MR-SIS can compete wth our model n terms of the quantzaton error, we reach better recognton accuracy than that of MR-SIS as shown n Table 4 because the proposed approach s a soft-thresholdng method whch s more stable and robust. There are two man dfferences between the MR-SIS model and the proposed MLC model: 1) Whle the dea of MR-SIS s frequently used of the hardthresholdng technque n the applcatons, t has no theoretcal support. We ntegrates margnal regresson and the lasso model wth a theoretcal support whch s a softthresholdng scheme. 2) We estmate the regularzaton parameter for each ndvdual nput feature and gan the lower quantzaton error than that of the MR-SIS model. 5.3 Actvty-Based Human Identfcaton The actvty-based human dentfcaton task ams to recognze the dentty of a person based on hs/her actvtes. We performed the experment on two publcly avalable databases, wezmann [13] and MOBISERV-AIIA [16], respectvely. In the dataset, each clp contans a person perforg one actvty. For each mage frame, we extract human body slhouette whch s smlar to the way n [25,12,16,22,15]. An ntal dctonary s learned va the onlne learnng method proposed n [23] to speed up the learnng process. The wezmann dataset contans 215 clps wth 9 persons perforg 9 dfferent actvtes. The bend actvty s excluded due to lack of samples. We randomly select one clp per acton per class as tranng data, and use the remanng as test data. The MOBISERV-AIIA dataset contans 12 persons perforg eatng and drnkng actvtes wth 2 clothng scenaros n four dfferent days. We adopt 2 knds of the actvtes, drnkng wth a cup and eatng wth a fork, wth 776 clps n total. We randomly choose one-day sequences as test samples and use the rest as the tranng samples.

14 Effcent Sparsty Estmaton va Margnal-Lasso Codng 591 The parameters are detered va cross valdaton. Table 5 shows the recognton accuracy based on the 10% self-rato sparsty constrant. In ths applcaton, the softthresholdng-type methods (MLC and MR-Soft) work better than the hard-thresholdngtype methods (MR-Hard and MR-SIS). In addton, MLC outperforms MR-Soft. Table 5. Comparson wth the exstng methods for the wezmann and the MOBISERV-AIIA datasets backslashboxmethoddataset wezmann MOBISERV-AIIA MR-Hard MR-Soft MR-SIS [6] MLC Concluson In ths paper, we have proposed a novel feature quantzaton approach for sparsty estmaton by explotng the advantages of the margnal regresson and the lasso smultaneously to provde more effcent and effectve solutons. The proposed approach has been evaluated on three vsual applcatons: mage classfcaton, acton recognton and actvty-based human dentfcaton. Expermental results have shown that our method can acheve excellent performance n terms of speed, quantzaton error and recognton accuracy. All these suffcently demonstrate the effcacy of our proposed methods. Acknowledgment. Ths work s supported by the research grant for the Human Cyber Securty Systems (HCSS) Program at the Advanced Dgtal Scences Center from the Agency for Scence, Technology and Research of Sngapore. References 1. Balasubramanan, K., Yu, K., Lebanon, G.: smooth sparse codng va margnal regresson for learnng sparse representatons. In: ICML (2013) 2. Chang, C.C., Ln, C.J.: Lbsvm: A lbrary for support vector machnes. ACM Trans. Intell. Syst. Technol. 2(3), 27:1 27:27 (2011) 3. Donoho, D.L.: For most large underdetered systems of lnear equatons the mal l1- norm soluton s also the sparsest soluton. Comm. Pure Appl. Math. 59, (2004) 4. Efron, B., Haste, T., Johnstone, I., Tbshran, R.: Least angle regresson. Annals of Statstcs (2004) 5. Elhamfar, E., Vdal, R.: Robust classfcaton usng structured sparse representaton. In: CVPR (2011) 6. Fan, J., Lv, J.: Sure ndependence screenng for ultrahgh dmensonal feature space. Journal of the Royal Statstcal Socety: Seres B (Statstcal Methodology) 70(5), (2008) 7. Fredman, J., Haste, T., Hoflng, H., Tbshran, R.: Pathwse coordnate optmzaton (2007) 8. Gao, S., Tsang, I., Cha, L.: Laplacan sparse codng, hypergraph laplacan sparse codng, and applcatons. IEEE Transactons on Pattern Analyss and Machne Intellgence(2012)

15 592 T.-Y. Hung et al. 9. Gao, S., Tsang, I.W.H., Cha, L.T., Zhao, P.: Local features are not lonely - laplacan sparse codng for mage classfcaton. In: 2013 IEEE Conference on Computer Vson and Pattern Recognton (2010) 10. van Gemert, J.C., Geusebroek, J.-M., Veenman, C.J., Smeulders, A.W.M.: Kernel codebooks for scene categorzaton. In: Forsyth, D., Torr, P., Zsserman, A. (eds.) ECCV 2008, Part III. LNCS, vol. 5304, pp Sprnger, Hedelberg (2008) 11. Genovese, C.R., Jn, J., Wasserman, L., Yao, Z.: A comparson of the lasso and margnal regresson. J. Mach. Learn. Res. 13(1), (2012) 12. Gkalels, N., Tefas, A., Ptas, I.: Human dentfcaton from human movements. In: ICIP, pp (2009) 13. Gorelck, L., Blank, M., Shechtman, E., Iran, M., Basr, R.: Acton as space-tme shapes. PAMI 29(12), (2007) 14. Hull, J.: A database for handwrtten text recognton research. PAMI 16(5), (1994) 15. Hung, T.-Y., Lu, J., Tan, Y.-P.: Graph-based sparse codng and embeddng for actvty-based human dentfcaton. In: ICME (2013) 16. Iosfds, A., Tefas, A., Ptas, I.: Actvty based person dentfcaton usng fuzzy representaton and dscrant learnng. TIFS 7(2), (2012) 17. Laptev, I.: On space-tme nterest ponts. Internatonal Journal of Computer Vson 64(2-3), (2005) 18. Laptev, I., Marszalek, M., Schmd, C., Rozenfeld, B.: Learnng realstc human actons from moves. In: CVPR (2008) 19. Lazebnk, S., Schmd, C., Ponce, J.: Beyond bags of features: Spatal pyramd matchng for recognzng natural scene categores. In: CVPR, pp (2006) 20. Lee, H., Battle, A., Rana, R., Ng, A.Y.: Effcent sparse codng algorthms. In: Schölkopf, B., Platt, J., Hoffman, T. (eds.) Advances n Neural Informaton Processng Systems, vol. 19, pp MIT Press, Cambrdge (2007) 21. L, F.F., Perona, P.: A bayesan herarchcal model for learnng natural scene categores. In: CVPR, pp (2005) 22. Lu, J., Hu, J., Zhou, X., Shang, Y.: Actvty-based human dentfcaton usng sparse codng and dscratve metrc learnng. In: ACM Multmeda (2012) 23. Maral, J., Bach, F., Ponce, J., Sapro, G.: Onlne learnng for matrx factorzaton and sparse codng. J. Mach. Learn. Res. 11, (2010) 24. Olva, A., Torralba, A.: Modelng the shape of the scene: A holstc representaton of the spatal envelope. IJCV 42(3), (2001) 25. Sarkar, S., Phllps, P.J., Lu, Z., Vega, I.R., Grother, P., Bowyer, K.W.: The humanid gat challenge problem: data sets, performance, and analyss. PAMI 27(2), (2005) 26. Schuldt, C., Laptev, I., Caputo, B.: Recognzng human actons: A local svm approach. In: ICPR, pp (2004) 27. Tbshran, R.: Regresson shrnkage and selecton va the lasso. Journal of the Royal Statstcal Socety. Seres B (Methodologcal) 58(1), (1996) 28. Wrght, J., Yang, A., Ganesh, A., Sastry, S., Ma, Y.: Robust face recognton va sparse representaton. IEEE Transactons on Pattern Analyss and Machne Intellgence 31(2), (2009) 29. Wu, T., Lange, K.: Coordnate descent algorthms for lasso penalzed regresson. Annals of Appled Statstcs (2008) 30. Yang, J., Yu, K., Gong, Y., Huang, T.: Lnear spatal pyramd matchng usng sparse codng for mage classfcaton. In: IEEE Conference on Computer Vson and Pattern Recognton, CVPR 2009, pp (June 2009) 31. Zheng, M., Bu, J., Chen, C., Wang, C., Zhang, L., Qu, G., Ca, D.: Graph regularzed sparse codng for mage representaton. IEEE Transactons on Image Processng 20(5), (2011)