Robust Dictionary Learning with Capped l 1 -Norm

Transcription

1 Proceedngs of the Twenty-Fourth Internatonal Jont Conference on Artfcal Intellgence (IJCAI 205) Robust Dctonary Learnng wth Capped l -Norm Wenhao Jang, Fepng Ne, Heng Huang Unversty of Texas at Arlngton Arlngton, Texas 7609, USA cswhjang@gmal.com, fepngne@gmal.com, heng@uta.edu Abstract Expressng data vectors as sparse lnear combnatons of bass elements (dctonary) s wdely used n machne learnng, sgnal processng, and statstcs. It has been found that dctonares learned from data are more effectve than off-the-shelf ones. Dctonary learnng has become an mportant tool for computer vson. Tradtonal dctonary learnng methods use quadratc loss functon whch s known senstve to outlers. Hence they could not learn the good dctonares when outlers exst. In ths paper, ag at learnng dctonares resstant to outlers, we proposed capped l -norm based dctonary learnng and an effcent teratve re-weghted algorthm to solve the problem. We provded theoretcal analyss and carred out extensve experments on real word datasets and synthetc datasets to show the effectveness of our method. Introducton Dctonary learnng [Tosc and Frossard, 20] and sparse representaton [Olshausen and Feld, 997] are mportant tools for computer vson. Dctonary learnng seeks to learn an adaptve set of bass elements (dctonary) from data nstead of predefned ones [Mallat, 999], so that each data sample s represented by sparse lnear combnaton of these bass vectors. It has acheved start-of-the-art performance for numerous mage processng tasks such as classfcaton [Rana et al., 2007; Maral et al., 2009b], denosng [Elad and Aharon, 2006], self-taught learnng [Wang et al., 203b], and audo processng [Grosse et al., 2007]. Unlke prncpal component analyss, dctonary learnng does not mpose that the dctonary be orthogonal, hence allow more flexblty to represent data. Usually, dctonary learnng s formulated as: X D C,A DA 2 F, s.t. A 0 γ. () In ths formulaton, X R d n s the data matrx whose columns represent samples, A R K n s the new representatons of data and the matrx D R d K contans K bass Correspondng Author. Ths work was partally supported by NSF IIS-7965, IIS , IIS-34452, DBI (a) Capped l -norm loss (ε = 2.5) (b) l -norm loss Fgure : Capped l -norm loss vs l -norm loss vectors to learn. To prevent the l 2 norm of D beng arbtrarly large whch would lead to arbtrarly small values n the columns of A, t s common to constrant the columns have l 2 norm less than or equal to. Hence, D s n the followng convex set of matrces: C = D R d K,, K}, d 2 }. (2) The dctonary learned s the foundaton for sparse representatons. Dctonary wth good expressve ablty s the key to acheve good performance. It s well known that the quadratc loss functon s not robust to outlers. To tran the dctonary, a large amount data wll be used, and outlers wll be ncluded n the tranng data unavodably. Hence t s necessary to learn dctonary resstant to outlers. To be robust to outlers, the quadratc loss functon could be replaced by an loss functon that are not senstve to outlers, e.g. l norm loss or Huber loss. In ths paper, we use capped l -norm based loss functon l ε (u) = ( u, ε), where ε s a parameter. It s llustrated n Fg.. Ths loss functon treat u equally f u s bgger than ε. Hence, t s more robust to outlers than l norm. Unfortunately, ths loss functon s not convex. In ths paper, we propose an effcent algorthm to fnd the local optmal solutons. And we wll also provde theoretcal analyss of ths method. 2 Related Work In ths secton, we present a bref revew on dctonary learnng methods as follows. Snce the orgnal dctonary learnng model () s NP-hard, the straghtforward way to fnd dctonary s to adopt greedy 3590

2 strategy. K-SVD [Aharon et al., 2006] tred to solve the followng model: D,A 2 X DA 2 2 s.t. d j =, for j =, 2,, K a 0 T 0, for =, 2,, n, (3) where d s the th column of matrx D. The K-SVD s an teratve method that alternates between sparse codng of the examples based on the current dctonary and a process of updatng the dctonary atoms. The columns of dctonary are updated wth SVD sequentally, and the correspondng coeffcents are updated wth any pursut algorthm. The K-SVD algorthm showed good performance on mage denosng. As extensons of K-SVD, LC-KSVD [Jang et al., 203] and dscratve K-SVD [Zhang and L, 200] ntroduced label nformaton nto the procedure of learnng dctonares. Hence the dctonares learned are more dscratve. Except for adoptng greedy strategy to solve problems wth l 0 constrants, the orgnal problem can be relaxed nto the followng tradtonal dctonary learnng problem wth l regularzaton: X D C,A DA 2 F + λ A. (4) It s convex separately wth respect to D and A, a l regularzed least squares problem and a l 2 constraned least squares problem are needed to be solved teratvely to fnd the solutons. In [Lee et al., 2007], an feature-sgn search algorthm for learnng coeffcents, and a Lagrange dual method for learnng the bases are proposed to compute the dctonary and correspondng representatons. An onlne algorthm was proposed to solve t effcently n [Maral et al., 2009a]. The method mentoned above all use quadratc loss functons to measure the reconstructon errors, hence these methods are not robust enough to outlers. In order to learn dctonary robustly, Lu et al. proposed onlne robust dctonary learnng (ORDL) [Lu et al., 203], whch uses l loss functon and l regularzaton term that could be expressed as D,A, d x Da + λ a. (5) It updates dctonary and representatons alternately n an onlne way. For vsual trackng, Wang proposed onlne robust non-negatve dctonary learnng [Wang et al., 203c], whch uses the Huber loss functon. In [Wang et al., 203a], the sem-supervsed robust dctonary learnng model was proposed by solvng the l p -norm based objectve. 3 Proposed New Algorthm To facltate the presentaton of our method, we descrbe the notatons n the followng subsecton. 3. Notatons Throughout ths paper, the followng defntons and notatons are used. We use bold upper letters for matrces, bold lower letters for vectors and regular lower letters for elements. For vector a = (a, a 2,, a m ) T R m, let a = m a be the l -norm of a. Smlarly, we defne A 0 as the number of nonzero elements n matrx A and A = a be the l norm of matrx A. 3.2 Robust dctonary learnng Capped l -norm has been used n [Zhang, 200; 203] as a better approxmaton of l 0 -norm regularzaton. An extenson capped-l, regularzaton was proposed n the mult-task feature learnng settng [Gong et al., 203]. In ths paper, we use capped l -norm as a robust loss functon. For dctonary learnng, our goal s to fnd a set of hgh qualty atoms. An straghtforward way s to use l loss functon. To provde better robustness, we go further to use capped l -norm loss functon. As llustrated n Fg, the objectve values of capped l -norm loss does not ncrease any more f u s larger than ε. Therefore, capped l -norm loss s more robust than l -norm loss. We use the same constrants for dctonary and the objectve functon of our roust dctonary learnng wth capped l -norm s expressed as: D,A, d ( x Da 2, ε) + λ A. (6) From ths objectve functon, we can see that f ε s set as, the above objectve functon becomes l 2, -norm wth the same constrants. We defne f(a) = A, whch s a convex functon. And defne a concave functon L(u) = ( u, ε) where u > 0 and g(d, a) = x Da 2 2. We have: L (u) = 2, f 0 < u < ε2 u 0, f u > ε 2 (7) Our objectve functon s sum of a convex functon and a concave functon. Accordng to the re-weghted method proposed n [Ne et al., 200; 204], t can be solved by updatng D, A and auxlary varables s wth followng updatng rules: [D, A] = arg s x Da λ A, s = D,A, d 2 x Da 2, f x Da 2 ε 0, otherwse The whole teratve re-weghted method s lsted n Alg.. The subproblem (8) s smlar to the tradtonal dctonary learnng. The only dfference les n the fact that samples are not treated equally. It s weghted dctonary learnng. From the updatng rule for s, we can see that the samples wth lower reconstructon errors have hgher weghts, whch s very ntutve for learnng dctonary robustly. The subproblem (8) s convex separately wth respect to D and A. We adopt the same strategy as solvng tradtonal dctonary learnng. We solve t by updatng D and A alternately. Wth A fxed, the objectve functon becomes: D, d (8) (9) s x Da 2 2, (0) 359

3 whch s equvalent to: D, d z Dc 2 2, () where z = s x and c = s a. To solve the above problem, we update D column by column whch s smlar to the tradtonal dctonary learnng. The algorthm we use s adopted from [Maral et al., 2009a], whch s descrbed n Alg. 3. Wth D fxed, the objectve of (8) becomes: A = arg A s x Da λ A, (2) whch can be decomposed nto n ndependent problems as follows: a = arg a s x Da λ a. (3) If s 0, ths problem can be transformed nto the followng LASSO problem: a = arg a x Da λ s a, (4) whch could be solved effcently. The algorthm to solve weghted dctonary learnng s summarzed n Alg. 2. Durng the stage of tranng dctonary D, f s s 0, the correspondng representaton a wll be a zero vector. Hence, when tranng D, our method does not fnd codngs for outlers. However, the outlers n tranng D stage mght not be outlers for classfcaton tasks. Hence, we let the classfcaton algorthm to decde how to use the codngs of the outlers. Wth the learned D, n order to get the codngs for outlers, we just run Alg. wth settng ε as and omt the D updatng step (step n Alg. 2) to learn the representatons for both tranng data and testng data wth the same λ. In fact, we are solvng a dctonary learnng problem wth l 2, -norm loss functon. Algorthm Robust dctonary learnng wth capped l -norm nput: Data matrx X, dctonary sze K, λ and ε. Intalze D, A and s = for =, 2,, n. repeat. Solve subproblem (8) wth algorthm 2 2. Update s for =,, n as (9) untl Converge output: D and A 3.3 Convergence analyss Theorem 3.. The algorthm descrbed n Alg. wll decrease the objectve value of (6) n each teraton untl t converges. Proof. We defne h(w) = w 2, and denote u = x Da 2 2 and L(u) = ( u, ε). The capped l -norm loss functon ( x Da 2, ε) could be re-wrtten as: ( x Da 2, ε) (5) = nf 2) L (s)], s 0 (6) Algorthm 2 Weghted dctonary learnng nput: Data matrx X, ntal dctonary matrx D 0 and representaton matrx A 0, λ and weght vector s. Intalze D = D 0 and A = A 0 repeat. Update D wth algorthm Update a for =,, n as arga x a = Da λ s a f s > 0 0 otherwse untl Converge output: D, A Algorthm 3 Dctonary update nput: Data matrx X, dctonary matrx D, representaton matrx A and weght vector s. Compute matrx G = [g,, g K ] R K K and H = [h,, h K ] R d K as G = s a a T, H = repeat for j = to K do Update the j-th column of D: end for untl Converge output: D s x a T. u j = (h j Dg j ) + d j G jj d j = max( u j 2, ) u j where L (s) s the concave dual of L(u) defned as: L [ ] (s) = nf su L(u). (7) u Plug n L(u) and t s easy to fnd that: L (s) = 4s, f u < ε sε 2 ε, f u ε. (8) Therefore, the capped l -norm loss could be expressed as: ( x Da 2, ε) = nf L x(s, D, a), (9) s 0 where L x (s, D, a) s x Da 2 = + 4s, f x Da 2 < ε s x Da 2 2 sε 2 + ε, f x Da 2 ε. (20) Hence, the objectve functon (6) s equvalent to the followng objectve: nf L x (s, D, a ) + λ a, (2) s 0 D,A, d 3592

4 whch can be re-wrtten as: where we denote: o(d, A, s) = o(d, A, s), (22) D,A, d,s 0 L x (s, D, a ) + λ a. (23) Therefore, the algorthm descrbed n Alg. can be seen as a two stage optmzaton method: D and A are updated by solv- Updatng D and A stage: ng: o(d, A, s), (24) D,A, d whch s equvalent to: s x Da λ a. (25) D,A, d The updated D and A decrease the objectve values of (25). Hence, they also decrease the objectve values of (24). Updatng s stage: solvng: The auxlary varable s are updated by o(d, A, s), (26) s 0 whch s equvalent to n ndependent problems L x (s, D, a ). (27) s 0 Both stages wll decrease the value of the objectve functon (2), hence our algorthm also decrease the value of the orgnal objectve functon (6) and s guaranteed to converge. 4 Expermental Results In ths secton, we analyze and llustrate the performance of our method on real word datasets. 4. Prelary study on natural mage patches Frst, we present a prelary analyss of our method on natural mage patches. In our method, λ manly controls the sparsty of representatons and ε manly controls the number of outlers dentfed by the algorthm. ε s related to the resduals of representatons. If the resdual of a sample s larger than ε, t s not used to tran the dctonary, snce the correspondng s s zero. We set ε as nfnty frst to get the dstrbuton of the resduals. We use ths dstrbuton as an estmaton of the dstrbuton of resduals wth optmal D and A. We set ε such that a fracton of samples are seen as outlers. In ths experment, we set the fracton of outlers as 0.05 and λ = 0. emprcally. The values of objectve functon durng teratons are presented n Fg. 2. We can see that our method converged n only 22 teratons. The bases learned on natural mage patches by our method s shown n Fg. 3. objectve value teraton Fgure 2: The values of objectve functon durng teratons when learnng dctonares on 0,000 natural mages patches. Fgure 3: Learned,024 bases mages bases (each 4 4 pxels) on 0,000 natural mages patches. 4.2 Face recognton wthout occluson In ths subsecton, we provde the expermental results on face recognton tasks wthout synthetc occluson on extended Yale B dataset [Georghades et al., 200] and AR face dataset [Martnez, 998]. Extended Yale B dataset The extended Yale B database [Georghades et al., 200] contans 2,44 mages of 38 human frontal faces under 64 lluaton condtons and expressons. There are about 64 mages for each person. The orgnal mages were cropped to pxels. Some samples are shown n Fg. 4(a). We do not perform any preprocess on the mages. Followng [Wrght et al., 2009], we project each face mage nto a 504-dmensonal feature vector usng a random matrx. We splt the database randomly nto two halves. One half whch contans about 32 mages for each person was used for tranng the dctonary. The other half was used for testng. 3593

5 (a) Sample mages from extended Yale B dataset Table : Average classfcaton accuraces(%) on extended Yale B dataset. Method Accuracy (%) Capped Norm 96.9 Tradtonal DL KSVD LC-KSVD 93.6 LC-KSVD D-KSVD ORDL 89.7 (b) Sample mages from AR face dataset Fgure 4: Sample mages In the practcal mplementaton of our method, we frst set ε as nfnty and choose the best λ wth cross valdaton. Then we fx λ and choose ε. For face recognton, we compared our method wth a few start-of-the-art methods, ncludng tradtonal dctonary learnng [Maral et al., 2009a], K-SVD [Aharon et al., 2006], LC-KSVD and LC-KSVD2 [Jang et al., 203], D-KSVD [Zhang and L, 200] and ORDL [Lu et al., 203]. The dctonary sze s 570 for all methods, whch means 5 tems for person on average. The parameters for these methods were selected by cross valdaton. We ran all methods on 0 dfferent splts of tranng and testng set. The results are summarzed n Table. We can see that our method acheved the best performance. The reason that ORDL dd not perform well mght be the dctonary and representatons traned wth l -norm loss functon are not sutable for classfcaton tasks. We show that our method s robust when tranng dctonary. Recall that, n our method s = 0 means that the th sample s recognzed as outler and not used n tranng the dctonary. In runnng on one random splt, our method found 20 outlers, whch are all shown n Fg. 5. We can see that most of the outlers found by our method are wth extreme lluaton, whch wll effect the qualty of bases. AR face dataset The AR face dataset [Martnez, 998] conssts of over 4,000 color mages from 26 ndvduals. For each ndvdual, 26 pctures were taken n two separate sessons. Compared to the extended Yale B dataset, as shown Fgure 5: The outlers found by our method when tranng dctonary on an randomly selected tranng set from extended Yale B dataset. n Fg. 4(b), the AR face dataset ncludes more facal varatons, ncludng dfferent lluaton condtons, dfferent expressons, and dfferent facal dsguses (sunglasses and scarves). Followng the standard evaluaton procedure from [Wrght et al., 2009], we used a subset of the database consstng of 2,600 mages from 50 male subjects and 50 female subjects. For each person, 20 mages were randomly selected for tranng and the remanng mages are for testng. Each face mage s cropped to and then projected nto a 540-dmensonal feature vector. Smlar to the experments on Yale B dataset, we also ran all methods n 0 dfferent splts of tranng and testng set. The results are presented n Table 2. We can see that our method acheved the best performance among all dctonary learnng algorthms. We show the 7 outlers found by our algorthm n Fg 6. We can see that these outlers are faces wth glasses or scarves. Our algorthm dentfed these faces as outlers that wll hurt the qualty of dctonary and dd not use them n the tranng process. 4.3 Face recognton wth occluson In order to study the property of robustness to outlers, we carred out experments on extended Yale B dataset wth syn- We use grayscale mages n our experments. 3594

6 Table 2: Average classfcaton accuraces(%) on AR face dataset. Method Accuracy (%) Capped Norm Tradtonal DL KSVD LC-KSVD LC-KSVD D-KSVD 88.8 ORDL 9.72 Fgure 7: Sample faces wth occluson from extended Yale B dataset. Table 3: Average classfcaton accuraces(%) on extended Yale B dataset wth block occluson of dfferent levels. Method 0% 20% 30% 40% Capped Norm Tradtonal DL KSVD LC-KSVD LC-KSVD D-KSVD ORDL Fgure 6: The 7 outlers found by our method when tranng dctonary on an randomly selected tranng set from AR face dataset. and are dffcult to recognze. In our method, the dctonares were traned wthout these samples, hence they were better than dctonares traned n other ways, whch can be proved from the comparsons of performances n Table 3. thetc outlers. If the samples for tranng the dctonary contan outlers, tradtonal dctonary learnng wll try to fnd dctonary that could express these outlers. But t s of no use to express the outlers, hence the qualty of dctonary wll be effected for tradtonal dctonary learnng. In ths experment, we generate outlers by addng block occluson. Other ways of generatng outlers wll also lead to the smlar phenomena descrbed below. We selected a fracton of mages (outler rato) from tranng set, then we added block occluson of sze nto these mages at random postons. Some samples are shown n Fg. 7. We set outler rato as 0%, 20%, 30% and 40% to get 4 dfferent datasets wth synthetc outlers n the tranng sets. The mages from testng sets are not added any occlusons. Under ths settng, the effect on the performance of classfcaton wll come from the qualty of dctonares. Smlar to the experments we have done above, we also ran experments on 0 dfferent datasets and the average accuraces are reported n Table 3. We can see that the classfcaton accuraces of all methods decreased as the outler rato ncreased. Compared wth performances wthout block occluson n Table, the performance of tradtonal dctonary learnng dropped a lot (from to wth 0% outlers ). But our method dd not drop so much (from 96.9 to ). And our method also performed the best on all datasets. Hence we can conclude that our capped l -norm loss does provde resstance to such knd of nose. The outlers found by our algorthm from an synthetc dataset wth 0% corrupton are shown n Fg. 8. We can see that most outlers are mages that are too dark or corrupted Fgure 8: The 93 outlers found by our method when tranng dctonary on an randomly selected tranng set wth 0% samples corrupted wth block occluson from extended Yale B dataset. 5 Concluson In ths paper, we presented a robust dctonary learnng model based on capped l -norm and an effcent algorthm to fnd local solutons. One mportant advantage of our method s ts robustness to outlers. By teratvely assgnng weghts to samples, our algorthm succeeded n fndng outlers and reduced ther effects on tranng the dctonares. The proposed method was extensvely evaluated on face recognton jobs on dfferent real word datasets and synthetc datasets. The expermental results demonstrated that our method outperforms prevous state-of-the-art dctonary learnng methods. 3595

7 References [Aharon et al., 2006] Mchal Aharon, Mchael Elad, and Alfred Brucksten. K-svd: An algorthm for desgnng overcomplete dctonares for sparse representaton. IEEE Transactons on Sgnal Processng, 54(): , [Elad and Aharon, 2006] Mchael Elad and Mchal Aharon. Image denosng va sparse and redundant representatons over learned dctonares. IEEE Transactons on Image Processng, 5(2): , [Georghades et al., 200] A.S. Georghades, P.N. Belhumeur, and D.J. Kregman. From few to many: Illuaton cone models for face recognton under varable lghtng and pose. IEEE Transactons on Pattern Analyss and Machne Intellgence, 23(6): , 200. [Gong et al., 203] Pnghua Gong, Jepng Ye, and Changshu Zhang. Mult-stage mult-task feature learnng. The Journal of Machne Learnng Research, 4(): , 203. [Grosse et al., 2007] Roger Grosse, Rajat Rana, Helen Kwong, and Andrew Y. Ng. Shft-Invarant Sparse Codng for Audo Classfcaton. In Uncertanty n Artfcal Intellgence, [Jang et al., 203] Zhuoln Jang, Zhe Ln, and Larry S Davs. Label consstent k-svd: learnng a dscratve dctonary for recognton. IEEE Transactons on Pattern Analyss and Machne Intellgence, 35(): , 203. [Lee et al., 2007] Honglak Lee, Alexs Battle, Rajat Rana, and Andrew Y. Ng. Effcent sparse codng algorthms. In B. Schölkopf, J. Platt, and T. Hoffman, edtors, Advances n Neural Informaton Processng Systems 9, pages MIT Press, Cambrdge, MA, [Lu et al., 203] Cewu Lu, Japng Sh, and Jaya Ja. Onlne robust dctonary learnng. In IEEE Conference on Computer Vson and Pattern Recognton (CVPR) 203, pages , 203. [Maral et al., 2009a] Julen Maral, Francs Bach, Jean Ponce, and Gullermo Sapro. Onlne dctonary learnng for sparse codng. In Proceedngs of the 26th Internatonal Conference on Machne Learnng, pages , Montreal, June Omnpress. [Maral et al., 2009b] Julen Maral, Francs Bach, Jean Ponce, Gullermo Sapro, and Andrew Zsserman. Supervsed dctonary learnng. In D. Koller, D. Schuurmans, Y. Bengo, and L. Bottou, edtors, Advances n Neural Informaton Processng Systems 2, pages [Mallat, 999] Stéphane Mallat. A wavelet tour of sgnal processng. Academc press, 999. [Martnez, 998] Alex M Martnez. The ar face database. CVC Techncal Report, 24, 998. [Ne et al., 200] Fepng Ne, Heng Huang, Xao Ca, and Chrs H Dng. Effcent and robust feature selecton va jont 2, -norms mzaton. In Advances n Neural Informaton Processng Systems, pages 83 82, 200. [Ne et al., 204] Fepng Ne, Janjun Yuan, and Heng Huang. Optmal mean robust prncpal component analyss. In Proceedngs of the 3st Internatonal Conference on Machne Learnng (ICML-4), pages , 204. [Olshausen and Feld, 997] Bruno A Olshausen and Davd J Feld. Sparse codng wth an overcomplete bass set: A strategy employed by v? Vson research, 37(23): , 997. [Rana et al., 2007] Rajat Rana, Alexs Battle, Honglak Lee, Benja Packer, and Andrew Y Ng. Self-taught learnng: transfer learnng from unlabeled data. In Proceedngs of the 24th nternatonal conference on Machne learnng, pages ACM, [Tosc and Frossard, 20] I. Tosc and P. Frossard. Dctonary learnng. IEEE Sgnal Processng Magazne, 28(2):27 38, March 20. [Wang et al., 203a] Hua Wang, Fepng Ne, Wedong Ca, and Heng Huang. Sem-supervsed robust dctonary learnng va effcent l 2,0+ -norms mzaton. Internatonal Conference on Computer Vson (ICCV 203), pages 45 52, 203. [Wang et al., 203b] Hua Wang, Fepng Ne, and Heng Huang. Robust and dscratve self-taught learnng. The 30th Internatonal Conference on Machne Learnng (ICML 203), Journal of Machne Learnng Research, W&CP, 28(3): , 203. [Wang et al., 203c] Nayan Wang, Jngdong Wang, and Dt- Yan Yeung. Onlne robust non-negatve dctonary learnng for vsual trackng. In IEEE Internatonal Conference on Computer Vson, ICCV 203, Sydney, Australa, December -8, 203, pages , 203. [Wrght et al., 2009] J. Wrght, A.Y. Yang, A. Ganesh, S.S. Sastry, and Y Ma. Robust face recognton va sparse representaton. IEEE Transactons onpattern Analyss and Machne Intellgence, 3(2):20 227, [Zhang and L, 200] Qang Zhang and Baoxn L. Dscratve k-svd for dctonary learnng n face recognton. In IEEE Conference on Computer Vson and Pattern Recognton (CVPR) 203, pages IEEE, 200. [Zhang, 200] Tong Zhang. Analyss of mult-stage convex relaxaton for sparse regularzaton. The Journal of Machne Learnng Research, :08 07, 200. [Zhang, 203] Tong Zhang. Mult-stage convex relaxaton for feature selecton. Bernoull, 9(5B): ,