Robust and Effective Metric Learning Using Capped Trace Norm

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1 Robust and Effectve Metrc Learnng Usng Capped Trace Norm Zhouyuan Huo Department of Computer Scence and Engneerng Unversty of Texas at Arlngton Texas, USA Fepng Ne Department of Computer Scence and Engneerng Unversty of Texas at Arlngton Texas, USA Heng Huang Department of Computer Scence and Engneerng Unversty of Texas at Arlngton Texas, USA ABSTRACT Metrc learnng ams at automatcally learnng a metrc from par or trplet based constrants n data, and t can be potentally benefcal whenever the noton of metrc between nstances plays a nontrval role. In Mahalanobs dstance metrc learnng, dstance matrx M s n symmetrc postve sem-defnte cone, and n order to avod overfttng and to learn a better Mahalanobs dstance from weakly supervsed constrants, the low-rank regularzaton has been often mposed on matrx M to learn the correlatons between features and samples. As the approxmatons of the rank mnmzaton functon, the trace norm and Fantope have been utlzed to regularze the metrc learnng objectves and acheve good performance. However, these low-rank regularzaton models are ether not tght enough to approxmate rank mnmzaton or tme-consumng to tune an optmal rank. In ths paper, we ntroduce a novel metrc learnng model usng the capped trace norm based regularzaton, whch uses a sngular value threshold to constrant the metrc matrx M as low-rank explctly such that the rank of matrx M s stable when the large sngular values vary. The capped trace norm regularzaton can also be vewed as the adaptve Fantope regularzaton. We mnmze sngular values whch are less than threshold value and the rank of M s not necessary to be k, thus our method s more stable and applcable n practce when we do not know the optmal rank of matrx M. We derve an effcent optmzaton algorthm to solve the proposed new model and the algorthm convergence proof s also provded n ths paper. We evaluate our method on a varety of challengng benchmarks, such as LFW and Pubfg datasets. Face verfcaton experments are performed and results show that our method consstently outperforms the state-of-the-art metrc learnng algorthms. To whom all correspondence should be addressed. Ths work was partally supported by US NSF-IIS , NSF-IIS , NSF-IIS , NSF-DBI , NIH R01 AG Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. Copyrghts for components of ths work owned by others than ACM must be honored. Abstractng wth credt s permtted. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Request permssons from permssons@acm.org. KDD 16, August 13 17, 2016, San Francsco, CA, USA. c 2016 ACM. ISBN /16/08... $15.00 DOI: Fgure 1: An llustraton of metrc learnng on Outdoor Scene Recognton OSR dataset. CCS Concepts Informaton systems Data mnng; Computng methodologes Machne learnng; Keywords Metrc Learnng; Capped Trace Norm; Low-Rank Approxmaton; Face Verfcaton 1. INTRODUCTION Metrc learnng has shown promsng results wth learnng the proper Mahalanobs dstance for many data mnng tasks.the goal of metrc learnng s to learn an optmal lnear or nonlnear projecton for orgnal hgh-dmenson features from supervsed or weakly supervsed constrants, and there have been a lot of works n ths feld [28, 1, 26, 9, 27, 24]. Metrc learnng has been wdely used n applcatons where metrc between samples plays an mportant role, such as mage classfcaton, face verfcaton and recognton n computer vson [6, 17, 13, 14], learnng to rank n nformaton retreval [16, 15], bonformatcs [25], etc. In mage mnng and retreval, there are many metrc learnng algorthms learnng an optmal Mahalanobs dstance from weakly supervsed constrants between mages. The man constrant paradgms nclude: par constrant [1], trplet constrant [27], and quadruplet constrant [13]. To avod overfttng and learn correlaton among samples, many regularzatons were proposed to mpose on the projecton matrx. Among these regularzatons, the low-rank regularzaton s proved to be effectve and effcent to learn potental correlatons from tranng data, e.g. trace norm and Fantope regularzaton[14]. In ths paper, we propose a novel metrc learnng model usng the capped trace norm as the low-rank regularzaton for Mahalanobs

2 dstance metrc learnng. Dfferent from trace norm whch mnmzes sum of all sngular values, or Fantope regularzaton whch mnmzes sum of k smallest sngular values, the capped trace norm penalzes the sngular values that are less than a threshold adaptvely learned n the optmzaton. As a result, the non-relevant nformaton assocated to smallest sngular values can be fltered out, such that the metrc learnng model s more robust. Meanwhle, we only need nput an approxmate rank value, thus our regularzaton term s tghter than trace norm and more stable and applcable n practcal problems. We also derve an effcent optmzaton algorthm wth rgorous convergence analyss. In the experments, we mpose our novel low-rank regularzaton on dfferent metrc learnng formulatons and compare wth other the state-ofthe-art metrc learnng methods. Expermental results show that our method outperforms other related methods on benchmark datasets. 2. RELATED WORK The goal of metrc learnng s to learn an adaptve dstance, such as Mahalanobs dstance d M x, x j = x x j T Mx x j, for the problem of nterest usng the nformaton brought by tranng examples. Most of metrc learnng methods use weakly-supervsed constrants. There are three manly paradgms of constrants, such as parwse, trplet, or quadruplet constrant. The parwse constrant contans nformaton whether two objects n a par are smlar or dssmlar, sometmes postve pars or negatve pars. Parwse constrant s represented by D and S as: S = {x, x j : x and x jare smlar} D = {x, x j : x and x jare dssmlar}. 1 The nformaton-theoretc Metrc Learnng ITML s one of many methods usng parwse constrant tranng examples n metrc learnng feld [28, 1, 11, 5], and t s formulated as follows: mn M S d s.t. γ,j ξ j D ld M, M 0 d 2 M x, x j u ξ j, x, x j S d 2 M x, x j l ξ j, x, x j D, 2 where u and l are upper bound and lower bound for smlar samples and dssmlar samples respectvely. ξ j s a safety margn dstance for each par and D ld M, M 0 s LogDet dvergence and D ld M, M 0 = TrMM 1 0 logdetmm 1 0 d where d s the dmenson of nput space and M 0 s a postve defnte matrx. Trplet constrant s also wdely used n metrc learnng, and t s denoted by R as: R = {x, x j, x k : x s more smlar to x j than to x k }. 3 Large Margn Nearest Neghbor LMNN [27] s one of the most wdely used metrc learnng methods whch uses trplet constrant on tranng examples. The LMNN model s to solve: mn M S d 1 µ x,x j S d 2 M x, x j µ,j,k R ξ jk 4 s.t. d 2 M x, x k d 2 M x, x j 1 ξ jk x, x j, x k R, where µ [0, 1] controls relatve weght between two terms, and S = {x, x j : y = y j and x j belongs to the k-neghborhood of x } R = {x, x j, x k : x, x j S, y y k }. 5 It s proved to be very effectve to learn Mahalanobs dstance n practce, and s extended to many methods for dfferent applcatons [20, 10, 9]. However, LMNN s prone to be overfttng sometmes, and t s also senstve to Eucldean dstance when t computes neghbors of each sample at the begnnng. In [13], a novel quadruplet constrant was proposed to model smlarty from complex semantc label relatons, for example, the degree of presence of smle, from least smlng to most smlng. The scheme of quadruplet constrant s as follows: A = {x, x j, x k, x l : d 2 M x k, x l d 2 M x, x j δ}, 6 where δ s a soft margn. Quadruplet s able to encompass par and trplet constrant. Parwse constrant can be represented as x, x, x, x j, and set δ = l, so d 2 M x, x j l and x, x j are from dssmlar set; or x, x j, x, x, set δ = u, then d 2 M x, x j u, x, x j are from smlar set. Smlarly, trplet constrant can also be represented as x, x j, x, x k. To solve the problem of overfttng, many regularzatons over matrx M were proposed n the past. In [22], they mpose squared Frobenus norm on M, and form an SVM lke structure to do metrc learnng: mn W M 2 F C,j,k ξ jk 7 s.t. d 2 M x, x k d 2 M x, x j 1 ξ jk, x, x j, x k R, where M = A T W A, matrx A s fxed and known, and the dagonal matrx W s learned. There are some works mposng low-rank structure on M. The most drect way s let M = L T L, where M R d d, L R k d and k s smaller than d. So, M s a matrx of rank k. In [16], they proposed a robust structure metrc learnng method, and used nuclear norm as convex approxmaton of low-rank regularzaton, and t can be expressed as a convex optmzaton problem: mn M S d TrM C n q X s.t. q X, y Y : ξ q M, φq, y q φq, y F y q, y ξ q, 8 where X R d s the tranng set of n ponts, Y s the set of all permutatons over X, C > 0 s a slack trade-off parameter, φ s a feature encodng of an nput-output par, and y q, y s the desgned margn. In [14], a tghter rank mnmzaton approxmaton, Fantope regularzaton, was proposed and mposed on M, and holds an explct control over rank of M. The formulaton s RegM = k σ M, where σ M are k smallest sngular values. 3. METRIC LEARNING USING CAPPED TRACE NORM In ths paper, we are gong to ntroduce a novel low-rank regularzaton based metrc learnng method, so that we can avod the problem of overfttng and learn an effectve structure from lmted tranng data. As we mentoned n last secton, there have been already many dfferent types of regularzaton over M S d n metrc learnng lterature. The Frobenus norm regularzaton proposed by [22] can avod overfttng, however, the defnton of M n ths paper restrcts the generaton of M and t cannot learn correlaton between features. In weakly supervsed metrc learnng, the algorthm does not have access to the labels of tranng data, and t s only provded wth sde nformaton whch s n the form of par or trplet constrants. In ths case, the low-rank regularzaton seems to be an

3 effectve way to learn correlatons between data. Trace norm also called as nuclear norm has been used as the convex relaxaton of the rank mnmzaton, however, there stll s a gap between trace norm and rank mnmzaton. Because trace norm s the sum of all sngular values, f one of the large sngular values changes, the trace norm wll also change correspondngly, but the rank of the orgnal matrx keeps constant. Settng M = L T L or mposng Fantope regularzaton are both explct way to control the rank of matrx M. The performance could be good f we can fnd a good ftted rank. However, n practce, we do not know the rank of a matrx accurately, and we have to tune ths parameter very carefully, because a small devaton of parameter k from optmal value may have large nfluence on the fnal performance. It s a tedous process to select the best k from a large range. In ths paper, we wll use the capped trace norm as low-rank regularzaton [29, 30]. It can be represented as RegM = where ε s a threshold value. In ths regularzaton, we only mnmze the sngular values that are smaller than ε, and we gnore other large sngular values. Thus, when large sngular values vary, our regularzaton behaves the same as low-rank regularzaton, and keeps constant too. In practcal problems, t s dffcult to estmate the rank of matrx M, but the ε value n capped trace norm can be easly decded [4, 8]. Because quadruplet constrant can encompass par and trplet constrants, n ths paper, we use quadruplet constrant to form a new robust metrc learnng model as: mn M S d [ ξ q M, x jx T j x kl x T kl ] γ mn{σ M, ε}. 9 2 We wll also use ths model n the optmzaton and convergence analyss sectons, but the conclusons are the same when we use par or trplet constrant. 3.1 Optmzaton Algorthm Objectve functon 9 s non-smooth and non-convex, and t s hard to optmze. In ths secton, at frst, we wll use re-weghted method to transform the orgnal objectve functon to a convex subproblem, then proxmal gradent method s appled to solve ths new subproblem. In next secton, we wll prove that our objectve functon wll converge, and the values of orgnal objectve functon 9 are non-ncreasng after each step, and a local optmum value s to be obtaned. Accordng to the re-weghted algorthm descrbed n [19, 18], let M = UΣV T and sngular value σ are n ascendng order. We defne: D = 1 2 u u T. 10 Therefore, orgnal problem 9 can be transformed to: [ ] mn ξ q M, x jx T M S d j x kl x T kl γ 2 TrM T DM. 11 of problem 11 wth respect to M s: M = Input: A, X R d n mn{σ M, ε}, Output: M S d whle not converge do Update D 1 k 2 σ 1 u u T. Update M Π S d M η M x jx T j x kl x T kl γdm 12 where A denotes the subset of constrants n A that s larger than 0 n functon 11. After each step, M s projected onto the postve semdefnte cone. M = Π S d M η M. 13 Optmzaton algorthm of our method s summarzed n Algorthm 1 below. Algorthm 1 Algorthm to solve problem 9. M = x jx T j x kl x T kl γdm end whle 3.2 Convergence Analyss Usng the algorthm above, we can solve our orgnal non-smooth and non-convex objectve functon 9. In ths secton, we prove the convergence of our optmzaton algorthm, and a local soluton can be obtaned n the end. In capped trace norm optmzaton algorthm, the number of thresholded sngular values vares for dfferent teratons, thus the convergence proof s dffcult. THEOREM 1. Through Algorthm 1, the objectve functon 9 wll converge, or the values of objectve functon 9 are non-ncreasng monotoncally. In order to prove Theorem 1, at frst, we need the followng Lemmas. LEMMA 1. Accordng to [23], any two hermtan matrces A, B R d d satsfy the nequalty σ A, σ B are sngular values sorted n the same order d σ A σ d 1 B Tr A T B d σ A σ B. 14 LEMMA 2. Let M = UΣV T, σ are sngular values of M n ascendng order, and there are k sngular values less than ε. ˆM = Û ˆΣ ˆV T, ˆσ are sngular values of ˆM n ascendng order, and there are ˆk sngular values less than ε. ˆM denotes the updated parameter after M. ε s a constant value. So t s true that, mn{ˆσ ˆM, ε} Tr ˆM T D ˆM where D s defned as 10. mn{σ M, ε} Tr M T DM, 15 When we fx D, ths problem s convex, and we use proxmal gradent method to optmze t teratvely. In each teraton, M s updated by performng a subgradent descent, and the subgradent descent Proof: It s obvous that σ 2 ˆσ ˆσ 2 = 1 σ σ ˆσ 2 0, 16

4 hence the followng nequalty holds: ˆσ 1 2 σ 1 ˆσ σ. 17 We know there are ˆk sngular values of ˆM less than ε and they are n ascendng order, the frst ˆk smallest sngular values ˆσ are less than ε, thus no matter whether ˆk k or ˆk < k, t holds that: ˆ ˆσ ˆkε ˆσ kε. 18 Combnng two nequaltes 17 and 18, we get: ˆ ˆσ 1 2 ˆσ 2 ˆkε 1 2 σ kε. 19 Then, summng dε on both sdes of nequalty 19, where d s the dmenson of matrx M, we are able to get the followng nequalty: that: ˆ ˆσ d ˆk ε 1 2 k σ d k ε 1 2 As per the defnton of matrx D = 1 2 Tr M T DM ˆσ 2 σ. 20 u u T, we know = 1 2 Tr u u T MM T = 1 2 Tr UΛU T UΣ 2 U T = 1 2 Tr UΛΣ 2 U T = 1 2 σ. 21 Va Lemma 1, we know that: 1 2 Tr u u T ˆM ˆM T = 1 2 Tr UΛU T Û ˆΣ 2 Û T 1 2 ˆσ Combnng Eq. 21, nequaltes 20 and 22, we have: ˆ ˆσ d ˆk ε 1 2 Tr σ d k ε 1 2 Tr u u T ˆM ˆM T u u T MM T. Fnally, nequalty holds that: mn{ˆσ ˆM, ε} Tr ˆM T D ˆM mn{σ M, ε} Tr M T DM LEMMA 3. Functon 11, s convex wth doman S d, and gradent of ths functon s clearly Lpschtz contnuous wth a large enough constant L. Secondly, postve semdefnte cone s a closed convex cone. As per [2], when we use proxmal gradent method and select a proper learnng rate, the value of ths functon converges n each teraton. Rght now, we are able to prove Theorem 1 by usng the Lemmas above. Proof: Va Lemma 3, after we use proxmal gradent descent method to mnmze functon 11 n Algorthm 1, t s guaranteed that: [ ] ξ q ˆM, xjx T j x kl x T kl γ 2 Tr ˆM T D ˆM [ ξ q ] M, x jx T j x kl x T kl γ M 2 Tr T DM Va Lemma 2, we can easly know that: γ mn{ˆσ ˆM, ε} Tr ˆM T D ˆM γ mn{σ M, ε} Tr M T DM. Fnally, we combne nequaltes 25 and 26 to acheve: [ ] ξ q ˆM, xjx T j x kl x T kl γ mn{ˆσ 2 ˆM, ε} [ ] ξ q M, x jx T j x kl x T kl γ mn{σ M, ε} So far, t s clear that the value of our proposed objectve functon wll not ncrease by usng our optmzaton algorthm, thus we prove the Theorem 1 that our optmzaton algorthm s nonncreasng monotoncally. Because M s a postve semdefnte matrx, we know that the objectve functon 9 s at least larger than zero. So our objectve functon s also lower bounded. Therefore we can conclude that our optmzaton algorthm converges, and a local optmum value s to be obtaned n the end. 4. EXPERIMENTAL RESULTS We evaluate our proposed model on dfferent datasets, ncludng synthetc dataset, wdely used face recognton datasets, and some other datasets n mage data mnng. There are two man goals n our experment: frst, we wll show that our model s able to outperform the state-of-the-art metrc learnng methods; second, our proposed capped trace norm s more stable to be appled to solve practcal problems than Fantope regularzaton. 4.1 Synthetc Data In ths experment, we evaluate our proposed metrc learnng on a synthetc dataset, and each constrant s quadruplet. Dataset: We follow the settng of experment n [14] to generate a synthetc dataset. We defne a target symmetrc postve semdefnte matrx T S d, and T =, where A S A e s a random symmetrc postve defnte matrx wth ranka = e and e < d. Matrx A s a multplcaton of one random symmetrc

5 matrx and ts transpose. So, rankt = ranka = e. X R d n s a feature matrx, each element s generated from gaussan dstrbuton n [0, 1], and each sample s a feature vector x R d. The Mahalanobs dstance between two feature vectors x and x j s gven by: d 2 T x, x j = x x j T T x x j. To buld a tranng constrant set A, we randomly sample pars of dstance usng quadruplets and get the ground truth usng d 2 T, so that: x, x j, x k, x l A, d 2 T x, x j < d 2 T x k, x l, and t denotes that dstance between sample par, j s smaller than sample par k, l. Tranng set A s used as tranng data to learn Mahalanobs metrc matrx M. Valdaton set V and test set T are generated n the same way as A, and they are used to tune parameters and test evaluaton respectvely. Settng: In the experment, we set e = 10, d = 100, n = 10 6, A = V = T = After we learn a metrc M, we evaluate these metrcs on test set T, and measure accuracy of satsfyng the constrants. In ths experment, we compare wth four other methods: metrc learnng wth no regularzaton, metrc learnng wth trace norm regularzaton and metrc learnng wth fantope regularzaton. Parameter γ are tuned n the range of {10 2, 10 1, 1, 10, 10 2 }, and rank of Mahalanobs metrc M are tuned from [5, 20]. Compared Methods: Because we use quadruplet constrants n ths experment, the general model we use to solve ths problem s: [ ] ξ q M, x jx T j x kl x T kl γ RegM Metrc learnng wth no regularzaton. We set γ = 0 n problem 28. Metrc learnng wth trace norm regularzaton. In ths model, γ > 0 and RegM = σ M, where σ M are sngular values of M. Metrc learnng wth Fantope regularzaton. In ths model, γ > 0 and RegM = k smallest sngular values of M. σ M, where σ M are k Metrc learnng wth capped trace norm regularzaton. In our model, γ > 0 and RegM = mn{σ M, ε}, where σ M are sngular values of M. Evaluaton Metrcs: After learnng a metrc matrx M from tranng constrants A, we evaluate t by computng the number of domnant sngular values, namely rankm. Then we test t on testng constrants T and compute the accuracy of satsfed constrants. Results: Table 1 shows the results of our experment. As we can see, metrc learnng wth Fantope regularzaton and capped norm regularzaton performs much better than other two methods. Our method can get a comparable results to metrc learnng wth Fantope regularzaton, and ths s the result when we tune parameter k for Fantope regularzaton very carefully. Fgure 2 represents the accuraces of metrc learnng wth Fantope regularzaton and our method when selecton of rank changes. It s obvous that our method always outperforms Fantope regularzaton when we select parameter rank randomly except 10. Our method performs more stable when we do not have enough tme to tune the rank of Mahalanobs metrc M. In practce, t common that we do not know the exact rank of a matrx, our method s more applcable to solve practcal problems. Method Accuracy rankm ML 85.62% 53 ML Trace 88.44% 41 ML Fantope 95.50% 10 ML capped 95.43% Table 1: Synthetc experment results. MLTrace MLFantope Fgure 2: Accuracy vs the number of rank k 4.2 Face Verfcaton In ths secton, we evaluate our method, parwse constrants wth capped norm regularzaton, on two challengng face recognton datasets: Labeled Faces n the Wld LFW [7] and Publc Fgures Face Database PubFg [12]. In our experments, we focus on face verfcaton task, namely decdng f two face mages are from the same person, and results show that our method can outperform the state-of-the-art metrc learnng algorthm Labeled Faces n The Wld Dataset: The Labeled Faces n the Wld dataset s consdered as the current state-of-the-art face recognton benchmark. It contans 13,233 unconstraned face mages of 5749 ndvduals, and 1680 of these pctured people appear n two or more dstnct photos n ths data set. There are two dfferent feature representatons n ths experment, LFW SIFT feature dataset and LFW Attrbute feature dataset. We use the face representaton proposed by [5], t extracts SIFT descrptors [6] at 9 automatcally detected facal landmarks over three scales. Each mage s a feature vector of sze 3,456. To make ths dataset tractable for dstance metrc learnng algorthm and save tme, we perform prncple component analyss to reduce the dmenson, and we select 100 largest prncple components n ths experment [11]. We also use hgh-level descrbable vsual attrbutes gender, race, age, har color n [12]. These features of face mage are nsenstve to pose, llumnaton, expresson and other magng condtons, and can avod some obvous mstakes, for example men are confused for women or chld for mddle-aged. Each mage s represented by a vector x R d where d s the number of attrbutes to descrbe the mage. Each entry n vector x means the score of presence of each specfc attrbute. Settng: To compare wth other state-of-the-art methods on face verfcaton, we follow the experment settng n [11]. Data are organzed n 10 folds, and each fold conssts of 300 smlar constrants two faces n a par are from the same person and 300 dssmlar constrants two faces n a par are from dfferent person. The average over results of 10 folds s used as fnal evaluaton metrc. In the experment, we an only access parwse constrants gven

6 by smlar or dssmlar pars, and labels or more tranng data are not allowed. In the experment, we tune parameter µ from range [10 2, 10 1, 1, 10, 102 ], and parameter rank k of matrx M from {30, 35, 40, 45, 50, 55, 60, 65, 70} n LFW SIFT dataset and from {10, 15, 20, 25, 30, 35, 40, 45, 50} n LFW Attrbute dataset. For other methods, we follow ther already tuned parameter settng n [11]. To prove the stableness of our method, we splt LFW SIFT feature dataset and LFW attrbute feature dataset nto 5 folds, and evaluate our metrc learnng wth capped trace norm model and metrc learnng wth Fantope norm on these datasets. We run experments 5 tmes usng dfferent tranng/testng splts and compute mean value and standard varance for valdaton. Compared Methods: IDENTITY: We compute Eucldean dstance drectly as a baselne. MAHALANOBIS: Tradtonal Mahalanobs dstance between mages n a par s computed, where the metrc matrx s nverse of covarance between two vectors. a Test results on LFW SIFT Feature dataset. Frst two rows face pars from the same person correctly verfed by our method but not by metrc learnng wth Fantope regularzaton. Last two rows are face pars from dfferent person correctly verfed by our method but not by metrc learnng wth Fantope regularzaton. KISSME: A metrc learnng methods based on a statstcal nference perspectve. It learns a dstance metrc from equvalence constrants and can be used n large scale dataset [11]. ITML: Metrc learnng methods proposedd n [1]. They use LogDet dvergence as regularzaton so that they do not need do explct postve sem-defnte projecton. LDML: [5] offers a metrc learnng method that uses logstc dscrmnant to learn a metrc from a set of labeled mage pars. Because of the settng of ths experment, for method Fantope and Cap, the general functon for ther models on ths task s: P P M, xj xtj u l M, xj xtj,j S,j D γ2 RegM, 29 where D and S denote dssmlar par set and smlar par set respectvely. u s the upper bound for Mahalanobs dstance between two samples n smlar par set, and l represents the dstance lower bound between two samples n dssmlar par set. MLFantope: It denotes metrc learnng wth Fantope reguk P larzaton [14]. As per ts defnton, RegM = σ M. MLCap: Parwse constrants metrc P learnng wth capped trace norm regularzaton. RegM = mn{σ M, ε}. Evaluaton Metrcs: To measure the face verfcaton accuracy for all these compared methods, we report a Recever Operator Characterstc ROC curve. To compare the performance of each method, we compute Equal Error Rate EER of the respectve method, and use 1 EER as evaluaton crteron, and the method wth the lowest EER, or the hghest 1 EER s the most accurate one. Results: We plot ROC curve for each method n Fgure 4, and 1 EER values are also computed to evaluate ther performance. Fgure 4a shows the results on LFW SIFT feature dataset. Mahalanobs dstance between two smlar pars performs qute well comparng wth Eucldean dstance, t ncreases the performance from 67.5% to 74.8%. KISSME method s the state-of-the-art b Test results on LFW Attrbute Feature dataset. Frst two rows face pars from the same person correctly verfed by our method but not by metrc learnng wth Fantope regularzaton. Last two rows are face pars from dfferent person correctly verfed by our method but not by metrc learnng wth Fantope regularzaton. Fgure 3: Sample results on LFW dataset. method on ths feature type and reaches an 1 EER at 80.06%, t outperforms wdely used metrc learnng methods ITML and LDML. It s clear that parwse contrant wth low-rank approxmaton methods, Fantope and CAP, perform better than KISSME, and reach 81.4%, and 81.7% respectvely. ncreases the performance of Eucldean dstance by 14.2% and KISSME method by 0.9%. Fgure 4b presents the performance of each method on LFW Attrbute feature dataset. outperforms KISSME method, and also works better than metrc learnng than Fantope regularzaton by 0.4%. 3 shows the result samples on ths dataset. We run 5-fold cross-valdaton experments on Fantope method and our method. In Fgure 5, we plot accuracy result wth respect to rank selecton. When we select rank parameter k roughly, t s clearly that our method can get better results than metrc learnng wth Fantope regularzaton. In large scale dataset, t not pratcal to tune parameters as carefully as n small dataset, and sometmes, there s not exact rank at all. So, our method s much more applcable to ths knd of stuaton, and performs better when we nputs an approxmaton of matrx rank.

7 1 ROC 1 ROC True Postve Rate TPR True Postve Rate TPR CAP FANTOPE KISSME ITML LDML IDENTITY MAHAL CAP FANTOPE KISSME ITML LDML IDENTITY MAHAL False Postve Rate FPR False Postve Rate FPR a ROC curves on SIFT Feature dataset b ROC curves on Attrbute Feature dataset Fgure 4: Face verfcaton results on LFW dataset: Fgure 4a ROC curve for dfferent methods on LFW SIFT feature dataset. Fgure 4b ROC curve for dfferent methods on LFW Attrbute dataset MLTrace MLFantope a Verfcaton accuracy vs Rank/SIFT Feature MLTrace MLFantope b Verfcaton accuracy vs Rank/Attrbute Feature Fgure 5: Verfcaton accuracy wth respect to rank selecton on LFW SIFT feature dataset 5a and LFW attrbute feature dataset 5b Publc Fgures Face Database Dataset: The PubFg database s a large, real-world face dataset consstng of 58,797 mages of 200 people collected from the nternet. Images n the data set are downloaded from the nternet usng search query on a varety of mage search engnes. Compared to LFW, there are larger number of mages per person, and more varaton on dfferent poses, lghtng condtons, and expressons. Served as a complementary to LFW dataset, PubFg dataset conssts hgh-level features of vsual face trats that are not senstve to pose, llumnaton or other magng condtons. Settng: Face verfcaton benchmark dataset conssts of 20,000 pars of mages of 140 people. The data s dvded nto 10 folds wth mutually dsjont sets of 14 people each, and each fold contans 1,000 ntra and 1,000 extra-personal pars. Smlar to experment settng n LFW dataset, to prove stableness of our method, we splt Pubfg feature dataset nto 5 folds, and evaluate our metrc learnng wth capped trace norm model and metrc learnng wth Fantope norm on these datasets. We run experments 5 tmes usng dfferent tranng/testng splts and compute average and standard varance. Compared Methods: We use the same compared methods n LFW experment. Evaluaton Metrcs: ROC fgure s plotted for each method and we also compare verfcaton accuracy when we tune parameter k for methods Fantope and Cap. Results: Fgure 6 represents the performance of each compared method on Pubfg dataset. We calculate Equal Error Rate for them, and t s clear that our method outperforms other correlated metrc learnng methods. By mposng capped trace norm regularzaton, our method ncreases the performance of tradtonal Mahalanobs dstance by about 6%, and the state-of-the-art KISSME method by over 1%. We also perform another experments to show the process of parameter tunng procedure. In 7, we tune parameter rank k from 5 to 40, and we can see that our method works more stable than metrc learnng wth Fantope regularzaton. The performance of Fantope regularzaton s greatly subject to the selecton of rank k. In ths fgure, we use trace norm regularzaton as baselne. 4.3 Image Classfcaton In ths secton, we evaluate our methods on mage classfcaton and the task s assgnng an mage to a predefned class. We can also look as ths task as object recognton. In the experments, we use mage dataset wth attrbute features. Attrbute s an mportant type of semantc propertes shared among dfferent objects or actvtes.

8 ROC 1 81 MLTrace MLFantope True Postve Rate TPR CAP FANTOPE KISSME ITML LDML IDENTITY MAHAL False Postve Rate FPR Fgure 9: Classfcaton accuracy vs rank k on Pubfg dataset. Fgure 6: ROC curves of dfferent methods on the Pubfg datasets MLTrace MLFantope a Results of 5 neareset neghbors when we query an mage on Pubfg dataset. Frst row s the results of our method, and second shows the results of metrc learnng wth Fantope regularzaton. Fgure 7: Verfcaton accuracy vs rank k on Pubfg dataset Fgure 8: Classfcaton accuracy of each method on Pubfg dataset. It s a representaton n a hgher level than the raw feature representaton drectly extracted from mages or vdeos. In recent years, several attrbute datasets are used by varous researchers for the study of utlzng attrbutes for dfferent vson applcatons. There are two mage classfcaton datasets, PubFg dataset and Outdoor Scene Recognton OSR dataset Publc Fgures Face Database Dataset: We use a subset face mages of Pubfg dataase, and there are 771 mages from 8 face categores. b Results of 5 neareset neghbors when we query an mage on Pubfg dataset. Frst row s the results of our method, and second shows the results of metrc learnng wth Fantope regularzaton. Fgure 10: Results of 5 neareset neghbors when we query an mage. Green lne means ths neghbor s n the same class wth query mage, and red lne denotes they are dfferent. Settng: We use the same experment setup as [14]. Each person contrbutes 30 mages as tranng data to learn Mahalanobs metrc matrx M and buld classfer, other mages are used as testng data to evaluate classfcaton performance. We run ths experment 5 tmes, and 30 mages per person n tranng data are selected ran-

9 domly each tme, and average performance s used as evaluaton crteron. Compared Methods: KNN: In ths method, we use k-nearest neghbor method as classfer and compute Eucldean dstance to measure the smlarty between any two mages. Ths method works as a baselne. SVM: Support vector machne SVM s a wdely used classfer. In ths method, we compare our method wth SVM, and the code s from LbLnear [3]. LMNN: It s one of the most wdely-used Mahalanobs dstance metrc learnng methods. In ths method, they use labeled nformaton to generate trplet constrants. We mpose low-rank constrant on matrx M n LMNN method, and the general form s, γ RegM 1 µ 2 M, xjxj T µ,j,k R,j S [ 1 M, xjx T j x k x T k ] 30 LMNNTrace norm: We mpose trace norm as low-rank regularzaton approxmaton and RegM = σ M, where σ M s sngular value of matrx M. LMNNFantope: Instead of usng trace norm regularzaton as low-rank approxmaton, we mpose rank of matrx explctly, and RegM = k σ M, where σ M are k smallest sngular values of M. LMNNCapped trace norm: We mnmze sngular value of matrx M smaller than a value learned n the optmzaton ε, so RegM = mn{σ M, ε}. Evaluaton Metrcs: In ths experment, we compute classfcaton accuracy for each method as evaluaton crteron. Results: Fgure 8 represents the performance of compared methods. 1NN method usng Eucldean dstance works really bad on ths task, and ts accuracy s just 55%. SVM method ncreases the performance of 1NN method greatly by about 20%. When we use LMNN method to learn Mahalanobs dstance for ths task, and use 1NN classfer, the accuracy reaches 77% and s better than SVM. We mpose low-rank regularzaton, trace norm, Fantope regularzaton, and capped trace norm respectvely, t s clear that our method works best and ncreases the performance of LMNN method by about 1.5%. Sample query results are presented n Fgure 10a, we plot 5 nearest neghbors for each query, and we can fnd out that our method does a better job than metrc learnng wth Fantope regularzaton n ths task. In Fgure 9, we shows the performance of metrc learnng wth Fantope regularzaton and capped trace norm method when we tune parameter rank k. The performance of Fantope regularzaton s very senstve to the choce of parameter rank k, t s obvous that sometmes, the performance of Fantope regularzaton s worse than trace norm regularzaton. It s clear that when we select k from 40 to 100, our method performs much stable than Fantope regularzaton, because t explctly controls the rank of matrx M to be k. can learn an adaptve threshold n the optmzaton and the rank of matrx M s not necessary to be k. Fgure 11: Classfcaton accuracy of each method on OSR dataset MLTrace MLFantope Fgure 12: Classfcaton accuracy vs rank k on OSR dataset Outdoor Scene Recognton Dataset Dataset: We use Outdoor Scene Recognton OSR dataset from [21], and there are 2688 mages from 8 scene categores, and t s descrbed by hgh level attrbute features. Settng: In the experment, we also use 30 mages for each category as tranng data, and other mages are used as testng data. Each tme, we select tranng data randomly and we repeat ths procedure 5 tmes, and use average accuracy as performance of each method. Compared Methods: There are sx compared methods as n last secton, namely KNN, SVM, LMNN, LMNNTrace norm, LMNNFantope and LMNNcapped trace norm. Evaluaton Metrcs: Classfcaton accuracy s compute as the performance of each method. Results: In Fgure 11, 1NN method reaches accuracy at about 65%, and SVM method perform a large ncrease by about 10%. Although the result of 1NN method usng Mahalanobs dstance learned by LMNN s a lttle worse than SVM method, LMNN method wth low-rank regularzaton always outperform the result of SVM. We can also fnd out that our LMNN wth capped trace regularzaton method works better than trace norm and Fantope regularzaton, and reaches an accuracy at about 76.5%. It mproves LMNN method by about 1%. Fgure 10b shows two sample query results of two compared methods. It s a lttle hard to dstngush coast and mountan when both of them has sky n the back, t s clear that our method learns a better Mahalanobs dstance to do classfcaton and mage search. We also plot fgure to compare the performance of LMNN method wth low-rank regularzaton, and we use LMNN wth trace norm regularzaton as a baselne. In Fgure 12, t s clear that our method

10 always works better than metrc learnng wth Fantope regularzaton. When we make k = 40, we can fnd that the performance of Fantope regularzaton s even worse than the baselne. 5. CONCLUSION In ths paper, we propose to use a novel low-rank regularzaton, capped trace norm regularzaton, to mpose on metrc learnng method. Capped trace norm regularzaton s a better rank mnmzaton approxmaton than trace norm. It works more stable than Fantope regularzaton and can be seen as an adaptve Fantope regularzaton as well. We also ntroduce an effcent optmzaton algorthm, and prove the convergence of our objectve functon. Expermental results show that our method outperforms the state-ofthe-art metrc learnng methods. 6. REFERENCES [1] J. V. Davs, B. Kuls, P. Jan, S. Sra, and I. S. Dhllon. Informaton-theoretc metrc learnng. In Proceedngs of the 24th nternatonal conference on Machne learnng, pages ACM, [2] L. V. EE236C. 6. proxmal gradent method. [3] R.-E. Fan, K.-W. Chang, C.-J. Hseh, X.-R. Wang, and C.-J. 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