Feature Extraction for Incomplete Data via Low-rank Tensor Decomposition with Feature Regularization

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1 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P Featue Extaction fo Incoplete Data via Low-ank Tenso Decoposition with Featue Regulaization Qiquan Shi, Student Mebe, IEEE, Yiu-Ming Cheung, Fellow, IEEE, Qibin Zhao, Senio Mebe, IEEE and Haiping Lu, Mebe, IEEE Abstact Multi-diensional data (i.e., tensos) with issing enties ae coon in pactice. Extacting featues fo incoplete tensos is an ipotant yet challenging poble in any fields such as achine leaning, patten ecognition and copute vision. Although the issing enties can be ecoveed by tenso copletion techniques, these copletion ethods focus only on issing data estiation instead of effective featue extaction. To the best of ou knowledge, the poble of featue extaction fo incoplete tensos has yet to be well exploed in the liteatue. In this pape, we theefoe tackle this poble within the unsupevised leaning envionent. Specifically, we incopoate low-ank Tenso Decoposition with featue Vaiance Maxiization (TDVM) in a unified faewok. Based on othogonal Tucke and CP decopositions, we design two TDVM ethods, TDVM-Tucke and TDVM-CP, to lean low-diensional featues viewing the coe tensos of the Tucke odel as featues and viewing the weight vectos of the CP odel as featues. TDVM exploes the elationship aong data saples via axiizing featue vaiance and siultaneously estiates the issing enties via low-ank Tucke/CP appoxiation, leading to infoative featues extacted diectly fo obseved enties. Futheoe, we genealize the poposed ethods by foulating a geneal odel that incopoates featue egulaization into low-ank tenso appoxiation. In addition, we develop a joint optiization schee to solve the poposed ethods by integating the altenating diection ethod of ultiplies with the block coodinate descent ethod. Finally, we evaluate ou ethods on six eal-wold iage and video datasets unde a newly designed ulti-block issing setting. The extacted featues ae evaluated in face ecognition, object/action classification and face/gait clusteing. Expeiental esults deonstate the supeio pefoance of the poposed ethods copaed with the state-of-the-at appoaches. Index Tes Incoplete tenso, featue extaction, othogonal tenso decoposition, low-ank tenso copletion, featue egulaization, vaiance axiization. I. INTRODUCTION Featue extaction is a fundaental and significant topic in any fields such as achine leaning, patten ecognition, data ining, and copute vision. In ecent decades, any ethods fo featue extaction have been developed, such as the classical pincipal coponent analysis (PCA) []. In eal-wold, Qiquan Shi is with Huawei Noah s Ak Lab, Hong Kong (e-ail: qiquan.shi@gail.co). Yiu-Ming Cheung is with the Depatent of Copute Science, Hong Kong Baptist Univesity, Hong Kong (e-ail: yc@cop.hkbu.edu.hk). Yiu-Ming Cheung is the coesponding autho. Qibin Zhao is with Tenso Leaning Unit, RIKEN AIP, Japan and School of Autoation, Guangdong Univesity of Technology, China (eail: qibin.zhao@iken.jp). Haiping Lu is with the Depatent of Copute Science, the Univesity of Sheffield, U.K. (e-ail: h.lu@sheffield.ac.uk). any data such as colo iages, videos and 4D fmri data ae ulti-diensional, i.e., tensos, and have becoe inceasingly popula and ubiquitous in any applications [2]. Tenso decoposition is a poweful coputational tool fo extacting valuable infoation fo tensoial data, which can effectively pefo diensionality eduction, featue extaction, etc.. To lean featues fo tensoial data, any ultilinea ethods have been poposed based on tenso decoposition [3], [4], [5], [6]. Thee ae two popula and fundaental decoposition odels: Tucke decoposition [7], which decoposes a tenso into a coe tenso ultiplied by a facto atix along each ode, and CANDECOMP/PARAFAC (CP) decoposition [8], [9], which factoizes a tenso into a weighted su of ank-one tensos. Based on the Tucke odel, fo exaple, ultilinea pincipal coponent analysis (MPCA) [3] is developed as a popula extension of PCA and can diectly extact featues fo highe-ode tensos. Futheoe, based on CP decoposition, a sei-othogonal ultilinea PCA with elaxed stat (SOMPCARS)[6] ipoves [0] by elaxing the othogonality constaint and initialization on factos. In addition, obust ethods such as obust tenso PCA (TRPCA) [] have been well studied fo leaning featues fo data with couptions (e.g., noise and outlies). In pactice, soe enties of tensos ae often issing in the acquisition pocess, costly expeients, etc. [2], [3]. The easons fo issing data ae nueous. Fo exaple, in social science, when data ae collected in suveys, it is likely that soe people efuse to answe a few questions elated to pesonal pivacy o sensitive topics, thus esulting in issing values with abitay pattens [4]. In industial applications, soe data, such as iages, ae coupted with iegula pattens due to the insufficient esolution of a device o the dysfunction of equipent [5]. Ove all, issing data ae coon in eal-wold [6]. In these scenaios, the featue leaning ethods entioned above cannot wok well due to issing data. How to coectly handle issing data is a fundaental yet challenging poble in any fields [7], [8], [5], which is citical to any eal-wold applications such as classification [2], [9], [6], iage inpainting [20] and clusteing [2], [22]. Howeve, to the best of ou knowledge, effectively extacting featues fo incoplete tensos has yet to be well exploed. Thee ae two possible appoaches to solving the poble of extacting featues fo incoplete tensos. One natual solution is to fill in the issing values and then view the

2 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P ecoveed tensos as the extacted featues. Many tenso copletion techniques have been extended fo atix copletion cases [23], [24], which ae widely used fo pedicting issing data given patially obseved enties and have dawn uch attention in any applications such as iage/video ecovey [25], [26]. Fo exaple, Liu et al. [25] defined the Tuckebased tenso nuclea no by cobining nuclea nos of all atices unfolded along each ode and poposed a high accuacy low-ank tenso copletion algoith (HaLRTC) fo estiating issing values in tenso visual data. Jain et al. [27] developed an altenating iniization algoith (TenALS) fo tensos with a fixed low-ank othogonal CP decoposition, which yields good copletion esults fo incoplete data unde cetain conditions (e.g., a good CP ank [28]). Futheoe, Liu et al. [26] poposed a nuclea no egulaized CP decoposition ethod (TNCP) fo tenso copletion by iposing the Tucke-based tenso nuclea no on facto atices. Although these tenso copletion ethods can ecove data well unde typical conditions, they focus only on tenso ecovey without consideing the elationship aong data saples fo effective featue extaction. In addition, by teating the ecoveed data as leaned featues, the diension of featues cannot be educed. Anothe staightfowad appoach is a two-step stategy: applying tenso copletion algoiths (e.g., HaLRTC) to estiate issing enties fist and then featue extaction ethods (e.g., MPCA) on the ecoveed tensos to lean the featues, i.e., tenso copletion ethods + featue extaction ethods. Fo exaple, LRANTD [4] eploys nonnegative Tucke decoposition (NTD) fo incoplete tensos by incopoating low-ank epesentation (LRA) with nonnegative featue extaction. LRANTD equies a tenso copletion algoith to estiate the issing enties in the peceding LRA step. This appoach likely aplifies the appoxiation eo as the issing data and the featues ae leaned in sepaate stages. Besides, the econstuction eo fo the tenso copletion step can deteioate the pefoance of featue extaction in the subsequent step. Moeove, the twostep stategy cobining two sepaate ethods is usually not coputationally efficient. On the othe hand, a few supevised ethods have been poposed fo classifying low-ank issing data [2], [6], and soe studies have integated a disciinant analysis citeion into low-ank atix/tenso copletion odels fo featue classification [29], [30]. Howeve, these ethods equie labels, which ae expensive and difficult to obtain in pactice, especially fo incoplete data. To solve the poble of featue extaction fo incoplete tensos, we incopoate low-ank Tenso Decoposition with featue Vaiance Maxiization (TDVM) into a unified faewok. In this pape, we focus on two popula tenso decopositions fo TDVM and design two ethods: TDVM- Tucke and TDVM-CP based on Tucke and CP odels, espectively. These two ethods ae essentially deployed unde a geneal unsupevised odel that incopoates low-ank Tenso Decoposition with Featue Regulaization (TDFR). TDFR siultaneously estiates issing data via low-ank appoxiation and exploes the elationship aong saples via featue egulaization. In othe wods, TDVM- Tucke and TDVM-CP specify TDFR by eploying low-ank Tucke/CP decoposition fo low-ank appoxiation and using featue vaiance axiization as the featue constaint. Specifically, TDVM-Tucke iposes the Tucke-based tenso nuclea no on the coe tensos of Tucke decoposition with othonoal facto atices (a.k.a., highe-ode singula value decoposition (HOSVD) [3]) while iniizing the appoxiation eo, and eanwhile axiizes the vaiance of coe tensos. Hee, the leaned coe tensos (analogous to the singula values of a atix) ae viewed as the extacted featues. TDVM-CP ealizes the low-ank CP appoxiation by iniizing the CP-based tenso nuclea no [32] of weight vectos and the econstuction eo based on othogonal CP decoposition, and eanwhile axiizes the vaiance of leaned featue vectos fo featue egulaization. The weight vecto of the othogonal CP decoposition of a tenso (analogous to the vecto of singula values of the SVD of a atix) is viewed as the featue vecto. TDVM incopoates Tucke- and CP-based tenso nuclea no egulaization with vaiance axiization on featues while estiating issing enties, which esults in infoative featues extacted diectly fo obseved enties. Moeove, TDVM-Tucke ais to lean low-diensional tensoial featues fo high-diensional incoplete tensos (i.e., tensoto-tenso pojection [0]), while TDVM-CP can extact lowdiensional vectoial featues (i.e., tenso-to-vecto pojection [0]). The poposed ethods diffe fo both tenso copletion ethods and two-step stategies as follows. ) Tenso copletion ethods ai to ecove the incoplete tensos only without exploing the elationship aong saples fo effective featue extaction. In contast, TDVM ethods focus on extacting low-diensional featues instead of estiating issing data. Moeove, TDVM utilizes a featue constaint (featue vaiance axiization) to captue the elationship aong saples fo extacting infoative featues; 2) unlike the two-step stategies, which lean the featues of incoplete data via two sepaate stages, TDVM siultaneously estiates issing enties and leans low-diensional featues diectly fo the obseved enties in the unified faewok. Thus, TDVM can extact oe infoative featues and educe coputational cost; 3) copaed with the supevised ethods, TDVM does not equie label infoation duing featue leaning, which is oe feasible in pactice. We eploy Altenating Diection Method of Multiplies (ADMM) [33] and Block Coodinate Descent (BCD) to optiize TDVM odels. Afte featue extaction via TDVM, we evaluate the extacted featues on six iage and video databases fo thee applications: face ecognition, object/action classification and face/gait clusteing. Patial wok petaining to TDVM-Tucke has been published in the confeence vesion [34] of this pape, and the ain contibutions of this wok ae theefold: ) We popose two unsupevised ethods, TDVM-Tucke and TDVM-CP, fo featue extaction of incoplete tensos. The TDVM ethods exploe the elationship aong tenso saples via featue vaiance axiization while estiating issing values by low-ank appoxiation,

3 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P leading to infoative featues extacted diectly fo obseved enties. Moeove, we discuss the genealization of TDVM by poposing the geneal odel TDFR. 2) We develop an ADMM-BCD joint optiization schee to solve the TDVM-CP odel, in which each subpoble of TDVM-CP can be solved in a closed fo although its oveall objective is non-convex and non-sooth. 3) We evaluate the poposed ethods on six tenso datasets with newly designed ulti-block issing settings. Tensos with ulti-block data issing ae not only oe geneal, as they cove the existing pixel-based and blockbased issing settings, but ae also oe difficult and pactical in eal-wold scenaios. Moe ipotantly, the expeiental esults show that the poposed ethods outpefo the state-of-the-at appoaches with significant ipoveents. The est of the pape is oganized as follows. We eview elated peliinaies and elated woks in Section II. Then, we pesent the poposed ethods and geneal odel in Section III. We epot the epiical esults in Section IV and conclude this pape in Section V. II. PRELIMINARIES AND RELATED WORK A. Notations and Opeations The nube of diensions of a tenso is the ode and each diension is a ode of it. A vecto is denoted by a bold lowe-case lette x R I and a atix is denoted by a bold capital lette X R I I2. A highe-ode (N 3) tenso is denoted by a calligaphic lette X R I I N. The ith enty of a vecto a R I is denoted by a(i), and the (i, j)th enty of a atix X R I I2 is denoted by X(i, j). The (i,, i N )th enty of an Nth-ode tenso X is denoted by X (i,, i N ), whee i n {,, I n } and n {,, N}. The Fobenius no of a tenso X is defined by X F = X, X /2. Ω R I I N is a binay index set: Ω(i,, i N ) = if X (i,, i N ) is obseved, and Ω(i,, i N ) = 0 othewise. P Ω is the associated sapling opeato which acquies only the enties indexed by Ω: { X (i,, i (P Ω (X ))(i,, i N ) = N ), if(i,, i N ) Ω, whee Ω c is the copleent of Ω. () 0, if(i,, i N ) Ω c Definition. Mode-n Poduct. A ode-n poduct of X R I I N and U R In Jn is denoted by Y = X n U R I In Jn In+ I N, with enties given by Y i i n j ni n+ i N = i n X i i n i ni n+ i N U in,j n, and Y = U T X [0]. Definition 2. Mode-n Unfolding. a.k.a., aticization, is the pocess of eodeing the eleents of a tenso into atices along each ode [2]. A ode-n unfolding atix of a tenso X R I I N is denoted as X R In Π n ni n. B. Tucke and CP Decoposition ) Tucke Decoposition: It epesents a tenso X R I I2 I N as a coe tenso with facto atices [2]: X = C U () 2U (2) N U (N), (2) whee {U R In Rn, n =, 2 N, and R n < I n } ae facto atices with othonoal coluns and C R R R2 R N is the coe tenso with lowe diension. The Fig.. The Tucke decoposition of a thid-ode tenso saple X, whee the coe tenso C consists of extacted featues fo X. Fig. 2. The CP decoposition of a thid-ode tenso saple X, whee the coe tenso D is supe-diagonal and its eleents {d, d 2,, d R } (i.e. featue vecto d ) ae viewed as extacted featues fo X. Tucke-ank of an N th-ode tenso X is an N-diensional vecto, denoted as = [R,, R n,, R N ], whose n-th enty R n is the ank of the ode-n unfolded atix X of X. Figue illustates this decoposition. In this pape, Tucke-ank is equivalent to the diension of featues (each coe tenso). Based on Tucke decoposition, Liu et al. [25] have defined Tucke-based Tenso Nuclea No, that is, Definition 3. Tucke-based Tenso Nuclea No [25] of a tenso X is defined as: X = N n= X = N Rn n= j= σ j, whee X is the ode-n unfolding atix of X and σ j is the singula values of the unfolded atix. 2) CP Decoposition: It decoposes a tenso X R I I N as the su of a set of weighted ank-one tensos: R X = d u () u (2) u (N) (3) = = D U () 2U (2) N U (N), whee each coon facto {u, n =,, N} is a unit vecto with a weight absobed into the weight vecto d = [d, d, d R ] R R, and denotes the oute poduct [2]. Figue 2 shows that CP decoposition can also be efoulated as Tucke decoposition whee the coe tenso D is supe-diagonal, i.e., D (,, ) = d. R is the CP-ank as the iniu nube of ank-one coponents. In this pape, CP-ank is equivalent to the diension of featues (each weight vecto). Based on othogonal CP decoposition, we have defined the CP-based Tenso Nuclea No: Definition 4. CP-based Tenso Nuclea No [32] of a tenso X is defined as the L no of the weight vecto d of its othogonal CP decoposition: X CP = d. C. Related Wok Consideing the taget poble of extacting featues fo incoplete data, thee ae fou categoies of elated appoaches, which ae biefly suaized as follows. ) Tenso Copletion Appoach: Tenso copletion appoach is extended fo the atix case [23] and widely used fo ecoveing issing data. Thee ae any successful tenso copletion ethods, such as HaLRTC [25], TenALS [27], TNCP [26] and [35], [20], [36], [37]. These copletion ethods can yield good ecovey esults unde typical conditions, but they focus only on estiating issing data instead of extacting infoative featues. 2) Featue Extaction Appoach: Many tenso ethods have been poposed fo featue extaction diectly fo ultilinea data, e.g., the classical MPCA [3] and [4], [5], [6],

4 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P [38], [], [39], [40], [4], [42]. These ethods can achieve state-of-the-at esults fo leaning featues fo coplete (and noisy) tensos, howeve, they cannot pefo well on data with issing values. 3) Supevised Classification Appoach: Soe classification algoiths have been well studied fo classifying low-ank issing data [2], [6]. Besides, a few studies have integated a disciinant analysis citeion into low-ank atix/tenso copletion odels fo classification [29], [30]. Howeve, these ethods equie labels which ae expensive and difficult to obtain in pactice, especially fo incoplete data. 4) Subspace Clusteing Appoach: Subspace clusteing odels such as spase subspace clusteing [43] wee applied in the pesence of issing data in [2], [22]. In addition, soe studies have incopoated atix copletion appoaches with subspace clusteing fo incoplete atices [44], [45]. Howeve, these algoiths do not yield good esults fo leaning featues fo incoplete tensos because they ae developed fo clusteing incoplete vectos/atices. In Sec. IV, we copae the poposed unsupevised ethods with elated state-of-the-at algoiths selected the aboveentioned categoy 3) as they ae supevised. III. THE PROPOSED: TDVM-TUCKER AND TDVM-CP A. Poble Definition Given a total M tenso saples {T,, T,, T M } with issing enties in each saple T R I I N. I n is the ode-n diension. We denote T = [T,, T, T M ] R I I N M, whee the M ae the nube of tenso saples concatenated along the ode-(n + ) of T. To achieve featue extaction (diension eduction) objective, we ai to diectly extact low-diensional featues fo the given highdiensional incoplete tensos T. Reak : This poble is diffeent fo the case of data with couptions (e.g., noise and outlies), which has been well studied in [], [46], [47], [43]. Missing data could be equivalent to the couption case only if the couptions ae abitay and the indices of couptions ae known. Howeve, in eality, the agnitudes of couptions ae not abitaily lage. In othe wods, hee we study a diffeent featue extaction poble that existing ethods cannot solve well. To solve this poble, we popose an unsupevised featue extaction appoach by incopoating low-ank Tenso Decoposition with featue Vaiance Maxiization (TDVM). Based on two widely used Tucke and CP decoposition odels, we develop two algoiths of TDVM as follows. B. TDVM-Tucke: Leaning Low-diensional Tenso Featues We fist popose a TDVM ethod based on othogonal Tucke decoposition: we ipose the Tucke-based tenso nuclea no on the coe tensos while iniizing the econstuction eo, and eanwhile axiize the vaiance of coe tensos (featues), i.e., incopoating low-ank Tucke Decoposition with featue Vaiance Maxiization, naely TDVM-Tucke: in X,C,U = 2 X C U() N U (N) 2 F + C = = 2 C C 2 F, s.t. P Ω (X ) = P Ω (T ), U U = I, n = N, whee {U R In Rn } N n= ae coon facto atices with othonoal coluns. I R Rn Rn is an identity atix. C R R R N is the coe tenso, which consists of the extacted featues of an incoplete tenso T with obseved enties in Ω. C is the Tucke-based tenso nuclea no of C. C = M M = C is the ean of coe tensos (extacted featues). To optiize the objective function of TDVM-Tucke using ADMM, we apply the vaiable splitting technique, intoduce a set of auxiliay vaiables {S R R R N, = M}, and then efoulate Eq. (4) as follows: in 2 X C U() N U (N) 2 F X,C,S,U = + S = = 2 C C 2 F, s.t. P Ω (X ) = P Ω (T ), S = C, U U = I. Reak 2: The objective function Eq. (5) integates thee tes into a unified faewok. The fist and second te lead to low-ank Tucke appoxiation, which ais to iniize the econstuction eo and obtains low-diensional featues. Because iposing the Tucke-based tenso nuclea no on a coe tenso C is equivalent to that on its oiginal tenso X [48], we obtain a low-ank solution, i.e., R n can be sall (R n < I n ). Theefoe, the featue subspace is natually low-diensional. Moeove, iposing nuclea no on coe tensos instead of oiginal ones educes the coputational cost. The thid te (iniizing M = 2 C C 2 F ) ais to axiize the vaiance of leaned featues inspied by PCA. TDVM-Tucke thus exploes the elationship aong tenso saples via featue vaiance axiization while estiating the issing data via low-ank Tucke appoxiation. ) Deivation of TDVM-Tucke by ADMM: To facilitate the deivation of Eq. (5), we efoulate the equation by unfolding each tenso vaiable along ode-n and absobing the constaints. Thus, we obtain the Lagange function as follows: N ( L = 2 X U C H 2 F = n= + S + Y n, C S + µ 2 C S 2 F 2 C C ) 2 F whee H = U (N) U (n+) U (n ) U () R j n Ij j n Rj, and µ and {Y n R Rn j n Rj, n =,, N, =,, M} ae the Lagange ultiplies. X R In j n Ij and {C, S, C } R Rn j n Rj ae the ode-n unfolded atices of X and {coe tenso C, auxiliay vaiable S, ean of featues C}, espectively. ADMM solves the poble (6) by successively iniizing L ove {S, U, C, X }, and then updating Y n. Fo siplicity, the iteation nube k is oitted in the updates of all vaiables in TDVM-Tucke and TDVM-CP optiization. (4) (5) (6)

5 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P Algoith Low-ank Tenso (Tucke) Decoposition with Featue Vaiance Maxiization (TDVM-Tucke) : Input: Incoplete tensos P Ω (T ), Ω, µ, and the axiu iteations K, featue diension D = [R,, R N ] (Tuckeank), and stopping toleance tol. 2: Initialization: Set P Ω (X ) = P Ω (T ), P Ω c(x ) = 0, =,, M; initialize {C } M = and {U } N n= andoly; ρ = 0, µ ax = e0. 3: fo k = to K do 4: fo = to M do 5: fo n = to N do 6: Update S, U espectively. and C by (8), () and (3) 7: Update Y n by Y n = Y n + µ(c S ). 8: end fo 9: Update X by (5). 0: end fo : If C S 2 F / C 2 F < tol, beak; othewise, continue. 2: Update µ k+ = in(ρµ k, µ ax). 3: end fo 4: Output: Tensoial featues: C = [C,, C, C M ] R R R N M. a) Update S : Eq. (6) with espect to S is, L S = N = n= ( S + µ 2 (C ) + Yn/µ) S 2 F, (7) whee S is coputed via soft-thesholding opeato [49]: S = pox /µ (C + Y n/µ) = Udiag(ax σ µ, 0)V, (8) whee pox is the soft-thesholding opeation and U diag(ax σ µ, 0)V is the SVD of (C + Y n /µ). b) Update U : Eq. (6) with espect to U is: N L U = 2 X = n= U C H 2 F, s.t. U U =I, (9) The iniization of (9) ove the atices {U (),, U (N) } with othonoal coluns is equivalent to the axiization of the following poble [50]: U = ag ax ( tace U X (C H ) ) (0) whee tace() is the tace of a atix, and we denote W = C H. The poble (0) is actually the well-known othogonal Pocustes poble [5], whose global optial solution is given by the SVD of X W, i.e., U = Û ( ˆV ), () whee Û and ˆV ae the left and ight singula vectos of SVD of X W, espectively. c) Update C : Eq. (6) with espect to C is: L C = N = n= + µ 2 C ( X U C H 2 F S 2 ( M )C M setting the patial deivative L C ( C = µs M 2 M 2 µ + 2M + Y n/µ 2 F j / C X H ( ( M M 2 ) M ) C j 2 F, Y n + U j (2) to zeo, we get: C ) ) j. (3) d) Update X : Eq. (5) with espect to X is: 2 X C U() 2U (2) N U (N) 2 F, = (4) s.t. P Ω (X ) = P Ω (T ), by deiving the Kaush-Kuhn-Tucke (KKT) conditions fo function (4), we can update X by: X = P Ω (X ) + P Ω c(c U () 2U (2) N U (N) ). (5) We suaize the poposed ethod, TDVM-Tucke, in Algoith. Reak 3: TDVM-Tucke exploes the elationship aong tenso saples via featue vaiance axiization while estiating the issing data via low-ank Tucke appoxiation, leading to low-diensional infoative featues diectly fo obseved enties. The poposed ethods diffe fo both tenso copletion ethods and two-step stategies as follows. Tenso copletion ethods ai to ecove incoplete tensos only without exploing the elationship aong saples fo effective featue extaction. In contast, TDVM-Tucke focuses on extacting low-diensional featues instead of estiating issing data. Moeove, TDVM utilizes a featue constaint (featue vaiance axiization) to captue the elationship aong saples fo extacting infoative featues. Unlike the two-step stategies, which lean the featues of incoplete data via two sepaate stages, TDVM- Tucke siultaneously estiates issing enties and leans low-diensional featues diectly fo the obseved enties in the unified faewok. The two-step stategies can aplify the appoxiation eo because the issing data and the featues ae leaned in sepaate stages, and the econstuction eo fo the tenso copletion step can deteioate the pefoance of featue extaction in the subsequent step. This clai has been veified by ou expeiental esults (as shown in the Tables I, II and III in Section IV-C). Theefoe, TDVM-Tucke and TDVM-CP (which is intoduced in the following) can extact oe infoative featues within the unified faewok. C. TDVM-CP: Leaning Low-diensional Vecto Featues We futhe popose anothe new TDVM ethod to lean low-diensional vectoial featues based on CP decoposition, i.e., incopoating low-ank CP decoposition with featue vaiance axiization, naely TDVM-CP. Because tenso decoposition with issing data is oe challenging than that with coplete data in taditional pobles, hee we conside incopoating othogonality into the CP odel fo TDVM-CP (i.e., iposing othogonality constaints on factos {u } in Eq. (3)) with the following two otivations: Like HOSVD [3], CP decoposition can be egaded as a genealization of SVD to tensos [52]. It appeas natual to inheit the othogonality of SVD in the CP odel. The othogonality constaint is consideed unnecessay in geneal o even ipossible in cetain cases in exact CP decoposition [53], [54], [55], but soe studies have

6 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P poved that iposing othogonality in CP decoposition can tansfo a non-unique tenso odel into a unique one with guaanteed optiality [54], [27], [56]. Like the othogonality used in TDVM-Tucke, we believe that iposing othogonality constaints can help TDVM-CP estiate issing values and extact featues bette. In addition, hee we do not use the Tucke-based nuclea no [25]; instead, we use a new CP-based tenso nuclea no 2 [32] to achieve low-ank CP appoxiation. In othe wods, TDVM-CP couples othogonal CP decoposition with the CP-based tenso nuclea no fo the low-ank appoxiation, while axiizing the vaiance of leaned featues as the featue egulaization te. Thus, the objective function of TDVM-CP is as follows: in X,d,u,R R 2 X = = = d u () = u (N) 2 F + λ d 2 d d 2 2, (6) s.t. P Ω (X ) = P Ω (T ), u u =, n = N, u u q = 0, q =, = R, whee d is the CP-based tenso nuclea no on each weight vecto, and we view the weight vecto d R R of the othogonal CP decoposition (analogous to the vecto of singula values of a atix) as the featue vecto extacted fo a tenso saple X. d = M M = d is the ean of the weight vectos (extacted featues). λ > 0 is a penalty paaete. Copaed with TDVM-Tucke, TDVM-CP can obtain uch lowe diensional featues because it leans vectoial featues fo each tenso saple. ) ADMM-BCD Joint Optiization fo TDVM-CP: To solve the objective function Eq. (6) which is non-convex and non-sooth, we design a ADMM-BCD joint optiization schee. We divide all the taget vaiables into M (R + ) goups: {{d, u (), u (2),, u (N) } R =, X } M =, whee we optiize a goup of vaiables while fixing the othe goups, and update one vaiable while fixing the othe vaiables in each goup. Afte updating the R + goups fo each saple using BCD, we jup to the outside loop to update all saples iteatively using ADMM. To apply ADMM, we intoduce a set of auxiliay vaiables {s R R } M = fo the weight vectos {d } M =, i.e., s = d R R, = M. Then, we foulate the Lagangian function of Eq. (6) as follows: L = ( R 2 X = = d u () u (N) 2 F + λ d 2 s s y, d s + γ 2 d s 2 2 η(u u ) q= µ qu u q, (7) whee γ, η, {µ q } q= and y ae the Lagange ultiplies. In the ADMM-BCD joint optiization, we fist update the vaiables {d, u (), u (2),, u (N) } R = of each data saple via BCD. Thus, we foulate the Eq. (7) with espect to the -th goup {d, u (), u (2), u (N) } as follows: 2 Fo easy eading, we use d instead of X CP in the deivation. ) L d,u = X du() u (2) 2 + γ 2 d + y/γ s 2 2 η(u u ) u (N) 2 F + λ d q= µ qu u q, (8) whee X = X q= d qu () q u (2) q u (N) q is the esidual of the appoxiation of each tenso saple. a) Update u : Eq. (8) with espect to u is, L u = X du 2 η(u u u (2) ) u (N) 2 F q= µ qu u q. (9) Then we set the patial deivative of L u to zeo and eliinate the Lagange ultiplies, and get: u u =(X j {u (j) } j n )/d ( q= u q (X j {u (j) } j n ) u q with espect to ) /d, (20) whee X j {u (j) } j n = X u () (n ) u (n ) (n+) u (n+) N u (N), j =, 2,, n, n +,, N, and we noalize u = u / u 2. Note that we only update the vaiable goups with non-zeo weights (i.e. d 0). b) Update d : Eq. (8) with espect to d is: L d = X du() u (2) u 2 F + λ d 2 + γ 2 d + y/γ s 2 2. (2) Setting the patial deivative L d / d to zeo, we obtain, d = ( γs y + X u () 2 u (2) ( + γ) N u (N) λ d / d ). (22) Accoding to the soft thesholding algoith [57] fo L egulaization, we update d by: Q t (Q > t) d = shink t(q) = 0 ( Q t). (23) Q + t (Q < t) whee shink is the shinkage opeato [57], and t = λ (+γ), Q = (+γ) (γs y + X u () 2 N u (N) ). Afte updating {d, u (), u (2),, u (N) } R = by the BCD ethod, we jup out of the inne loop fo each tenso saple and update the vaiables {s, X } M = fo all tenso saples iteatively via ADMM. c) Update s : Eq. (7) with espect to s is, L s = = 2 γ d + y/γ s 2 2 = 2 s s 2 2, (24) whee y consists of Lagange ultiplies. Then we set the patial deivative L s / s to zeo and obtain, M 2 ( ) s = γd + y γm 2 + 2M + M 2 + M γm 2 + 2M + M 2 (25) s j j

7 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P Algoith 2 Low-ank Tenso (CP) Decoposition with Featue Vaiance Maxiization (TDVM-CP) : Input: Incoplete tensos P Ω (T ), Ω, λ, featue diension D = R (CP-ank), axiu iteations K, and tol. 2: Initialization: Set P Ω (X ) = P Ω (T ), P Ω c(x ) = 0, γ = 0; Initialize {u (), u (2), u (N) } R =, {d } M = andoly. 3: fo k =,..., K do 4: fo =,..., M do 5: X = X ; 6: fo =,..., R do 7: if d 0 then 8: Update u and d by (20) and (23),espectively. 9: X = X d u () u (2) u (N). 0: end if : end fo 2: Update s and X by (24) and (27) espectively. 3: Update y = y + γ(d s ) 4: end fo 5: If d s 2 2/ d 2 2 < tol, beak; othewise, continue. 6: end fo 7: output: Vectoial featues D = [d, d, d M ] R R M. d) Update X : Eq. (6) with espect to X is, R in X 2 X d u () u (2) u (N) 2 F, = (26) s.t. P Ω (X ) = P Ω (T ), by deiving KKT conditions fo Eq. (26), X is updated by: R X = P Ω (X ) + P Ω c( d u () u (2) u (N) ). (27) = Using the ADMM-BCD joint optiization, we solve each subpoble of Eq. (6) in a closed-fo. Finally, we suaize the poposed TDVM-CP in Algoith 2. Reak 4: TDVM-CP is siila in spiit to TDVM-Tucke, but it can yields featues with lowe diension than TDVM- Tucke: the foe extacts low-diensional vecto featues fo each data saple, while the latte ais to lean lowdiensional tenso featues fo each saple. Thus, using TDVM-CP to extact featues can educe the coputational cost and eoy equieents fo futhe applications such as classification and clusteing. D. Coputational Coplexity Analysis Fo TDVM-Tucke, we set the featue diensions (Tuckeank) R = R 2 = R N = R fo siplicity. In each iteation, the tie coplexity of coputing the soft-thesholding opeato (8) is O(MNR N+ ). The tie coplexities of ultiplications in ()/(3) and (5) ae O(MNR( N j= I j)) and O(MR( N j= I j)), espectively. Thus, the total tie coplexity of TDVM-Tucke is O(M(N + )R( N j= I j)) in each iteation. Fo TDVM-CP, the tie coplexity of pefoing the shinkage opeato in (23) is O(R( N j= I j). This is also the tie coplexity of coputing {u } N n= and Eq. (27). Hence, the total tie coplexity of TDVM-CP is O(MR( N j= I j) in each iteation. E. Discussion: Geneal Model TDFR The poposed TDVM-Tucke and TDVM-CP essentially can be suaized into a geneal odel, i.e., low-ank Tenso Decoposition with Featue Regulaization (TDFR): in X,Z F (X, Z) + G(Z) s.t. P Ω(X ) = P Ω (T ), (28) whee F (X, Z) efes to a low-ank tenso decoposition odel and G(Z) is a egulaization of taget featues Z. X R I I2 I N M is the appoxiation of incoplete data (tensos) T based on obseved enties indexed by Ω. Z is a coponent of X and could be a lowe-diensional tenso (e.g., a coe tenso of Tucke odel) o vecto (e.g., a weight vecto of CP odel) that consists of all featues extacted fo T. In this pape, we specify TDFR by TDVM-Tucke and TDVM-CP. In addition, we biefly discuss oe specific cases of TDFR. Fo exaple, consideing the whole dataset as a tenso including all saples along the last ode, we can specify TDFR as follows: in X,C,U,Z 2 X C U() 2U (2) N U (N) N+Z 2 F + C 2 Z 2 F, s.t. P Ω (X ) =P Ω (T ), U U = I, n = N, (29) whee the (N +)th facto atix Z R M R (N+) ae viewed as the extacted featues fo T = [T,, T, T M ] R I I N M. Such usage of teating the (N + )th facto atix as featues can also be found in [58], [59]. Inspied by PCA, the thid te: in 2 Z 2 F = ax tace(zz ), ais to axiize the vaiance of extacted featues. Due to space liitations, oe specific cases ae discussed in Appendix A of the Suppleentay Mateial 3. Reak 5: As TDFR siultaneously estiates issing data via low-ank tenso appoxiation and exploes the elationship aong saples via featue egulaization (e.g., axiizing vaiance of featues in TDVM), we assue that TDFR can solve the poble of extacting featues fo incoplete tensos. In addition to the two poposed ethods, thee ae any vaiants of specific cases of the geneal odel TDFR: ) Fo the low-ank appoxiation F (X, Z) of Eq. (28), we can not only use Tucke and CP decopositions in conjunction with the Tucke- and CP-based tenso nuclea no, but can also conside othe tenso decoposition odels such as Tenso SVD [60], [], Tenso-tain decoposition [6], [62], etc., coupled with othe constaints such as tenso nuclea no [60], [] to achieve low-ank tenso appoxiation; 2) Fo the featue egulaization te G(Z), we can use not only vaiance axiization fo egulaization such as TDVM but also othe constaints such as uncoelation o othogonality, etc., to lean infoative featues. IV. EXPERIMENTS We evaluate the pefoance of the poposed TDVM- Tucke and TDVM-CP on six eal-wold tenso datasets with 30% 90% issing enties unde ulti-block issing settings. MR efes to the Missing Ratio. We ipleent the poposed ethods in MATLAB, and all expeients ae pefoed on a PC (Intel Xeon(R) 4.0 GHz, 64 GB eoy). 3 Suppleentay Mateial: zbqyofzwc5lsd0w/aabidjvamuwwvfgud-uvofa?dl=0

8 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P A. Expeiental Setup ) Data: We evaluate TDVM-Tucke and TDVM-CP on six eal-wold datasets fo thee applications, including fou thid-ode tensos and two fouth-ode tensos 4 : Fo face ecognition, we use two face datasets: one is a subset of the Facial Recognition Technology database (FERET) 5 [63], which has 72 face saples fo 70 subjects. Each subject has 8 to 3 face iages with at ost 5 degees of pose vaiation, and each face iage is noalized to an gay iage. The othe dataset is a subset of the extended Yale Face Database B (YaleB) 6 [64], which has 244 face saples fo 38 subjects. Each subject has 59 to 64 nea fontal iages unde diffeent illuination and each iage is noalized to a gay iage. Fo object/action classification tasks, we evaluate two datasets: one is a subset of the COIL-00 iage database, which contains 00 diffeent objects, each viewed fo 72 diffeent angles 7 [65]. The size of each saple (totally 000 saples) is noalized to a gay iage following [66]. The othe dataset is a subset of the Weizann action dataset 8 [67], which consists of 80 videos of 8 actos pefoing ten diffeent actions: bending, juping, juping jacks, juping in place, unning, galloping sideways, skipping, walking, one hand-waving, and two hands waving. Each video is esized to Fo face/gait clusteing tests, we also test two datasets: one is a subset of the AR face database [68], which contains 200 face iages with size of 00 subjects including iages of non-occluded faces, and face occluded by scaves/glasses following [66] 9 ; the othe dataset is the galley set (73 saples fo 7 subjects) of the USF HuanID Gait Challenge database 0 [69]. Each gait video saple is esized to ) Copaed Methods: We copae TDVM-Tucke and TDVM-CP with 7 ethods in fou categoies : (i) Thee Tucke-/CP- based tenso copletion ethods: HaLRTC [25], TenALS [27] and TNCP [26]. (ii) Nine {tenso copletion ethods + featue extaction ethods} (i.e., two-step stategies): HaLRTC + MPCA [3], TenALS + MPCA, TNCP + MPCA, HaLRTC + SOMPCARS [6], TenALS + SOMPCARS, TNCP + 4 Fo fast evaluation, we use esized tenso saples with salle diensions, while the poposed ethods ae applicable to oiginal (lage) tensos without subsapling (esizing). Refe to Appendix D of the Suppleentay Mateial fo esults on lage tensos. 5 haiping/msl.htl vision/spacetieactions.htl haiping/msl.htl We have also copaed with the state-of-the-at tenso singula value decoposition (t-svd) ethods such as [70] and its cobined two-step stategies. Although the t-svd based tenso copletion ethods slightly outpefo the Tucke and CP based ethods (such as HaLRTC and TNCP), they still give uch pooe featue extaction esults than ou TDVM ethods. Because the poposed ethods ae based on the Tucke and CP odels, we do not pesent the copaison against t-svd ethods hee fo siplicity and please efe to Appendix C of the Suppleentay Mateial fo these esults. Oiginal FERET saple Missing 30% enties Missing 50% enties Missing 70% enties Missing 90% enties (a) FERET with {20 5, 6 8, 4 4, } ulti-block issing enties. Oiginal video fae Missing 30% enties Missing 50% enties Missing 70% enties Missing 90% enties Oiginal video fae 0 Oiginal video fae 5 Oiginal video fae 20 (b) USF gait with { , , 4 3 4, } ulti-block issing enties. Fig. 3. Exaples of (a) one saple of FERET database (b) fou faes of the fist video saple (20 faes) of USF gait database, with {30%, 50%, 70%, 90%} issing enties geneated by MbM settings. SOMPCARS, HaLRTC + LRANTD [4], TenALS + LRANTD, TNCP + LRANTD. (iii) One obust tenso featue leaning ethod: TRPCA []. (iv) Fou clusteing ethods (used fo the copaison of clusteing with issing data): Spase subspace clusteing (SSC) [43], Zeo-Fill + SSC (ZF + SSC) [2], SSC by Colun-wise Expectation-based Copletion (SSC-CEC) [2], Spase Repesentation with Missing Enties and Matix Copletion (SRME-MC) [22]. We copae the 3 ethods of the fist thee categoies with espect to face ecognition and object/action classification tasks, and copae all 7 ethods in face/gait clusteing tests. Afte featue extaction, we use the Neaest Neighbos Classifie (NNC) to evaluate the extacted featues fo face ecognition and object/action classification. Fo face/gait clusteing tests, we use the K-eans [7] to cluste the featues extacted by the fist 3 ethods and use a spectal clusteing technique as a post-pocessing step fo the fou subspace clusteing ethods. 3) Multi-block Missing (MbM) Setting: In this pape, we design a Multi-block Missing (MbM) setting to geneate ando issing pattens of tensos. Accoding to the data saple size, we use a set of tensoial blocks with diffeent sizes as issing blocks to geneate issing enties andoly in each tenso saple. We pogess fo the lagest issing blocks to the sallest issing blocks to geneate issing pattens until the equied atio of issing enties is achieved. Fo exaple, we can use a ando set of issing blocks { , , 4 3 4, } to obtain an incoplete USF gait database (saple size ) with 50% issing enties. We fist use the k lagest ( ) blocks to ceate issing enties andoly until the (k + )th lagest block exceeds the equied issing atio (e.g., k = 5); then we use the p second lagest (20 5 5) blocks until the (p + )th second lagest block exceeds the equied issing atio (e.g., p = 2). We continue by using the s thid lagest

9 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P (4 3 4) blocks (e.g., s = 25) to geneate the issing data successively. Finally, we use the q sallest issing blocks with sallest size (e.g., q = 70) to ake up the eaining issing egion. Thus, we use (k + p + s + q) issing blocks of diffeent sizes to geneate an incoplete USF gait tenso saple with 50% issing enties. Hee, these issing blocks can be ovelapped (i.e., the values of k, p, s, q ae diffeent in diffeent saples) and the issing blocks ae distibuted andoly in each tenso saple. Hence, the iegula issing shapes (positions of issing data, i.e., Ω) ae diffeent in each tenso saple, while the total nube of issing enties is the sae. Nevetheless, one can set any types of MbM sets with ultiple blocks of diffeent sizes unde the MbM setting. Figue 3 illustates the data saples with issing enties geneated by the poposed MbM setting. Reak 6: The Multi-block Missing setting geneates diffeent iegula issing shapes (issing pattens) in tenso saples, which is oe geneal and pactical in eal-wold applications. MbM setting with only one type of block (with size = ) is equivalent to the pixel-based issing (unifoly selecting MR (e.g., MR = 50%) pixels (enties) fo each tenso saple as issing at ando) which is widely used in atix/tenso copletion fields. MbM setting with only one type of block (with size > ) is equivalent to the block-based issing setting (andoly selecting a single block enties of each tenso saple as issing) which is also coonly used in issing data iputation. In othe wods, existing issing data settings ae special cases of ou MbM setting. Intuitively, handling data with geneal ulti-block issing is oe difficult than that with pixel-based issing and block-based issing if the nube of issing enties is the sae. The eason is that the MbM setting is soehow close to the non-ando issing setting especially when MR is highe (e.g., when MR= 90%, soe whole ows/coluns of iages/videos ae issing as shown in Figue 3), although the MbM setting is essentially ando block issing with ovelapping. 4) Paaete Settings: We set the axiu iteations K = 500, tol = e 5 fo all ethods, although ou ethods usually convege within 0 iteations. Fo Tucke decoposition-based ethods, naely, TDVM-Tucke and LRANTD, we set the featue diension D = [R, R 2,, R N ] (Tucke-ank) = ound (/2 ([I, I 2, I N ])) fo each tenso saple. Fo CP decoposition-based ethods, naely, TDVM-CP, TenALS and TNCP, we set D = R (CP-ank) = ound (in{/2 ean([i, I 2,, I N ]), in([i, I 2,, I N ])}) fo each saple. Fo othe paaetes of the copaed ethods, we have tuned the paaetes based on the oiginal papes to obtain the best esults unde sae expeiental settings. On the othe hand, we futhe evaluate extacted featues fo classification via NNC, in which we andoly select L = {, 7} extacted featue saples fo each subject of FERET fo taining in NNC. Siilaly, we set L = {5, 50}, {, 8} and {, 7} on the YaleB, COIL-00 and Weizann datasets, espectively. B. Analysis of Diffeent (Paaete) Settings and Convegence ) Effect of Diffeent Multi-block Missing Settings: Hee, we study the effect of applying TDVM-Tucke and TDVM- CP to datasets with diffeent MbM settings. We andoly set Classification Accuacy (%) MbM set : {8 6 3} MbM set 2: {5 4 8, } MbM set 3: {8 5 3, 3 5 2, 2 2 2} MbM set 4: {0 8 6, 4 7 5, 3 3 3, } 50 MbM set 5: {5 7 3, 3 3 9, 2 2 4, 2 2 2} MbM set 6: {2 6 0, 8 5 4, 4 7 5, 2 3 4, 2 2 2} MbM set 7: { } 40 30% 40% 50% 60% 70% 80% 90% Missing Ratio (TDVM-Tucke) (a) TDVM-Tucke on Weizann with MbM settings Classification Accuacy (%) MbM set : {8 6 3} MbM set 2: {5 4 8, } MbM set 3: {8 5 3, 3 5 2, 2 2 2} 65 MbM set 4: {0 8 6, 4 7 5, 3 3 3, } MbM set 5: {5 7 3, 3 3 9, 2 2 4, 2 2 2} 60 MbM set 6: {2 6 0, 8 5 4, 4 7 5, 2 3 4, 2 2 2} MbM set 7: { } 55 30% 40% 50% 60% 70% 80% 90% Missing Ratio (TDVM-CP) (b) TDVM-CP on Weizann with MbM settings Fig. 4. Classification esults of Weizann with 30% 90% issing enties geneated by seven diffeent MbM settings via TDVM-Tucke and TDVM-CP (featue diension D = {6 5, 0} espectively). Classification Accuacy (%) D = D2 = D3 = D4 = 6 5 D5 = D6 = D7 = % 40% 50% 60% 70% 80% 90% Missing Ratio (TDVM-Tucke) (a) TDVM-Tucke with diffeent featue diensions Classification Accuacy (%) D = 32 D2 = 2 60 D3 = 5 D4 = 0 50 D5 = 8 D6 = 5 D7 = % 40% 50% 60% 70% 80% 90% Missing Ratio (TDVM-CP) (b) TDVM-CP with diffeent featue diensions Fig. 5. Classification esults on Weizann with 30% 90% issing enties (MbM set ={0 8 6, 4 7 5, 3 3 3, }) via TDVM-Tucke and TDVM-CP with seven diffeent featue diensions. seven MbM sets using diffeent types of issing blocks to geneate issing patten on the Weizann database to obtain incoplete Weizann data, i.e., MbM set using only one type of block with size {8 6 3}, which also efes to the coonly used block-based issing setting; MbM set 2 using two types of blocks:{5 4 8, 7 3 2}; MbM set 3 using thee types of blocks: {8 5 3, 3 5 2, 2 2 2}; MbM set 4 using fou types of blocks: {0 8 6, 4 7 5, 3 3 3, }; MbM set 5 using fou types of blocks: {5 7 3, 3 3 9, 2 2 4, 2 2 2}; MbM set 6 using five types of blocks: {2 6 0, 8 5 4, 4 7 5, 2 3 4, 2 2 2}; and MbM set 7 using only one type of block with size = ( ), which is equivalent to the pixel-based issing setting widely used in atix/tenso copletion. Using the seven MbM sets, we geneate an incoplete Weizann database ( ) with 30% 90% issing enties. TDVM-Tucke and TDVM- CP diectly extact and 0 80 featues fo these incoplete tensos, espectively, and these featues ae futhe evaluated via NNC using L = 7 video featue saples pe subject (each subject has 8 saples) as taining. Figue 4 shows that on the Weizann dataset with vaious issing pattens using diffeent ando MbM sets, both TDVM-Tucke and TDVM-CP consistently yield good esults. Two cases ae paticulaly woth entioning. On the Weizann dataset with MbM set and set 7, TDVM-Tucke and TDVM-CP can achieve bette classification esults than othe cases (MbM set 2-6) especially when MR > 70%. This veifies ou clai entioned in Reak 6: handling data with the MbM setting which uses ultiple issing blocks (MbM set 2-6) is oe difficult than that with existing block-based issing (MbM set ) and pixel-based issing (MbM set 7) settings. Fo geneal MbM settings (MbM set 2-6), TDVM-Tucke and TDVM-CP can obtain siila esults with acceptable deviation of classification accuacy. On the othe hand, using these

10 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS. ACCEPTED: TNNLS-208-P Classification Accuacy (%) = = 5 = 0 = 5 = 20 = 25 = 30 = 35 = 40 = 45 = L (Taining Saple Pe Subject) (a) TDVM-Tucke with diffeent µ Classification Accuacy (%) L (Taining Saple Pe Subject) (b) TDVM-CP with diffeent λ Fig. 6. Classification esults on Weizann with 50% issing enties (MbM set ={0 8 6, 4 7 5, 3 3 3, }) via TDVM-Tucke and TDVM-CP with diffeent values of µ and λ, espectively. Relative Eo TDVM-Tucke Nube of Inteations (a) TDVM-Tucke Relative Eo = = 5 = 0 = 5 = 20 = 25 = 30 = 35 = 40 = 45 = 50 TDVM-CP Nube of Inteations (b) TDVM-CP Fig. 7. Convegence cuves of TDVM-Tucke and TDVM-CP in tes of Relative Eo: G S 2 F / G 2 F and d s 2 2 / d 2 2 espectively, on Weizann with 50% issing enties (MbM set = {0 8 6, 4 7 5, 3 3 3, }). MbM sets with diffeent types of issing blocks, the achieved issing atios ae likely slightly diffeent, especially fo these MbM sets without size = block to ake up the eaining issing enties. Fo exaple, with MbM set 5, we actually obtain Weizann with 29.94% 89.92% instead of exact 30% 90% issing enties because of the sizes of the issing blocks. This slight diffeence of actual nube of issing enties can also slightly affect the classification esults, leading to an incease in the deviation of classification accuacy unde diffeent MbM settings. In shot, the poposed TDVM-Tucke and TDVM-CP ae ae not highly sensitive to the issing pattens oveall and consistently yield good esults unde vaious MbM settings. Thus, in the following tests, we test datasets with fou types of issing blocks in MbM settings fo siplicity, and each MbM set includes the size = block to ensue the total nube of issing enties is the sae unde diffeent MbM settings. 2) Effect of Diffeent Featue Diensions: We study the effect of diffeent featue diensions used in TDVM-Tucke and TDVM-CP fo featue extaction on an incoplete Weizann database. Figue 5 shows that, with diffeent diensions of featues, TDVM-Tucke and TDVM-CP yield siila classification esults stably on the whole, except in the case of TDVM-Tucke with D7 = (i.e., only 8 featues ae extacted fo each video) whee the nube of featues is too liited to achieve good esults. TDVM-CP obtains uch fewe leaned featues, but it consistently achieve good esults. On the othe hand, as TDVM-Tucke and TDVM- CP ae based on the othogonal Tucke and CP odels espectively, the diension of effective featues fo TDVM-Tucke is uppe-bound by the data diension in each ode, and that of TDVM-CP is liited by the iniu saple diension. Thus, setting {D = > } fo TDVM- Tucke and {D=32, D2=2, D3=5>0=in [32, 22, 0]} fo TDVM-CP leads to a slight deteioation of classification pefoance especially in the cases of Weizann with highe issing atio (e.g., MR=90%), as seen fo Figs. 5(a) and 5(b) espectively. In shot, the poposed ethods ae not sensitive to the featue diensions. Since a lage featue diension will lead to highe coputational costs and eoy equieents, and we ai to lean low-diensional featues, we thus set D = ound (/2 ([I, I 2, I N ])) and D = ound (in{/2 ean([i, I 2,, I N ]), in([i, I 2,, I N ])}) fo TDVM- Tucke and TDVM-CP by default, espectively. 3) Sensitivity Analysis of Paaete µ and λ: Figue 6 shows the classification esults given featues extacted by TDVM-Tucke and TDVM-CP with diffeent values fo the penalty paaetes µ and λ, espectively, on Weizann videos with 50% issing enties via an MbM set. Figue 6(a) show that TDVM-Tucke yields good esults stably with diffeent values of µ. Figue 6(b) shows that the featue extaction pefoance of TDVM-CP is also stable and not sensitive to the values of λ, except fo the case in which λ =. In othe wods, the poposed ethods ae not sensitive to the paaetes oveall. In addition, as the paaetes ρ and γ can be fixed (fix ρ = 0, γ = ) within Algoith and Algoith 2 espectively based on peliinay studies, we thus do not exaine the hee. In shot, we do not need to caefully tune the paaetes µ and λ fo TDVM-Tucke and TDVM-CP, espectively. In this pape, we fix µ = λ = 0 fo all tests. 4) Convegence: We study the convegence of TDVM in tes of the elative eo on a Weizann dataset with 50% issing enties via an MbM set. Figue 7 shows that: TDVM-Tucke conveges within 0 iteations while TDVM- CP equies oe iteations (about 20) to each convegence. If set tol = e 5, ou ethods convege fast with aound 5-0 iteations. C. Evaluation of Extacted Featues fo Incoplete Tensos To save space, we epot the esults of six eal tenso datasets with {30%, 50%, 70%, 90%} issing pixels unde ando MbM settings in Tables I, II and III 2. We highlight the best esults in bold font and undeline the second best esults, and we use to indicate that the ethod diveges (e.g., TenALS) in soe cases. The aveage esults of 0 uns ae epoted. ) Face Recognition: Table I shows that TDVM-Tucke and TDVM-CP consistently outpefo all the ethods copaed in all cases. Specifically, TDVM-Tucke and TDVM-CP diectly lean featues and featues fo FERET ( ) with {30%, 50%, 70%, 90%} issing pixels via a ando MbM set ({32 32, 0 4, 8 6, }). As shown in the left half of Table I, TDVM- Tucke and TDVM-CP shae the two best ecognition esults, 2 The poposed ethods ae based on low-ank decopositions and thus can yield good esults on tensos with good low-ank stuctue even when the issing atio eaches 90%. Howeve, if too any (e.g., 95%, 99%) enties ae issing, the pefoance of ou ethods will dop daatically.