Social Trust Prediction via Max-norm Constrained 1-bit Matrix Completion

Social T rust Pred iction via Max-norm Constrained 1-bit Matrix Completion Jing W ang 1 , Jie Shen 2 , Huan Xu 3 1 Hefei Univ ersity of T echnology , China wangjing@mail.hfut.edu .cn 2 Rutgers Univ ersity , USA js2007@rutgers.edu 3 National Univ ersity of Singapore, S ingapore mpexuh@nus.edu.sg Abstract Social trust prediction addresses the significant problem of exploring interactions among users in social networks. Natu rally , this problem can be for- mulated in the matrix comp letion fr amew ork, with each entry indicating the trustne ss o r distrustness. Howe ver, there are two challenges for the social trust pr oblem: 1) the observed data are with sign (1-bit) measur ements; 2) the y are typ ically sampled non-u niform ly . Most of the previous matrix com- pletion metho ds do no t well handle the two issues. Motiv ated by the recen t progr ess of max-no rm, we propo se to solv e the problem with a 1-bit max- norm constrained f ormulatio n. Since max -norm is not easy to o ptimize, we utilize a refor mulation of max-no rm which facilitates an efficient projec ted gradient decent alg orithm. W e d emonstrate the superiority of ou r f ormulatio n on two be nchmark datasets. 1 Intr oduction W ith the increa sing pop ularity of social networks, there ex- ist many interesting and difficult prob lems, such as friends recommen dation, informatio n prop agation, etc. In this paper, we study the problem o f social trust prediction, which aims to esti mate the positiv e or negati ve relation ship amon g the users based on the existing tru stness informatio n associated with them. This pro blem play s a n importan t ro le in social networks as the system can b lock the in vitation fro m some- one that the user does not trust, or reco mmend new frien ds who enjoys a high reputation. Naturally , the social trust problem can be formu lated wit hin the matrix completion framework [ Liben-Nowell and Kleinberg, 2007 ] [ Billsus and Pazzani, 1998 ] . That is, the ( i, j ) -th entry of the o bserved data matrix Z ∈ R n × n is a 1- bit c ode implying that th e i -th user trusts th e j -th user if Z ij = 1 . Here, n deno tes the number of users. Howe ver, what we observe is only a small fra ction o f the entr ies, whose values are zero . And our goal is to estimate the m issing entries accordin g to the 1-bit measuremen ts in Z . Note that the problem is ill-posed if no assumption is im- posed on the s tructure of the data. T o solve the prob lem, a numb er of method s are p ropo sed. Generally , existing so- cial trust p rediction methods fall into three categories. The first category is based on similarity measures or the struc- tural context similarity [ Newman, 2001 ] [ Chowdhury , 2010 ] [ Katz, 1953 ] [ Jeh and W id om, 2002 ] , motiv ated by the in- tuition that an individual tends to trust their neighb ors, o r the ones with similar trusted people. The second is based on low rank matrix com pletion [ Billsus and Pazzani, 1998 ] [ Cai et al. , 2010 ] [ Huang et al. , 2013 ] , which assumes tha t the underlying data matrix is low-rank or can be approximate by a lo w-rank matr ix. The third o ne mo dels the problem as a binary classification one a nd utilizes tech niques such as lo - gistic regression [ Leskovec e t al. , 2010 ] . Challenges. Howe ver, the re are two issues emerging in social trust pr ediction which are no t well ch aracterized by the algorithms in pr evious works. First, th e value of the observed entry is either 1 o r − 1 , which is analogous to the binary c lassification problem. But in our problem, we are h andling much more com plex matrix data. For - tunately , [ Srebro et al. , 2004 ] presented a maximum margin matrix factorization framework that un ifies the binar y prob - lem fo r vector case and m atrix case. The key idea in their work is a low-norm m atrix factorization , which w ill also be utilized in this pap er . Sec ond, th e lo cations of the entries are sampled non- unifor mly , which gaps the theory and practice for a lo t of matrix completion algorithms. T o tackle this chal- lenge, we sugg est using the max -norm as a conve x surrogate for the rank func tion, which is shown to be sup erior to the well-known nuc lear norm wh en addressing the n on-un iform data [ Salakhutdin ov and Sreb ro, 2010 ] . Our contrib utions are two-folds: 1) T o the be st of our knowledge, we are the first to addr ess the social trust pred ic- tion pro blem by utilizing a max -norm constrain ed formula- tion. 2) Althou gh a m ax-nor m co nstrained pro blem can be solved by SDP solvers and an accu rate enou gh solution c an be achieved, we here utilize a pro jected g radient alg orithm that is scalable to large scale datasets. W e empirically show the imp rovement of our formu lation for the no n-un iform 1-bit benchm arks compared to state-of-the- art solvers. 2 Related W ork Social interaction is in vestigated intensi vely in the last decades. The social interactio n indicates the fr iendship, sup- port, en emy or disapproval as shown in Figure 1. On line users Figure 1 : Illustration of the data matr ix of social trust. Each column of th e d ata matrix is a rating sample. Each entry is a tag tha t a u ser assign s to ano ther user . Th e sym bol “+” de - notes the “trust” relationship , “- ” denotes “d istrust” and “? ” means unk nown re lationship (that we aim to predict). Most of the r elationships are unk nown, so the data is sparse. T ypi- cally , each individual has own preference and fr iendship net- work, making the d ata non-un iform. Also n oted that the d ata matrix may not be symmetric. rely on the tr ustworthiness information to filter inf ormation , establish collabora tion or b u ild socia l b onds. Social ne tworks rely o n the trust information to make r ecommen dation, at- tract u sers fro m other circles, or in fluencing pu blic opin ions. Thus, the exp loration of social trust has a wide range of ap- plications, and has em erged as an important to pic in social network research. A numb er of method s are proposed. One kind o f methods are based on the similar- ity me asurement. Specifically , Jaccar d’ s coefficient is co mmon ly u sed to me asure th e p robability two items that have a relation ship. Inspired by the met- ric, Jeh and W idom [ Jeh and W id om, 2002 ] propo sed a domain- indepen dent similar ity measure ment, SimRank. [ Newman, 2001 ] directly define d a score to verify th e corre- lation between co mmon neig hbors. Some meth ods are based on relational data modeling, stru ctural proximity measures and stochastic relational mod el [ Getoor and Diehl, 2005 ] [ Liben-Nowell and Kleinberg, 2007 ] [ Y u et al. , 2009 ] . The above men tioned methods a re mainly derived from th e solutions of link prediction . The link pred iction is or i- ented to n etwork-level pred iction, wher eas social trust prediction focu ses on person-level. Another class of methods are d erived fr om the c ollaborative filtering m eth- ods, such as clu stering technique s [ Sarwar et al. , 2001 ] , model-b ased methods [ Hofmann and Puzicha, 199 9 ] , and the matrix factorization models [ Srebro et al. , 2003 ] [ Mnih and Salakhutdinov , 200 7 ] . Howe ver , the data ma trix of trust has som e structure pro perties d ifferent from th e user-item matrix , such as transitivity . Meanwhile, th e soc ial trust in reality is extremely sparse. For instance, Facebook has hundred s of millions of users, but most of them ha ve l ess than 1,000 friend s. Besides, the peop le with similar p erson- ality tend to behave similarly . T o sum up, the d ata matrix of social trust has bo th sparse and low-rank structure. Thus, the social trust pred iction problem is especially suitable for the matrix completio n model. Th at is the focus of our paper . The problem of matrix completion is to recover a lo w -rank matrix from a sub set of entries [ Cand ` es and Recht, 2009 ] , which is given b y: min X rank ( X ) s.t. P Ω ( X ) = P Ω ( Z ) (2.1) where Z ∈ R p × n is the d ata matrix , X is the recovered matrix, and Ω is the in dex set o f o bserved entries. Th op- timization problem ( 2.1) is n ot only NP-h ard, b ut requires double exponential tim e complexity with the number of sam- ples [ Recht et al. , 2010 ] . T o solve the ab ove pr oblem, one alternative is to use n uclear norm as a relaxation to the rank function : min X || X || ∗ s.t. P Ω ( X ) = P Ω ( M ) (2.2) where || X || ∗ denotes th e sum o f singular values of m atrix X . [ Cai et al. , 2010 ] developed a first-ord er pro cedure to solve the co n vex problem (2.2), namely singu lar value thresholding (SVT). [ Jain et al. , 2010 ] minimized the rank min imization b y th e singular value p rojection (SVP) algor ithm. [ Kes hav an et al. , 2010 ] solved the problem by first tr imming e ach r ow and colu mn with too few e ntries, then compu te th e truncated SVD of the trimmed matrix. U nder certain conditio ns, it sh owed accurate r ecovery on the order of nd log n samp les ( n is the number of samples, d is the rank of recovered ma- trix). W ith the r apid development o f ma trix completion problem , some more efficient metho ds have been p roposed [ Cand ` es and T ao, 201 0 ][ Gross, 2011 ][ W an g and Xu, 2012 ][ Huang et al. , 2 0 1 3 ] . Howe ver, all the m ethods men tioned ab ove use the nuclear n orm a s the surrog ate to the rank, who se ex- act recovery can be gu aranteed only when the data are sampled unifor mly , w hich is not pr actical in real world applications. On the other h and, recent empiri- cal o n max -norm [ Srebro et al. , 2004 ] shows promising re- sults fo r non -unifo rm data if o ne utilize the m ax-no rm as a surr ogate [ Salakhutdin ov and Sreb ro, 2010 ] . No tably , for some specific problems, such as collaborative filter- ing, [ Srebro and Shraibman , 2005 ] proved that the general- ization error bou nd for max-nor m is better tha n the nuclear norm. More r ecently , [ Shen et al. , 201 4 ] reported encou rag- ing results on the sub space r ecovery task (which is closely relev ant to matr ix com pletion). Since the social trust data is non-u niform ly sampled, we belie ve that a max-n orm regu- larized formulation can better handle th e challenge tha n the nuclear norm. Our fo rmulation is also inspir ed by a re- cent th eoretical study on m atrix comp letion with 1-bit mea- surement [ Cai and Zhou, 2013 ] , which establishe d a minimax lower bou nd on the gener al sampling model and derived the optimal c onv ergence rate in terms of Frobeniu s norm loss. Furthermo re, there are several practical algo rithms to solve max-no rm regularized or m ax-nor m constrained prob lems, see [ Lee et al. , 2010 ] and [ Shen et al. , 2014 ] for example. 2.1 Over view After revie w of related work in Section 2, we intro duce the notations and formulate the problem in Section 3. Then we giv e algorith m to solve the m ax-nor m co nstrained 1 -bit matrix comp letion (MMC) problem in Sectio n 4. Mean - while, we also provide an equi valent SDP formulation for the MMC, which ca n be accurately solved at the expense of effi- ciency . T hen we r eport the emp irical study on two benchmark datasets in Section 5. Section 6 concludes this paper and dis- cusses possible future work. 3 Notations and Prob lem Setup In this section, we introdu ce the notation s th at will be used in this pap er . Capital letters s uch as M are used for matrices an d lowercase bold letters s uch as v den otes vectors. T he i -th row and j -th column of a m atrix M is den oted b y m ( i ) and m j respectively , and the ( i, j ) -th en try is d enoted by M ij . For a vector v , we use v i to d enote its i -th elemen t. W e denote the ℓ 2 norm of a vector v by k v k 2 . For a matrix M ∈ R p × n , we denote th e Fro benius no rm by k M k F and k M k 2 , ∞ denotes the maximum ℓ 2 row norm of M , i.e., k M k 2 , ∞ := p max i =1 k m ( i ) k 2 . W e furth er define the max-norm of M [ Linial et al. , 2007 ] , k M k max = min U,V ,M = U V ⊤ max {k U k 2 2 , ∞ , k V k 2 2 , ∞ } , (3 .1) where we enum erate a ll possible factorizations to obtain the minimum. Intuition on max-norm. At a first sight, th e ma x-nor m is hard to under stand. W e simply e xplain why it is a tighter approx imation to the rank f unction than th e n uclear norm. Again, we write the nuclear norm of M as a factorization form [ Recht et al. , 2010 ] : k M k ∗ := min U,V ,M = U V ⊤ 1 2  k U k 2 F + k V k 2 F  . Note th at the Frobenius n orm is the sum o f th e squar e of th e ℓ 2 row norm. Thus, a nuclear nor m regularizer actually con- strains the average of th e ℓ 2 row norm , while the max- norm constrains the maximu m of the ℓ 2 row norm! Giv en th e ob served data Z ∈ R p × n , we are interested in approx imating Z with a low-rank m atrix X , wh ich can be formu lated by , minimize 1 2 kP Ω ( Z − X ) k F , s.t. rank ( X ) ≤ d, where Ω is a n index set of ob served entries a nd d is some expected rank. P Ω ( M ) is a pro jection operator on a ma- trix M such that P Ω ( m ij ) = m ij if ( i , j ) ∈ Ω and zero otherwise. H owe ver, it is u sually intractable to optim ize the above pr ogram as the r ank fu nction is non- conv ex and non- continuo us [ Recht et al. , 2010 ] . One com mon appr oach is to use the nuclear n orm as a con vex surrogate to the rank f unc- tion. Howe ver , it is well kn own that the nuclear no rm c an- not w ell handle the non- unifor m data. Mo tiv ated by th e re- cent progress in max-nor m [ Salakhutdin ov and Sreb ro, 2010; Cai and Zhou, 2013; Sh en et al. , 2014 ] , we use the max - norm as a n alternative conv ex relaxation, which giv es the fol- lowing formu lation: min X 1 2 kP Ω ( Z − X ) k 2 F , s.t. k X k max ≤ λ 2 , (3.2) where λ is some tunable parameter . 4 Algorithm The m ax-no rm is co n vex an d m oreover , it can be solved by any SDP solver . Formally , we have the following lemma: Lemma 4.1 ( [ Srebro et al. , 2004 ] ) . F or an y matrix X ∈ R p × n and λ ∈ R , k X k max ≤ λ if and only if ther e e x- ist A ∈ R p × p and B ∈ R n × n , such th at  A X X ⊤ B  is semi- definite po sitive and ea ch d iagonal element in A and B is upper bounde d by λ . W ith Lemma 4.1 on hand, on e can formu late Problem 3 .2 as an SDP: min X,A,B 1 2 kP Ω ( Z − X ) k 2 F , s.t. A ii ≤ λ 2 , B j j ≤ λ 2 , ∀ i ∈ [ p ] , j ∈ [ n ] ,  A X X ⊤ B   0 . (4.1) And this pro gram can b e solved by any SDP solver to obtain accurate enough solution. Howe ver, SDP solvers are n ot scalable to large matrices. Thus, in this paper, we ap ply a projected g radient meth od to solve Pr oblem (3.2), which is due to [ Lee et al. , 2010 ] . A key technique is the ref ormulatio n of the ma x-nor m (3.1). As- sume that the rank of the optimal solution X ∗ produ ced by the SDP (4.1) is at most d . Then we can safely f actorize X = U V ⊤ , with U ∈ R p × d and V ∈ R n × d . Com bining the factor ization and the definition, we obtain th e fo llowing equiv alent progr am : min U,V 1 2   P Ω ( Z − U V ⊤ )   2 F , s.t. k U k 2 , ∞ ≤ λ, k V k 2 , ∞ ≤ λ. (4.2) Note that the gradient o f the ob jectiv e fun ction w . r .t. U and V can be easily computed. Th at is, ∇ U f ( Z, U, V ) = P Ω  ( U V ⊤ − Z ) V  , ∇ V f ( Z, U, V ) = P Ω  ( V U ⊤ − Z ⊤ ) U  . (4.3) Here, for simplicity we define f ( Z, U, V ) = 1 2   P Ω ( Z − U V ⊤ )   2 F . Algorithm 1 Max-n orm Constrained 1-Bit Matrix Comple- tion (MMC) Input: Z ∈ R p × n (observed sam ples), pa rameters λ , initial solution ( U 0 , V 0 ) , maximu m iteration τ . Output: Optimal solution ( U ∗ , V ∗ ) . 1: for t = 1 to τ do 2: Compute the gradient by Eq. (4.3): U ′ t = ∇ U f ( Z, U, V t − 1 ) | U = U t − 1 , V ′ t = ∇ V f ( Z, U t − 1 , V ) | V = V t − 1 . 3: Compute the step size α t accordin g to Armijo rule. 4: Compute the new iterate: U t = Π( U t − 1 − α t U ′ t ) , V t = Π( V t − 1 − α t V ′ t ) . 5: end for The inequality con straints can be addressed by a pro jection step. T hat is, when we have a new iterate ( U t , V t ) a t the t - th iteration, we can check if they v iolate the constrain ts. If not, we can proc eed to the next iteration. Otherwise, we can scale the r ows of U an d/or V by λ k U k 2 , ∞ and/or λ k V k 2 , ∞ re- spectiv ely . I n this way , we ha ve the projection operator: Π( M ) = ( λ k M k 2 , ∞ M , if k M k 2 , ∞ > λ , M , otherwise. (4.4) If we further p ick the step size α t via the Armijo rule [ Armijo and others, 1966 ] , it c an be shown that the sequ ence of ( U t , V t ) will co n verge to a station ary point [ Bertsekas, 1999 ] . Th e algorithm is sum marized in Al- gorithm 1. The b enefits o f applyin g the factorization on the max-n orm are tw o-folds: 1) the memory c ost ca n be sig nificantly r e- duced fro m O ( pn ) of SDP to O ( d ( p + n )) . 2) it facil- itates the projected gradien t algorithm, which is compu ta- tionally efficient whe n working on large matrices (see Sec- tion 5 ). Howe ver, note that Problem (4.2) is non- conv ex. For - tunately , [ Burer and Monteiro, 2005 ] proved that as lon g a s we p ick a sufficiently large value for d , then any loc al m ini- mum of Eq . (4 .2) is a global op timum. In Sectio n 5 , we will report the influence of d on th e performance. Actually , in Algorithm 1, the stoppin g criteria is set to b e a maximum it- eration. One may also check if it reaches a local min ima as the stopping criteria, as discussed in [ Cai and Zhou, 2013 ] . 4.1 Heuristic on λ The λ is the only tuna ble parameter in ou r algo rithm. For our prob lem, note that the data is of 1-b it measurements, i.e. , | Z ij | = 1 for ( i , j ) ∈ Ω . Also no te that Z ij = u ( i ) v ( j ) ⊤ . Thus,   u ( i ) v ( j ) ⊤   = 1 . So we have λ ≥ 1 . Howe ver , if we choose a large λ , the estimation | X ij | may de v iate away fro m 1 . W e find that λ = 1 . 2 lead to satisfactory improvement. 5 Experiments In th is section, we empirica lly evaluate o ur method for the matrix completion perform ance. W e will first introduce the used datasets. I n the exper imental settings, we present the compara ti ve metho ds and e valuation metr ics. Then we report encour aging results on two b enchma rk datasets. W e also ex- amine the influence of matrix rank d . 5.1 Datasets W e c onduc t the experime nts o n two b enchmar k datasets: Epinions and Slashdot. In these two datasets, the u sers are connected by explicit positiv e (trust) o r negative (d istrust) links ( i.e., the 1-bit measur ements in Z ). Th e first dataset con- tains 119,21 7 nod es (u sers) an d 8 41,00 0 edges (links), 85 .0% of which are positive. The Slashdot dataset co ntains 82, 144 users a nd 549,2 02 link s, and 77.4 % of the edges are lab eled as positive. T able 1 giv es a summary descr iption about the subset used in our experiment. It is clear that th e distribution o f link s are not uniform since each u ser has his/her individual preference and own friendship network . Following [ Huang et al. , 2013 ] , we se- lect 2,000 users with the highest degrees f rom each dataset to form the observation matrix Z . T able 1: Description of 2 datasets Dataset Epinions Slashdot # of Users 2,000 2,000 # of T rust 171,7 31 68,9 32 # of Distrust 18,91 6 20,03 2 5.2 Experimental Settings Baselines. W e choose fou r state-of-th e-art metho ds as baselines, including SVP [ Jain et al. , 2010 ] , SVT [ Cai et al. , 2010 ] , OPTSp ace [ Kes hav an et al. , 2010 ] and RRMC [ Huang et al. , 2013 ] . Since SVT and RRMC need a sp ecified rank, we tune the ran k fo r these method s and choose the best perfor mance as the final result. Evaluation Metric. Let T be the index set of all observed entries. W e use two ev aluation m etrics to measure the per- forman ce, mean av erage error (MAE) an d root mean square error (RMSE), compute d as follows: M AE = X ( i,j ) ∈ T \ Ω ( X ij − M ij ) / ( | T | − | Ω | ) , RM S E = s X ( i,j ) ∈ T \ Ω ( X ij − M ij ) 2 / ( | T | − | Ω | ) , where | T | d enotes the cardinality of T . T raining and T esting. W e random ly split the dataset for training a nd testing. In particu lar , the num ber of observation measuremen ts Ω fo r trainin g r anges f rom 1 0% to 60%, with step size 10 %. For each split, we run all the algorithms fo r 20 trials, with the training da ta in each tr ail being rand omly sampled. Then , we report the mean and standa rd de v iation of MAE and RMSE over all 20 trials. T able 2: MAE Results o n Epinions Dataset Observed entries (%) Methods SVT OPTSpace SVP RRMC MMC 10 0.359 ± 0.00 4 0 .289 ± 0.019 0.450 ± 0.0 08 0.576 ± 0.00 1 0.2 54 ± 0. 003 20 0.394 ± 0.02 2 0 .236 ± 0.005 0. 294 ± 0 .002 0.5 18 ± 0. 002 0 .212 ± 0.003 30 0.360 ± 0.05 7 0 .219 ± 0.009 0. 248 ± 0 .001 0.4 60 ± 0. 002 0 .201 ± 0.002 40 0.410 ± 0.09 9 0 .205 ± 0.008 0. 224 ± 0 .001 0.4 18 ± 0. 002 0 .193 ± 0.001 50 0.471 ± 0.12 9 0 .197 ± 0.007 0. 210 ± 0 .001 0.3 86 ± 0. 001 0 .190 ± 0.002 60 0.476 ± 0.14 6 0 .197 ± 0.003 0. 199 ± 0 .001 0.3 62 ± 0. 002 0 .206 ± 0.003 T able 3: RMSE Results on Ep inions Dataset Observed entries (%) Methods SVT OPTSpace SVP RRMC MMC 10 0.513 ± 0.01 0 0 .530 ± 0.021 0. 610 ± 0 .010 0.6 50 ± 0. 001 0 .466 ± 0.004 20 0.559 ± 0.03 1 0 .456 ± 0.005 0. 459 ± 0 .002 0.6 06 ± 0. 002 0 .406 ± 0.004 30 0.532 ± 0.08 2 0 .422 ± 0.011 0. 415 ± 0 .002 0.5 63 ± 0. 002 0 .383 ± 0.002 40 0.620 ± 0.17 1 0 .406 ± 0.015 0. 394 ± 0 .002 0.5 33 ± 0. 002 0 .371 ± 0.002 50 0.719 ± 0.22 5 0 .398 ± 0.016 0. 381 ± 0 .001 0.5 09 ± 0. 001 0 .364 ± 0.001 60 0.728 ± 0.28 8 0 .403 ± 0.009 0. 371 ± 0 .002 0.4 91 ± 0. 002 0 .365 ± 0.002 5.3 Experimental Results W e r eport detailed r esults from T able 2 to T able 5. Fro m the results in T ab les 2 and 3 , we observe that MMC out- perfor ms the o ther method s in terms of b oth evaluation met- rics on the Epin ions dataset most of th e time. In particular, when ther e are fe w observations 10 % (wh ich indicates a hard task), MMC o btains th e RMSE o f 0.46 6, mu ch b etter th an OPTSpace (0. 530), SVP (0 .610) and RRMC (0.6 50). Ex cept on the case with 60% observed entries, OPTSpace obtains the smallest MSE with 0.19 7, but our algorithm is compar ativ e with 0.2 06. In a nut o f shell, the gap between MM C an d the baselines becomes larger as the fraction of observed entries decreases. Similarly , our me thod ach iev es the le ast MAE and RMSE on the S lashdot dataset (see T able 4 and 5). For instance with 30 % ob served entries, MMC o btains the MAE with less than 0.4, much better than the co mparative methods, such as SVT (0.513 ), OPTSpace (0.427), SVP (0.50 1) and RRMC ( 0.686 ). In terms o f RMSE, in the case of 20 % observed entries, th e RMSE values of other meth ods are all above 0. 7 while ou r method reaches 0.6 79. In su m, our method is superior than the comparative me thods on tw o real-life datasets in terms of MSE and RMSE most of the time. Since we have studied the effectiv eness of ou r m ethod, here we examine the computational efficiency in T able 6, which is importan t for p ractical ap plications. T o test the time com- plexity of the method s, we report the av eraged time cost on the Epin ions d ataset with 10% observed entries and Slash- dot with 20 % observed entries. T o illustrate the trad e-off be- tween ac curacy and efficiency , we also rep ort the MAE and RMSE. As we see, SVP is the most efficient metho d, wh ose runnin g tim e is 0.84 seconds on Epinion s while ou rs is 1 .92 seconds. On Slashd ot, it also ach iev es the best per forman ce in terms of e fficiency . Howe ver , our me thod enjoys a sign ifi- cant impr ovement of MAE and RMSE comp ared to all base- 0 30 60 90 120 150 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Epoch MAE d=1 d=5 d=50 d=100 d=300 d=500 0 30 60 90 120 150 0.4 0.5 0.6 0.7 0.8 0.9 1 Epoch RMSE d=1 d=5 d=50 d=100 d=300 d=500 Figure 2: MAE and RMSE in ter ms of different rank d o n the Epinions dataset. T able 4: MAE Results on Slashdot Dataset Observed entries (%) Methods SVT OPTSpace SVP RRMC MMC 10 0.679 ± 0.00 8 0 .554 ± 0.017 0. 755 ± 0 .005 0.7 15 ± 0. 001 0 .546 ± 0.008 20 0.562 ± 0.00 4 0 .458 ± 0.008 0. 582 ± 0 .007 0.7 04 ± 0. 001 0 .437 ± 0.008 30 0.513 ± 0.03 0 0 .427 ± 0.009 0. 501 ± 0 .003 0.6 86 ± 0. 003 0 .395 ± 0.006 40 0.506 ± 0.04 1 0 .395 ± 0.009 0. 460 ± 0 .002 0.6 47 ± 0. 006 0 .380 ± 0.009 50 0.495 ± 0.06 0 0 .376 ± 0.004 0. 432 ± 0 .002 0.6 09 ± 0. 002 0 .366 ± 0.003 60 0.520 ± 0.06 5 0 .362 ± 0.011 0. 413 ± 0 .002 0.5 85 ± 0. 002 0 .350 ± 0.002 T able 5: RMSE Results on Slashdot Dataset Observed entries (%) Methods SVT OPTSpace SVP RRMC MMC 10 0.788 ± 0.00 8 0 .826 ± 0.021 0. 873 ± 0 .004 0.8 29 ± 0. 001 0 .774 ± 0.008 20 0.718 ± 0.02 0 0 .704 ± 0.013 0. 746 ± 0 .007 0.8 21 ± 0. 001 0 .679 ± 0.009 30 0.680 ± 0.04 3 0 .652 ± 0.006 0. 680 ± 0 .003 0.8 07 ± 0. 002 0 .633 ± 0.008 40 0.670 ± 0.05 6 0 .620 ± 0.009 0. 647 ± 0 .003 0.7 78 ± 0. 005 0 .615 ± 0.011 50 0.642 ± 0.05 5 0 .596 ± 0.009 0. 624 ± 0 .003 0.7 49 ± 0. 002 0 .581 ± 0.006 60 0.699 ± 0.08 4 0 .577 ± 0.011 0. 609 ± 0 .002 0.7 30 ± 0. 002 0 .566 ± 0.002 T able 6: Th e trade-off between accuracy and ef ficiency . Dataset SVT OPTSpace SVP RRMC MMC MAE RMSE Time MAE RMSE T ime M AE RMSE Time M AE RMSE Time MAE RMSE T ime Epinions 0.46 6 0.61 8 2 6.76 0 .288 0.531 15.26 0 .450 0.61 0 0.84 0.5 76 0. 651 43.11 0.2 62 0. 481 1.92 Slashdot 0.618 0.728 50.73 0.458 0.705 9.23 0.5 82 0.746 0.83 0 .715 0 .830 44.12 0 .437 0 .679 2.41 lines. Also, our algorithm is ord ers of magnitu de faster than other three baselines. This implies that MMC fav ors a good trade-off between the accura cy an d efficiency . 5.4 Examine The Influence of d The no n-conve x re formu lation (4.2) requires an exp licit rank estimation d o n th e true m atrix. In this section, we in vestigate the influence of d on t he Epinions dataset as an e xample. The rank d is chosen from [ 1, 5, 50 ,100, 300, 500] and the results are plot in Figure 2. W e observe th at the rank has little in- fluence on the perfo rmance. Th is is possibly because that the actual d ata has a low-rank structure ( close to r ank one). And from [ Burer and Monteiro, 2005 ] , we k now that if d is larger than the actual rank, an y local minimum of Eq. (4.2) is also a global optima. 6 Conclusion and Futur e W ork In this pap er , we form ulated the social tru st pre diction in th e matrix com pletion framework. In par ticular, due to the spe- cial structure of the social trust pro blem, i.e., the measu re- ments are 1-bit a nd th e observed en tries are no n-un iformly sampled, we presen ted a max-norm con strained 1 -bit ma trix completion (MMC) algorithm. Sin ce SDP solvers are not scalable to large scale matrices, we u tilized a n on-co n vex re- formu lation of the max-no rm, wh ich facilitates an efficient projected gradien t decent algorithm. W e empirically exam- ined ou r algorithm on two ben chmark datasets. Compared to other state- of-the- art m atrix co mpletion fo rmulation s, MMC consistently ou tperfor med them , which meets with recently developed theories on max -nor m. 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