Incomplete Multi-Modal Visual Data Grouping

Transcription

1 Proeedings of the Twenty-ifth International Joint Conferene on Artifiial Intelligene (IJCAI-16) Inomplete Multi-Modal Visual Data Grouping Handong Zhao, Hongfu Liu and Yun u Department of Eletrial and Computer Engineering, Northeastern University, Boston, USA, 115 College of Computer and Information Siene, Northeastern University, Boston, USA, 115 {hdzhao, hfliu, yunfu}@ee.neu.edu Abstrat Nowadays multi-modal visual data are muh easier to aess as the tehnology develops. Nevertheless, there is an underlying problem hidden behind the emerging multi-modality tehniques: What if one/more modal data fail? Motivated by this question, we propose an unsupervised method whih well handles the inomplete multi-modal data by transforming the original and inomplete data to a new and omplete representation in a latent spae. Different from the existing efforts that simply projet data from eah modality into a ommon subspae, a novel graph Laplaian term with a good probabilisti interpretation is proposed to ouple the inomplete multi-modal samples. In suh a way, a ompat global struture over the entire heterogeneous data is well preserved, leading to a strong grouping disriminability. As a non-trivial ontribution, we provide the optimization solution to the proposed model. In experiments, we extensively test our method and ompetitors on one syntheti data, two RGB-D video datasets and two image datasets. The superior results validate the benefits of the proposed method, espeially when multimodal data suffer from large inompleteness. 1 Introdution In reent years, a large volume of tehniques emerge in artifiial intelligene field thanks to the easy aessibility of multimodal data aptured from multiple sensors [Cai et al., 13; Zhao and u, 15; Zhang et al., 15; Liu et al., 16]. Working in an unsupervised manner, multi-modal grouping (or lustering) offers a general view of the heterogeneous data grouping struture, whih has been drawing extensive attention [Bikel and Sheffer, 4; Ding and u, 14; Blashko and Lampert, 8; Chaudhuri et al., 9; red and Jain, 5; Singh and Gordon, 8; Cao et al., 15]. While beneath the prosperous studies of the multi-modal data grouping problem, there is an underlying problem, i.e., when the data from one modality/more modalities are inaessible beause of sensor failure or other reasons, most methods mentioned above would inevitably degenerate or even fail. RGB Modality Depth Modality Missing Missing Compat Global Struture Data Projetion Grouping in Latent Spae igure 1: ramework of the proposed method. Take RGB-D video sequene as an example, to solve the IMG problem, we projet the inomplete RGB-D data into a latent spae as well as preserve the ompat global struture simultaneously. In this paper, we fous on this hallenging, i.e., Inomplete multi-modality Grouping (IMG) problem. To solve IMG problem, a natural thought is to reuse the existing tehniques by remedying the inomplete data. In [Li et al., 14], two strategies are applied to failitate to fit IMG problem, i.e., remove samples suffering from missing information or preproess the inomplete samples to fill in the missing information. Obviously, the first strategy hanges the number of samples, whih essentially disobeys the goal of the original problem. The seond strategy has been experimentally tested to be not good enough [Shao et al., 15]. Most reently, there are few attempts proposed to solve IMG problem. [Li et al., 14] proposed a pioneer work to handle two-modal inomplete data ase, by projeting the partial data into a ommon latent subspae via nonnegative matrix fatorization (NM) and `1 sparse regularizer. ollowing this line, a similar idea of weighted NM and `,1 regularizer was proposed in [Shao et al., 15]. However, both methods [Li et al., 14; Shao et al., 15] overlook the global struture over the entire data samples. Inspired by this, we propose a novel method integrating the latent subspae generation and the ompat global struture into a unified framework as shown in igure 1. More speifially, a novel graph Laplaian term oupling the omplete visual data samples is introdued in latent spae, where the similar samples are more likely to be grouped together

2 Compared with the existing approahes, the ontributions of our method are three folds: We propose a novel method to deal with IMG problem for visual data with the onsideration of the ompat global struture in the low-dimensional latent spae. The pratie is ahieved through a Laplaian graph on omplete data instanes, bridging the omplete-modal samples and partial-modal samples. Nontrivially, we provide the optimization solution to our proposed objetive, where three auxiliary variables are introdued to make the optimization of the proposed graph Laplaian term happen under the inomplete multi-modality setting. The superior results on six datasets, i.e., one syntheti and four visual datasets, validate the effetiveness of the proposed method. Speifially, when data suffer from large inompleteness, we raise the performane bar by more than 3% and 1% for the syntheti and realworld visual data, respetively. Method We start with the introdution of some basi operator notations used in this paper. tr( ) is the operator to alulate the trae of matrix. ha, Bi is the inner produt of two matrixes alulated as tr(a T B). k k denotes the robenius norm. Operator (A) + works as max(,a) to make the matrix (or vetor) non-negative. Other variable and parameter notations are introdued later in the manusript..1 Inomplete multi-modality Grouping or the ease of disussion, we use two-modal ase for illustration. Given a set of data samples = [x 1,...,x i,...,x N ], i = 1,...,N, where N is the total number of samples. Eah sample has two modalities, i.e., x i =[x i,x i ]. or IMG problem, we follow the setting in [Li et al., 14], the input data is separated as an inomplete modal sample set ˆ = { ˆ(1,), ˆ, ˆ } instead of the omplete multi-modal data, where ˆ (1,), ˆ, and ˆ denote the data samples presented in both modalities, modal-1 and modal-, respetively. The feature dimensions of modal-1 and modal- data are d 1 and d, and the numbers of shared samples and unique samples in modal-1 and modal- are, m and n, respetively. Aordingly, we have ˆ (1,) R (d1+d), ˆ R m d1, ˆ R n d, N = + m + n. Same as traditional multi-view lustering, the goal of IMG is to group the samples into their orresponding lusters. Previous methods, like MultiNM [Liu et al., 13] and PVC [Li et al., 14], pursuit a ommon latent spae via nonnegative matrix fatorization (NM) where samples from different views an be well grouped. In this work, we follow this line to find a latent ommon subspae for heterogenous multi-modal visual data. However differently, we get rid of the non-negative onstraint to make the optimization muh easier. Besides, the major ontribution of this paper is to demonstrate the orretness and effetiveness of the proposed global onstraint in IMG problem. Given the latent dimension of projetive subspae k, we denote P R k and P R k as the latent representations of ˆ (1,) =[ ; ] from two different modalities. Note that and are the samples existing in both modalities, thus P and P are expeted to be lose, i.e., P! P P. Consequently, we have the basi inomplete multi-modality grouping formulation as min P, ˆP, ˆP U,U ˆ P ˆP U ˆ P ˆP U + + (ku k +ku k ), where is a trade-off parameter, P is the shared latent representation from and, U R k d1, U R k d are known as the bases in matrix deomposition, ˆP R m k, ˆP R n k are the latent low-dimensional oeffiients for missing modal samples orresponding to ˆ and ˆ. Regularizers ku k and ku k are used to prevent the trivial solution.. Complete Graph Laplaian By onatenating the projetive oeffiients in latent subspae P =[P ; ˆP ; ˆP ] R N k, we an diretly apply lustering method on P to get the luttering result. However, it is deserved to know that the learned oeffiients P is without global property whih is ruial in subspae lustering. or the traditional multi-modality learning problem, global onstraints are easy to be inorporated beause of the omplete modality setting, suh as low-rank onstraint in [Ding and u, 14]. While in IMG problem, this annot be easily ahieved. To takle this, we propose to learn a Laplaian graph inorporating all the samples in the latent spae. Integrating the idea of graph Laplaian and Eq. into the unified objetive funtion, we have our formulation with the omplete graph Laplaian term G as Here, min P, ˆP, ˆP U,U,A ˆ P ˆP U ˆ P ˆP U s.t. 8i A T i 1 =1,A i. + + G(P, A)+R(U, A). G(P, A) = tr(p T L A P ), (3) R(U, A) = (ku k + ku k )+ kak, (4) where L A R N N is the Laplaian matrix of similarity matrix A R N N, defined by L A = D A, in whih the degree matrix is the diagonal matrix with D ii = P N j=1 A ij. Several remarks are made here. 393

3 Remark 1: Thanks to the graph Laplaian term L A, we bridge the sample onnetion between omplete modal samples and partial modal samples. In suh a way, the global onstraint on the omplete set of data samples is integrated into the objetive, whih in turn influenes the projeted oeffiients in low-dimensional spae with the global struture. The pratie of adding a graph term on the omplete set of data gives the name of omplete graph Laplaian. Remark : A is the affinity graph, with eah element denoting the similarity between two data samples in latent subspae. We normalize eah olumn as the summation equals to 1 as well as all the elements are nonnegative, making A have a probability interpretation. This naturally provides us with an opportunity to do spetral lustering on the optimized A, whih none of the existing partial multi-view methods have. Remark 3: As shown in Eq. (4), the regularizers we add are all robenius norms for simpliity. Aording to [Lu et al., 1], other regularizers suh as `1-norm or trae (nulear) norm are also good hoies for preserving the global struture that benefits lustering performane..3 Optimization As seen in Eq., in order to learn a meaningful affinity matrix in a unified framework, our proposed objetive inludes several matrix fatorization terms, regularizers and onstraints. It is obviously not jointly onvex w.r.t. all the variables. Instead, we plan to update eah variable at a time via augmented Lagrange Multiplier (ALM) with alternating diretion minimizing strategy [Lin et al., 11]. However, it is noted that P, ˆP and ˆP are diffiult to be optimized beause of the following reasons: Laplaian graph L A is the graph measuring the affinities among all sample points, that is, we have to update them as a whole; There is no way to diretly ombine P, ˆP and ˆP for optimization sine ˆP and ˆP do not share the same basis, and even the size of input data in modal-1 is not the same as that in modal- ([ ; ˆ ] R (+m) d1 for modal- 1, [ ; ˆ ] R (+n) d for modal-). This dilemma makes the variables P, ˆP and ˆP impossible to be optimized individually nor together as P. To solve this hallenge, we propose to introdue three auxiliary variables Q R k, ˆQ R m k and ˆQ R n k for P, ˆP and ˆP respetively. In this way, we separately update the affinity matrix A (Laplaian L A ) and matrix fatorization with the bridges of P = Q, ˆP = ˆQ and ˆP = ˆQ. Correspondingly, the augmented Lagrangian funtion of Eq. with three auxiliary variables is written as C (8i A T i 1=1;A i ) = ˆ P ˆP U ˆ P ˆP U + + (ku k + ku k ) + tr(q T L A Q)+ kak + hy,p Qi + µ kp Qk, (5) where Y =[Y ; Ŷ ; Ŷ ] is the lagrangian multiplier, and µ > is a penalty parameter. Speifially, variables P, Q, U, U, A in the +1iteration are updated as follows: Update U &U. ixing P, Q and A, the Lagrangian funtion w.r.t. U ( +1) is written as: C(U )=k P U k + ku k. (6) This is a standard least square problem with regularization, with its solution as U ( +1) =(P ( ) T P ( ) + I k ) 1 P ( ) T. (7) Here I k is the identity matrix with k-dimension. Similarly, we have the following funtion to update U ( +1) : U ( +1) =(P ( ) T P ( ) + I k ) 1 P ( ) T. (8) Update P. This part inludes three subproblems, i.e., update P ( +1), ˆP ( +1) and ˆP ( +1). or P ( +1), by fixing other variables, the orresponding Lagrangian funtion is C(P )=k P U k + k + hy,p Q i + µ kp Q k. P U k With the help of KKT ondition ))/@(P )=, we have the following solver for P ( +1) =: U T ( +1) + U T ( +1) Y( ) +µq ( ) R 1 ( +1), (9) where R ( +1) =U ( +1) U T ( +1) +U ( +1) U T ( +1) + µik. Similarly, we obtain the solutions for ˆP ( +1) as ( U T ( +1) Y1( ) +µ and ˆP ( +1) as ( U T ( +1) Y( ) +µ ˆQ ( ))(U( +1) U T ( +1) +µik ) 1, (11) ˆQ ( ))(U( +1) U T ( +1) +µik ) 1. (1) ( +1). Update Q. Reall that the motivation of introduing auxiliary Q = [Q ; ˆQ ; ˆQ ] is to bridge the gap of global representation of all the data samples in different modalities. Therefore, instead of individually updating Q ( +1), ˆQ ( +1) and ˆQ We update Q ( +1) as a whole, with the Lagrangian funtion written as C(Q) = tr(q T L A Q)+hY,P Qi + µ kp Qk. (13) Correspondingly, the solver of Q ( +1) via KKT ondition is (L A T ( ) + L A( ) )+µi N 1 (Y( ) + µp ( +1) ), (14) where I N is the identity matrix with N-dimension. 394

4 Update A (L A ). ixing other variables, the graph A- problem is in the following form min tr(q T L A Q)+ kak A s.t. 8i A T i 1 = 1; A i. (15) As disussed in Remark, A has the probability interpretation with eah element onsidered as the similarity probability between two data samples. Therefore, we divide problem (15) into a set of subproblems A i ( +1) aording to sample index i as A i ( +1) = argmin ka i + S( +1) i k, (16) A i { T 1=1; } where S( +1) i is a olumn vetor with its element j defined as S ij ( +1) = kqi ( +1) Q j ( +1) k. A detailed dedution an 4 be found in [Guo, 15]. To sum up, for the omplete algorithm, we initialize the variables and parameters (in the iteration #, denoted as () ) in ALM as follows: penalty parameter µ () = 1 3, =1.1, the max penalty parameter µ max = 1 6, stopping threshold = 1 6, P () = Q () = Y () = R N k, P () = Q () = Y () = R k, ˆP () = ˆQ () = Ŷ () = R m k, ˆP () = ˆQ () = Ŷ () = Rn k, U () = R k d1, U () = Rk d, A () = L A() = R N N. Then we update eah variable one by one as disussed above until onvergene..4 Complexity Analysis Note that with different partial example ratios, the dimensions of ˆ (1,), ˆ and ˆ are different, i.e.,, m, n vary. or simpliity, we onsider the extreme ase that no inomplete data exist, that is, omplete (traditional) multi-view lustering ase. The feature dimensions of different modalities are all d. In Algorithm 1, the most time-onsuming parts are the matrix multipliation and inverse operations when updating U, U, P, Q, A. or eah iteration, the inverse operations in Eqs. (7)(8)(11)(1) ost O(k 3 ) due to the k k size matrix. While the inverse on graph in Eq. (14) takes time of O(N 3 ). Usually k N, then the asymptoti upper-bound for inverse operation an be expressed as O(N 3 ). The multipliation operations take O(dkN) when updating U, U, P, Q. It osts O(N 3 ) when updating A. However, it is noted that the number of operations of O(N 3 ) in eah iteration is only, the major omputations are of order O(dkN). Suppose M is the number of operations onsuming O(dkN), L is the iteration time. In sum, the time omplexity of our algorithm is O(MLdkN +LN 3 ). 3 Experiment Syntheti data are omprised of two modalities. We first hoose the luster i eah sample belongs to, and then generate eah of the modalities x i and x i from a two-omponent Gaussian mixture model. Two modalities are ombined to form the sample (x i,x i, i ). We sample 1 points from eah modality. The luster means in modal-1 are µ 1 = (1 1), µ = (3 4), in modal- are µ 1 = (1 ), µ = ( ). The ovariane for modal-1 are =.5 1.5,.3 =.1 1 =.1,.1 =.1.5 Real-world visual data: (a) MSR Ation Pairs dataset [Oreifej and Liu, 13] is a RGB-D ation dataset ontaining 1 types of ativities performed by 1 subjets. Eah ator provides 36 videos for eah modality. (b) MSR Daily Ativity dataset [Wang et al., 1] ontains 16 types of ativities performed by 1 subjets. Eah ator repeats an ation twie, providing 3 videos for eah of the RGB and depth hannels. or the above two RGBD video sequenes, we temporally normalize eah video lip to 1 frames with spatial resolution of Histograms of gradient oriented feature is extrated from both depth and RGB videos with path size 8 8. Thus, a total of 3 pathes are extrated from eah video, with the feature dimensionality of 31. We will larify this in our final version. () BUAA NirVis [Huang et al., 1] ontains two types of data, i.e., visual spetral (VIS) and near infrared (NIR) data. The first 1 subjets with 18 images are used. To fasten the omputation, we resize the images to 1 1, and vetorize them. (d) UCI handwritten digit 1 onsists of -9 handwritten digits data from UCI repository. It inludes examples, with one modality being the 76 ourier oeffiients and modal- being the 4 pixel averages in 3 windows. or the ompared methods, we onsider the following algorithms as the baselines. BSV (Best Single View): Due to the missing samples in eah modality, we annot diretly perform k-means lustering on eah modality data. ollowing [Shao et al., 15], we firstly fill in all the missing data with the average features for eah modality, and then perform lustering on eah modality, and report the best result. Conat: eature onatenation is a straightforward way to deal with multi-modal data, whih serves as our seond baseline. Same as BSV, we firstly fill in all the missing data with the average features for eah modality, and then onatenate all modal features into one. (3) MultiNM: Multi-view NM [Liu et al., 13] seeks a ommon latent subspae based on joint NM, whih an be approximately regarded as the omplete-view ase of PVC. or the syntheti data, there are few data points ontaining negative values. In order to suessfully run the ode, we make the input data nonnegative as preproessing. (4) PVC: Partial multi-view lustering [Li et al., 14] is one of the most reent works in dealing with inomplete multi-modal data. This work an be onsidered as our proposed model without the omplete graph Laplaian. One important parameter on regularizer is hosen from the parameter grid of {1e-4, 1e-3, 1e-, 1e-1, 1e, 1e1, 1e}, inluding the default 1e- used in the original paper

5 Table 1: /Preision results on syntheti data under different PER settings. Method \ PER BSV 19 / / / / /.519 Conat 644 / 9 19 / / / /.5965 MultiNM.5767 / / / / / 985 PVC 194 / / / / / 833 Ours 781 / / / / / 947 Table : /Preision results on MSR Ation Pairs dataset under different PER settings. Method \ PER BSV 87 / / / / /.185 Conat 7 / / / / / 68 MultiNM 33 / / / / / 67 PVC 917 / / / / /.369 Ours 859 / / / / /.3734 Table 3: /Preision results on MSR Daily Ativity dataset under different PER settings. Method \ PER BSV 1 / / / / /.66 Conat 499 / / / / /.758 MultiNM 77 / / / / /.674 PVC 65 / / / / /.9 Ours 87 / / / / /.97 or the evaluation metri, we follow [Li et al., 14], using Normalized Mutual Information (). Besides, preision of lustering result is also reported to give a omprehensive view. Same as [Li et al., 14], we test all the methods under different partial/inomplete example ratio (PER) varying from.1 to.9 with an interval of. 3.1 Experimental Result or eah dataset, we randomly selet samples from eah modality as the missing ones. Note that our method not only learns a better low-dimensional representation but also learns a similarity matrix among samples iteratively. This naturally gives us two opportunities to do lustering, i.e., k-means lustering on the latent representation P, and spetral lustering on the learned affinity graph A. To make the fair omparison, k-means results on P are reported. or eah experiment, we repeats 1 times to get the average performane, the standard deviations are omitted here beause it is observed that these values are usually small. Table 1,,3 and igure report the values and preision on syntheti, video and image datasets with different PER ratio settings. rom these tables and bar graphs, the following observations and disussions are made. The proposed method performs superiorly to the other baselines in almost all the settings; Espeially for the hallenging syntheti data, we raise the performane bar by around 31.83% in. With more missing modal samples (PER ratio inreases), the performane of all the methods drops. With more missing modal samples, our method improves more ompared with the state-of-the-art baseline. Speifially, our improvement reahes 1.34% (PER=.9) from 4.58% (PER=.1) for five real-world video/image datasets. or the real-world data, as PER ratio grows, the extent that performane drops is less than that of syntheti data. Disussion: The first observation experimentally demonstrates that the proposed omplete graph Laplaian term works in both syntheti and video/image multi-modal data, espeially when some modal data are missing to a large extent. Note that for the matrix fatorization part, we use the simplest way with only robenius norm regularization on basis matrix. However, we still outperform the ompetitors with the help of omplete graph Laplaian term. With better matrix fatorization tehniques, e.g. NM in [Li et al., 14] or weighted NM in [Shao et al., 15], we believe that a better performane will be ahieved. With no doubt, the problem beomes more hallenging when the number of shared samples is fewer. However, one may be urious why our method performs muh better than others in syntheti data. The possible reason is that ompared with modal-1 data, modal- data are diffiult to separate. When points in modal-1 are missing, the existing methods annot do a good job with only modal- data even in latent spae. However, thanks to the affinity graph built on all data points, the data points from different lusters are iteratively pulled towards their orresponding luster enters by the influene of global onstraint. This enlightens us that in real-world multi-modal visual data, if one modal data perform poorly (e.g. people blend in the bakground visually) than the others, our proposed omplete graph Laplaian term is apable to make it up from other disriminative modal data (e.g. disriminative depth information between people and bakground). One may be also interested in 396

6 Preision (a) BUAA-NirVis dataset Preision (b) UCI handwritten digits dataset Ours PVC MultiNM Conat BSV igure : /Preision results on (a) BUAA-NirVis dataset and (b) UCI handwritten digits dataset. the reason why our method has a onsiderable improvement when data suffer from a large inompleteness. We believe, as PER ratio inreases, the state-of-the-art method PVC degenerates dramatially beause the ommon projetion P beomes harder to be aurately estimated simply from the less shared multi-modal data. Nevertheless, our proposed omplete graph Laplaian remedies the deviation by onsidering the global struture of inomplete multi-modal data in the latent spae, whih further leads to a robust grouping struture. 3. Convergene Study To show the onvergene property, we ondut an experiment on syntheti data with PER ratio set as.3 and parameters {,, } set as {1e-, 1e, 1e}. The relative error of stop riterion kp Q k 1 is omputed in eah iteration. The red urve in igure 3(a) plots the onvergene urve of our model. It is observed that after the first several iterations bump, the relative error drops steadily, and then meets the onvergene at around #4 iteration. The value during eah iteration is drawn in blak. It an be seen that there are three stages before onverging: the first stage (from #1 to #4), the value grows dramatially; the seond stage (from #5 to #4), the bumps in a ertain range but grows; the final stage (from #41 to the end), the ahieves the best at the onvergene point. 3.3 Parameter Study There are three major parameters in our approah, i.e.,, and. Same as onvergene study, we ondut the parameter analytial experiments on syntheti data with PER ratio set as.3. igure 3(b) shows the experiment of result w.r.t. the parameter under two settings {[ =1e, =1e]; [ =1e, =1e]}. We selet the parameter in the grid of {1e-3, 1e-, 1e-1, 1e, 1e1, 1e, 1e3}. It is observed that our method has a relatively good performane when is in the range of [1e-3, 1e-1], and drops when beomes larger. The experiments shown in igure 3(,d) are designed to test the robustness of our model w.r.t. the trade-off parameters and on the proposed graph Laplaian term. As we Relative Error Iteration 1 (a) λ = 1e, γ = 1e 1 λ = 1e, γ = 1e 1e 3 1e 1e 1 1e 1e1 1e 1e3 Parameter β () 1 β = 1e, γ = 1e β = 1e, γ = 1e 1e 3 1e 1e 1 1e 1e1 1e 1e3 Parameter λ (b) λ = 1e, β = 1e λ = 1e, β = 1e1 1e 3 1e 1e 1 1e 1e1 1e 1e3 Parameter γ igure 3: Convergene and parameter studies on syntheti data. (a) shows the relative error and result w.r.t. iteration times. (b-d) plot the results in terms of parameters, and respetively. or eah parameter analysis, we run two different settings shown in red irle and blak ross. observe, the s under different settings reah a relatively good performane when = [1e-1, 1e, 1e1] and =1e. 4 Conlusion In this paper, we proposed a method dealing with inomplete multi-modal visual data grouping problem with the onsideration of the ompat global struture via a novel graph Laplaian term. This pratie bridged the onnetion of missing samples data from different modalities. Superior experimental results on syntheti data and four real-world multi-modal visual datasets ompared with several baselines validated the effetiveness of our method. (d) 397

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