COMPLETER: Incomplete Multi-View Clustering via ...

1 COMPLETER: Incomplete Multi-View Clustering via Contrastive PredictionYijie Lin1, Yuanbiao Gou1, Zitao Liu2, Boyun Li1, Jiancheng Lv1, Xi Peng1 1 College of Computer Science, Sichuan University, Education Group, this paper, we study two challenging problems in in-complete Multi-View Clustering analysis, namely, i) how tolearn an informative and consistent representation amongdifferent views without the help of labels and ii) how to re-cover the missing views from data. To this end, we proposea novel objective that incorporates representation learningand data recovery into a unified framework from the viewof information theory. To be specific, the informative andconsistent representation is learned by maximizing the mu-tual information across different views through contrastivelearning, and the missing views are recovered by minimiz-ing the conditional entropy of different views through dualprediction.

2 To the best of our knowledge, this could be thefirst work to provide a theoretical framework that unifies theconsistent representation learning and cross-view data re-covery. Extensive experimental results show the proposedmethod remarkably outperforms 10 competitive multi-viewclustering methods on four challenging datasets. The codeis available IntroductionIn the real world, Multi-View data, which often exhibitheterogeneous properties, is collected from diverse sensorsor obtained from various feature extractors. As one of themost important unsupervised Multi-View methods, Multi-View Clustering (MVC) aims to separate data points into dif-ferent clusters in an unsupervised fashion [11,17,20,29,40,54]. To achieve the end, the key is exploring the consistencyacross different views so that a common/shared representa-tion is learned [5,12,14,21,33,47]. Behind the consistencylearning, the implicit assumption is that the views are com-plete, , all data points will present in all possible practice, however, some views of data points might bemissing due to the complexity in data collection and trans-mission, leading to so-calledincomplete Multi-View prob- Corresponding authorFigure 1.

3 Our basic observation and theoretical results from theperspective of information theory. In the figure, the solid and dot-ted rectangles denote the information contained in view 1 (X1) andview 2 (X2), respectively. In mathematical, the mutual informa-tionI(Z1,Z2)(grey area) quantifies the amount of informationshared byZ1andZ2, whereZ1andZ2are the representationsofX1andX2, respectively. To learn consistent representations,it is encouraged to maximizeI(Z1,Z2). In addition, minimiz-ing the conditional entropyH(Zi|Zj)(blue area) will encouragethe recovery of missing view becauseZiis fully determined byZjif and only if the conditional entropyH(Zi|Zj) = 0, wherei= 1, j= 2ori= 2, j= 1. Subtly, on the one hand,the maximization ofI(Z1,Z2)could increase the amount of theshared information, thus the data recoverability could be benefited, , it is easier to recover one view from the other. On the otherhand, asH(Zi|Zj)quantifies the amount of information ofZiconditioned onZj, the minimization ofH(Zi|Zj)will encour-age to discard the inconsistent information across-views, and thusthe consistency could be further improved.

4 With the above ob-servation, cross-view consistency and data recovery are treated astwo sides of one coin under the above unified information (IMP). For example, in online meetings, some videoframes might lose the visual or audio signal due to thebreakdown of sensors. To solve IMP, some incompletemulti-view Clustering algorithms (IMC) have been proposedby employing numerous data recovery methods to com-plete the missing data, , matrix factorization based meth-ods [10,22,35,46,53] and generative adversarial networksbased methods [16,41,45]. These works have attemptedto overcome the following two challenges: i) how to learninformative and consistent representations across differentviews? and ii) how to eliminate the influence of the miss-11174ing views? Although some promising results have beenachieved, almost all existing works treat these two chal-lenges as two independent problems and a unified theoreti-cal understanding is still from existing IMC studies, we theoreticallyshow that cross-view consistency learning and data recov-ery could be treated as two sides of one coin and thesetwo challenging tasks could mutually boost.

5 Our motiva-tion comes from [38], as shown in It should bepointed out that, [38] utilizes predictive learning to enhancethe performance of contrastive learning, while we aim atrecovering the missing data through dual prediction. More-over, another difference lies on our theoretical result, ,the data recovery and consistency learning could mutuallyboost through contrastive learning and dual on our observations and theoretical results,we propose a novel Incomplete Multi-View clusteringmethod, termed inCOMP lete Multi-View Clustering viaconTrastivE pRediction (COMPLETER). In detail, COM-PLETER projects a given dataset into a feature spacewherein information consistency and data restorability areguaranteed using three jointly learning objectives. Morespecifically, a within-view reconstruction loss is used tolearn a view-specific representation so that the trivial solu-tion is avoided. In the latent feature space, a contrastive lossis introduced to learn the cross-view consistency by maxi-mizing mutual informationI(Z1,Z2), and a dual predictionloss is used to recover the missing view by minimizing con-ditional entropyH(Z1|Z2)andH(Z2|Z1).

6 It should bepointed out that the data recovery referred in this paper istask-oriented, , only the shared instead of all informationwould be recovered to facilitate the downstream tasks likeMVC. To summarize: We provide a novel insight to the community, , thedata recovery and consistency learning of incompletemulti-view Clustering are with intrinsic connections,which could be elegantly unified into the framework ofinformation theory. Such a theoretical view is remark-ably different from existing works which treat consis-tency learning and data recovery as two separate prob-lems. The proposed COMPLETER method is with a novelloss function which achieves the information consis-tency and data restorability using a contrastive loss anda dual prediction loss. Extensive experiments verifythe effectiveness of the proposed loss Related WorkIn this section, we briefly review some recent develop-ments in two related topics, namely, Incomplete multi-viewclustering and contrastive Incomplete Multi-View ClusteringBased on the way of utilizing the Multi-View informa-tion, most existing IMC methods could be roughly classifiedinto three categories, , matrix factorization (MF) basedIMC [10,22,35,53], spectral Clustering based IMC [39], andkernel learning based IMC [26].

7 In brief, MF based meth-ods project the Incomplete data into a common subspaceby utilizing the low-rankness. For example, DAIMC [10]establishes a consensus basis matrix with the help of 2,1-norm and IMG [53] utilizes the F-norm to reduce the in-fluence of missing data. As a typical spectral clusteringbased method, PIC [39] learns the common representationusing a consistent Laplacian graph constructed from incom-plete views. EERIMVC [26] proposes using a multi-kernelmethod to achieve IMC in an iterative optimization , the methods like [16,41] utilize cycleGAN [55]to generate the missing view from the complete views andCDIMC-net [44] incorporates the view-specific encodersand the graph embedding strategy to handle the incompletemulti-view differences between this study and existing worksare given below. First, we aim to infer the missing datarather than the missing similarity, thus enjoying higher in-terpretability [26].

8 Second, our method is a deep ratherthan shallow model [10,19,22,26,35,39,53], thus nat-urally embracing the capacity of handling complex andlarge-scale dataset. Third, almost all existing IMC meth-ods [10,16,22,26,35,39,41,53] treat data recovery and con-sistency learning as two independent problems/steps, whilelacking a theoretical understanding. In contrast, we pro-posed that data recovery and consistency learning could beunified into the framework of information theory [36]. Bothdata recovery and consistency learning could be of benefitto learning the common Contrastive LearningAs one of most effective unsupervised learningparadigms, contrastive learning [2,4,8,23,28,30,37,38]has achieved state-of-the-art performance in representationlearning. The basic idea of contrastive learning is learninga feature space from raw data by maximizing the similar-ity between positive pairs while minimizing that of neg-ative pairs.

9 In recent, some studies show that the suc-cess of contrastive learning could attribute to the maximiza-tion of mutual information. For example, MoCo [9] andCPC [30] minimize the InfoNCE loss that can be regardedas maximizing a lower bound on mutual information, ,I(Z1,Z2) log(N) LNCE, whereNis the number ofnegative pairs,Z1andZ2are the latent representations ofmulti-view dataX1andX2, differences between this work and existing con-trastive learning studies are as below. First, most exist-ing contrastive learning methods [2,8,9,28] aim to handle11175 Figure 2. Overview of COMPLETER. In the figure, bi-view data is used as a showcase. As shown, our method contains three joint learningobjectives, , within-view reconstruction, cross-view contrastive learning, and cross-view dual prediction. To be specific, the within-viewreconstruction objective aims to project all views into view-specific spaces with the minimal reconstruction loss.

10 The cross-view contrastivelearning objective is implemented by maximizing the mutual information betweenZ1andZ2. The cross-view dual prediction objectiveutilizes two mappingG(1)andG(2)to recover one view from another one by minimizing the conditional entropyH(Zi|Zj).single-view data and exhaustively explore a variety of dataaugmentations to build different views/augmentations. Incontrast, our method aims to learn consistency from a givenmulti-view dataset. To the best of our knowledge, this couldbe one of the first studies on Multi-View contrastive learn-ing. Second, our method is specifically designed for han-dling missing data, whereas the existing contrastive learningworks ignore this practical problem. Third, although exist-ing contrastive learning studies have shown that the con-sistency could be learned by maximizing the mutual infor-mation of different augmentations, they ignore the incon-sistency learning.