jmlr jmlr2013 jmlr2013-15 knowledge-graph by maker-knowledge-mining

15 jmlr-2013-Bayesian Canonical Correlation Analysis

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Author: Arto Klami, Seppo Virtanen, Samuel Kaski

Abstract: Canonical correlation analysis (CCA) is a classical method for seeking correlations between two multivariate data sets. During the last ten years, it has received more and more attention in the machine learning community in the form of novel computational formulations and a plethora of applications. We review recent developments in Bayesian models and inference methods for CCA which are attractive for their potential in hierarchical extensions and for coping with the combination of large dimensionalities and small sample sizes. The existing methods have not been particularly successful in fulﬁlling the promise yet; we introduce a novel efﬁcient solution that imposes group-wise sparsity to estimate the posterior of an extended model which not only extracts the statistical dependencies (correlations) between data sets but also decomposes the data into shared and data set-speciﬁc components. In statistics literature the model is known as inter-battery factor analysis (IBFA), for which we now provide a Bayesian treatment. Keywords: Bayesian modeling, canonical correlation analysis, group-wise sparsity, inter-battery factor analysis, variational Bayesian approximation

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We review recent developments in Bayesian models and inference methods for CCA which are attractive for their potential in hierarchical extensions and for coping with the combination of large dimensionalities and small sample sizes. [sent-10, score-0.142]

2 Keywords: Bayesian modeling, canonical correlation analysis, group-wise sparsity, inter-battery factor analysis, variational Bayesian approximation 1. [sent-13, score-0.175]

3 Introduction Canonical correlation analysis (CCA), originally introduced by Hotelling (1936), extracts linear components that capture correlations between two multivariate random variables or data sets. [sent-14, score-0.165]

4 While the analysis part of Browne (1979) is limited to the special case of CCA, the generic IBFA model describes not only the correlations between the data sets but provides also components explaining the linear structure within each of the data sets. [sent-27, score-0.147]

5 If the analysis focuses on only the correlating components, or equivalently the latent variables shared by both data sets, the solution becomes equivalent to CCA. [sent-29, score-0.259]

6 Using the term CCA emphasizes ﬁnding of the correlations and shared components, whereas IBFA emphasizes the decomposition into shared and data source-speciﬁc components. [sent-35, score-0.203]

7 , 2011), and in particular provide two efﬁcient inference algorithms, a variational approximation and a Gibbs sampler, that automatically learn the structure of the model that is, in the general case, unidentiﬁable. [sent-37, score-0.183]

8 The model is solved as a generic factor analysis (FA) model with a speciﬁc group-wise sparsity prior for the factor loadings or projections, and an additional constraint tying the residual variances within each group to be the same. [sent-38, score-0.155]

9 At the core of the generative process is an unobserved latent variable z ∈ RK×1 , which is transformed via linear mappings to the observation spaces to represent the two multivariate random variables x(1) ∈ RD1 ×1 and x(2) ∈ RD2 ×1 . [sent-51, score-0.145]

10 1 Inter-battery Factor Analysis In the latent variable model studied in this work, z ∼ N(0, I), z(m) ∼ N(0, I), (2) x(m) ∼ N(A(m) z + B(m) z(m) , Σ(m) ), following the probabilistic interpretation of inter-battery factor analysis by Browne (1979). [sent-61, score-0.16]

11 The shared latent variables z capture the variation common to both data sets, and they are transformed to the observation 1. [sent-67, score-0.233]

12 The shaded nodes x(m) denote the two observed random variables, and the latent variables z capture the correlations between them. [sent-70, score-0.152]

13 The variation speciﬁc to each view is modeled with view-speciﬁc latent variables z(m) . [sent-71, score-0.15]

14 The remaining variation is modeled by the latent variables z(m) ∈ RKm ×1 speciﬁc to each data set, transformed to the observation space by another linear mapping B(m) z(m) , where B(m) ∈ RDm ×Km . [sent-74, score-0.15]

15 The process starts by integrating out the view-speciﬁc latent variables z(m) , to reach a model that has explicit components only for the shared variation similarly to how CCA only explains the correlations. [sent-83, score-0.343]

16 The latent representation of this model is simpler, only containing the z instead of three separate sets T of latent variables, but the diagonal covariance of the IBFA model is replaced with B(m) B(m) +Σ(m) . [sent-85, score-0.328]

17 The model comes with three separate sets of latent variables with component numbers K, K1 and K2 . [sent-105, score-0.198]

18 However, individual components can be moved between these sets without inﬂuencing the likelihood of the observed data; removal of a shared component can always be compensated by introducing two view-speciﬁc components, one for each data set, that have the same latent variables. [sent-106, score-0.273]

19 This is done either by explicitly parameterizing the noise through a covariance matrix Ψ(m) as in (3) or by the separate view-speciﬁc components B(m) z(m) as in (2). [sent-113, score-0.148]

20 It may be possible to identify these components as view-speciﬁc in a post-processing step to reach interpretation similar to CCA, but directly modeling the view-speciﬁc variation as separate components has obvious advantages. [sent-117, score-0.213]

21 In particular, such models are likely to misinterpret strong view-speciﬁc variation as a shared effect, since they have no means of explaining it otherwise. [sent-124, score-0.139]

22 Inference For learning the IBFA model we need to infer both the latent signals z and z(m) as well as the linear projections A(m) and B(m) from data. [sent-127, score-0.165]

23 3 The posterior of the model then becomes (m) one where the number of components is automatically selected by pushing αk of unnecessary (m) components towards inﬁnity. [sent-141, score-0.208]

24 971 K LAMI , V IRTANEN AND K ASKI to the ARD prior updates being more efﬁcient in the variational framework, whereas the latter is easier to extend, as demonstrated by Klami and Kaski (2007) by using the Bayesian CCA as part of a non-parametric hierarchical model. [sent-156, score-0.187]

25 Hence the direct Bayesian treatment m of CCA needs to resort to either using very strong priors (for example, favoring diagonal covariance matrices and hence regularizing the model towards Bayesian PCA), or it will end up doing inference over a very wide posterior. [sent-162, score-0.153]

26 3, is that it requires learning three separate sets of components and the solution is unidentiﬁable with respect to allocating components to the three groups. [sent-172, score-0.18]

27 We solve the BIBFA model by doing inference directly for (5) and learn the structure of W by imposing group-wise sparsity for the components (columns of W), which results in the model automatically converging to a solution that matches (4) (up to an arbitrary re-ordering of the columns). [sent-187, score-0.25]

28 (2010) introduced a similar sparsity constraint for learning factorized latent spaces; our approach can be seen as a Bayesian realization of the same idea, applied to canonical correlation analysis. [sent-192, score-0.207]

29 Similarly to how ARD has earlier been used to choose the number of (m) components, the group-wise ARD makes unnecessary components wk inactive for each of the (m) views separately. [sent-196, score-0.153]

30 The components needed for modeling the shared response will have small αk (m) (that is, large variance) for both views, whereas the view-speciﬁc components will have small αk for the active view and a large one for the inactive one. [sent-197, score-0.27]

31 Finally, the model still selects automatically the total number of components by making both views inactive for unnecessary components. [sent-198, score-0.193]

32 Similarly, the explorative data analysis experiment illustrated in Figures 8 and 9 is invariant to the actual components and only relies on the total amount of contribution each feature has on the shared variation. [sent-208, score-0.153]

33 The ARD prior efﬁciently pushes the variance of inactive components towards zero, and hence selecting the threshold is often easy in practice. [sent-211, score-0.145]

34 3, the model is unidentiﬁable with respect to linear transformations of y, and only the prior p(y) is inﬂuenced by the allocation of the components into shared and view-speciﬁc ones. [sent-220, score-0.24]

35 The prior, in turn, assumes independent latent variables, implying that the optimal solution will result in an R that makes the latent variables of the posterior approximation also maximally independent. [sent-226, score-0.26]

36 We prefer to use the phrase independence as it better ﬁts the notion of assuming independent latent variables and may become more precise in extensions; for other priors and inference algorithms independence need not equal orthogonality. [sent-229, score-0.191]

37 the classical CCA solution is; the latent variables are assumed orthogonal, instead of assuming orthogonal projections (like in PCA). [sent-238, score-0.192]

38 That property also allows deriving a more efﬁcient algorithm for optimizing the variational approximation, following the idea of parameter-expanded variational Bayes (Qi and Jaakkola, 2007; Luttinen and Ilin, 2010). [sent-239, score-0.184]

39 In summary, the above model formulation with the associated variational approximation provides a fully Bayesian treatment for the IBFA model. [sent-243, score-0.151]

40 Here we brieﬂy discuss possible alternatives, and derive one practical implementation that uses sampling-based inference instead of the variational approximation described above. [sent-253, score-0.143]

41 As general properties, such priors allow continuous inference procedures that are often efﬁcient, but it is not always trivial to separate low-activity components from inactive ones for interpretative purposes. [sent-261, score-0.193]

42 Efﬁcient inference is done in the factor analysis model for x = [x(1) ; x(2) ] with group-wise sparsity prior for the projection matrix: y ∼ N(0, I), x ∼ N(Wy, Σ), (7) W(m) ∼ ARD(α0 , β0 ). [sent-284, score-0.166]

43 An alternative inference scheme replaces the above ARD prior with the group-wise spike-andslab prior of (6) and draws samples from the posterior using Gibbs sampling. [sent-288, score-0.173]

44 The full model is speciﬁed as z ∼ N(0, I), (m) x ∼ N(A(m) z, Ψ(m) ), (8) A(m) ∼ ARD(α0 , β0 ), Ψ(m) ∼ IW(S0 , ν0 ), and inference follows the variational updates provided by Wang (2007). [sent-292, score-0.209]

45 We initialize the model by sampling the latent variables from the prior, and recommend running the algorithm multiple times and choosing the solution with the best variational lower bound. [sent-295, score-0.268]

46 However, since the ARD prior (or the spike-and-slab prior for the Gibbs sampler variant) automatically shuts down components that are not needed, the parameter can safely be set large enough; the only drawback of using too large K is in increased computation time. [sent-298, score-0.208]

47 Illustration In this section we demonstrate the BIBFA model on artiﬁcial data, in order to illustrate the factorization into shared and data set-speciﬁc components, as well as to show that the inference proceduree converge to the correct solution. [sent-302, score-0.174]

48 The results are illustrated primarily from the point-of-view of the variational inference solution; the variational approximation is easier to visualize and compare with alternative methods. [sent-304, score-0.235]

49 1 Artiﬁcial Example First, we validate the model on artiﬁcial data drawn from a model from the same model family, with parameters set up so that it contains all types of components (view-speciﬁc and shared components). [sent-307, score-0.273]

50 The latent signals y were manually constructed to produce components that can be visually matched with the true ones for intuitive assessment of the results. [sent-308, score-0.166]

51 Also the α(m) parameters, controlling the activity of each latent component in both views, were manually speciﬁed. [sent-309, score-0.139]

52 The left column of Figure 3 illustrates the data generation, showing the four latent components, two of which are shared between the two views. [sent-311, score-0.179]

53 We generated N = 100 samples with D1 = 50 and D2 = 40 dimensions, and applied the BIBFA model with K = 6 components to show that it learns the correct latent components and automatically discards the excess ones. [sent-312, score-0.276]

54 The results of the variational inference are shown in the middle column of Figure 3; the Gibbs sampler produces virtually indistinguishable results. [sent-313, score-0.187]

55 The learned matrix of α-values (and the corresponding elements in W) reveals that the model extracted exactly four components, correctly identifying two of them as shared components and two as view-speciﬁc ones (one for each data set). [sent-314, score-0.193]

56 The actual latent components also correspond to the ones used for generating the data. [sent-315, score-0.166]

57 We also see how the model is invariant to the sign of y, but that it gives the actual components instead of a linear combination of those, demonstrating that the variational approximation indeed solves the rotational disambiguity that would remain for instance in the maximum likelihood solution. [sent-317, score-0.202]

58 We see that the true parameter values fall nicely within the posterior and both the variational approximation and the Gibbs sampler provide almost the same posterior. [sent-320, score-0.164]

59 We generated data with 4 true correlating components drawn from the prior, drawing N independent samples of D1 = D2 dimensions, and measure the performance by comparing the average of the four largest correlations ρk , normalized by the ground truth correlations. [sent-332, score-0.147]

60 We compare the two variants of the Bayesian CCA, denoting by BCCA a model parameterized with full covariance matrices (8) and by BIBFA the fully factorized model (7), with both classical CCA and a regularized CCA (RCCA). [sent-336, score-0.166]

61 The left column shows four components in the generated data, the ﬁrst two being shared between the two views and the last two being speciﬁc to just one view. [sent-346, score-0.208]

62 Components 5 and 6 are shared, revealed by non-zero variance for both views, components 2 and 4 are the two view-speciﬁc components, and the unnecessary components 1 and 3 have been suppressed to the prior in the sense that their mean and variance match those of the prior. [sent-349, score-0.187]

63 The small lines depict one standard deviation, revealing that the model is more conﬁdent on its predictions for the shared components, due to more data (D1 + D2 features compared to just D1 or D2 ) available for inferring them. [sent-350, score-0.146]

64 To further illustrate the behavior of BIBFA, we plot in Figure 6 the estimated number of shared and view-speciﬁc components for both the variational and Gibbs sampling variants. [sent-367, score-0.245]

65 We see that both inference algorithms are conservative in the sense that for very small sample sizes they miss some of the components, using the residual noise term to model the variation that cannot be reliably explained with so limited data. [sent-369, score-0.148]

66 For reasonable number of samples both the variational approximation (VB) and the spike-and-slab sampler (Gibbs) learn the correct number of both shared components (red lines) and total components (black lines) for all three dimensionalities (subplots). [sent-409, score-0.381]

67 The solid lines correspond to the results of the variational approximation, averaged over 10 random initializations, whereas the dashed line shows the mean of the posterior samples for the Gibbs sampler and the shaded region covers the values between the 5% and 95% quantiles. [sent-411, score-0.164]

68 1 Modifying the Generative Model Since the latent variable model is described through a generative process, it is straightforward to change the distributional assumptions in the model to arrive at alternatives designed for speciﬁc purposes. [sent-420, score-0.206]

69 That is, the model automatically learns topics that are shared between the views as well as topics speciﬁc to each view, using a hierarchical Dirichlet process (HDP; Teh et al. [sent-440, score-0.2]

70 They introduced sparsity priors and associated variational approximations for Bayesian PCA and the full IBFA model, but did not provide empirical experiments with the latter. [sent-444, score-0.145]

71 (2009), using an element-wise ARD prior to obtain sparsity, though the method is actually not a proper CCA model since it does not model view-speciﬁc variation at all. [sent-446, score-0.162]

72 (2008) extended the probabilistic formulation to create Gaussian process latent variable models (GP-LVM) for modeling dependencies between two data sets. [sent-452, score-0.201]

73 (2012) extended the approach to a Bayesian multi-view model that uses group-wise sparsity to identify shared and view-speciﬁc latent manifolds for a GP-LVM model, using an ARD prior very similar to the one used by Virtanen et al. [sent-457, score-0.294]

74 (2010) presented clustering models that capture the dependencies between two views with the cluster structure while modeling view-speciﬁc variation with another set of clusters. [sent-462, score-0.152]

75 However, the exact deﬁnition of the noise additive to the factorization is crucial; BIBFA includes explicit components for modeling view-speciﬁc variation (or they are modeled with full covariance matrices as in the earlier Bayesian CCA solutions). [sent-473, score-0.183]

76 (2011) extended CMFs to localized factor model (LMF) that allows separate latent variables z(m) for the views and models them as a linear combination of global latent proﬁles u. [sent-477, score-0.346]

77 By iteratively treating the view-speciﬁc and shared components as the explanatory factors they can learn the maximum likelihood solution of IBFA (and hence CCA) through eigen-decompositions, but their general formulation also applies to other data analysis scenarios. [sent-481, score-0.193]

78 Already Archambeau and Bach (2009) mention that the generative model directly generalizes to more than two views, but they do not show that their inference solution would provide meaningful results for multiple views. [sent-483, score-0.142]

79 (2011), in turn, created a hierarchical topic trajectory model (HTTM) by using CCA as the observation model in a hidden Markov model (HMM). [sent-504, score-0.142]

80 Direct inspection of the weights in the shared components then reveals the cancer-associated genes; a high weight implies an association between the copy number and gene expression, relating the gene to the cancer under study. [sent-551, score-0.238]

81 We repeated their experimental setup to obtain results directly comparable with their study, and measured the performance by the same measure, the area under curve (AUC) for retrieving known cancer genes (37 out of 4247 genes in Pollack, and 47 out of 7363 genes in Hyman). [sent-554, score-0.187]

82 We ran the BIBFA model for Kc between 5 and 60 components and chose the model with the best variational lower bound, resulting in Kc = 15 for Hyman and Kc = 40 for Pollack. [sent-555, score-0.242]

83 The BIBFA model ranks the genes based on the weight of that gene in both W(1) and W(2) , and ﬁnds the cancer genes with better accuracy than any of the methods studied in the recent comparison by Lahti et al. [sent-576, score-0.203]

84 In a sense, the model can be seen as a combination of purely unsupervised and supervised learning; both of the views can be considered as supervising the other view, yet the model is (here) deﬁned as a generative description of the whole data collection. [sent-590, score-0.165]

85 In other words, the model squeezes the prediction through the shared latent variables z, the lower-dimensional representation capturing all the information ﬂow from one data set to another. [sent-609, score-0.238]

86 Similar observation holds also for the Gibbs sampler variant using the spike-and-slab prior (6); again the mean prediction only depends on the shared components. [sent-611, score-0.174]

87 When making predictions for new samples, we need to store all of the posterior samples and then run the sampler again for each of those estimate the latent variables and the predicted X(1) . [sent-613, score-0.187]

88 Due to this extra computational overhead for the sampler, we will next demonstrate the BIBFA in multi-label prediction tasks using only the variational inference variant. [sent-614, score-0.143]

89 We compare the BIBFA model (7) with both classical CCA and the standard Bayesian CCA with full covariance matrices (8), using the variational approximation of Wang (2007). [sent-624, score-0.196]

90 For BCCA we set the maximal number of components to K = min(D1 , D2 , 50), and for classical CCA we chose the number of components by 10-fold cross-validation within the training set. [sent-625, score-0.167]

91 Discussion In this paper we have reviewed the works on probabilistic and Bayesian canonical correlation analysis, with particular focus on the extensions made possible by the probabilistic interpretation. [sent-697, score-0.157]

92 The key is to make a low-rank assumption for the noise speciﬁc to each data set, which results in re-formulation of CCA as a more complex latent variable model called inter-battery factor analysis (IBFA) in the statistics literature (Tucker, 1958). [sent-704, score-0.158]

93 The computational difﬁculties stemming from introducing the extra latent variables are solved by clever usage of group-wise sparsity assumption. [sent-706, score-0.143]

94 Instead of explicitly instantiating several latent variables, we re-cast the IBFA model as a straightforward joint factor analysis model with a speciﬁc prior driving the component group-wise sparse, showing how the resulting model is equivalent to IBFA. [sent-707, score-0.287]

95 We use mean ﬁeld variational approximation to approximate the posterior, with the factorization 2 Q(Θ) = q(Y) ∏ q(W(m) )q(α(m) )q(τm ), m=1 where Θ denotes all of the parameters and latent variables. [sent-717, score-0.188]

96 For the latent variables we further assume column-wise independence (that is, the latent variables of observations are independent) and for projections row-wise independence (that is, each component is independent). [sent-718, score-0.283]

97 BIBFA with Spike-and-slab Prior For inference with the spike-and-slab prior (6) we use Gibbs sampling, following closely the updates given for element-wise sparse FA model by Knowles and Ghahramani (2011). [sent-758, score-0.164]

98 Below we show the sampling equations for yn as an example; the updates for α(m) and τ(m) can be easily modiﬁed from the variational updates given in Appendix A. [sent-773, score-0.168]

99 A Gaussian process latent variable model formulation of canonical correlation analysis. [sent-963, score-0.238]

100 Learning shared latent structure for image synthesis and robotic imitation. [sent-1029, score-0.179]

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Packaged as one jar ﬁle, the GUI module runs automatically. 2.3 Functional Interaction Between R and Java We adopt the open-source program RCaller (http://code.google.com/p/rcaller) to implement the interaction between R and Java modules (Fig. 2), supported by explicitly designed R scripts such as Java-runCAM-CM.R. Speciﬁcally, ﬁve featured Java classes are introduced to interact with R for importing data or parameters, running algorithms, passing on or recording results, displaying ﬁgures, and handing over error messages. The examples of these classes include guiModel.MyRCaller.java, guiModel.MyRCaller.readResults(), and guiView.MyRPlotViewer. 3. Case Studies and Experimental Results The CAM package has been successfully applied to various data types. Using dynamic contrastenhanced magnetic resonance imaging data set of an advanced breast cancer case (Chen, et al., 2011b),“double click” (or command lines under Ubuntu) activated execution of CAM-Java.jar reveals two biologically interpretable vascular compartments with distinct kinetic patterns: fast clearance in the peripheral “rim” and slow clearance in the inner “core”. These outcomes are consistent with previously reported intratumor heterogeneity (Fig. 3). Angiogenesis is essential to tumor development beyond 1-2mm3 . It has been widely observed that active angiogenesis is often observed in advanced breast tumors occurring in the peripheral “rim” with co-occurrence of inner-core hypoxia. This pattern is largely due to the defective endothelial barrier function and outgrowth blood supply. In another application to natural image mixtures, CAM algorithm successfully recovered the source images in a large number of trials (see Users Manual). 4. Summary and Acknowledgements We have developed a R-Java CAM package for blindly separating mixed nonnegative sources. The open-source cross-platform software is easy-to-use and effective, validated in several real-world applications leading to plausible scientiﬁc discoveries. The software is freely downloadable from http://mloss.org/software/view/437/. We intend to maintain and support this package in the future. This work was supported in part by the US National Institutes of Health under Grants CA109872, CA 100970, and NS29525. We thank T.H. Chan, F.Y. Wang, Y. Zhu, and D.J. Miller for technical discussions. 2902 T HE CAM S OFTWARE IN R-JAVA References T.H. Chan, W.K. Ma, C.Y. Chi, and Y. Wang. A convex analysis framework for blind separation of non-negative sources. IEEE Transactions on Signal Processing, 56:5120–5143, 2008. L. Chen, T.H. Chan, P.L. Choyke, and E.M. Hillman et al. Cam-cm: a signal deconvolution tool for in vivo dynamic contrast-enhanced imaging of complex tissues. Bioinformatics, 27:2607–2609, 2011a. L. Chen, P.L. Choyke, T.H. Chan, and C.Y. Chi et al. Tissue-speciﬁc compartmental analysis for dynamic contrast-enhanced mr imaging of complex tumors. IEEE Transactions on Medical Imaging, 30:2044–2058, 2011b. D. Child. The essentials of factor analysis. Continuum International, 2006. S.A. Cruces-Alvarez, Andrzej Cichocki, and Shun ichi Amari. From blind signal extraction to blind instantaneous signal separation: criteria, algorithms, and stability. IEEE Transactions on Neural Networks, 15:859–873, 2004. N. Gillis. Sparse and unique nonnegative matrix factorization through data preprocessing. Journal of Machine Learning Research, 13:3349–3386, 2012. E.M.C. Hillman and A. Moore. All-optical anatomical co-registration for molecular imaging of small animals using dynamic contrast. Nature Photonics, 1:526–530, 2007. A. Hyvarinen, J. Karhunen, and E. Oja. Independent Component Analysis. John Wiley, New York, 2001. N. Keshava and J.F. Mustard. Spectral unmixing. IEEE Signal Processing Magazine, 19:44–57, 2002. D.D. Lee and H.S. Seung. Learning the parts of objects by non-negative matrix factorization. Nature, 401:788–791, 1999. E. Oja and M. Plumbley. Blind separation of positive sources by globally convergent gradient search. Neural Computation, 16:1811–1825, 2004. F.Y. Wang, C.Y. Chi, T.H. Chan, and Y. Wang. Nonnegative least-correlated component analysis for separation of dependent sources by volume maximization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32:857–888, 2010. 2903

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