Presented By Wanchen Lu 2/25/2013 Multi-view Clustering via Canonical Correlation Analysis Kamalika Chaudhuri et al. ICML 2009. Presented By Wanchen Lu 2/25/2013

Introduction

Assumption in Multi-View problems The input variable (a real vector) can be partitioned into two different views, where it is assumed that either view of the input is sufficient to make accurate predictions --- essentially the co-training assumption. e.g. Identity recognition with one view being a video stream and the other an audio stream; Web page classification where one view is the text and the other is the hyperlink structure; Object recognition with pictures from different camera angles; A bilingual parallel corpus, with each view presented in one language.

Intuition in Multi-View problems Many multi-view learning algorithms force agreement between the predictors based on either view. (usually force the predictor on view 1 to equal to the predictor based on view 2) The complexity of the learning problem is reduced by eliminating hypothesis from each view that do not agree with each other.

Background

Canonical correlation analysis CCA is a way of measuring the linear relationship between two multidimensional variables. Find two basis vectors, one for x and one for y, such that the correlations between the projections of the variables onto these basis vectors are maximized. x = \mathbf{x^T\hat{w}}_x , y = \mathbf{y^T\hat{w}}_y \vspace{5mm} \\ \rho = \frac{E[xy]}{\sqrt{E[x^2]E[y^2]}} = \frac{E[\mathbf{\hat{w}}^T_x \mathbf{x y}^T \mathbf{\hat{w}}_y]}{\sqrt{E[\mathbf{\hat{w}}^T_x \mathbf{x x}^T \mathbf{\hat{w}}_x]E[\mathbf{\hat{w}}^T_y \mathbf{y y}^T \mathbf{\hat{w}}_y]}} =\frac{\mathbf{w}^T_x\mathbf{C}_{xy}\mathbf{w}_y}{\sqrt{\mathbf{w}^T_x\mathbf{C}_{xx}\mathbf{w}_x \mathbf{w}^T_y\mathbf{C}_{yy}\mathbf{w}_y}}

Calculating Canonical correlations Consider the total covariance matrix of random variables x and y with zero mean: The canonical correlations between x and y can be found by solving the eigenvalue equations x = \mathbf{x^T\hat{w}}_x , y = \mathbf{y^T\hat{w}}_y \vspace{5mm} \\ \rho = \frac{E[xy]}{\sqrt{E[x^2]E[y^2]}} = \frac{E[\mathbf{\hat{w}}^T_x \mathbf{x y}^T \mathbf{\hat{w}}_y]}{\sqrt{E[\mathbf{\hat{w}}^T_x \mathbf{x x}^T \mathbf{\hat{w}}_x]E[\mathbf{\hat{w}}^T_y \mathbf{y y}^T \mathbf{\hat{w}}_y]}} =\frac{\mathbf{w}^T_x\mathbf{C}_{xy}\mathbf{w}_y}{\sqrt{\mathbf{w}^T_x\mathbf{C}_{xx}\mathbf{w}_x \mathbf{w}^T_y\mathbf{C}_{yy}\mathbf{w}_y}}

Relation to other linear subspace methods Formulate the problems in one single eigenvalue equation

Principal component analysis The principal components are the eigenvectors of the covariance matrix. The projection of data onto the principal components is an orthogonal transformation that diagonalizes the covariance matrix.

Partial least squares PLS is basically the singular value decomposition (SVD) of a between-sets covariance matrix. In PLS regression, the principal vectors corresponding to the largest principal values are used as basis. A regression of y onto x is then performed in this basis.

ALGORITHM

The basic idea Use CCA to project the data down to the subspace spanned by the means to get an easier clustering problem, then apply standard clustering algorithms in this space. When the data in at least one of the views is well separated, this algorithm clusters correctly with high probability.

Algorithm Input: a set of samples S, the number of clusters k Randomly partition S into two subsets A and B of equal size. Let C_12(A) be the covariance matrix between views 1 and 2, computed from the set A. Compute the top k-1 left singular vectors of C_12(A), and project the samples in B on the subspace spanned by these vectors. Apply clustering algorithm (single linkage clustering, K-means) to the projected examples in view 1.

Experiments

Speaker identification Dataset 41 speakers, speaking 10 sentences each Audio features 1584 dimensions Video feature 2394 dimensions Method 1: use PCA project into 40 D Method 2: use CCA (after PCA into 100 D for images and 1000 D for audios) Cluster into 82 clusters (2 / speaker) using K-means

Speaker identification Evaluation Conditional perplexity = the mean # of speakers corresponding to each cluster

clustering Wikipedia articles Dataset 128 K Wikipedia articles, evaluated on 73 K articles that belong to the 500 most frequent categories. Link structure feature L is a concatenation of ``to`` and ``from`` vectors. L(i) is the number of times the current article links to/from article i. Text feature is a bag-of-words vector. Methods: compared PCA and CCA Used a hierarchical clustering procedure, iteratively pick the largest cluster, reduce the dimensionality using PCA or CCA, and use k-means to break the cluster into smaller ones, until reaching the total desired number of clusters.

Results

Thank you

APPENDIX: A note on correlation Correlation between x_i and x_i is the covariance normalized by the geometric mean of the variances of x_i and x_j

Affine transformations An affine transformation is a map F:\mathbb{R}^n \rightarrow \mathbb{R}^n \center { $F(\mathbf{p}) = \mathbf{Ap}+\mathbf{q}$, \forall \mathbf{p} \in \mathbb{R}^n \\ } \mbox{where } \mathbf{A} \mbox{ is a linear tranformation of } \mathbb{R}^n \mbox{ and }\\ \mathbf{q} \mbox{ is a translation vector in } \mathbb{R}^n.$