Algorithms for two dimensional multi set canonical correlation analysis☆
Introduction
Canonical correlation analysis (CCA) [1] is a data driven technique that has been successfully applied to capture the linear relationship amongst various types of multivariate data. Basic objective of CCA is to determine the coordinate system, which represents the best possible linear relationship between the given multivariate datasets [1] by maximizing the mutual correlation between the datasets. Formally, CCA is defined as the problem of finding two sets of projection vectors a and b for the two sets of multivariate data X and Y, such that correlation between projected values of X on a and Y on b is maximized. Multi-set canonical correlation analysis (mCCA) extended the applicability of CCA to more than two datasets [2]. Various approaches to compute mCCA have been developed. They aim to optimize an objective function of the correlation matrix of the data-sets to be analyzed. It has been successfully used in many applications such as genomic data integration to identify the relationship amongst multiple phenotypic measures [3], cross-language document retrieval [4], etc. The equivalence between linear discriminant analysis (LDA) and CCA is proved in [5]. A new variant, called within-class coupling CCA, is proposed that is applicable in case of data whose samples are implicitly indicative of their class membership. An efficient investigation of CCA and generalized CCA for a text document classification task is presented in [6].
mCCA finds applications in image data as well. Rapid advancements in imaging devices in the last decade have enabled technologies such as remote sensing, medical imaging to generate large amounts of image data. mCCA, for instance, has been used previously in the analysis of image data in applications such as feature extraction and classification [7], and analysis of remote sensing data [8]. Kernel canonical correlation analysis (KCCA) has been used in fMRI analysis in the estimation of correlated subspaces datasets from multiple subjects of a particular medical imaging modality [9]. Conventional mCCA theory is directly applicable to only multivariate vector data. Therefore, the image data has to be vectorized before it can be analysed using mCCA. However, CCA on vectorized data does not consider the spatial structure of the images. In addition, vectorization results in a large covariance matrix that may be ill-conditioned, which makes the solution unstable or non-existent and also, it increases the computational complexity. A two dimensional CCA, directly applicable to image data, was first proposed in [10]. It defines two separate projection vectors that operate along the row and column directions of the image data and therefore, does not require the image to be vectorized. However, it is limited to only two datasets and cannot be used for the analysis of more than two datasets. To overcome these drawbacks, we propose a 2DmCCA algorithm that can be used to analyze multiple (more than two) image datasets, simultaneously [11].
The paper is organized as follows: Section 2 describes the background work on the mCCA and 2DCCA. Section 3 discusses the proposed 2DmCCA framework and a new iterative procedure is described to implement the proposed 2DmCCA algorithm. Section 4 contains the performance comparisons of the proposed 2DmCCA algorithm with conventional mCCA [2] in face recognition experiments. Experiments against block paradigm right finger tapping fMRI are also included in section to demonstrate the applicability of the proposed approach in multisubject medical image analysis. Concluding remarks are given in Section 5.
Section snippets
Two dimensional canonical correlation analysis
Canonical correlation analysis directly applicable to matrix data, known as two dimensional CCA (2DCCA) was first developed in [10]. Given two sets of matrices where and where 2DCCA defines left linear transforms and right linear transforms that operate along the rows and columns of the matrices, respectively. Let and be the mean matrices of dataset X and Y respectively. Centered datasets and can be obtained as
Proposed work
This section details the development of the proposed 2DmCCA framework that can be directly applied to more than two matrix datasets. 2DmCCA objectives are defined and optimization problems are formulated, which are then solved by placing constraints on the canonical coefficient values. A new iterative algorithm is described to compute the canonical coefficient vectors.
Results
Face recognition experiment is carried out using 2DmCCA and conventional mCCA [2] to compare the performances of the proposed 2DmCCA and regular mCCA approaches. Results for conventional mCCA, using constraint 2, are not provided due to its high memory space requirements and a high time complexity. Therefore, we were unable to implement it on our current hardware (64-bit system equipped with an Intel®i7-4790 CPU running at 3.60GHz), as it showed an out of memory error. An application of the
Conclusion
Two dimensional CCA is analyzed in the context of mCCA to develop new 2DmCCA algorithms. The conventional mCCA requires image data to be reshaped into vectors. The proposed algorithms overcome this limitation and can be efficiently applied to image data. One of the future extensions is to induce sparsity into this framework to develop a sparse 2DmCCA framework that helps account for the parsimony in the biomedical data [18].
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This work was supported by the Australian Research Council through Grant FT. 130101394.