Exponential And Randomized Exponential Canonical Correlation Analysis Algorithms

Canonical correlation analysis(CCA)is data processing method that has been successfully used in data analysis.CCA extracts meaningful information from a pair of data sets by seeking pairs of linear combinations from two sets of variables with maximum correlation.Mathematically,CCA resolves to solving a generalized eigenvalue problem.However,as the dimension of the data sets is much larger than the number of samples,CCA often suffers from the small-sample-size(SSS)and the over-fitting problems.In order to overcome these difficulties,regularized technique is often utilized in CCA,but the optimal regularized parameter is difficult to choose in advance.As an alternative,we propose an Exponential Canonical Correlation Analysis(ECCA)based on matrix exponential,which is parameter-free and can overcome the over-fitting and SSS problem fundamentally.Unfortunately,the computational overhead of standard CCA and ECCA is very large for practical problems.Based on the randomized singular value decomposition(RSVD),we further propose a Randomized Canonical Correlation Analysis(RSVD CCA)and Random Exponential Canonical Correlation Analysis(RECCA)for data analysis and dimensionality reduction.Theoretical results are established to show the relationship between the RECCA and the ECCA methods.Numerical experiments are performed on some high-dimensional and large-sample data sets as well as nonlinear data sets,which illustrate the superiority of the proposed algorithms over some popular CCA algorithms.