CMSA Talk: Zongming Ma

Title: Sparse Canonical Correlation Analysis Abstract: Canonical correlation analysis (CCA) is a widely used multivariate statistical technique for exploring the relation between two sets of variables. In this talk we consider the problem of estimating the leading canonical correlation directions in high dimensional settings. Recently, under the assumption that the leading canonical correlation directions are sparse, various procedures have been proposed for many high dimensional applications involving massive data sets. However, there has been few theoretical justification available in the literature. In this talk, we establish rate-optimal non-asymptotic minimax estimation with respect to an appropriate loss function for a wide range of model spaces. Two interesting phenomena are observed. First, the minimax rates are not affected by the presence of nuisance parameters, namely the covariance matrices of the two sets of random variables, though they need to be estimated in the canonical correlation analysis problem. Second, we allow the presence of the residual canonical correlation directions. However, they do not influence the minimax rates under a mild condition on eigengap. A generalized sin-theta theorem and an empirical process bound for Gaussian quadratic forms under rank constraint are used to establish the minimax upper bounds, which may be of independent interest. Time permitting, we will also discuss some efficient algorithm for sparse CCA. This talk is based on joint works with Chao Gao, Zhao Ren and Harrison Zhou. (Random Matrix and Probability Theory seminar: flyer)