Optimal Rates of Convergence of Transelliptical Component Analysis

Han and Liu (2012) proposed a method named transelliptical component analysis (TCA) for conducting scale-invariant principal component analysis on high dimensional data with transelliptical distributions. The transelliptical family assumes that the data follow an elliptical distribution after unspecified marginal monotone transformations. In a double asymptotic framework where the dimension $d$ is allowed to increase with the sample size $n$, Han and Liu (2012) showed that one version of TCA attains a "nearly parametric" rate of convergence in parameter estimation when the parameter of interest is assumed to be sparse. This paper improves upon their results in two aspects: (i) Under the non-sparse setting (i.e., the parameter of interest is not assumed to be sparse), we show that a version of TCA attains the optimal rate of convergence up to a logarithmic factor; (ii) Under the sparse setting, we also lay out venues to analyze the performance of the TCA estimator proposed in Han and Liu (2012). In particular, we provide a "sign subgaussian condition" which is sufficient for TCA to attain an improved rate of convergence and verify a subfamily of the transelliptical distributions satisfying this condition.