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In this paper, we investigate the optimal statistical performance and theimpact of computational constraints for independent component analysis (ICA).Our goal is twofold. On the one hand, we characterize the precise role ofdimensionality on sample complexity and statistical accuracy, and howcomputational consideration may affect them. In particular, we show that theoptimal sample complexity is linear in dimensionality, and interestingly, thecommonly used sample kurtosis-based approaches are necessarily suboptimal.However, the optimal sample complexity becomes quadratic, up to a logarithmicfactor, in the dimension if we restrict ourselves to estimates that can becomputed with low-degree polynomial algorithms. On the other hand, we developcomputationally tractable estimates that attain both the optimal samplecomplexity and minimax optimal rates of convergence. We study the asymptoticproperties of the proposed estimates and establish their asymptotic normalitythat can be readily used for statistical inferences. Our method is fairly easyto implement and numerical experiments are presented to further demonstrate itspractical merits.

Large Dimensional Independent Component Analysis: Statistical Optimality and Computational Tractability