A0279
Title: Edgeworth expansion and bootstrap for entrywise eigenvectors statistics of low-rank random matrices
Authors: Fangzheng Xie - Indiana University (United States) [presenting]
Abstract: Understanding the distributions of spectral estimators in low-rank random matrix models, also known as signal-plus-noise matrix models, is fundamentally important in various statistical machine-learning problems, including network analysis, matrix denoising, and matrix completion. The distributions of entrywise eigenvector statistics are studied for a broad range of signal-plus-noise matrix models by establishing their Edgeworth expansion formulae. The key to the approach is a sharp higher-order entrywise eigenvector stochastic expansion. It is shown that the first-order term in the expansion is a linear function of the noise matrix, while the second-order term is a linear function of the squared noise matrix. Furthermore, under mild conditions, it is shown that Cramer's condition on the smoothness of noise distribution is not required, thanks to the self-smoothing effect of the second-order term in the eigenvector stochastic expansion. This phenomenon is unusual in low-dimensional problems. The Edgeworth expansion result is further applied to justify the higher-order accuracy of the residual bootstrap for approximating the distributions of the studentized entrywise eigenvector statistics.
