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Hypothesis testing for eigenspaces of covariance matrix

Eigenspaces of covariance matrices play an important role in statistical machine learning, arising in variety of modern algorithms. Quantitatively, it is convenient to describe the eigenspaces in terms of spectral projectors. This work focuses on hypothesis testing for the spectral projectors, both in one- and two-sample scenario. We present new tests, based on a specific matrix norm developed in order to utilize the structure of the spectral projectors. A new resampling technique of independent interest is introduced and analyzed: it serves as an alternative to the well-known multiplier bootstrap, significantly reducing computational complexity of bootstrap-based methods. We provide theoretical guarantees for the type-I error of our procedures, which remarkably improve the previously obtained results in the field. Moreover, we analyze power of our tests. Numerical experiments illustrate good performance of the proposed methods compared to previously developed ones.