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The eigendecomposition of a matrix is the central procedure in probabilisticmodels based on matrix factorization, for instance principal component analysisand topic models. Quantifying the uncertainty of such a decomposition based ona finite sample estimate is essential to reasoning under uncertainty whenemploying such models. This paper tackles the challenge of computing confidencebounds on the individual entries of eigenvectors of a covariance matrix offixed dimension. Moreover, we derive a method to bound the entries of theinverse covariance matrix, the so-called precision matrix. The assumptionsbehind our method are minimal and require that the covariance matrix exists,and its empirical estimator converges to the true covariance. We make use ofthe theory of U-statistics to bound the $L_2$ perturbation of the empiricalcovariance matrix. From this result, we obtain bounds on the eigenvectors usingWeyl's theorem and the eigenvalue-eigenvector identity and we derive confidenceintervals on the entries of the precision matrix using matrix inversionperturbation bounds. As an application of these results, we demonstrate a newstatistical test, which allows us to test for non-zero values of the precisionmatrix. We compare this test to the well-known Fisher-z test for partialcorrelations, and demonstrate the soundness and scalability of the proposedstatistical test, as well as its application to real-world data from medicaland physics domains.

On confidence intervals for precision matrices and the eigendecomposition of covariance matrices