Bootstrapping the error of Oja's algorithm
This provides a method for uncertainty quantification in streaming PCA, which is incremental as it builds on existing statistical tools for a specific algorithm.
The paper tackles the problem of quantifying uncertainty in the estimation error of the leading eigenvector from Oja's algorithm in streaming PCA, establishing a weighted chi-squared approximation for the error and proposing an online multiplier bootstrap algorithm for inference.
We consider the problem of quantifying uncertainty for the estimation error of the leading eigenvector from Oja's algorithm for streaming principal component analysis, where the data are generated IID from some unknown distribution. By combining classical tools from the U-statistics literature with recent results on high-dimensional central limit theorems for quadratic forms of random vectors and concentration of matrix products, we establish a weighted $χ^2$ approximation result for the $\sin^2$ error between the population eigenvector and the output of Oja's algorithm. Since estimating the covariance matrix associated with the approximating distribution requires knowledge of unknown model parameters, we propose a multiplier bootstrap algorithm that may be updated in an online manner. We establish conditions under which the bootstrap distribution is close to the corresponding sampling distribution with high probability, thereby establishing the bootstrap as a consistent inferential method in an appropriate asymptotic regime.