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Principal Component Analysis (PCA) is a popular tool in data analysis,especially when the data is high-dimensional. PCA aims to find subspaces,spanned by the so-called \textit{principal components}, that best explain thevariance in the dataset. The deflation method is a popular meta-algorithm --used to discover such subspaces -- that sequentially finds individual principalcomponents, starting from the most important one and working its way towardsthe less important ones. However, due to its sequential nature, the numericalerror introduced by not estimating principal components exactly -- e.g., due tonumerical approximations through this process -- propagates, as deflationproceeds. To the best of our knowledge, this is the first work thatmathematically characterizes the error propagation of the inexact deflationmethod, and this is the key contribution of this paper. We provide two mainresults: $i)$ when the sub-routine for finding the leading eigenvector isgeneric, and $ii)$ when power iteration is used as the sub-routine. In thelatter case, the additional directional information from power iteration allowsus to obtain a tighter error bound than the analysis of the sub-routineagnostic case. As an outcome, we provide explicit characterization on how theerror progresses and affects subsequent principal component estimations forthis fundamental problem.

On the Error-Propagation of Inexact Deflation for Principal Component Analysis