Asymptotic Matrix-Variate Bingham Distributions and Privacy-Preserving Principal Component Analysis

The matrix Bingham distribution is a measure on the set of low-dimensional frames that has found applications in mixture models, shape analysis, statistical signal processing, and control. Although the density has a natural shape, it presents analytical and computational challenges such as approximating the normalizing constant and computing the probability of level sets. This paper uses a novel representation of this distribution that yields asymptotic characterizations of these quantities as the dimension of the ambient space becomes large. By using random matrix theory, these in turn lead to an asymptotic result on the sample complexity for differentially private principal component analysis. Applications of the developed asymptotic analysis to several other areas of applied science are also discussed.