The Stochastic Complexity of Principal Component Analysis

PCA (principal component analysis) and its variants are ubiquitous techniques for matrix dimension reduction and reduced-dimension latent-factor extraction. For an arbitrary matrix, they cannot, on their own, determine the size of the reduced dimension, but rather must be given this as an input. NML (normalized maximum likelihood) is a universal implementation of the Minimal Description Length principle, which gives an objective compression-based criterion for model selection. This work applies NML to PCA. A direct attempt to do so would involve the distributions of singular values of random matrices, which is difficult. A reduction to linear regression with a noisy unitary covariate matrix, however, allows finding closed-form bounds on the NML of PCA.