Probabilistic interpolative decomposition

Interpolative decomposition (ID) is a low-rank matrix decomposition where the data matrix is expressed via a sub-set of its own columns. In this work, we propose a novel probabilistic method for ID where it is expressed as a statistical model within a Bayesian framework. The proposed method considerably differs from other ID methods in the literature: It handles the model selection automatically and enables the construction of problem-specific interpolative decompositions. We derive the analytical solution for the normal distribution and we provide a numerical solution for the generic case. Simulation results on synthetic data are provided to illustrate that the method converges to the true decomposition, independent of the initialization; and it can successfully handle noise. In addition, we apply probabilistic ID to the problem of automatic polyphonic music transcription to extract important information from a huge dictionary of spectrum instances. We supply comparative results with the other proposed techniques in the literature and show that it performs better. Probabilistic interpolative decomposition serves as a promising feature selection and de-noising tool to be exploited in big data problems.