Bayesian non-parametric approaches to reconstructing oscillatory systems and the Nyquist limit

Reconstructing continuous signals from discrete time-points is a challenging inverse problem encountered in many scientific and engineering applications. For oscillatory signals classical results due to Nyquist set the limit below which it becomes impossible to reliably reconstruct the oscillation dynamics. Here we revisit this problem for vector-valued outputs and apply Bayesian non-parametric approaches in order to solve the function estimation problem. The main aim of the current paper is to map how we can use of correlations among different outputs to reconstruct signals at a sampling rate that lies below the Nyquist rate. We show that it is possible to use multiple-output Gaussian processes to capture dependences between outputs which facilitate reconstruction of signals in situation where conventional Gaussian processes (i.e. this aimed at describing scalar signals) fail, and we delineate the phase and frequency dependence of the reliability of this type of approach. In addition to simple toy-models we also consider the dynamics of the tumour suppressor gene p53, which exhibits oscillations under physiological conditions, and which can be reconstructed more reliably in our new framework.