Belief Propagation for Probabilistic Slow Feature Analysis

Slow feature analysis (SFA) is a time-series analysis method for extracting slowly-varying latent features from multidimensional data. A recent study proposed a probabilistic framework of SFA using the Bayesian statistical framework. However, the conventional probabilistic framework of SFA can not accurately extract the slow feature in noisy environments since its marginal likelihood function was approximately derived under the assumption that there exists no observation noise. In this paper, we propose a probabilistic framework of SFA with rigorously derived marginal likelihood function. Here, we rigorously derive the marginal likelihood function of the probabilistic framework of SFA by using belief propagation. We show using numerical data that the proposed probabilistic framework of SFA can accurately extract the slow feature and underlying parameters for the latent dynamics simultaneously even under noisy environments.