Efficient Kalman smoothing for harmonic state-space models

Harmonic probabilistic models are common in signal analysis. Framed as a linear-Gaussian state-space model, smoothed inference scales as O(TH2) where H is twice the number of frequencies in the model and T is the length of the time-series. Due to their central role in acoustic modelling, fast effective inference in this model is of some considerable interest. We present a form of `rotation-corrected' low-rank approximation for the backward pass of the Rauch-Tung-Striebel smoother. This provides an effective approximation with computation complexity O(TSH) where S is the rank of the approximation.