  We introduce the probabilistic sequential matrix factorization (PSMF) method
for factorizing time-varying and non-stationary datasets consisting of
high-dimensional time-series. In particular, we consider nonlinear Gaussian
state-space models where sequential approximate inference results in the
factorization of a data matrix into a dictionary and time-varying coefficients
with potentially nonlinear Markovian dependencies. The assumed Markovian
structure on the coefficients enables us to encode temporal dependencies into a
low-dimensional feature space. The proposed inference method is solely based on
an approximate extended Kalman filtering scheme, which makes the resulting
method particularly efficient. PSMF can account for temporal nonlinearities
and, more importantly, can be used to calibrate and estimate generic
differentiable nonlinear subspace models. We also introduce a robust version of
PSMF, called rPSMF, which uses Student-t filters to handle model
misspecification. We show that PSMF can be used in multiple contexts: modeling
time series with a periodic subspace, robustifying changepoint detection
methods, and imputing missing data in several high-dimensional time-series,
such as measurements of pollutants across London.
