  Practical tensor data is often along with time information. Most existing
temporal decomposition approaches estimate a set of fixed factors for the
objects in each tensor mode, and hence cannot capture the temporal evolution of
the objects' representation. More important, we lack an effective approach to
capture such evolution from streaming data, which is common in real-world
applications. To address these issues, we propose Streaming Factor Trajectory
Learning for temporal tensor decomposition. We use Gaussian processes (GPs) to
model the trajectory of factors so as to flexibly estimate their temporal
evolution. To address the computational challenges in handling streaming data,
we convert the GPs into a state-space prior by constructing an equivalent
stochastic differential equation (SDE). We develop an efficient online
filtering algorithm to estimate a decoupled running posterior of the involved
factor states upon receiving new data. The decoupled estimation enables us to
conduct standard Rauch-Tung-Striebel smoothing to compute the full posterior of
all the trajectories in parallel, without the need for revisiting any previous
data. We have shown the advantage of SFTL in both synthetic tasks and
real-world applications. The code is available at
{https://github.com/xuangu-fang/Streaming-Factor-Trajectory-Learning}.
