  Tensor decomposition is a fundamental framework to analyze data that can be
represented by multi-dimensional arrays. In practice, tensor data is often
accompanied by temporal information, namely the time points when the entry
values were generated. This information implies abundant, complex temporal
variation patterns. However, current methods always assume the factor
representations of the entities in each tensor mode are static, and never
consider their temporal evolution. To fill this gap, we propose NONparametric
FActor Trajectory learning for dynamic tensor decomposition (NONFAT). We place
Gaussian process (GP) priors in the frequency domain and conduct inverse
Fourier transform via Gauss-Laguerre quadrature to sample the trajectory
functions. In this way, we can overcome data sparsity and obtain robust
trajectory estimates across long time horizons. Given the trajectory values at
specific time points, we use a second-level GP to sample the entry values and
to capture the temporal relationship between the entities. For efficient and
scalable inference, we leverage the matrix Gaussian structure in the model,
introduce a matrix Gaussian posterior, and develop a nested sparse variational
learning algorithm. We have shown the advantage of our method in several
real-world applications.
