  Tensor ring (TR) decomposition is a powerful tool for exploiting the low-rank
nature of multiway data and has demonstrated great potential in a variety of
important applications. In this paper, nonnegative tensor ring (NTR)
decomposition and graph regularized NTR (GNTR) decomposition are proposed,
where the former equips TR decomposition with local feature extraction by
imposing nonnegativity on the core tensors and the latter is additionally able
to capture manifold geometry information of tensor data, both significantly
extend the applications of TR decomposition for nonnegative multiway
representation learning. Accelerated proximal gradient based methods are
derived for NTR and GNTR. The experimental result demonstrate that the proposed
algorithms can extract parts-based basis with rich colors and rich lines from
tensor objects that provide more interpretable and meaningful representation,
and hence yield better performance than the state-of-the-art tensor based
methods in clustering and classification tasks.
