  We present a new method for online prediction and learning of tensors
($N$-way arrays, $N >2$) from sequential measurements. We focus on the specific
case of 3-D tensors and exploit a recently developed framework of structured
tensor decompositions proposed in [1]. In this framework it is possible to
treat 3-D tensors as linear operators and appropriately generalize notions of
rank and positive definiteness to tensors in a natural way. Using these notions
we propose a generalization of the matrix exponentiated gradient descent
algorithm [2] to a tensor exponentiated gradient descent algorithm using an
extension of the notion of von-Neumann divergence to tensors. Then following a
similar construction as in [3], we exploit this algorithm to propose an online
algorithm for learning and prediction of tensors with provable regret
guarantees. Simulations results are presented on semi-synthetic data sets of
ratings evolving in time under local influence over a social network. The
result indicate superior performance compared to other (online) convex tensor
completion methods.
