We present a method for tensor completion using optimization on low-rank matrix manifolds. Our notion of tensorrank is based on the recently proposed framework of tensorSingular Value Decomposition (t-SVD) in [1], [2]. In contrast to convex optimization methods used in [1] that operate in a highdimensional space, in the manifold setting, one works directly in the reduced dimensionality space and is thus able to significantly reduce the computational costs [3], [4]. In this paper we focus on 3-D data and under the tensor algebraic framework of [1], [2] we show that a 3-D tensor of fixed tubal-rank can be seen as an element of the product manifold of fixed low-rank matrices in the Fourier domain. The tensor completion problem then reduces to finding the best approximation to the sampled data on this product manifold.Further, for 3-D data we consider and compare recovery performance under two approaches. In the first approach one samples entire mode-3 fibers of the tensor, which we refer to as tubal-sampling. The second approach employs element-wise sampling and we simply refer to this method as sampling. For these two types of sampling approaches, we present simulation results for surveillance video data and show that recovery under random sampling has better performance compared to the random tubal-sampling.

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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.

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