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Tensor completion via optimization on the product of matrix manifolds

We present a method for tensor completion using optimization on low-rank matrix manifolds. Our notion of tensor-rank is based on the recently proposed framework of tensor- Singular Value Decomposition (t-SVD) in [1], [2]. In contrast to convex optimization methods used in [1] that operate in a high-dimensional 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.