Incoherent Tensor Norms and Their Applications in Higher Order Tensor Completion

Abstract

In this paper, we investigate the sample size requirement for a general class of nuclear norm minimization methods for higher order tensor completion. We introduce a class of tensor norms by allowing for different levels of coherence, which allows us to leverage the incoherence of a tensor. In particular, we show that a $k$ th-order tensor of multilinear rank $r$ and dimension $d \times \cdots \times d$ can be recovered perfectly from as few as $O((r^{(k-1)/2}d^{3/2}+r^{k-1}d)(\log (d))^{2})$ uniformly sampled entries through an appropriate incoherent nuclear norm minimization. Our results demonstrate some key differences between completing a matrix and a higher order tensor: they not only point to potential room for improvement over the usual nuclear norm minimization but also highlight the importance of explicitly accounting for incoherence, when dealing with higher order tensors. Although our focus is primarily on the theoretical guarantees for nuclear norm minimization, such insights may prove useful for understanding performance of other related methods and developing improved practical algorithms.