Provable Tensor Ring Completion

Tensor completion recovers a multi-dimensional array from a limited number of measurements. Using the recently proposed tensor ring (TR) decomposition, in this paper we show that a d-order tensor of size n × ⋅⋅⋅ × n and TR rank [r, ... , r] can be exactly recovered with high probability by solving a convex optimization program, given O(n⌈d/2⌉r2ln 7(n⌈d/2⌉)) samples. In the optimization model, a weighted sum of nuclear norms of factors surrogates the TR rank. The proposed TR incoherence condition under which the result holds is similar to the matrix incoherence condition. The experiments on synthetic data verify the recovery guarantee for TR completion. Moreover, the experiments on real-world data show that our method improves the recovery performance compared with the state-of-the-art methods.