Interpolating Convex and Non-Convex Tensor Decompositions via the Subspace Norm

We consider the problem of recovering a low-rank tensor from its noisy observation. Previous work has shown a recovery guarantee with signal to noise ratio O(nlceil K/2 rceil /2) for recovering a Kth order rank one tensor of size n× cdots × n by recursive unfolding. In this paper, we first improve this bound to O(nK/4) by a much simpler approach, but with a more careful analysis. Then we propose a new norm called the subspace norm, which is based on the Kronecker products of factors obtained by the proposed simple estimator. The imposed Kronecker structure allows us to show a nearly ideal O(√n+√HK-1) bound, in which the parameter H controls the blend from the non-convex estimator to mode-wise nuclear norm minimization. Furthermore, we empirically demonstrate that the subspace norm achieves the nearly ideal denoising performance even with H=O(1).