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(n {\ceil{K/2}/2}) for recovering a K th order rank one tensor of size n\times \cdots \times n by recursive unfolding. In this paper, we first improve this bound to O(n {K/4}) by a much simpler approach, but with a more careful analysis. Then we propose a new norm called the \textit{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(\sqrt{n} \sqrt{H {K-1}}) bound, in which the parameter H controls the blend from the non-convex estimator to mode-wise nuclear norm minimization.