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We consider the problem of recovering a low-rank tensor from its noisyobservation. Previous work has shown a recovery guarantee with signal to noiseratio $O(n^{\lceil K/2 \rceil /2})$ for recovering a $K$th order rank onetensor 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, butwith a more careful analysis. Then we propose a new norm called the subspacenorm, which is based on the Kronecker products of factors obtained by theproposed simple estimator. The imposed Kronecker structure allows us to show anearly 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 normminimization. Furthermore, we empirically demonstrate that the subspace normachieves the nearly ideal denoising performance even with $H=O(1)$.

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