We consider the problem of recovering a low-rank tensor from its noisy observation.

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.

We consider the problem of recovering a low-rank tensor from its noisy observation.

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.

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. Furthermore, we empirically demonstrate that the subspace norm achieves the nearly ideal denoising performance even with $H=O(1)$.