Interpolating Convex and Non-Convex Tensor Decompositions via the Subspace Norm
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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^ K/2 /2) for recovering a Kth order rank one tensor of size n×...× 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 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)+√(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).
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