Interpolating Convex and NonConvex Tensor Decompositions via the Subspace Norm
Abstract
We consider the problem of recovering a lowrank tensor from its noisy observation. Previous work has shown a recovery guarantee with signal to noise ratio $O(n^{\lceil K/2 \rceil /2})$ 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 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 $O(\sqrt{n}+\sqrt{H^{K1}})$ bound, in which the parameter $H$ controls the blend from the nonconvex estimator to modewise nuclear norm minimization. Furthermore, we empirically demonstrate that the subspace norm achieves the nearly ideal denoising performance even with $H=O(1)$.
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 DOI:
 10.48550/arXiv.1503.05479
 arXiv:
 arXiv:1503.05479
 Bibcode:
 2015arXiv150305479Z
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Artificial Intelligence;
 Statistics  Machine Learning