Interpolating Convex and Non-Convex Tensor Decompositions via the Subspace Norm
This work addresses tensor recovery, a key challenge in machine learning and signal processing, by providing improved theoretical bounds and a novel norm for better denoising, though it is incremental in building on existing methods.
The paper tackles the problem of recovering low-rank tensors from noisy observations, improving the previous recovery guarantee from O(n^(ceil(K/2)/2)) to O(n^(K/4)) with a simpler method and introducing a subspace norm that achieves a nearly ideal O(sqrt(n) + sqrt(H^(K-1))) bound, with empirical results showing near-ideal denoising performance even with H=O(1).
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 $O(n^{\lceil K/2 \rceil /2})$ 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 $O(\sqrt{n}+\sqrt{H^{K-1}})$ 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 $H=O(1)$.