Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entries
This work addresses noisy tensor completion, a problem in machine learning and statistics, by providing both statistical guarantees and computational efficiency, representing a significant advancement over existing methods.
The paper tackles the problem of estimating a low-rank tensor from noisy, partially observed entries, establishing minimax optimal convergence rates and proposing a polynomial-time algorithm that achieves these rates.
In this article, we develop methods for estimating a low rank tensor from noisy observations on a subset of its entries to achieve both statistical and computational efficiencies. There have been a lot of recent interests in this problem of noisy tensor completion. Much of the attention has been focused on the fundamental computational challenges often associated with problems involving higher order tensors, yet very little is known about their statistical performance. To fill in this void, in this article, we characterize the fundamental statistical limits of noisy tensor completion by establishing minimax optimal rates of convergence for estimating a $k$th order low rank tensor under the general $\ell_p$ ($1\le p\le 2$) norm which suggest significant room for improvement over the existing approaches. Furthermore, we propose a polynomial-time computable estimating procedure based upon power iteration and a second-order spectral initialization that achieves the optimal rates of convergence. Our method is fairly easy to implement and numerical experiments are presented to further demonstrate the practical merits of our estimator.