Nonconvex Low-Rank Tensor Completion from Noisy Data
This addresses a noisy tensor completion problem with broad practical applications, offering a computationally efficient and statistically optimal solution.
The paper tackles the problem of reconstructing a low-rank tensor from incomplete and noisy observations, proposing a two-stage nonconvex algorithm that achieves near-optimal statistical guarantees and nearly linear computational time.
We study a noisy tensor completion problem of broad practical interest, namely, the reconstruction of a low-rank tensor from highly incomplete and randomly corrupted observations of its entries. While a variety of prior work has been dedicated to this problem, prior algorithms either are computationally too expensive for large-scale applications, or come with sub-optimal statistical guarantees. Focusing on "incoherent" and well-conditioned tensors of a constant CP rank, we propose a two-stage nonconvex algorithm -- (vanilla) gradient descent following a rough initialization -- that achieves the best of both worlds. Specifically, the proposed nonconvex algorithm faithfully completes the tensor and retrieves all individual tensor factors within nearly linear time, while at the same time enjoying near-optimal statistical guarantees (i.e. minimal sample complexity and optimal estimation accuracy). The estimation errors are evenly spread out across all entries, thus achieving optimal $\ell_{\infty}$ statistical accuracy. We have also discussed how to extend our approach to accommodate asymmetric tensors. The insight conveyed through our analysis of nonconvex optimization might have implications for other tensor estimation problems.