Nonconvex Low-Rank Tensor Completion from Noisy Data

This paper investigates a problem of broad practical interest, namely, the reconstruction large-dimensional low-rank tensor from highly incomplete and randomly corrupted observations its entries. Although number papers have been dedicated to this completion problem, prior algorithms either are computationally too expensive for large-scale applications or come with suboptimal statistical performance. Motivated by this, we propose fast two-stage nonconvex algorithm—a gradient method following rough initialization—that achieves best both worlds: optimal accuracy computational efficiency. Specifically, proposed algorithm provably completes retrieves all factors within nearly linear time, while at same time enjoying near-optimal guarantees (i.e., minimal sample complexity estimation accuracy). The insights conveyed through our analysis optimization might implications broader family problems beyond completion.