Nonconvex Low-Rank Tensor Completion from Noisy Data

Changxiao Cai, Gen Li, H. Vincent Poor, Yuxin Chen

Abstract: We study a problem of broad practical interest, namely, the reconstruction of a low-rank tensor from highly incomplete and randomly corrupted observations of its entries. While various works have been dedicated to this tensor completion problem, prior algorithms either are computationally too expensive, or come with sub-optimal statistical guarantees. Focusing on ``incoherent'' 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 provably completes the tensor and retrieves all low-rank 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 entrywise statistical accuracy.

CMT Num: 1072