期刊
OPERATIONS RESEARCH
卷 70, 期 2, 页码 1219-1237出版社
INFORMS
DOI: 10.1287/opre.2021.2106
关键词
tensor completion; nonconvex optimization; gradient descent; spectral methods; entrywise statistical guarantees; minimaxity
资金
- Air Force Office of Scientific Research [FA9550-19-1-0030]
- Office of Naval Research [N00014-19-1-2120]
- Army Research Office [W911NF-18-1-0303, W911NF-20-1-0097]
- National Science Foundation (NSF) [CCF-1907661, IIS-1900140, IIS-2100158]
- Princeton SEAS Innovation Award
- NSF [DMS-1736417, PHY-1748958]
- Gordon Y. S. Wu Fellowship in Engineering
This study focuses on a noisy tensor completion problem and introduces a two-stage nonconvex algorithm that can find all individual tensor factors within nearly linear time while enjoying near-optimal statistical guarantees. The insights from nonconvex optimization in this analysis may have implications for other tensor estimation problems.
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. Whereas 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 suboptimal statistical guarantees. Focusing on incoherent and well -conditioned tensors of a constant canonical polyadic 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 l(infinity) statistical accuracy. We also discuss 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.
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