Nonconvex Low-Rank Tensor Completion from Noisy Data
Nonconvex Low-Rank Tensor Completion from Noisy Data
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DOI：
10.1287/opre.2021.2106
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发表时间：
2021-06-03
影响因子：
2.7
通讯作者：
Chen, Yuxin
中科院分区：
文献类型：
--
作者：
Cai, Changxiao;Li, Gen;Chen, Yuxin
我们研究了一个嘈杂的张量完成问题，即广泛的实践兴趣，即从高度不完整和随机损坏的参赛作品中重建了低排名的张量。尽管已经针对此问题进行了各种先前的工作，但先前的算法在计算上对于大规模应用来说太昂贵了，或者具有次优统计保证。为了关注恒定规范多边形等级的“不一致”和良好的条件张量，我们提出了一个两阶段的非convex算法 - （香草）梯度下降，因为初始化了初始化，这实现了两全其美。具体而言，提出的非凸算法忠实地完成了张量，并在几乎线性的时间内检索所有单个张量因子，同时享受近乎最佳的统计保证（即，最小的样本复杂性和最佳估计精度）。估计误差均匀分布在所有条目中，从而达到了最佳的L（Infinity）统计准确性。我们还讨论了如何扩展我们的方法以适应不对称的张量。通过我们对非凸优化的分析传达的见解可能对其他张量估计问题有影响。
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.