We introduce Wedge Sampling, a new non-adaptive sampling scheme for low-rank tensor completion. We study recovery of an order-$k$ low-rank tensor of dimension $n \times \cdots \times n$ from a subset of its entries. Unlike the standard uniform entry model (i.e., i.i.d. samples from $[n]^k$), wedge sampling allocates observations to structured length-two patterns (wedges) in an associated bipartite sampling graph. By directly promoting these length-two connections, the sampling design strengthens the spectral signal that underlies efficient initialization, in regimes where uniform sampling is too sparse to generate enough informative correlations. Our main result shows that this change in sampling paradigm enables polynomial-time algorithms to achieve both weak and exact recovery with nearly linear sample complexity in $n$. The approach is also plug-and-play: wedge-sampling-based spectral initialization can be combined with existing refinement procedures (e.g., spectral or gradient-based methods) using only an additional $\tilde{O}(n)$ uniformly sampled entries, substantially improving over the $\tilde{O}(n^{k/2})$ sample complexity typically required under uniform entry sampling for efficient methods. Overall, our results suggest that the statistical-to-computational gap highlighted in Barak and Moitra (2022) is, to a large extent, a consequence of the uniform entry sampling model for tensor completion, and that alternative non-adaptive measurement designs that guarantee a strong initialization can overcome this barrier.

翻译：本文提出楔形采样，一种用于低秩张量补全的新型非自适应采样方案。我们研究从部分条目中恢复维度为 $n \times \cdots \times n$ 的 $k$ 阶低秩张量的问题。与标准的均匀条目采样模型（即从 $[n]^k$ 中独立同分布采样）不同，楔形采样将观测分配到关联二分采样图中的结构化长度为二的模式（楔形）上。通过直接增强这些长度为二的连接，该采样设计在均匀采样过于稀疏而无法产生足够信息性相关性的情况下，强化了支撑高效初始化的谱信号。我们的主要结果表明，这种采样范式的改变使得多项式时间算法能够以 $n$ 的近线性样本复杂度同时实现弱恢复与精确恢复。该方法还具有即插即用特性：基于楔形采样的谱初始化可与现有细化流程（例如谱方法或基于梯度的方法）相结合，仅需额外 $\tilde{O}(n)$ 个均匀采样条目，这显著优于均匀条目采样下高效方法通常所需的 $\tilde{O}(n^{k/2})$ 样本复杂度。总体而言，我们的结果表明，Barak 与 Moitra (2022) 所强调的统计-计算鸿沟在很大程度上是张量补全中均匀条目采样模型的后果，而能够保证强初始化的替代性非自适应测量设计可以克服这一障碍。