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：我们提出楔形采样（Wedge Sampling），一种用于低秩张量补全的新型非自适应采样方案。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。与标准均匀条目采样（即从 $[n]^k$ 中独立同分布采样）不同，楔形采样将观测值分配至关联二分采样图中结构化的长度为二的模式（楔形）。直接促进这些长度为二的连接能够增强支撑高效初始化的谱信号，特别是在均匀采样过于稀疏而无法产生足够信息相关性的场景中。我们的主要结果表明，这种采样范式的转变使得多项式时间算法能够以 $n$ 的近线性样本复杂性实现弱恢复与精确恢复。该方法具有即插即用特性：基于楔形采样的谱初始化可与现有精炼过程（如谱方法或梯度方法）结合，仅需额外 $\tilde{O}(n)$ 个均匀采样条目，显著优于高效方法在均匀条目采样下通常所需的 $\tilde{O}(n^{k/2})$ 样本复杂性。总体而言，我们的结果揭示，Barak 与 Moitra（2022）强调的统计-计算鸿沟很大程度上源于张量补全的均匀条目采样模型，而保证强初始化的替代性非自适应测量设计可突破此障碍。