Wedge Sampling: Efficient Tensor Completion with Nearly-Linear Sample Complexity

2026-02-05

AI Summary

This work addresses the fundamental gap between statistical and computational complexity in recovering low-rank tensors under uniform sampling, where sparse observations hinder efficient recovery. To overcome this limitation, the authors propose wedge sampling—a non-adaptive, structured sampling strategy that enhances spectral initialization by collecting length-two paths in an associated bipartite graph. Departing from the conventional uniform sampling paradigm, wedge sampling achieves both weak and exact recovery in polynomial time using only nearly linear (Õ(n)) samples, substantially improving upon the Õ(n^{k/2}) sample complexity required by traditional approaches. Moreover, the method seamlessly integrates with existing gradient-based or spectral refinement algorithms, offering a practical and theoretically grounded solution for low-rank tensor recovery.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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.

Problem

• tensor completion
• sample complexity
• low-rank tensor
• non-adaptive sampling
• statistical-to-computational gap

Innovation

• Wedge Sampling
• Tensor Completion
• Sample Complexity
• Spectral Initialization
• Low-rank Tensor