Wedge Sampling: New Method Achieves Nearly-Linear Sample Complexity for Tensor Completion
Researchers have introduced Wedge Sampling, a non-adaptive sampling scheme for low-rank tensor completion that achieves nearly linear sample complexity in the tensor dimension. Unlike standard uniform entry sampling, wedge sampling allocates observations to structured length-two patterns in a bipartite graph, strengthening the spectral signal needed for efficient initialization. The work, accepted at COLT 2026, suggests that the known statistical-to-computational gap in tensor completion is largely an artifact of the uniform sampling model rather than a fundamental barrier.
The paper, authored by Yizhe Zhu and collaborators and accepted at COLT 2026, presents Wedge Sampling as a new paradigm for recovering a low-rank tensor of order k and dimension n×⋯×n from a subset of its entries. Standard uniform entry sampling requires on the order of n^(k/2) samples for computationally efficient methods, a cost that grows rapidly with tensor order. Wedge sampling instead structures observations as length-two paths (wedges) in an associated bipartite graph, directly amplifying the spectral correlations that initialization algorithms rely on. This design enables polynomial-time algorithms to achieve both weak and exact recovery with only nearly linear sample complexity in n. The method is described as plug-and-play: the wedge-based spectral initialization can be paired with existing refinement procedures—spectral or gradient-based—using only an additional Õ(n) uniformly sampled entries. The authors argue that the statistical-to-computational gap identified by Barak and Moitra (2022) is, to a large extent, a consequence of the uniform sampling assumption, and that structured non-adaptive measurement designs can circumvent it. The preprint spans 65 pages with 3 figures and was updated in June 2026.
What's missing
The paper does not report empirical benchmarks on real-world tensor datasets; it is unclear how wedge sampling performs in practice when the low-rank assumption is only approximately satisfied or when the tensor dimensions are heterogeneous (non-square).
What different sources said
- arXiv cs.LGCenter
Wedge Sampling: Efficient Tensor Completion with Nearly-Linear Sample Complexity
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