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

TL;DR

This paper introduces Wedge Sampling, a novel non-adaptive sampling method for low-rank tensor completion that improves sample efficiency and computational feasibility by focusing on structured patterns rather than uniform sampling.

Contribution

The paper proposes Wedge Sampling, a new structured sampling scheme that enhances spectral initialization and reduces sample complexity for tensor completion.

Findings

Achieves nearly linear sample complexity in tensor dimension n.

Enables polynomial-time recovery algorithms with improved sample efficiency.

Demonstrates that alternative sampling designs can overcome the statistical-to-computational gap.

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.