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

TL;DR

This work studies tensor completion for order-$k$ tensors of size $n\times\cdots\times n$ under a non-adaptive sampling scheme called Wedge Sampling. Wedge Sampling allocates observations to structured wedges in a bipartite sampling graph to strengthen the spectral signal used for initialization, enabling recovery with nearly linear sample complexity in $n$ and a plug-and-play pipeline with existing refinement methods. The authors establish concentration and perturbation tools tailored to wedge sampling, derive spectral guarantees for initialization on unfolded tensors, and show that gradient-descent refinement from this initialization achieves exact recovery with modest additional uniform samples, effectively closing much of the statistical-to-computational gap under uniform sampling. Collectively, the results provide a broad, practical framework for efficient tensor completion with non-adaptive, nonuniform measurements and introduce new analytic techniques for handling wedge-induced dependencies in the data.

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.

Figures (3)

• Figure 1: Illustration of uniform entry sampling versus wedge sampling on the bipartite sampling graph for an order-3 tensor. Under uniform sampling, the edges $(i,\ell)$ and $(j,\ell)$ are observed independently; under wedge sampling, we sample the length-two path (wedge) $(i,\ell,j)$ uniformly from the wedge space $\{(i,\ell,j): 1\leq i\leq j\leq n, \ell\in [n^2] \}$, and $\{A_{i\ell}, A_{j\ell}\}$ are both observed for each wedge $(i,\ell,j)$.
• Figure 2: Comparison of wedge sampling versus uniform sampling across different tasks.
• Figure 3: Comparison of tensor completion via GD when using Wedge Sampling and Uniform Sampling for subspace estimation. Each panel corresponds to a different number of samples, ignoring constants and log factors, (left) $n^2$ (middle) $n^{1.5}$, and (right) $n^{1.25}$

Theorems & Definitions (76)

• Theorem 1: Tensor completion with wedge sampling, informal
• Definition 2: Incoherence
• Definition 3: Tensor incoherence
• Lemma 4
• Theorem 5: Concentration of random matrices under wedge sampling
• Theorem 6: Left singular subspace recovery
• Theorem 7: Spectral method, Frobenius-norm bound
• Theorem 8: Wedge sampling with gradient descent
• Theorem 9: Concentration of sparse random tensors
• proof