Solving exact and noisy rank-one tensor completion with semidefinite programming
Diego Cifuentes, Zhuorui Li
TL;DR
This work addresses exact and robust recovery for rank-one tensors from partial observations by formulating semidefinite programming relaxations that capitalize on combinatorial propagation conditions of the observation mask $\Omega$. The authors establish four deterministic propagation notions (GS, S, SR, A) with precise implications and demonstrate that, under SR-propagation, a compact SDP with a matrix variable of size $(N{+}1)\times(N{+}1)$ exactly recovers the tensor in the noiseless case and stably recovers it under noise. They further show that, for random masks, these propagation conditions hold whp with sample complexity $O\big(\sqrt{N}\,\mathrm{polylog}\,N\big)$, matching the best known results in the random setting while accommodating structured masks. The approach is extended via weighted and SOS-based variants to broaden the regime of exact recovery, and experiments indicate substantial gains over existing methods in both exact and noisy regimes, including a scalable solver for larger problems and a low-rank extension for practical applications like image inpainting.
Abstract
Consider recovering a rank-one tensor of size $n_1 \times \cdots \times n_d$ from exact or noisy observations of a few of its entries. We tackle this problem via semidefinite programming (SDP). We derive deterministic combinatorial conditions on the observation mask $Ω$ (the set of observed indices) under which our SDPs solve the exact completion and achieve robust recovery in the noisy regime. These conditions can be met with as few as $\bigl(\sum_{i=1}^d n_i\bigr) - d + 1$ observations for special $Ω$. When $Ω$ is uniformly random, our conditions hold with $O\!\bigl((\prod_{i=1}^d n_i)^{1/2}\,\mathrm{polylog}(\prod_{i=1}^d n_i)\bigr)$ observations. Prior works mostly focus on the uniformly random case, ignoring the practical relevance of structured masks. For $d=2$ (matrix completion), our propagation condition holds if and only if the completion problem admits a unique solution. Our results apply to tensors of arbitrary order and cover both exact and noisy settings. In contrast to much of the literature, our guarantees rely solely on the combinatorial structure of the observation mask, without incoherence assumptions on the ground-truth tensor or uniform randomness of the samples. Preliminary computational experiments show that our SDP methods solve tensor completion problems using significantly fewer observations than alternative methods.