Robust Completion for Rank-1 Tensors with Noises
Jiawang Nie, Xindong Tang, Jinling Zhou
TL;DR
This work proposes a robust approach to rank-1 tensor completion for cubic tensors under noisy observations by formulating a biquadratic optimization in $(a,b)$ with $||a||=||b||=1$ that eliminates $c$. It develops an efficient convex relaxation based on a moment/SOS-like framework using a linear map $G[y]$, and provides completeness results: when $G[y^*]$ is separable (notably rank 1), the relaxation is tight and yields exact minimizers; otherwise, a spectral decomposition yields approximate candidates for $a,b$. A retrieval step recovers $c$ via linear least squares, producing a rank-1 completing tensor and offering stability guarantees for small noise. Numerical experiments show the method’s efficiency and reliability across different noise regimes, densities, and separability scenarios, and compare it favorably against a traditional nonlinear least-squares approach. The work advances practical tools for noisy rank-1 tensor completion and opens questions for higher-order tensors and singular minimizers.
Abstract
This paper studies the rank-1 tensor completion problem for cubic tensors when there are noises for observed tensor entries. First, we propose a robust biquadratic optimization model for obtaining rank-1 completing tensors. When the observed tensor is sufficiently close to be rank-1, we show that this biquadratic optimization produces an accurate rank-1 tensor completion. Second, we give an efficient convex relaxation for solving the biquadratic optimization. When the optimizer matrix is separable, we show how to get optimizers for the biquadratic optimization and how to compute the rank-1 completing tensor. When that matrix is not separable, we apply its spectral decomposition to obtain an approximate rank-1 completing tensor. Numerical experiments are given to explore the efficiency of this biquadratic optimization model and the proposed convex relaxation.