Robust Completion for Rank-1 Tensors with Noises

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. The software SDPNAL+ is applied to solve the resulting large size semidefinite programs. Numerical experiments are given to explore the efficiency of this biquadratic optimization model and the proposed convex relaxation.