Guarantees of Augmented Trace Norm Models in Tensor Recovery

Ziqiang Shi, Jiqing Han, Tieran Zheng, Ji Li

Abstract: This paper studies the recovery guarantees of the models of minimizing ||χ||* + 1/2α||χ||F2 where χ is a tensor and ||χ||* and ||χ||F are the trace and Frobenius norm of respectively. We show that they can efficiently recover low-rank tensors. In particular, they enjoy exact guarantees similar to those known for minimizing ||χ||* under the conditions on the sensing operator such as its null-space property, restricted isometry property, or spherical section property. To recover a low-rank tensor χ0, minimizing ||χ||* + 1/2α||χ||F2 returns the same solution as minimizing ||χ||* almost whenever α ≥ 10 max i ||X(i)0||2.