Guarantees of Augmented Trace Norm Models in Tensor Recovery

Abstract

This paper studies the recovery guarantees of the models of minimizing $\|\mathcal{X}\|_*+\frac{1}{2\alpha}\|\mathcal{X}\|_F^2$ where $\mathcal{X}$ is a tensor and $\|\mathcal{X}\|_*$ and $\|\mathcal{X}\|_F$ 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 $\|\mathcal{X}\|_*$ 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 $\mathcal{X}^0$, minimizing $\|\mathcal{X}\|_*+\frac{1}{2\alpha}\|\mathcal{X}\|_F^2$ returns the same solution as minimizing $\|\mathcal{X}\|_*$ almost whenever $\alpha\geq10\mathop {\max}\limits_{i}\|X^0_{(i)}\|_2$.