Guarantees of Augmented Trace Norm Models in Tensor Recovery

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 $\alpha\geq10\mathop {\max}\limits{i}|X⁰{(i)}|_2$.