Guarantees of Augmented Trace Norm Models in Tensor Recovery

Abstract

This paper studies the recovery guarantees of the models of minimizing $\Vert \mathcal{X}{\Vert }_{\ast }+\frac{1}{2\alpha }\Vert \mathcal{X}{\Vert }_F^2$ where $\mathcal{X}$ is a tensor and $\Vert \mathcal{X}{\Vert }_{\ast }$ and $\Vert \mathcal{X}{\Vert }_F$ 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 $\Vert \mathcal{X}{\Vert }_{\ast }$ 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 $\Vert \mathcal{X}{\Vert }_{\ast }+\frac{1}{2\alpha }\Vert \mathcal{X}{\Vert }_F^2$ returns the same solution as minimizing $\Vert \mathcal{X}{\Vert }_{\ast }$ almost whenever $\alpha \ge 10\underset{i}{\max }\Vert X_{(i)}^0{\Vert }_2$.