Guarantees of Augmented Trace Norm Models in Tensor Recovery
This provides theoretical guarantees for tensor recovery methods, which is incremental for researchers in machine learning and signal processing.
This paper tackles the problem of recovering low-rank tensors by analyzing the augmented trace norm model, showing it efficiently recovers tensors with exact guarantees similar to standard trace norm minimization under conditions like null-space or restricted isometry properties, with a specific threshold for the parameter α.
This paper studies the recovery guarantees of the models of minimizing $\|\mathcal{X}\|_*+\frac{1}{2α}\|\mathcal{X}\|_F^2$ 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 $\|\mathcal{X}\|_*+\frac{1}{2α}\|\mathcal{X}\|_F^2$ returns the same solution as minimizing $\|\mathcal{X}\|_*$ almost whenever $α\geq10\mathop {\max}\limits_{i}\|X^0_{(i)}\|_2$.