Guarantees of Augmented Trace Norm Models in Tensor Recovery

This paper studies the recovery guarantees of the models of minimizing ‖X‖∗ + 1 2α‖X‖F where X is a tensor and ‖X‖∗ and ‖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 ‖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 X , minimizing ‖X‖∗ + 1 2α‖X‖F returns the same solution as minimizing ‖X‖∗ almost whenever α ≥ 10max i ‖X0 (i)‖2. Index Terms Tensor norms, augmented model, convex optimization, low-rank recovery, restricted isometry property.