On non-uniqueness of the Student's t-formulation for linear inverse problems

We review the statistical interpretation of inverse problem formulations, and the motivations for selecting non-convex penalties for robust behaviour with respect to measurement outliers or artifacts in the data. An important downside of using non-convex formulations such as the Student's t is the potential for non-uniqueness, and we present a simple example where the Student's t penalty can be made to have many local minima by appropriately selecting the degrees of freedom parameter. On the other hand, the non-convexity of the Student's t is precisely what gives it the ability to ignore artifacts in the data. We explain this idea, and present a stylized imaging experiment, where the Student's t is able to recover a velocity perturbation from data contaminated by a very peculiar artifact –- data from a different velocity perturbation. The performance of Student's t inversion is investigated empirically for different values of the degrees of freedom parameter, and different initial conditions.