To be fair the paper looks interesting, the news title is garbage but it's not the fault of the authors who did a pretty cool job. Anyway, it seems like a nice application of a very well-known idea, which is cool.

By the way, is measuring the perturbation influence on the loss a common idea? Because I am mostly aware of using it to see how the regression value or class probabilities change - and the perturbation is done on the inputs, not params (edit ** incorrect, they do the perturbation on the inputs).

edit: "We follow the results of the studies [Koh and Liang, 2017; Bis et al., 2021] to approximate the perturbation effect directly through the model’s parameters when executing Leaving-One-Out experiments on the input. The influence function estimating the perturbation  of an input z is then derived as:" - seems like I misunderstood it due to their notation. Seems like a pretty regular method.