Theoretical Aspect:

Not a lot of work has been done on Metric Repair. Preliminary work was on a special case of the problem when $G$ is complete weighted graph in Fan et al and Gilbert and Jain. One of the major results is that Fan et al. which states that the problem is NP Hard. A detailed analysis of the more detailed problem can be found here. In particular it provides some theoretical results about the hardness of the problem, structural results about the solution, and approximating algorithms. 

Questions that still need to be answered:

1. We can see that the problem in NP Hard, but it is not clear where it lies within NP. In particular if we parametrize by the size of the optimal solution, then we are interested in where the problem lies in the $W$ hierarchy. More specifically, we can Gilbert and Sonthalia showed that if $G$ is restricted to chordal graphs then the problem is fixed parameter tractable, but we don't know where the general problem lies. 

2. There is a very clear relationship between set cover or hitting set and metric repair. Here we think of each  broken cycle as an element in the universe and the bottom edges in these broken cycles are set that contains these elements (cycles). Hence, we would expect the approximating ratios for set cover to sort of translate over to metric repair. Specifically, we want to know if there is $O(\log(n))$ approximating algorithm.

Applications to Machine Learning

As we said before many algorithms depend on the data living in a metric space. In particular Gilbert and Sonthalia showed that the metric repair can be used to provide an extension to many dimensionality reduction algorithms such as Isomap, LLE, Laplacian EigenMaps to handle the situation when we have missing data or corrupted distances. 

An implementation of their algorithm in Julia along with some example can be found here. We can see some of the examples of the projections produced: