In my weekend research time, I worked out a fully data-driven method for measuring radial velocities in an extreme-precision (or even normal-precision) spectroscopic survey. The idea is to simultaneously fit for the spectrum of the star and its radial-velocity offset; you need multiple epochs of observations to get both (at least at high signal-to-noise). Because the model is fully data-driven, it won't give absolute radial velocities; it will only give relative velocities. That's always the cost of being data driven—the loss of interpretability.
I also added to the model some flexibility to capture spectral variations with time, especially those that might project onto the radial-velocity direction or measurement. That would permit us to discover and characterize spectral changes that co-vary with surface radial-velocity perturbations or jitter. I am trying to write something down that would be practical to apply to HARPS data, but I'm all theory right now.