is the red-giant branch a folded line?

Today Gaby Contardo (Flatiron) and I discussed the following problem, which (I find) is hard to even formulate: The red-giant branch (above the red clump) is supposed to be a place where most stars are “going up” (getting brighter and bigger) and a minority of stars are “going down”, heading back towards the red clump. And yet, all the stars on the red-giant branch look very similar spectrally (though not asteroseismically!). Can we see in the spectra (I'm thinking APOGEE here), in a purely data-driven way (that is, without a supervisory training set), that the red-giant branch is a folded locus that projects to a simple line, and not just a simple line? This paper by Ting, Hawkins, & Rix gives me hope that we can.

I'm interested in all this because it gets at what can be just observed in the data, as compared to what we project onto the data from theory. This project is a toy, in some sense, because we do know about the asteroseismic differences, and we can use them, without being theoretically supervised.