The other thing is, in some areas (primarily rural areas), HSR could use the median, and share right-of-way with a freeway. (Won't work in many areas, but that can reduce cost over long distances.)
Or, any rail could (and in some areas (primarily cities), it does, for local rail).
You could also use a system that runs high speed on dedicated high speed rail, and can also run on existing freight rail, to use what little existing infrastructure there is, or use cheaper infrastructure where the high speed infrastructure would cost too much.