It seems that the local minima aren't actually a problem in these high dimensional spaces because most of them are either very close to the global minima or are

It seems that the local minima aren't actually a problem in these high dimensional spaces because most of them are either very close to the global minima or are saddle points, which can be escaped. There was a lot of work on this out of Bengio's lab. One such paper was Dauphin's "Identifying and attacking the saddle point..."