This is actually a nice illustration of the central problem with this argument: the more personally identifiable a piece of information is, the less recoverable it'll be, and vice-versa. If all of the points of data are on some n-dimensional line, then obviously all of them can easily be recovered, but knowing all those things about a person doesn't actually tell you any more about them than knowing just one of those things. Conversely, if the points of data are very random then it'll only require a handful of points to uniquely identify a person and find the entry in the original data set with all their other information, but dimensionality reduction will have to throw that data away - you simply won't be able to recover that information from the model. (We actually know from the literature on de-anonymization that a lot of data falls into the second category.)