Lecture 17

Claim of Johnson-Lindenstrauss Theorem: The Euclidean metric on any finite set X (a bunch of high dimensional points) can be embedded with distortion D = 1 + ε in R for k = O(ε−2 log n). If we lose ε (ε = 0), it becomes almost impossible to do better than that in R. Nevertheless, it is not hard to construct a counter example to this: a simplex of n+ 1 points. The Johnson-Lindenstrauss theorem gives us an interesting result: if we project x to a random subspace, the projection y give us an approximate length of x for some fixed multiplication factor c, i.e. ‖x‖ ∼ c · ‖y‖. And c · y is embedded with distortion D = 1 + ε.