Cs 229r: Algorithms for Big Data Lecture October 13th 3 Main Section 3.1 Lower Bounds on Dimensionality Reduction 3.1.1 Lower Bounds on Distributional Johnson Lindenstrauss Lemma

The first lower bound was proved by Jayram and Woodruff [1] and then by Kane,Meka,Nelson [2]. The lower bound tells that any ( , δ)-DJL for , δ ∈ (0, 1/2) must have m = Ω(min{n, −2 log(1δ )}). The second proof builds on the following idea: Since for all x we have the probabilistic guarantee PΠ∼D ,δ [|‖Πx‖2−1| < max{ , 2}] < δ, then it is true also for any distribution over x. We are going to pick x according to the uniform distribution over the sphere. Then this implies that there exists a matrix Π such that Px[|‖Πx‖2 − 1| < ] < δ. It is shown that this cannot happen for any fixed matrix Π ∈ Rm×n unless m satisfies the lower bound.