Johnson-Lindenstrauss notes

PrA∼D [∣∣‖Ax‖22 − 1∣∣ > ε] < δ. Lemma 2 implies Lemma 1, since we can set δ = 1/n2 then apply a union bound on the vectors (xi − xj)/‖xi − xj‖2 for all i < j. The first proof of Lemma 2 was given by [15]. Since then, several proofs have been given where D can be taken as a distribution over matrices with independent Gaussian or Bernoulli entries, or even more generally, Ω(log(1/δ))-wise independent entries which each have mean 0, variance 1/k, and a subGaussian tail1; see the proofs in [1, 6, 7, 9, 11, 14, 16, 18]).