For any integers d,n≥2 and 1/(min{n,d})0.4999<ε<1, we show the existence of a set of n vectors X⊂Rd such that any embedding f:X→Rm satisfying
∀x,y∈X,(1−ε)∥x−y∥22≤∥f(x)−f(y)∥22≤(1+ε)∥x−y∥22 must have m=Ω(ε−2lgn). This lower bound matches the upper bound given by the Johnson-Lindenstrauss
lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of
ε of interest, since there is always an isometric embedding into
dimension min{d,n} (either the identity map, or projection onto
span(X)).
Previously such a lower bound was only known to hold against linear maps f,
and not for such a wide range of parameters ε,n,d [LN16]. The
best previously known lower bound for general f was m=Ω(ε−2lgn/lg(1/ε)) [Wel74, Lev83, Alo03], which
is suboptimal for any ε=o(1).Comment: v2: simplified proof, also added reference to Lev8