Entropy-based bounds on dimension reduction in L ¹

Abstract

We show that for every large enough integer N, there exists an N-point subset of L ₁ such that for every D > 1, embedding it into ℓ ^d₁ with distortion D requires dimension d at least ${N^{\Omega (1/{D^2})}}$, and that for every ɛ > 0 and large enough integer N, there exists an N-point subset of L ₁ such that embedding it into ℓ ^d₁ with distortion 1 + ɛ requires dimension d at least ${N^{\Omega (1/{D^2})}}$). These results were previously proven by Brinkman and Charikar [JACM, 2005] and by Andoni, Charikar, Neiman and Nguyen [FOCS 2011]. We provide an alternative and arguably more intuitive proof based on an entropy argument.