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 a"" (1) (d) with distortion D requires dimension d at least , and that for every E > > 0 and large enough integer N, there exists an N-point subset of L (1) such that embedding it into a"" (1) (d) with distortion 1 + E > requires dimension d at least ). 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.