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.
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Supported by the Israel Science Foundation and by a European Research Council (ERC) Starting Grant.
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Regev, O. Entropy-based bounds on dimension reduction in L 1 . Isr. J. Math. 195, 825–832 (2013). https://doi.org/10.1007/s11856-012-0137-6
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DOI: https://doi.org/10.1007/s11856-012-0137-6