Abstract:Nonlinear Poincaré inequalities are indispensable tools in the study of dimension reduction and low-distortion embeddings of graphs into metric spaces, and have found remarkable algorithmic applications. A basic open problem, posed by Jon Kleinberg (2013), asks whether the optimal nonlinear Poincaré constant for maps between two independent $3$-regular random graphs is dimension-free, i.e., independent of vertex-set sizes. We give a complete and affirmative resolution to Kleinberg's problem, also allowing for arbitrary graph degrees. As a corollary, we obtain a stochastic construction of $O(1)\text{-universal}$ approximators for random graphs, answering a question of Mendel and Naor.
From: Pandelis Dodos [view email]
[v1]
Fri, 20 Jun 2025 18:55:09 UTC (23 KB)
[v2]
Wed, 30 Jul 2025 15:36:50 UTC (24 KB)
[v3]
Mon, 17 Aug 2026 07:42:47 UTC (26 KB)
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