From the 1 of 12 linked papers with an AI index.
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Metric Poincaré inequalities for graphs
Dylan J. Altschuler, Pandelis Dodos, Konstantin Tikhomirov +1
This article obtains purely metric counterparts of cornerstone results in the theory of embedding graphs into normed spaces. Our first main result is a metric analogue of MatouÅ¡ek…
Metric dimension reduction modulus for superlogarithmic distortion
Dylan J. Altschuler, Konstantin Tikhomirov
The metric dimension reduction modulus is the smallest such that every --point metric space can be embedded into some -dimensional normed space, wit…
Discrete Poincaré inequalities and universal approximators for random graphs
Dylan J. Altschuler, Pandelis Dodos, Konstantin Tikhomirov +1
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 al…
A combinatorial approach to nonlinear spectral gaps
Dylan J. Altschuler, Pandelis Dodos, Konstantin Tikhomirov +1
A seminal open question of Pisier and Mendel--Naor asks whether every degree-regular graph which satisfies the classical discrete Poincaré inequality for scalar functions, also sa…
Universal geometric non-embedding of random regular graphs
Dylan J. Altschuler, Konstantin Tikhomirov
Let be fixed, be a large integer. It is a classical result that --regular expanders on vertices are not embeddable as geometric (distance) graphs int…