A combinatorial approach to nonlinear spectral gaps
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912400298475520 |
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| author | Altschuler, Dylan J. Dodos, Pandelis Tikhomirov, Konstantin Tyros, Konstantinos |
| author_facet | Altschuler, Dylan J. Dodos, Pandelis Tikhomirov, Konstantin Tyros, Konstantinos |
| contents | 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 satisfies an analogous inequality for functions taking values in \textit{any} normed space with non-trivial cotype. Motivated by applications, it is also greatly important to quantify the dependence of the corresponding optimal Poincaré constant on the cotype $q$. Works of Odell--Schlumprecht (1994), Ozawa (2004), and Naor (2014) make substantial progress on the former question by providing a positive answer for normed spaces which also have an unconditional basis, in addition to finite cotype. However, little is known in the way of quantitative estimates: the mentioned results imply a bound on the Poincaré constant depending super-exponentially on $q$.
We introduce a novel combinatorial framework for proving quantitative nonlinear spectral gap estimates. The centerpiece is a property of regular graphs that we call \emph{long range expansion}, which holds with high probability for random regular graphs. Our main result is that any regular graph with the long-range expansion property satisfies a discrete Poincaré inequality for any normed space with an unconditional basis and cotype $q$, with a Poincaré constant that depends \emph{polynomially} on $q$, which is optimal. As an application, any normed space with an unconditional basis which admits a low distortion embedding of an $n$-vertex random regular graph, must have cotype at least polylogarithmic in $n$. This extends a celebrated lower-bound of Matoušek for low distortion embeddings of random graphs into $\ell_q$ spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04394 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A combinatorial approach to nonlinear spectral gaps Altschuler, Dylan J. Dodos, Pandelis Tikhomirov, Konstantin Tyros, Konstantinos Metric Geometry Combinatorics Functional Analysis Probability 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 satisfies an analogous inequality for functions taking values in \textit{any} normed space with non-trivial cotype. Motivated by applications, it is also greatly important to quantify the dependence of the corresponding optimal Poincaré constant on the cotype $q$. Works of Odell--Schlumprecht (1994), Ozawa (2004), and Naor (2014) make substantial progress on the former question by providing a positive answer for normed spaces which also have an unconditional basis, in addition to finite cotype. However, little is known in the way of quantitative estimates: the mentioned results imply a bound on the Poincaré constant depending super-exponentially on $q$. We introduce a novel combinatorial framework for proving quantitative nonlinear spectral gap estimates. The centerpiece is a property of regular graphs that we call \emph{long range expansion}, which holds with high probability for random regular graphs. Our main result is that any regular graph with the long-range expansion property satisfies a discrete Poincaré inequality for any normed space with an unconditional basis and cotype $q$, with a Poincaré constant that depends \emph{polynomially} on $q$, which is optimal. As an application, any normed space with an unconditional basis which admits a low distortion embedding of an $n$-vertex random regular graph, must have cotype at least polylogarithmic in $n$. This extends a celebrated lower-bound of Matoušek for low distortion embeddings of random graphs into $\ell_q$ spaces. |
| title | A combinatorial approach to nonlinear spectral gaps |
| topic | Metric Geometry Combinatorics Functional Analysis Probability |
| url | https://arxiv.org/abs/2410.04394 |