Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel


CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
Source Normalized Impact per Paper (SNIP) 2017: 0.833

Mathematical Citation Quotient (MCQ) 2017: 0.86

Open Access
Online
ISSN
2299-3274
See all formats and pricing
More options …

Comparison of Metric Spectral Gaps

Assaf Naor
Published Online: 2014-02-28 | DOI: https://doi.org/10.2478/agms-2014-0001

Abstract

Let A = (aij) ∊ Mn(ℝ) be an n by n symmetric stochastic matrix. For p ∊ [1, ∞) and a metric space (X, dX), let γ(A, dpx) be the infimum over those γ ∊ (0,∞] for which every x1, . . . , xn ∊ X satisfy

Thus γ (A, dpx) measures the magnitude of the nonlinear spectral gap of the matrix A with respect to the kernel dpX : X × X →[0,∞). We study pairs of metric spaces (X, dX) and (Y, dY ) for which there exists Ψ: (0,∞)→(0,∞) such that γ (A, dpX) ≤Ψ (A, dpY ) for every symmetric stochastic A ∊ Mn(ℝ) with (A, dpY ) < ∞. When Ψ is linear a complete geometric characterization is obtained.

Our estimates on nonlinear spectral gaps yield new embeddability results as well as new nonembeddability results. For example, it is shown that if n ∊ ℕ and p ∊ (2,∞) then for every f1, . . . , fn ∊ Lp there exist x1, . . . , xn ∊ L2 such that

and

This statement is impossible for p ∊ [1, 2), and the asymptotic dependence on p in (0.1) is sharp. We also obtain the best known lower bound on the Lp distortion of Ramanujan graphs, improving over the work of Matoušek. Links to Bourgain-Milman-Wolfson type and a conjectural nonlinear Maurey-Pisier theorem are studied.

Keywords: Metric embeddings; nonlinear spectral gaps; expanders; nonlinear type

References

  • [1] N. Alon, R. Boppana, and J. Spencer. An asymptotic isoperimetric inequality. Geom. Funct. Anal., 8(3), 411-436, 1998.CrossrefGoogle Scholar

  • [2] N. Alon and Y. Roichman. Random Cayley graphs and expanders. Random Structures Algorithms, 5(2), 271-284, 1994.Google Scholar

  • [3] S. Arora, J. R. Lee, and A. Naor. Euclidean distortion and the sparsest cut. J. Amer. Math. Soc., 21(1), 1-21 (electronic), 2008.Google Scholar

  • [4] S. Arora, S. Rao, and U. Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2), Art. 5, 37, 2009.Google Scholar

  • [5] P. Assouad. Plongements lipschitziens dans Rn. Bull. Soc. Math. France, 111(4), 429-448, 1983.Google Scholar

  • [6] K. Ball. Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal., 2(2), 137-172, 1992.CrossrefGoogle Scholar

  • [7] K. Ball, E. A. Carlen, and E. H. Lieb. Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math., 115(3), 463-482, 1994.Google Scholar

  • [8] Y. Bartal, N. Linial, M. Mendel, and A. Naor. On metric Ramsey-type phenomena. Ann. ofMath. (2), 162(2), 643-709, 2005.Google Scholar

  • [9] Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis. Vol. 1, volume 48 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2000.Google Scholar

  • [10] P. Biswal, J. R. Lee, and S. Rao. Eigenvalue bounds, spectral partitioning, and metrical deformations via flows. J. ACM, 57(3), Art. 13, 23, 2010.Google Scholar

  • [11] B. Bollobás and W. Fernandez de la Vega. The diameter of random regular graphs. Combinatorica, 2(2), 125-134, 1982.CrossrefGoogle Scholar

  • [12] J. Bourgain. On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math., 52(1-2), 46-52, 1985.Google Scholar

  • [13] J. Bourgain, V. Milman, and H. Wolfson. On type of metric spaces. Trans. Amer. Math. Soc., 294(1), 295-317, 1986.Google Scholar

  • [14] M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.Google Scholar

  • [15] A. Z. Broder and E. Shamir. On the second eigenvalue of random regular graphs. In 28th Annual Symposium on Foundations of Computer Science, pages 286-294, 1987.Google Scholar

  • [16] A.-P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math., 24, 113-190, 1964.Google Scholar

  • [17] F. Chaatit. On uniform homeomorphisms of the unit spheres of certain Banach lattices. Pacific J.Math., 168(1), 11-31, 1995.Google Scholar

  • [18] I. Chavel. Eigenvalues in Riemannian geometry, volume 115 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1984. Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk.Google Scholar

  • [19] S. Chawla, A. Gupta, and H. Räcke. Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut. ACM Trans. Algorithms, 4(2), Art. 22, 18, 2008.Google Scholar

  • [20] J. Cheeger. A lower bound for the smallest eigenvalue of the Laplacian. In Problems in analysis (Papers dedicated to Salomon Bochner, 1969), pages 195-199. Princeton Univ. Press, Princeton, N. J., 1970.Google Scholar

  • [21] D. Christofides and K. Markström. Expansion properties of random Cayley graphs and vertex transitive graphs via matrix martingales. Random Structures Algorithms, 32(1), 88-100, 2008.Google Scholar

  • [22] F. Chung. Diameters and eigenvalues. J. Amer. Math. Soc., 2(2), 187-195, 1989.Google Scholar

  • [23] M. Cwikel and S. Reisner. Interpolation of uniformly convex Banach spaces. Proc. Amer. Math. Soc., 84(4), 555-559, 1982.CrossrefGoogle Scholar

  • [24] M. de la Salle. Towards Banach space strong property (T) for SL(3, R). Preprint available at http://arxiv.org/abs/1307.2475, 2013.Google Scholar

  • [25] N. R. Devanur, S. A. Khot, R. Saket, and N. K. Vishnoi. Integrality gaps for sparsest cut and minimum linear arrangement problems. In STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 537-546. ACM, New York, 2006.Google Scholar

  • [26] M. M. Deza and M. Laurent. Geometry of cuts and metrics, volume 15 of Algorithms and Combinatorics. Springer-Verlag, Berlin, 1997.Google Scholar

  • [27] J. Ding, J. R. Lee, and Y. Peres. Markov type and threshold embeddings. Geom. Funct. Anal., 23(4), 1207-1229, 2013.CrossrefGoogle Scholar

  • [28] J. Elton. Sign-embeddings of ln1 . Trans. Amer. Math. Soc., 279(1), 113-124, 1983.Google Scholar

  • [29] P. Enflo. Uniform homeomorphisms between Banach spaces. In Séminaire Maurey-Schwartz (1975-1976), Espaces, Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. No. 18, page 7. CentreMath., École Polytech., Palaiseau, 1976.Google Scholar

  • [30] J. Fakcharoenphol and K. Talwar. An improved decomposition theorem for graphs excluding a fixed minor. In Approximation, randomization, and combinatorial optimization, volume2764 of Lecture Notes in Comput. Sci., pages 36-46. Springer, Berlin, 2003.Google Scholar

  • [31] U. Feige and G. Schechtman. On the optimality of the random hyperplane rounding technique for MAX CUT. Random Structures Algorithms, 20(3), 403-440, 2002. Probabilistic methods in combinatorial optimization. [32] M. Fiedler. Laplacian of graphs and algebraic connectivity. In Combinatorics and graph theory (Warsaw, 1987), volume 25 of Banach Center Publ., pages 57-70. PWN, Warsaw, 1989.Google Scholar

  • [33] T. Figiel. On the moduli of convexity and smoothness. Studia Math., 56, 121-155, 1976.Google Scholar

  • [34] T. Figiel, W. B. Johnson, and G. Schechtman. Random sign embeddings from lnr , 2 < r < 1. Proc. Amer. Math. Soc., 102(1), 102-106, 1988.Google Scholar

  • [35] J. Friedman. A proof of Alon’s second eigenvalue conjecture and related problems. Mem. Amer. Math. Soc., 195(910), viii+100, 2008.Google Scholar

  • [36] E. D. Gluskin, A. Pietsch, and J. Puhl. A generalization of Khintchine’s inequality and its application in the theory of operator ideals. Studia Math., 67(2), 149-155, 1980.Google Scholar

  • [37] F. Göring, C. Helmberg, and S. Reiss. Graph realizations associated with minimizing themaximumeigenvalue of the Laplacian. Math. Program., 131(1-2, Ser. A), 95-111, 2012.Google Scholar

  • [38] F. Göring, C. Helmberg, and M. Wappler. Embedded in the shadow of the separator. SIAM J. Optim., 19(1), 472-501, 2008.CrossrefGoogle Scholar

  • [39] M. Gromov. Random walk in random groups. Geom. Funct. Anal., 13(1), 73-146, 2003.CrossrefGoogle Scholar

  • [40] M. Grötschel, L. Lovász, and A. Schrijver. Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics. Springer-Verlag, Berlin, second edition, 1993.CrossrefGoogle Scholar

  • [41] A. Gupta, R. Krauthgamer, and J. R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Symposium on Foundations of Computer Science, pages 534-543, 2003.Google Scholar

  • [42] L. H. Harper. Optimal numberings and isoperimetric problems on graphs. J. Combinatorial Theory, 1, 385-393, 1966.Google Scholar

  • [43] I. Haviv and O. Regev. The Euclidean distortion of flat tori. J. Topol. Anal., 5(2), 205-223, 2013.Google Scholar

  • [44] J. Heinonen. Lectures on analysis on metric spaces. Universitext. Springer-Verlag, New York, 2001.Google Scholar

  • [45] T. Hytönen and A. Naor. Pisier’s inequality revisited. Studia Math., 215(3), 221-235, 2013.Google Scholar

  • [46] M. Jerrum and A. Sinclair. Conductance and the rapid mixing property for markov chains: the approximation of the permanent resolved (preliminary version). In Proceedings of the 20th Annual ACM Symposium on Theory of Computing, pages 235-244, 1988.Google Scholar

  • [47] W. B. Johnson and J. Lindenstrauss. Extensions of Lipschitzmappings into a Hilbert space. In Conference in modern analysis and probability (New Haven, Conn., 1982), volume 26 of Contemp. Math., pages 189-206. Amer. Math. Soc., Providence, RI, 1984.Google Scholar

  • [48] D. M. Kane and R. Meka. A PRG for Lipschitz functions of polynomials with applications to sparsest cut. In STOC’13: Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pages 1-10, 2013.Google Scholar

  • [49] G. Kasparov and G. Yu. The coarse geometric Novikov conjecture and uniform convexity. Adv. Math., 206(1), 1-56, 2006.Google Scholar

  • [50] S. Khot and A. Naor. Nonembeddability theorems via Fourier analysis. Math. Ann., 334(4), 821-852, 2006.Google Scholar

  • [51] M. D. Kirszbraun. Über die zusammenziehenden und Lipschitzchen Transformationen. Fundam. Math., 22, 77-108, 1934.Google Scholar

  • [52] P. N. Klein, S. A. Plotkin, and S. Rao. Excluded minors, network decomposition, and multicommodity flow. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pages 682-690, 1993.Google Scholar

  • [53] T. Kondo. CAT(0) spaces and expanders. Mathematische Zeitschrift, pages 1-13, 2011.Google Scholar

  • [54] R. Krauthgamer and Y. Rabani. Improved lower bounds for embeddings into L1. SIAM J. Comput., 38(6), 2487-2498, 2009.CrossrefGoogle Scholar

  • [55] E. Kushilevitz, R. Ostrovsky, and Y. Rabani. Eficient search for approximate nearest neighbor in high dimensional spaces. SIAM J. Comput., 30(2), 457-474 (electronic), 2000.CrossrefGoogle Scholar

  • [56] V. Lafiorgue. Un renforcement de la propriété (T). Duke Math. J., 143(3), 559-602, 2008.Google Scholar

  • [57] V. Lafiorgue. Propriété (T) renforcée Banachique et transformation de Fourier rapide. J. Topol. Anal., 1(3), 191-206, 2009.Google Scholar

  • [58] U. Lang and T. Schlichenmaier. Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions. Int.Math. Res. Not., 58, 3625-3655, 2005.Google Scholar

  • [59] G. F. Lawler and A. D. Sokal. Bounds on the L2 spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Amer. Math. Soc., 309(2), 557-580, 1988.Google Scholar

  • [60] M. Ledoux. The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.Google Scholar

  • [61] J. R. Lee. On distance scales, embeddings, and eficient relaxations of the cut cone. In Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 92-101 (electronic), New York, 2005. ACM.Google Scholar

  • [62] J. R. Lee and A. Naor. Extending Lipschitz functions via random metric partitions. Invent. Math., 160(1), 59-95, 2005.Google Scholar

  • [63] J. R. Lee and A. Sidiropoulos. Genus and the geometry of the cut graph. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 193-201, 2010.Google Scholar

  • [64] B. Liao. Strong Banach property (T) for simple algebraic groups of higher rank. Preprint available at http://arxiv.org/abs/ 1301.1861, 2013.Google Scholar

  • [65] J. Lindenstrauss. On the modulus of smoothness and divergent series in Banach spaces. Michigan Math. J., 10, 241-252, 1963.Google Scholar

  • [66] N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2), 215-245, 1995.CrossrefGoogle Scholar

  • [67] A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3), 261-277, 1988.CrossrefGoogle Scholar

  • [68] R. Lyons and Y. Peres. Probability on Trees and Networks. Forthcoming book, available at http://mypage.iu.edu/~rdlyons/ prbtree/book.pdf, 2013. [69] G. A. Margulis. Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii, 24(1), 51-60, 1988.Google Scholar

  • [70] J. Matoušek. On embedding expanders into `p spaces. Israel J. Math., 102, 189-197, 1997.Google Scholar

  • [71] B.Maurey. Type, cotype and K-convexity. In Handbook of the geometry of Banach spaces, Vol. 2, pages 1299-1332. North- Holland, Amsterdam, 2003.Google Scholar

  • [72] B.Maurey and G. Pisier. Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Studia Math., 58(1), 45-90, 1976.Google Scholar

  • [73] S. Mazur. Une remarque sur l’homéomorphie des champs fonctionels. Studia Math., 1, 83-85, 1929.Google Scholar

  • [74] M. Mendel and A. Naor. Metric cotype. Ann. of Math. (2), 168(1):247-298, 2008. Preliminary version in SODA ’06.Google Scholar

  • [75] M. Mendel and A. Naor. Nonlinear spectral calculus and super-expanders. To appear in Inst. Hautes Études Sci. Publ. Math., available at http://arxiv.org/abs/1207.4705, 2012.Google Scholar

  • [76] M. Mendel and A. Naor. Expanders with respect to Hadamard spaces and random graphs. Preprint available at http://arxiv.org/abs/1306.5434, 2013.Google Scholar

  • [77] M. Mendel and A. Naor. Spectral calculus and Lipschitz extension for barycentric metric spaces. Anal. Geom. Metr. Spaces, 1, 163-199, 2013.Google Scholar

  • [78] A. Naor. An introduction to the Ribe program. Jpn. J. Math., 7(2), 167-233, 2012.Google Scholar

  • [79] A. Naor. On the Banach-space-valued Azuma inequality and small-set isoperimetry of Alon-Roichman graphs. Comb. Probab. Comput., 21(4), 623-634, 2012.CrossrefGoogle Scholar

  • [80] A. Naor, Y. Peres, O. Schramm, and S. Shefield. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1), 165-197, 2006.Google Scholar

  • [81] A. Naor, Y. Rabani, and A. Sinclair. Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs. J. Funct. Anal., 227(2), 273-303, 2005.Google Scholar

  • [82] A. Naor and G. Schechtman. Remarks on non linear type and Pisier’s inequality. J. Reine Angew. Math., 552, 213-236, 2002.Google Scholar

  • [83] A. Naor and L. Silberman. Poincaré inequalities, embeddings, and wild groups. Compos. Math., 147(5), 1546-1572, 2011.Google Scholar

  • [84] I. Newman and Y. Rabinovich. Hard metrics from Cayley graphs of abelian groups. Theory Comput., 5, 125-134, 2009.Google Scholar

  • [85] E. Odell and T. Schlumprecht. The distortion problem. Acta Math., 173(2), 259-281, 1994.Google Scholar

  • [86] S.-i. Ohta. Extending Lipschitz and Hölder maps between metric spaces. Positivity, 13(2), 407-425, 2009.CrossrefGoogle Scholar

  • [87] S.-i. Ohta. Markov type of Alexandrov spaces of non-negative curvature. Mathematika, 55(1-2), 177-189, 2009.Google Scholar

  • [88] N. Ozawa. A note on non-amenability of B(lp) for p = 1, 2. Internat. J. Math., 15(6), 557-565, 2004.Google Scholar

  • [89] G. Pisier. Sur les espaces de Banach qui ne contiennent pas uniformément de l1 n. C. R. Acad. Sci. Paris Sér. A-B, 277, A991-A994, 1973.Google Scholar

  • [90] G. Pisier. Martingales with values in uniformly convex spaces. Israel J. Math., 20(3-4), 326-350, 1975.Google Scholar

  • [91] G. Pisier. La méthode d’interpolation complexe: applications aux treillis de Banach. In Séminaire d’Analyse Fonctionnelle (1978-1979), pages Exp. No. 17, 18. École Polytech., Palaiseau, 1979.Google Scholar

  • [92] G. Pisier. Probabilistic methods in the geometry of Banach spaces. In Probability and analysis (Varenna, 1985), volume 1206 of Lecture Notes in Math., pages 167-241. Springer, Berlin, 1986.Google Scholar

  • [93] G. Pisier. Complex interpolation between Hilbert, Banach and operator spaces. Mem. Amer. Math. Soc., 208(978), vi+78, 2010.Google Scholar

  • [94] Y. Rabinovich. On average distortion of embedding metrics into the line. Discrete Comput. Geom., 39(4), 720-733, 2008.CrossrefGoogle Scholar

  • [95] S. Rao. Small distortion and volume preserving embeddings for planar and Euclidean metrics. In Proceedings of the Fifteenth Annual Symposium on Computational Geometry (Miami Beach, FL, 1999), pages 300-306 (electronic), New York, 1999. ACM.Google Scholar

  • [96] Y. Raynaud. On ultrapowers of non commutative Lp spaces. J. Operator Theory, 48(1), 41-68, 2002.Google Scholar

  • [97] C. A. Rogers. A note on coverings and packings. J. London Math. Soc., 25, 327-331, 1950.CrossrefGoogle Scholar

  • [98] J. Sun, S. Boyd, L. Xiao, and P. Diaconis. The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem. SIAM Rev., 48(4), 681-699, 2006.CrossrefGoogle Scholar

  • [99] M. Talagrand. An isoperimetric theorem on the cube and the Kintchine-Kahane inequalities. Proc. Amer.Math. Soc., 104(3), 905-909, 1988.Google Scholar

  • [100] P. Wojtaszczyk. Banach spaces for analysts, volume 25 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1991 Google Scholar

About the article

Received: 2013-08-19

Accepted: 2013-12-23

Published Online: 2014-02-28

Published in Print: 2014-01-01


Citation Information: Analysis and Geometry in Metric Spaces, Volume 2, Issue 1, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2014-0001.

Export Citation

© 2014 Naor Assaf. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[2]
Assaf Naor and Robert Young
Annals of Mathematics, 2018, Volume 188, Number 1, Page 171

Comments (0)

Please log in or register to comment.
Log in