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A Spectral Gap Precludes Low-Dimensional Embeddings

Author Assaf Naor

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Assaf Naor

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Assaf Naor. A Spectral Gap Precludes Low-Dimensional Embeddings. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.SoCG.2017.50

Abstract

We prove that if an n-vertex O(1)-expander embeds with average distortion D into a finite dimensional normed space X, then necessarily the dimension of X is at least n^{c/D} for some universal constant c>0. This is sharp up to the value of the constant c, and it improves over the previously best-known estimate dim(X)> c(log n)^2/D^2 of Linial, London and Rabinovich, strengthens a theorem of Matousek, and answers a question of Andoni, Nikolov, Razenshteyn and Waingarten.
Keywords
  • Metric embeddings
  • dimensionality reduction
  • expander graphs
  • nonlinear spectral gaps
  • nearest neighbor search
  • complex interpolation
  • Markov type.

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References

  1. Noga Alon, Peter Frankl, and Vojtech Rödl. Geometrical realization of set systems and probabilistic communication complexity. In 26th Annual Symposium on Foundations of Computer Science, pages 277-280, 1985. URL: http://dx.doi.org/10.1109/SFCS.1985.30.
  2. Noga Alon and Yuval Roichman. Random Cayley graphs and expanders. Random Structures Algorithms, 5(2):271-284, 1994. URL: http://dx.doi.org/10.1002/rsa.3240050203.
  3. A. Andoni, A. Nikolov, I. Razenshteyn, and E. Waingarten. Approximate near neighbors for general symmetric norms. To appear in STOC 2017. Preprint, available at https://arxiv.org/pdf/1611.06222, 2016.
  4. Juan Arias-de Reyna and Luis Rodríguez-Piazza. Finite metric spaces needing high dimension for Lipschitz embeddings in Banach spaces. Israel J. Math., 79(1):103-111, 1992. URL: http://dx.doi.org/10.1007/BF02764804.
  5. K. Ball. Markov chains, Riesz transforms and Lipschitz maps. Geom. Funct. Anal., 2(2):137-172, 1992. URL: http://dx.doi.org/10.1007/BF01896971.
  6. Keith Ball. The Ribe programme. Astérisque, (352):Exp. No. 1047, viii, 147-159, 2013. Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043-1058. Google Scholar
  7. Keith Ball, Eric A. Carlen, and Elliott H. Lieb. Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math., 115(3):463-482, 1994. URL: http://dx.doi.org/10.1007/BF01231769.
  8. Yair Bartal, Nathan Linial, Manor Mendel, and Assaf Naor. On metric Ramsey-type phenomena. Ann. of Math. (2), 162(2):643-709, 2005. URL: http://dx.doi.org/10.4007/annals.2005.162.643.
  9. S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman. Affine approximation of Lipschitz functions and nonlinear quotients. Geom. Funct. Anal., 9(6):1092-1127, 1999. URL: http://dx.doi.org/10.1007/s000390050108.
  10. Jöran Bergh and Jörgen Löfström. Interpolation spaces. An introduction. Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. Google Scholar
  11. J. Bourgain. On Lipschitz embedding of finite metric spaces in Hilbert space. Israel J. Math., 52(1-2):46-52, 1985. URL: http://dx.doi.org/10.1007/BF02776078.
  12. J. Bourgain. The metrical interpretation of superreflexivity in Banach spaces. Israel J. Math., 56(2):222-230, 1986. URL: http://dx.doi.org/10.1007/BF02766125.
  13. A.-P. Calderón. Intermediate spaces and interpolation, the complex method. Studia Math., 24:113-190, 1964. Google Scholar
  14. Michael Cwikel and Shlomo Reisner. Interpolation of uniformly convex Banach spaces. Proc. Amer. Math. Soc., 84(4):555-559, 1982. URL: http://dx.doi.org/10.2307/2044034.
  15. T. Figiel. On the moduli of convexity and smoothness. Studia Math., 56(2):121-155, 1976. Google Scholar
  16. Tadeusz Figiel and Gilles Pisier. Séries aléatoires dans les espaces uniformément convexes ou uniformément lisses. C. R. Acad. Sci. Paris Sér. A, 279:611-614, 1974. Google Scholar
  17. M. Fréchet. Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo, 22:1-74, 1906. URL: http://dx.doi.org/10.1007/BF03018603.
  18. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.), 43(4):439-561 (electronic), 2006. URL: http://dx.doi.org/10.1090/S0273-0979-06-01126-8.
  19. Fritz John. Extremum problems with inequalities as subsidiary conditions. In Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pages 187-204. Interscience Publishers, Inc., New York, N. Y., 1948. Google Scholar
  20. William B. Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings 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. URL: http://dx.doi.org/10.1090/conm/026/737400.
  21. William B. Johnson, Joram Lindenstrauss, and Gideon Schechtman. On Lipschitz embedding of finite metric spaces in low-dimensional normed spaces. In Geometrical aspects of functional analysis (1985/86), volume 1267 of Lecture Notes in Math., pages 177-184. Springer, Berlin, 1987. URL: http://dx.doi.org/10.1007/BFb0078145.
  22. Nigel J. Kalton. The nonlinear geometry of Banach spaces. Rev. Mat. Complut., 21(1):7-60, 2008. URL: http://dx.doi.org/10.5209/rev_REMA.2008.v21.n1.16426.
  23. Subhash Khot and Assaf Naor. Nonembeddability theorems via Fourier analysis. Math. Ann., 334(4):821-852, 2006. URL: http://dx.doi.org/10.1007/s00208-005-0745-0.
  24. R. Krauthgamer, J. R. Lee, M. Mendel, and A. Naor. Measured descent: a new embedding method for finite metrics. Geom. Funct. Anal., 15(4):839-858, 2005. URL: http://dx.doi.org/10.1007/s00039-005-0527-6.
  25. James R. Lee, Manor Mendel, and Assaf Naor. Metric structures in L₁: dimension, snowflakes, and average distortion. European J. Combin., 26(8):1180-1190, 2005. URL: http://dx.doi.org/10.1016/j.ejc.2004.07.002.
  26. J. R. Lee and A. Naor. Embedding the diamond graph in L_p and dimension reduction in L₁. Geom. Funct. Anal., 14(4):745-747, 2004. URL: http://dx.doi.org/10.1007/s00039-004-0473-8.
  27. Joram Lindenstrauss and Lior Tzafriri. Classical Banach spaces. II, volume 97 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin-New York, 1979. Function spaces. Google Scholar
  28. Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995. URL: http://dx.doi.org/10.1007/BF01200757.
  29. Jiří Matoušek. Note on bi-Lipschitz embeddings into normed spaces. Comment. Math. Univ. Carolin., 33(1):51-55, 1992. Google Scholar
  30. Jiří Matoušek. On the distortion required for embedding finite metric spaces into normed spaces. Israel J. Math., 93:333-344, 1996. URL: http://dx.doi.org/10.1007/BF02761110.
  31. Jiří Matoušek. On embedding expanders into l_p spaces. Israel J. Math., 102:189-197, 1997. URL: http://dx.doi.org/10.1007/BF02773799.
  32. Jiří Matoušek. Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. URL: http://dx.doi.org/10.1007/978-1-4613-0039-7.
  33. Manor Mendel and Assaf Naor. Markov convexity and local rigidity of distorted metrics. J. Eur. Math. Soc. (JEMS), 15(1):287-337, 2013. URL: http://dx.doi.org/10.4171/JEMS/362.
  34. Manor Mendel and Assaf Naor. Spectral calculus and Lipschitz extension for barycentric metric spaces. Anal. Geom. Metr. Spaces, 1:163-199, 2013. URL: http://dx.doi.org/10.2478/agms-2013-0003.
  35. Manor Mendel and Assaf Naor. Nonlinear spectral calculus and super-expanders. Publ. Math. Inst. Hautes Études Sci., 119:1-95, 2014. URL: http://dx.doi.org/10.1007/s10240-013-0053-2.
  36. Manor Mendel and Assaf Naor. Expanders with respect to Hadamard spaces and random graphs. Duke Math. J., 164(8):1471-1548, 2015. URL: http://dx.doi.org/10.1215/00127094-3119525.
  37. J. Milnor. On the Betti numbers of real varieties. Proc. Amer. Math. Soc., 15:275-280, 1964. Google Scholar
  38. Assaf Naor. An introduction to the Ribe program. Jpn. J. Math., 7(2):167-233, 2012. URL: http://dx.doi.org/10.1007/s11537-012-1222-7.
  39. Assaf Naor. On the Banach-space-valued Azuma inequality and small-set isoperimetry of Alon-Roichman graphs. Combin. Probab. Comput., 21(4):623-634, 2012. URL: http://dx.doi.org/10.1017/S0963548311000757.
  40. Assaf Naor. Comparison of metric spectral gaps. Anal. Geom. Metr. Spaces, 2:1-52, 2014. URL: http://dx.doi.org/10.2478/agms-2014-0001.
  41. Assaf Naor, Yuval Peres, Oded Schramm, and Scott Sheffield. Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J., 134(1):165-197, 2006. URL: http://dx.doi.org/10.1215/S0012-7094-06-13415-4.
  42. Assaf Naor and Lior Silberman. Poincaré inequalities, embeddings, and wild groups. Compos. Math., 147(5):1546-1572, 2011. URL: http://dx.doi.org/10.1112/S0010437X11005343.
  43. Ilan Newman and Yuri Rabinovich. Hard metrics from Cayley graphs of abelian groups. Theory Comput., 5:125-134, 2009. URL: http://dx.doi.org/10.4086/toc.2009.v005a006.
  44. Mikhail I. Ostrovskii. Metric embeddings. Bilipschitz and coarse embeddings into Banach spaces, volume 49 of De Gruyter Studies in Mathematics. De Gruyter, Berlin, 2013. URL: http://dx.doi.org/10.1515/9783110264012.
  45. Gilles Pisier. Complex interpolation between Hilbert, Banach and operator spaces. Mem. Amer. Math. Soc., 208(978):vi+78, 2010. URL: http://dx.doi.org/10.1090/S0065-9266-10-00601-0.
  46. Yuri Rabinovich. On average distortion of embedding metrics into the line. Discrete Comput. Geom., 39(4):720-733, 2008. URL: http://dx.doi.org/10.1007/s00454-007-9047-5.
  47. M. Ribe. On uniformly homeomorphic normed spaces. Ark. Mat., 14(2):237-244, 1976. Google Scholar
  48. Marcel Riesz. Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires. Acta Math., 49(3-4):465-497, 1927. URL: http://dx.doi.org/10.1007/BF02564121.
  49. Elias M. Stein. Interpolation of linear operators. Trans. Amer. Math. Soc., 83:482-492, 1956. Google Scholar
  50. René Thom. Sur l'homologie des variétés algébriques réelles. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pages 255-265. Princeton Univ. Press, Princeton, N.J., 1965. Google Scholar
  51. G. O. Thorin. Convexity theorems generalizing those of M. Riesz and Hadamard with some applications. Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], 9:1-58, 1948. Google Scholar
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