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Nearly Optimal Embeddings of Flat Tori

Authors Ishan Agarwal, Oded Regev, Yi Tang



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Author Details

Ishan Agarwal
  • Courant Institute of Mathematical Sciences, New York University, NY, USA
Oded Regev
  • Courant Institute of Mathematical Sciences, New York University, NY, USA
Yi Tang
  • Courant Institute of Mathematical Sciences, New York University, NY, USA

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Ishan Agarwal, Oded Regev, and Yi Tang. Nearly Optimal Embeddings of Flat Tori. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 43:1-43:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.43

Abstract

We show that for any n-dimensional lattice ℒ ⊆ ℝⁿ, the torus ℝⁿ/ℒ can be embedded into Hilbert space with O(√{nlog n}) distortion. This improves the previously best known upper bound of O(n√{log n}) shown by Haviv and Regev (APPROX 2010, J. Topol. Anal. 2013) and approaches the lower bound of Ω(√n) due to Khot and Naor (FOCS 2005, Math. Ann. 2006).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • Lattices
  • metric embeddings
  • flat torus

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References

  1. Ishay Haviv and Oded Regev. The Euclidean distortion of flat tori. J. Topol. Anal., 5(2):205-223, 2013. Preliminary version in APPROX 2010. URL: https://doi.org/10.1142/S1793525313500064.
  2. Piotr Indyk. Algorithmic applications of low-distortion geometric embeddings. In 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), pages 10-33. IEEE Computer Soc., Los Alamitos, CA, 2001. Google Scholar
  3. Subhash Khot and Assaf Naor. Nonembeddability theorems via Fourier analysis. Math. Ann., 334(4):821-852, 2006. Preliminary version in FOCS 2005. URL: https://doi.org/10.1007/s00208-005-0745-0.
  4. John Milnor and Dale Husemoller. Symmetric bilinear forms. Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. Google Scholar
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