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Nearly Optimal Embeddings of Flat Tori

Authors Ishan Agarwal, Oded Regev, Yi Tang

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Subject Classification

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

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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).

Cite As Get BibTex

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

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

Funding

  • Agarwal, Ishan: Research supported by National Science Foundation (NSF) under Grant No. CCF-1814524.
  • Regev, Oded: Research supported by the Simons Collaboration on Algorithms and Geometry, a Simons Investigator Award, and by the National Science Foundation (NSF) under Grant No. CCF-1814524.

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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