Nearly Optimal Embeddings of Flat Tori
Agarwal, Ishan, Oded Regev, Tang, Yi · DROPS (Schloss Dagstuhl – Leibniz Center for Informatics) · 2020
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).