Graph-Based Time-Space Trade-Offs for Approximate Near Neighbors

Author Thijs Laarhoven

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Thijs Laarhoven

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Thijs Laarhoven. Graph-Based Time-Space Trade-Offs for Approximate Near Neighbors. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 57:1-57:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We take a first step towards a rigorous asymptotic analysis of graph-based methods for finding (approximate) nearest neighbors in high-dimensional spaces, by analyzing the complexity of randomized greedy walks on the approximate nearest neighbor graph. For random data sets of size n = 2^{o(d)} on the d-dimensional Euclidean unit sphere, using near neighbor graphs we can provably solve the approximate nearest neighbor problem with approximation factor c > 1 in query time n^{rho_{q} + o(1)} and space n^{1 + rho_{s} + o(1)}, for arbitrary rho_{q}, rho_{s} >= 0 satisfying (2c^2 - 1) rho_{q} + 2 c^2 (c^2 - 1) sqrt{rho_{s} (1 - rho_{s})} >= c^4. Graph-based near neighbor searching is especially competitive with hash-based methods for small c and near-linear memory, and in this regime the asymptotic scaling of a greedy graph-based search matches optimal hash-based trade-offs of Andoni-Laarhoven-Razenshteyn-Waingarten [Andoni et al., 2017]. We further study how the trade-offs scale when the data set is of size n = 2^{Theta(d)}, and analyze asymptotic complexities when applying these results to lattice sieving.
  • approximate nearest neighbor problem
  • near neighbor graphs
  • locality-sensitive hashing
  • locality-sensitive filters
  • similarity search


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