Graph-based time-space trade-offs for approximate near neighbors

We take a first step towards a rigorous asymptotic analysis of graph-based approaches for finding (approximate) nearest neighbors in high-dimensional spaces, by analyzing the complexity of (randomized) greedy walks on the approximate near neighbor graph. For random data sets of size n = 2o(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 nρq + o(1) and space n1 + ρs + o(1), for arbitrary ρq, ρs ≥ 0 satisfying beginalign (2c2 - 1) ρq + 2 c2 (c2 - 1) √ρs (1 - ρs) ≥ c4. endalign 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 the recent optimal hash-based trade-offs of Andoni-Laarhoven-Razenshteyn-Waingarten [SODA'17]. We further study how the trade-offs scale when the data set is of size n = 2Theta(d), and analyze asymptotic complexities when applying these results to lattice sieving.