We consider tradeoffs between the query and update complexities for the
(approximate) nearest neighbor problem on the sphere, extending the recent
spherical filters to sparse regimes and generalizing the scheme and analysis to
account for different tradeoffs. In a nutshell, for the sparse regime the
tradeoff between the query complexity nρq and update complexity
nρu for data sets of size n is given by the following equation in
terms of the approximation factor c and the exponents ρq and ρu:
c2ρq+(c2−1)ρu=2c2−1.
For small c=1+ϵ, minimizing the time for updates leads to a linear
space complexity at the cost of a query time complexity n1−4ϵ2.
Balancing the query and update costs leads to optimal complexities
n1/(2c2−1), matching bounds from [Andoni-Razenshteyn, 2015] and [Dubiner,
IEEE-TIT'10] and matching the asymptotic complexities of [Andoni-Razenshteyn,
STOC'15] and [Andoni-Indyk-Laarhoven-Razenshteyn-Schmidt, NIPS'15]. A
subpolynomial query time complexity no(1) can be achieved at the cost of a
space complexity of the order n1/(4ϵ2), matching the bound
nΩ(1/ϵ2) of [Andoni-Indyk-Patrascu, FOCS'06] and
[Panigrahy-Talwar-Wieder, FOCS'10] and improving upon results of
[Indyk-Motwani, STOC'98] and [Kushilevitz-Ostrovsky-Rabani, STOC'98].
For large c, minimizing the update complexity results in a query complexity
of n2/c2+O(1/c4), improving upon the related exponent for large c of
[Kapralov, PODS'15] by a factor 2, and matching the bound nΩ(1/c2)
of [Panigrahy-Talwar-Wieder, FOCS'08]. Balancing the costs leads to optimal
complexities n1/(2c2−1), while a minimum query time complexity can be
achieved with update complexity n2/c2+O(1/c4), improving upon the
previous best exponents of Kapralov by a factor 2.Comment: 16 pages, 1 table, 2 figures. Mostly subsumed by arXiv:1608.03580
[cs.DS] (along with arXiv:1605.02701 [cs.DS]