Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
149 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Tradeoffs for nearest neighbors on the sphere (1511.07527v2)

Published 24 Nov 2015 in cs.DS, cs.CG, and cs.IR

Abstract: 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 $n{\rho_q}$ and update complexity $n{\rho_u}$ for data sets of size $n$ is given by the following equation in terms of the approximation factor $c$ and the exponents $\rho_q$ and $\rho_u$: $$c2\sqrt{\rho_q}+(c2-1)\sqrt{\rho_u}=\sqrt{2c2-1}.$$ For small $c=1+\epsilon$, minimizing the time for updates leads to a linear space complexity at the cost of a query time complexity $n{1-4\epsilon2}$. Balancing the query and update costs leads to optimal complexities $n{1/(2c2-1)}$, 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 $n{o(1)}$ can be achieved at the cost of a space complexity of the order $n{1/(4\epsilon2)}$, matching the bound $n{\Omega(1/\epsilon2)}$ 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 $n{2/c2+O(1/c4)}$, improving upon the related exponent for large $c$ of [Kapralov, PODS'15] by a factor $2$, and matching the bound $n{\Omega(1/c2)}$ of [Panigrahy-Talwar-Wieder, FOCS'08]. Balancing the costs leads to optimal complexities $n{1/(2c2-1)}$, while a minimum query time complexity can be achieved with update complexity $n{2/c2+O(1/c4)}$, improving upon the previous best exponents of Kapralov by a factor $2$.

Citations (21)

Summary

We haven't generated a summary for this paper yet.