On the construction of low-pass filters on the unit sphere

ABSTRACT This paper considers the problem of construction of low-pass filters on the unit sphere, which has wide ranging applications in the processing of signals on the unit sphere. We propose a design criterion for the construction of strictly bandlimited low-pass filters in the spectral domain with optimal concentration in the specified polar cap region in the spatial domain. Our approach uses the weighted sum of the first optimally concentrated eigenfunctions from appropriately formulated Slepian concentration problems on the sphere. Furthermore, in order to reduce the computational complexity of the proposed algorithm, we develop a closed-form expression to accurately model these eigenfunctions. We illustrate the construction of low-pass filters using the proposed approach and demonstrate the advantage of our method approach compared to a diffusion based approach in the literature in terms of control over both bandwidth in the spectral domain and concentration in the spatial domain

Tradeoffs for nearest neighbors on the sphere

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 $\rho_u$: $$c^2\sqrt{\rho_q}+(c^2-1)\sqrt{\rho_u}=\sqrt{2c^2-1}.$$ For small $c=1+\epsilon$, minimizing the time for updates leads to a linear space complexity at the cost of a query time complexity $n^{1-4\epsilon^2}$. Balancing the query and update costs leads to optimal complexities $n^{1/(2c^2-1)}$, 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 $n^{1/(4\epsilon^2)}$, matching the bound $n^{\Omega(1/\epsilon^2)}$ 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 $n^{2/c^2+O(1/c^4)}$, improving upon the related exponent for large $c$ of [Kapralov, PODS'15] by a factor $2$, and matching the bound $n^{\Omega(1/c^2)}$ of [Panigrahy-Talwar-Wieder, FOCS'08]. Balancing the costs leads to optimal complexities $n^{1/(2c^2-1)}$, while a minimum query time complexity can be achieved with update complexity $n^{2/c^2+O(1/c^4)}$, 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]

Faster tuple lattice sieving using spherical locality-sensitive filters

To overcome the large memory requirement of classical lattice sieving algorithms for solving hard lattice problems, Bai-Laarhoven-Stehl\'{e} [ANTS 2016] studied tuple lattice sieving, where tuples instead of pairs of lattice vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017] recently improved upon their results for arbitrary tuple sizes, for example showing that a triple sieve can solve the shortest vector problem (SVP) in dimension $d$ in time $2^{0.3717d + o(d)}$, using a technique similar to locality-sensitive hashing for finding nearest neighbors. In this work, we generalize the spherical locality-sensitive filters of Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near neighbor searching on dense data sets, and we apply these techniques to tuple lattice sieving to obtain even better time complexities. For instance, our triple sieve heuristically solves SVP in time $2^{0.3588d + o(d)}$. For practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this shows that a triple sieve uses less space and less time than the current best near-linear space double sieve.Comment: 12 pages + references, 2 figures. Subsumed/merged into Cryptology ePrint Archive 2017/228, available at https://ia.cr/2017/122