Andrei Osipov

A Randomized Approximate Nearest Neighbors Algorithm

We present a randomized algorithm for the approximate nearest neighbor problem in d-dimensional Euclidean space. Given N points {xj} in , the algorithm attempts to find k nearest neighbors for each of xj, where k is a user-specified integer parameter. The algorithm is iterative, and its running time requirements are proportional to T·N·(d·(log d) + k·(d + log k)·(log N)) + N·k2·(d + log k), with T the number of iterations performed. The memory requirements of the procedure are of the order N·(d + k). A by-product of the scheme is a data structure, permitting a rapid search for the k nearest neighbors among {xj} for an arbitrary point . The cost of each such query is proportional to T·(d·(log d) + log(N/k)·k·(d + log k)), and the memory requirements for the requisite data structure are of the order N·(d + k) + T·(d + N). The algorithm utilizes random rotations and a basic divide-and-conquer scheme, followed by a local graph search. We analyze the scheme’s behavior for certain types of distributions of {xj} and illustrate its performance via several numerical examples.

Certain inequalities involving prolate spheroidal wave functions and associated quantities

Abstract Prolate spheroidal wave functions (PSWFs) play an important role in various areas, from physics (e.g. wave phenomena, fluid dynamics) to engineering (e.g. signal processing, filter design). Even though the significance of PSWFs was realized at least half a century ago, and they frequently occur in applications, their analytical properties have not been investigated as much as those of many other special functions. In particular, despite some recent progress, the gap between asymptotic expansions and numerical experience, on the one hand, and rigorously proven explicit bounds and estimates, on the other hand, is still rather wide. This paper attempts to improve the current situation. We analyze the differential operator associated with PSWFs, to derive fairly tight estimates on its eigenvalues. By combining these inequalities with a number of standard techniques, we also obtain several other properties of the PSFWs. The results are illustrated via numerical experiments.

Analysis of the Integral Operator

This chapter contains several properties of the PSWFs and related quantities, whose derivation is based on an analysis of the integral operator F c defined via (2.7) in Sect. 2.4 (see also [46–48]).

Analysis of a Differential Operator

This chapter contains several properties of prolate spheroidal wave functions (PSWFs) and related quantities, whose derivation is based on an analysis of the prolate differential operator L c defined via (1.1) in Chap. 1 (see also Theorem 2.5 in Sect. 2.4, and [43–45]).

Miscellaneous Properties of PSWFs

Prolate spheroidal wave functions possess a rich set of properties. In this chapter, we list some of those properties. Some of the identities below can be found in [13, 33, 64]; others are easily derivable from the former (see also [73]).

Mathematical and Numerical Preliminaries

In this chapter, we introduce notation and summarize several facts to be used in the rest of the book.

Quadrature Rules and Interpolation via PSWFs

In this chapter, we describe several classes of prolate spheroidal wave function (PSWF)-based quadrature rules and interpolation formulas, designed for band-limited functions with a specified band limit c > 0 over the interval [ − 1, 1] (see also [49, 50, 53–55, 73]).

Asymptotic Analysis of PSWFs

In this chapter, we construct several asymptotic expansions for prolate spheroidal wave functions and related eigenvalues via straightforward implementation of the inverse power method in Mathematica. We present several examples of these formulas, and illustrate our results via numerical experiments.

Rational Approximations of PSWFs

In this chapter, we construct rational approximations of PSWFs. More specifically, we approximate the reciprocal of ψ n in the interval ( −1,1) by a rational function having n poles (these poles happen to be precisely the n roots of ψ n in (−1,1)). Also, we derive explicit bounds on the error of such approximations. The underlying analysis is based on a detailed investigation of certain properties of PSWFs outside the interval (−1,1) (see also [49, 50]).