Consider a dataset of n(d) points generated independently from R^d according to a common p.d.f. f_d with support(f_d) = [0,1]^d and sup f_d([0,1]^d) growing sub-exponentially in d. We prove that: (i) if n(d) grows sub-exponentially in d, then, for any query point q^d in [0,1]^d and any epsilon>0, the ratio of the distance between any two dataset points and q^d is less that 1+epsilon with probability -->1 as d-->infinity; (ii) if n(d)>[4(1+epsilon)]^d for large d, then for all q^d in [0,1]^d (except a small subset) and any epsilon>0, the distance ratio is less than 1+epsilon with limiting probability strictly bounded away from one. Moreover, we provide preliminary results along the lines of (i) when f_d=N(mu_d,Sigma_d).
A vast literature exists on the development of data structures and algorithms for nearest neighbor search, for brevity, see the discussion and citations in @cite_10 .
@cite_10
1502916507 2772756366 2071866949 2099812803
The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50). Nearest neighbor search is a fundamental problem in various domains, such as computer vision, data mining, and machine learning. With the explosive growth of data on the Internet, many new data structures using spatial partitions and recursive hyperplane decomposition (e.g., k-d trees) are proposed to speed up the nearest neighbor search. However, these data structures are facing big data challenges. To meet these challenges, binary hashing-based approximate nearest neighbor search methods attract substantial attention due to their fast query speed and drastically reduced storage. Since the most notably locality sensitive hashing was proposed, a large number of binary hashing methods have emerged. In this paper, we first illustrate the development of binary hashing research by proposing an overall and clear classification of them. Then we conduct extensive experiments to compare the performance of these methods on five famous and public data sets. Finally, we present our view on this topic. Given a set of n points in d-dimensional Euclidean space, S ⊂ E, and a query point q ∈ E, we wish to determine the nearest neighbor of q, that is, the point of S whose Euclidean distance to q is minimum. The goal is to preprocess the point set S, such that queries can be answered as efficiently as possible. We assume that the dimension d is a constant independent of n. Although reasonably good solutions to this problem exist when d is small, as d increases the performance of these algorithms degrades rapidly. We present a randomized algorithm for approximate nearest neighbor searching. Given any set of n points S ⊂ E, and a constant ǫ > 0, we produce a data structure, such that given any query point, a point of S will be reported whose distance from the query point is at most a factor of (1 + ǫ) from that of the true nearest neighbor. Our algorithm runs in O(log n) expected time and requires O(n log n) space. The data structure can be built in O(n) expected time. The constant factors depend on d and ǫ. Because of the practical importance of nearest neighbor searching in higher dimensions, we have implemented a practical variant of this algorithm, and show empirically that for many point distributions this variant of the algorithm finds the nearest neighbor in moderately large dimension significantly faster than existing practical approaches. We introduce a new probabilistic proximity search algorithm for range and A"-nearest neighbor (A"-NN) searching in both coordinate and metric spaces. Although there exist solutions for these problems, they boil down to a linear scan when the space is intrinsically high dimensional, as is the case in many pattern recognition tasks. This, for example, renders the A"-NN approach to classification rather slow in large databases. Our novel idea is to predict closeness between elements according to how they order their distances toward a distinguished set of anchor objects. Each element in the space sorts the anchor objects from closest to farthest to it and the similarity between orders turns out to be an excellent predictor of the closeness between the corresponding elements. We present extensive experiments comparing our method against state-of-the-art exact and approximate techniques, both in synthetic and real, metric and nonmetric databases, measuring both CPU time and distance computations. The experiments demonstrate that our technique almost always improves upon the performance of alternative techniques, in some cases by a wide margin.