Fixed-radius near neighbors

About: Fixed-radius near neighbors is a research topic. Over the lifetime, 549 publications have been published within this topic receiving 47430 citations. The topic is also known as: fixed-radius near neighbor problem.

Nearest neighbor pattern classification

Abstract: The nearest neighbor decision rule assigns to an unclassified sample point the classification of the nearest of a set of previously classified points. This rule is independent of the underlying joint distribution on the sample points and their classifications, and hence the probability of error R of such a rule must be at least as great as the Bayes probability of error R^{\ast} --the minimum probability of error over all decision rules taking underlying probability structure into account. However, in a large sample analysis, we will show in the M -category case that R^{\ast} \leq R \leq R^{\ast}(2 --MR^{\ast}/(M-1)) , where these bounds are the tightest possible, for all suitably smooth underlying distributions. Thus for any number of categories, the probability of error of the nearest neighbor rule is bounded above by twice the Bayes probability of error. In this sense, it may be said that half the classification information in an infinite sample set is contained in the nearest neighbor.

Approximate nearest neighbors: towards removing the curse of dimensionality

Abstract: We present two algorithms for the approximate nearest neighbor problem in high-dimensional spaces. For data sets of size n living in R d , the algorithms require space that is only polynomial in n and d, while achieving query times that are sub-linear in n and polynomial in d. We also show applications to other high-dimensional geometric problems, such as the approximate minimum spanning tree. The article is based on the material from the authors' STOC'98 and FOCS'01 papers. It unifies, generalizes and simplifies the results from those papers.

An Algorithm for Finding Best Matches in Logarithmic Expected Time

Abstract: An algorithm and data structure are presented for searching a file containing N records, each described by k real valued keys, for the m closest matches or nearest neighbors to a given query record. The computation required to organize the file is proportional to kNlogN. The expected number of records examined in each search is independent of the file size. The expected computation to perform each search is proportional to logN. Empirical evidence suggests that except for very small files, this algorithm is considerably faster than other methods.

An optimal algorithm for approximate nearest neighbor searching fixed dimensions

Abstract: Consider a set of S of n data points in real d-dimensional space, Rd, where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data structure, so that given any query point q∈ Rd, is the closest point of S to q can be reported quickly. Given any positive real ϵ, data point p is a (1 +ϵ)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ϵ) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in Rd in O(dn log n) time and O(dn) space, so that given a query point q ∈ Rd, and ϵ > 0, a (1 + ϵ)-approximate nearest neighbor of q can be computed in O(cd, ϵ log n) time, where cd,ϵ≤d ⌈1 + 6d/ϵ⌉d is a factor depending only on dimension and ϵ. In general, we show that given an integer k ≥ 1, (1 + ϵ)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.

Asymptotic Properties of Nearest Neighbor Rules Using Edited Data

Abstract: The convergence properties of a nearest neighbor rule that uses an editing procedure to reduce the number of preclassified samples and to improve the performance of the rule are developed. Editing of the preclassified samples using the three-nearest neighbor rule followed by classification using the single-nearest neighbor rule with the remaining preclassified samples appears to produce a decision procedure whose risk approaches the Bayes' risk quite closely in many problems with only a few preclassified samples. The asymptotic risk of the nearest neighbor rules and the nearest neighbor rules using edited preclassified samples is calculated for several problems.