This paper describes a novel algorithm for approximate nearest neighbor searching. For solving this problem especially in high dimensional spaces, one of the best-known algorithm is Locality-Sensitive Hashing (LSH). This paper presents a variant of the LSH algorithm that outperforms previously proposed methods when the dataset consists of vectors normalized to unit length, which is often the ca...

Given a cone C and a set S of n points in R, we want to preprocess S into a data structure so that we can find fast an approximate nearest neighbor to a query point q with respect to the points of S contained in the translation of C with apex at q. We develop an approximate conic Voronoi diagram of Õ(n/ε) size that supports conic nearest neighbor queries in O(log(n/ε)) time. Our preprocessing u...

Let n points be placed independently in d−dimensional space according to the density f(x) = Ade −λ‖x‖α , λ > 0, x ∈ Rd, d ≥ 2. Let dn be the longest edge length of the nearest neighbor graph on these points. We show that (λ−1 log n)dn−bn converges weakly to the Gumbel distribution where bn ∼ (d−1) λα log log n. We also prove the following strong law result for the normalized nearest neighbor di...

The nearest-neighbor and potential function decision rules are nonparametric techniques that partition the feature space based on a set of labelled sample points. Determining whether the partitions of the two rules are identical for a given set of points is an interesting problem in computational geometry. Here, a relationship between the two methods in terms of subclasses and composite classes...

The famous “spooky action at a distance” in the EPR-szenario is shown to be a local interaction, once entanglement is interpreted as a kind of “nearest neighbor” relation among quantum systems. Furthermore, the wave function itself is interpreted as encoding the “nearest neighbor” relations between a quantum system and spatial points. This interpretation becomes natural, if we view space and di...

Self-Organising Maps (SOM) are Artificial Neural Networks used in Pattern Recognition tasks. Their major advantage over other architectures is human readability of a model. However, they often gain poorer accuracy. Mostly used metric in SOM is the Euclidean distance, which is not the best approach to some problems. In this paper, we study an impact of the metric change on the SOM’s performance ...

Different aspects of the curse of dimensionality are known to present serious challenges to various machine-learning methods and tasks. This paper explores a new aspect of the dimensionality curse, referred to as hubness, that affects the distribution of k-occurrences: the number of times a point appears among the k nearest neighbors of other points in a data set. Through theoretical and empiri...

In this paper, we show a construction of locality-sensitive hash functions without false negatives, i.e., which ensure collision for every pair of points within a given radius R in d dimensional space equipped with lp norm when p ∈ [1,∞]. Furthermore, we show how to use these hash functions to solve the c-approximate nearest neighbor search problem without false negatives. Namely, if there is a...

We propose simple and fast methods based on nearest neighbors that order objects from high-dimensional data sets from typical points to untypical points. On the one hand, we show that these easy-to-compute orderings allow us to detect outliers (i.e. very untypical points) with a performance comparable to or better than other often much more sophisticated methods. On the other hand, we show how ...

The spatial clustering of points from two or more classes (or species) has important implications in many fields and may cause segregation or association, which are two major types of spatial patterns between the classes. These patterns can be studied using a nearest neighbour contingency table (NNCT) which is constructed using the frequencies of nearest neighbour types. Three new multivariate ...