Consider a set S of n data points in real d-dimensional space, R d , 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 2 R d , the closest point of S to q can be reported quickly. Given any positive real ffl, a data point p is a (1 + ffl)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ffl) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in R d in O(dn log n) time and O(dn) space, so that given a query point q 2 R d , and ffl ? 0, a (1 + ffl)-approximate nearest neighbor of q can be computed in O(c d;ffl log n) time, where c d;ffl d d1 + 6d=ffle d is a factor depending only on dimension and ffl. In general, we show that given an integer k 1, (1 + ffl)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.

The nearest neighbor problem is the following: Given a set of n points P = fp 1 ; : : : ; png in some metric space X, preprocess P so as to efficiently answer queries which require finding the point in P closest to a query point q 2 X. We focus on the particularly interesting case of the d-dimensional Euclidean space where X = ! d under some l p norm. Despite decades of effort, the current solutions are far from satisfactory; in fact, for large d, in theory or in practice, they provide little improvement over the brute-force algorithm which compares the query point to each data point. Of late, there has been some interest in the approximate nearest neighbors problem, which is: Find a point p 2 P that is an ffl-approximate nearest neighbor of the query q in that for all p 0 2 P , d(p; q) (1 + ffl)d(p 0 ; q). We present two algorithmic results for the approximate version that significantly improve the known bounds: (a) preprocessing cost polynomial in n and d, and a trul...

We consider the following two instances of the projective clustering problem: Given a set S of n points in R d and an integer k ? 0; cover S by k hyper-strips (resp. hyper-cylinders) so that the maximum width of a hyper-strip (resp., the maximum diameter of a hyper-cylinder) is minimized. Let w be the smallest value so that S can be covered by k hyper-strips (resp. hyper-cylinders), each of width (resp. diameter) at most w : In the plane, the two problems are equivalent. It is NP-Hard to compute k planar strips of width even at most Cw ; for any constant C ? 0 [50]. This paper contains four main results related to projective clustering: (i) For d = 2, we present a randomized algorithm that computes O(k log k) strips of width at most 6w that cover S. Its expected running time is O(nk 2 log 4 n) if k 2 log k n; it also works for larger values of k, but then the expected running time is O(n 2=3 k 8=3 log 4 n). We also propose another algorithm that computes a c...

We address the problem of designing data structures that allow efficient search for approximate nearest neighbors. More specifically, given a database consisting of a set of vectors in some high dimensional Euclidean space, we want to construct a space-efficient data structure that would allow us to search, given a query vector, for the closest or nearly closest vector in the database. We also address this problem when distances are measured by the L 1 norm, and in the Hamming cube. Significantly improving and extending recent results of Kleinberg, we construct data structures whose size is polynomial in the size of the database, and search algorithms that run in time nearly linear or nearly quadratic in the dimension (depending on the case; the extra factors are polylogarithmic in the size of the database). Computer Science Department, Technion --- IIT, Haifa 32000, Israel. Email: eyalk@cs.technion.ac.il y Bell Communications Research, MCC-1C365B, 445 South Street, Morristown, NJ ...

Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) -- a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.

In this article, we give an overview of efficient algorithms for the approximate and exact nearest neighbor problem. The goal is to preprocess a dataset of objects (e.g., images) so that later, given a new query object, one can quickly return the dataset object that is most similar to the query. The problem is of significant interest in a wide variety of areas.

High dimensional data has always been a challenge for clustering algorithms because of the inherent sparsity of the points. Recent research results indicate that in high dimensional data, even the concept of proximity or clustering may not be meaningful. We discuss very general techniques for projected clustering which are able to construct clusters in arbitrarily aligned subspaces of lower dimensionality. The subspaces are specific to the clusters themselves. This definition is substantially more general and realistic than currently available techniques which limit the method to only projections from the original set of attributes. The generalized projected clustering technique may also be viewed as a way of trying to rede ne clustering for high dimensional applications by searching for hidden subspaces with clusters which are created by inter-attribute correlations. We provide a new concept of using extended cluster feature vectors in order to make the algorithm scalable for very large databases. The running time and space requirements of the algorithm are adjustable, and are likely to tradeoff with better accuracy.

A classic result of Johnson and Lindenstrauss asserts that any set of n points in d-dimensional Euclidean space can be embedded into k-dimensional Euclidean space | where k is logarithmic in n and independent of d | so that all pairwise distances are maintained within an arbitrarily small factor. All known constructions of such embeddings involve projecting the n points onto a random k-dimensional hyperplane. We give a novel construction of the embedding, suitable for database applications, which amounts to computing a simple aggregate over k random attribute partitions.

Random projections have recently emerged as a powerful method for dimensionality reduction. Theoretical results indicate that the method preserves distances quite nicely; however, empirical results are sparse. We present experimental results on using random projection as a dimensionality reduction tool in a number of cases, where the high dimensionality of the data would otherwise lead to burdensome computations. Our application areas are the processing of both noisy and noiseless images, and information retrieval in text documents. We show that projecting the data onto a random lower-dimensional subspace yields results comparable to conventional dimensionality reduction methods such as principal component analysis: the similarity of data vectors is preserved well under random projection. However, using random projections is computationally signicantly less expensive than using, e.g., principal component analysis. We also show experimentally that using a sparse random matrix gives additional computational savings in random projection.

matrix A. It is often of interest to nd a low-rank approximation to A, i.e., an approximation D to the matrix A of rank not greater than a speci ed rank k, where k is much smaller than m and n. Methods such as the Singular Value Decomposition (SVD) may be used to nd an approximation to A which is the best in a well de ned sense. These methods require memory and time which are superlinear in m and n; for many applications in which the data sets are very large this is prohibitive. Two simple and intuitive algorithms are presented which, when given an m n matrix A, compute a description of a low-rank approximation D to A, and which are qualitatively faster than the SVD. Both algorithms have provable bounds for the error matrix A D . For any matrix X , let kXk and kXk 2 denote its Frobenius norm and its spectral norm, respectively. In the rst algorithm, c = O(1) columns of A are randomly chosen. If the m c matrix C consists of those c columns of A (after appropriate rescaling) then it is shown that from C C approximations to the top singular values and corresponding singular vectors may be computed. From the computed singular vectors a description D of the matrix A may be computed such that rank(D ) k and such that holds with high probability for both = 2; F . This algorithm may be implemented without storing the matrix A in Random Access Memory (RAM), provided it can make two passes over the matrix stored in external memory and use O(m + n) additional RAM memory. The second algorithm is similar except that it further approximates the matrix C by randomly sampling r = O(1) rows of C to form a r c matrix W . Thus, it has additional error, but it can be implemented in three passes over the matrix using only constant ...