Network alignment aims at inferring a set of anchor links matching the shared entities between different information networks, which has become a prerequisite step for effective fusion of multiple information networks. In this paper, we will study the network alignment problem to fuse online social networks specifically. Social network alignment is extremely challenging to address due to several reasons, i.e., lack of training data, network heterogeneity and one-to-one constraint. Existing network alignment works usually require a large number of training data, but such a demand can hardly be met in applications, as manual anchor link labeling is extremely expensive. Significantly different from other homogeneous network alignment works, information in online social networks is usually of heterogeneous categories, the incorporation of which in model building is not an easy task. Furthermore, the one-to-one cardinality constraint on anchor links renders their inference process intertwistingly correlated. To resolve these three challenges, a novel network alignment model, namely ActiveIter, is introduced in this paper. ActiveIter defines a set of inter-network meta diagrams for anchor link feature extraction, adopts active learning for effective label query and uses greedy link selection for anchor link cardinality filtering. Extensive experiments are conducted on real-world aligned networks datasets, and the experimental results have demonstrated the effectiveness of ActiveIter compared with other state-of-the-art baseline methods.
Similarity measure based on heterogeneous networks has been widely studied. Sun introduces the concept of in @cite_2 , where a is a path consisting of a sequence of relations. However, the suffers from two disadvantages. On one hand, cannot describe rich semantics effectively. On the other hand, once numerious are defined, it's challenging to assemble them. Some methods to resolve these deficiencies are proposed later. @cite_25 applys meta-graph to similarity measure problem, but entities are constrained to be of the same type. Zhao @cite_16 proposes the concept of and extends the idea to recommendation problems which require that entities belong to different types. However, and are proposed for single non-attribute networks. In our definition, not only regular node types but also attribute types are involved, and it can be applied to the similarity measure across networks.
@cite_16 @cite_25 @cite_2
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Similarity search is a primitive operation in database and Web search engines. With the advent of large-scale heterogeneous information networks that consist of multi-typed, interconnected objects, such as the bibliographic networks and social media networks, it is important to study similarity search in such networks. Intuitively, two objects are similar if they are linked by many paths in the network. However, most existing similarity measures are defined for homogeneous networks. Different semantic meanings behind paths are not taken into consideration. Thus they cannot be directly applied to heterogeneous networks. In this paper, we study similarity search that is defined among the same type of objects in heterogeneous networks. Moreover, by considering different linkage paths in a network, one could derive various similarity semantics. Therefore, we introduce the concept of meta path-based similarity, where a meta path is a path consisting of asequence of relations defined between different object types (i.e., structural paths at the meta level). No matter whether a user would like to explicitly specify a path combination given sufficient domain knowledge, or choose the best path by experimental trials, or simply provide training examples to learn it, meta path forms a common base for a network-based similarity search engine. In particular, under the meta path framework we define a novel similarity measure called PathSim that is able to find peer objects in the network (e.g., find authors in the similar field and with similar reputation), which turns out to be more meaningful in many scenarios compared with random-walk based similarity measures. In order to support fast online query processing for PathSim queries, we develop an efficient solution that partially materializes short meta paths and then concatenates them online to compute top-k results. Experiments on real data sets demonstrate the effectiveness and efficiency of our proposed paradigm. Similarity search is an important function in many applications, which usually focuses on measuring the similarity between objects with the same type. However, in many scenarios, we need to measure the relatedness between objects with different types. With the surge of study on heterogeneous networks, the relevance measure on objects with different types becomes increasingly important. In this paper, we study the relevance search problem in heterogeneous networks, where the task is to measure the relatedness of heterogeneous objects (including objects with the same type or different types). A novel measure HeteSim is proposed, which has the following attributes: (1) a uniform measure: it can measure the relatedness of objects with the same or different types in a uniform framework; (2) a path-constrained measure: the relatedness of object pairs are defined based on the search path that connects two objects through following a sequence of node types; (3) a semi-metric measure: HeteSim has some good properties (e.g., self-maximum and symmetric), which are crucial to many data mining tasks. Moreover, we analyze the computation characteristics of HeteSim and propose the corresponding quick computation strategies. Empirical studies show that HeteSim can effectively and efficiently evaluate the relatedness of heterogeneous objects. Motivation: Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics, such as the degree distribution, clustering coefficient, diameter, and relative graphlet frequency distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. Results: We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions of graphlets with 2--5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets), if necessary, and the extensions are limited only by the available CPU. Furthermore, we show a way to combine the 73 graphlet degree distributions into a network 'agreement' measure which is a number between 0 and 1, where 1 means that networks have identical distributions and 0 means that they are far apart. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, resulting from various high-throughput experimental techniques, as well as from curated databases, are better modeled by geometric random graphs than by Erdos--Reny, random scale-free, or Barabasi--Albert scale-free networks. Availability: Software executables are available upon request. Contact: natasha@ics.uci.edu As a fundamental task, document similarity measure has broad impact to document-based classification, clustering and ranking. Traditional approaches represent documents as bag-of-words and compute document similarities using measures like cosine, Jaccard, and dice. However, entity phrases rather than single words in documents can be critical for evaluating document relatedness. Moreover, types of entities and links between entities words are also informative. We propose a method to represent a document as a typed heterogeneous information network (HIN), where the entities and relations are annotated with types. Multiple documents can be linked by the words and entities in the HIN. Consequently, we convert the document similarity problem to a graph distance problem. Intuitively, there could be multiple paths between a pair of documents. We propose to use the meta-path defined in HIN to compute distance between documents. Instead of burdening user to define meaningful meta paths, an automatic method is proposed to rank the meta-paths. Given the meta-paths associated with ranking scores, an HIN-based similarity measure, KnowSim, is proposed to compute document similarities. Using Freebase, a well-known world knowledge base, to conduct semantic parsing and construct HIN for documents, our experiments on 20Newsgroups and RCV1 datasets show that KnowSim generates impressive high-quality document clustering. Although hashing techniques have been popular for the large scale similarity search problem, most of the existing methods for designing optimal hash functions focus on homogeneous similarity assessment, i.e., the data entities to be indexed are of the same type. Realizing that heterogeneous entities and relationships are also ubiquitous in the real world applications, there is an emerging need to retrieve and search similar or relevant data entities from multiple heterogeneous domains, e.g., recommending relevant posts and images to a certain Facebook user. In this paper, we address the problem of comparing apples to oranges'' under the large scale setting. Specifically, we propose a novel Relation-aware Heterogeneous Hashing (RaHH), which provides a general framework for generating hash codes of data entities sitting in multiple heterogeneous domains. Unlike some existing hashing methods that map heterogeneous data in a common Hamming space, the RaHH approach constructs a Hamming space for each type of data entities, and learns optimal mappings between them simultaneously. This makes the learned hash codes flexibly cope with the characteristics of different data domains. Moreover, the RaHH framework encodes both homogeneous and heterogeneous relationships between the data entities to design hash functions with improved accuracy. To validate the proposed RaHH method, we conduct extensive evaluations on two large datasets; one is crawled from a popular social media sites, Tencent Weibo, and the other is an open dataset of Flickr(NUS-WIDE). The experimental results clearly demonstrate that the RaHH outperforms several state-of-the-art hashing methods with significant performance gains. We describe an approach for performing entity type recognition in heterogeneous semantic graphs in order to reduce the computational cost of performing coreferenceresolution. Our research specifically addresses the problem of working with semi-structured text that uses ontologies that are not informative or not known. This problem is similar to co reference resolution in unstructured text, where entities and their types are identified using contextual information and linguistic-based analysis. Semantic graphs are semi-structured with very little contextual information and trivial grammars that do not convey additional information. In the absence of known ontologies, performing co reference resolution can be challenging. Our work uses a supervised machine learning algorithm and entity type dictionaries to map attributes to a common attribute space. We evaluated the approach in experiments using data from Wikipedia, Freebase and Arnetminer. This paper presents a new approach for measuring semantic similarity distance between words and concepts. It combines a lexical taxonomy structure with corpus statistical information so that the semantic distance between nodes in the semantic space constructed by the taxonomy can be better quantified with the computational evidence derived from a distributional analysis of corpus data. Specifically, the proposed measure is a combined approach that inherits the edge-based approach of the edge counting scheme, which is then enhanced by the node-based approach of the information content calculation. When tested on a common data set of word pair similarity ratings, the proposed approach outperforms other computational models. It gives the highest correlation value (r = 0.828) with a benchmark based on human similarity judgements, whereas an upper bound (r = 0.885) is observed when human subjects replicate the same task. Lexical-semantic resources, including thesauri and WORDNET, have been successfully incorporated into a wide range of applications in Natural Language Processing. However they are very difficult and expensive to create and maintain, and their usefulness has been severely hampered by their limited coverage, bias and inconsistency. Automated and semi-automated methods for developing such resources are therefore crucial for further resource development and improved application performance. Systems that extract thesauri often identify similar words using the distributional hypothesis that similar words appear in similar contexts. This approach involves using corpora to examine the contexts each word appears in and then calculating the similarity between context distributions. Different definitions of context can be used, and I begin by examining how different types of extracted context influence similarity. To be of most benefit these systems must be capable of finding synonyms for rare words. Reliable context counts for rare events can only be extracted from vast collections of text. In this dissertation I describe how to extract contexts from a corpus of over 2 billion words. I describe techniques for processing text on this scale and examine the trade-off between context accuracy, information content and quantity of text analysed. Distributional similarity is at best an approximation to semantic similarity. I develop improved approximations motivated by the intuition that some events in the context distribution are more indicative of meaning than others. For instance, the object-of-verb context wear is far more indicative of a clothing noun than get. However, existing distributional techniques do not effectively utilise this information. The new context-weighted similarity metric I propose in this dissertation significantly outperforms every distributional similarity metric described in the literature. Nearest-neighbour similarity algorithms scale poorly with vocabulary and context vector size. To overcome this problem I introduce a new context-weighted approximation algorithm with bounded complexity in context vector size that significantly reduces the system runtime with only a minor performance penalty. I also describe a parallelized version of the system that runs on a Beowulf cluster for the 2 billion word experiments. To evaluate the context-weighted similarity measure I compare ranked similarity lists against gold-standard resources using precision and recall-based measures from Information Retrieval, An alternative form to multidimensional projections for the visual analysis of data represented in multidimensional spaces is the deployment of similarity trees, such as Neighbor Joining trees. They organize data objects on the visual plane emphasizing their levels of similarity with high capability of detecting and separating groups and subgroups of objects. Besides this similarity-based hierarchical data organization, some of their advantages include the ability to decrease point clutter; high precision; and a consistent view of the data set during focusing, offering a very intuitive way to view the general structure of the data set as well as to drill down to groups and subgroups of interest. Disadvantages of similarity trees based on neighbor joining strategies include their computational cost and the presence of virtual nodes that utilize too much of the visual space. This paper presents a highly improved version of the similarity tree technique. The improvements in the technique are given by two procedures. The first is a strategy that replaces virtual nodes by promoting real leaf nodes to their place, saving large portions of space in the display and maintaining the expressiveness and precision of the technique. The second improvement is an implementation that significantly accelerates the algorithm, impacting its use for larger data sets. We also illustrate the applicability of the technique in visual data mining, showing its advantages to support visual classification of data sets, with special attention to the case of image classification. We demonstrate the capabilities of the tree for analysis and iterative manipulation and employ those capabilities to support evolving to a satisfactory data organization and classification. Domain specific (dis-)similarity or proximity measures used e.g. in alignment algorithms of sequence data are popular to analyze complicated data objects and to cover domain specific data properties. Without an underlying vector space these data are given as pairwise (dis-)similarities only. The few available methods for such data focus widely on similarities and do not scale to large datasets. Kernel methods are very effective for metric similarity matrices, also at large scale, but costly transformations are necessary starting with non-metric (dis-) similarities. We propose an integrative combination of Nystrom approximation, potential double centering and eigenvalue correction to obtain valid kernel matrices at linear costs in the number of samples. By the proposed approach effective kernel approaches become accessible. Experiments with several larger (dis-)similarity datasets show that the proposed method achieves much better runtime performance than the standard strategy while keeping competitive model accuracy. The main contribution is an efficient and accurate technique, to convert (potentially non-metric) large scale dissimilarity matrices into approximated positive semi-definite kernel matrices at linear costs. HighlightsWe propose a linear time and memory efficient approach for converting low rank dissimilarity matrices to similarity matrices and vice versa.Our approach is applicable for proximities obtained from non-metric proximity measures (indefinite kernels, non-standard dissimilarity measures).The presented approach also comprises a generalization of Landmark MDS - the presented approach is in general more accurate and flexible than Landmark MDS.We provide an alternative derivation of the Nystrom approximation together with a convergence proof, also for indefinite kernels not given in the workshop paper as a core element of the approach. The performance of Nearest Neighbor (NN) classifier is known to be sensitive to the distance (or similarity) function used in classifying a test instance. Another major disadvantage of NN is that it uses all training instances in the generalization phase. This can cause slow execution speed and high storage requirement when dealing with large datasets. In the past research, many solutions have been proposed to handle one or both of the above problems. In the scheme proposed in this paper, we tackle both of these problems by assigning a weight to each training instance. The weight of a training instance is used in the generalization phase to calculate the distance (or similarity) of a query pattern to that instance. The basic NN classifier can be viewed as a special case of this scheme that treats all instances equally (by assigning equal weight to all training instances). Using this form of weighted similarity measure, we propose a learning algorithm that attempts to maximize the leave-one-out (LV1) classification rate of the NN rule by adjusting the weights of the training instances. At the same time, the algorithm reduces the size of the training set and can be viewed as a powerful instance reduction technique. An instance having zero weight is not used in the generalization phase and can be virtually removed from the training set. We show that our scheme has comparable or better performance than some recent methods proposed in the literature for the task of learning the distance function and or prototype reduction. Abstract When dealing with the registration of information from different image sources, the de facto similarity measure used is Mutual Information (MI). Although MI gives good performance in many image registration applications, recent works in thermal–visible registration have shown that other similarity measures can give results that are as accurate, if not more than MI. Furthermore, some of these measures also have the advantage of being calculated independently from each image to register, which allows them to be integrated more easily in energy minimization frameworks. In this article, we investigate the accuracy of similarity measures for thermal–visible image registration of human silhouettes, including MI, Sum of Squared Differences (SSD), Normalized Cross-Correlation (NCC), Histograms of Oriented Gradients (HOG), Local Self-Similarity (LSS), Scale-Invariant Feature Transform (SIFT), Speeded-Up Robust Features (SURF), Census, Fast Retina Keypoint (FREAK), and Binary Robust Independent Elementary Feature (BRIEF). We tested the various similarity measures in dense stereo matching tasks over 25,000 windows to have statistically significant results. To do so, we created a new dataset in which one to five humans are walking in a scene in various depth planes. Results show that even if MI is a very strong performer, particularly for large regions of interest (ROI), LSS gives better accuracies when ROI are small or segmented into small fragments because of its ability to capture shape. The other tested similarity measures did not give consistently accurate results. Measuring the similarity between semantic relations that hold among entities is an important and necessary step in various Web related tasks such as relation extraction, information retrieval and analogy detection. For example, consider the case in which a person knows a pair of entities (e.g. Google, YouTube), between which a particular relation holds (e.g. acquisition). The person is interested in retrieving other such pairs with similar relations (e.g. Microsoft, Powerset). Existing keyword-based search engines cannot be applied directly in this case because, in keyword-based search, the goal is to retrieve documents that are relevant to the words used in a query -- not necessarily to the relations implied by a pair of words. We propose a relational similarity measure, using a Web search engine, to compute the similarity between semantic relations implied by two pairs of words. Our method has three components: representing the various semantic relations that exist between a pair of words using automatically extracted lexical patterns, clustering the extracted lexical patterns to identify the different patterns that express a particular semantic relation, and measuring the similarity between semantic relations using a metric learning approach. We evaluate the proposed method in two tasks: classifying semantic relations between named entities, and solving word-analogy questions. The proposed method outperforms all baselines in a relation classification task with a statistically significant average precision score of 0.74. Moreover, it reduces the time taken by Latent Relational Analysis to process 374 word-analogy questions from 9 days to less than 6 hours, with an SAT score of 51 . Graph clustering has been widely applied in exploring regularities emerging in relational data. Recently, the rapid development of network theory correlates graph clustering with the detection of community structure, a common and important topological characteristic of networks. Most existing methods investigate the community structure at a single topological scale. However, as shown by empirical studies, the community structure of real world networks often exhibits multiple topological descriptions, corresponding to the clustering at different resolutions. Furthermore, the detection of multiscale community structure is heavily affected by the heterogeneous distribution of node degree. It is very challenging to detect multiscale community structure in heterogeneous networks. In this paper, we propose a novel, unified framework for detecting community structure from the perspective of dimensionality reduction. Based on the framework, we first prove that the well-known Laplacian matrix for network partition and the widely-used modularity matrix for community detection are two kinds of covariance matrices used in dimensionality reduction. We then propose a novel method to detect communities at multiple topological scales within our framework. We further show that existing algorithms fail to deal with heterogeneous node degrees. We develop a novel method to handle heterogeneity of networks by introducing a rescaling transformation into the covariance matrices in our framework. Extensive tests on real world and artificial networks demonstrate that the proposed correlation matrices significantly outperform Laplacian and modularity matrices in terms of their ability to identify multiscale community structure in heterogeneous networks. We consider the problem of classification using similarity distance functions over data. Specifically, we propose a framework for defining the goodness of a (dis)similarity function with respect to a given learning task and propose algorithms that have guaranteed generalization properties when working with such good functions. Our framework unifies and generalizes the frameworks proposed by [1] and [2]. An attractive feature of our framework is its adaptability to data - we do not promote a fixed notion of goodness but rather let data dictate it. We show, by giving theoretical guarantees that the goodness criterion best suited to a problem can itself be learned which makes our approach applicable to a variety of domains and problems. We propose a landmarking-based approach to obtaining a classifier from such learned goodness criteria. We then provide a novel diversity based heuristic to perform task-driven selection of landmark points instead of random selection. We demonstrate the effectiveness of our goodness criteria learning method as well as the landmark selection heuristic on a variety of similarity-based learning datasets and benchmark UCI datasets on which our method consistently outperforms existing approaches by a significant margin. This work addresses graph-based semi-supervised classification and betweenness computation in large, sparse, networks (several millions of nodes). The objective of semi-supervised classification is to assign a label to unlabeled nodes using the whole topology of the graph and the labeling at our disposal. Two approaches are developed to avoid explicit computation of pairwise proximity between the nodes of the graph, which would be impractical for graphs containing millions of nodes. The first approach directly computes, for each class, the sum of the similarities between the nodes to classify and the labeled nodes of the class, as suggested initially in [1,2]. Along this approach, two algorithms exploiting different state-of-the-art kernels on a graph are developed. The same strategy can also be used in order to compute a betweenness measure. The second approach works on a trellis structure built from biased random walks on the graph, extending an idea introduced in [3]. These random walks allow to define a biased bounded betweenness for the nodes of interest, defined separately for each class. All the proposed algorithms have a linear computing time in the number of edges while providing good results, and hence are applicable to large sparse networks. They are empirically validated on medium-size standard data sets and are shown to be competitive with state-of-the-art techniques. Finally, we processed a novel data set, which is made available for benchmarking, for multi-class classification in a large network: the U.S. patents citation network containing 3M nodes (of six different classes) and 38M edges. The three proposed algorithms achieve competitive results (around 85 classification rate) on this large network-they classify the unlabeled nodes within a few minutes on a standard workstation. Abstract Similarity networks contain important topological features and patterns critical to understanding interactions among samples in a large dataset. To create a comprehensive view of the interactions within a dataset, the Similarity Network Fusion (SNF) technique has been proposed to fuse the similarity networks based on different data types into one similarity network that represents the full spectrum of underlying data. In this paper, a modified version of SNF, which is named as Contextual Information based SNF (CI-SNF), is proposed. In CI-SNF, first, modified Jaccard distance is performed on the SNF fused similarity to utilize the contextual information contained in the fused similarity network. Second, the local consistency of samples from the same category is enhanced by speculating that the samples which are located high in the Jaccard distance based ranking list of a specific query are from the same category as the query. Third, the inverted index technique is introduced to utilize the sparsity property of the locally consistent similarity network to enhance the computational efficiency. To verify the effectiveness and efficiency of CI-SNF model, it is applied in four different tasks, Cover Song Identification (CSI), image classification, cancer subtype identification, and drug taxonomy, respectively. Extensive experiments on thirteen challenging datasets demonstrate that CI-SNF scheme outperforms state-of-the-art similarity fusion algorithms including SNF in all four tasks. It is also verified that utilizing the contextual information contained in the SNF-based similarity network helps to enhance the performance of the SNF-based scheme, further. The increasing availability of location-acquisition technologies (GPS, GSM networks, etc.) enables people to log the location histories with spatio-temporal data. Such real-world location histories imply, to some extent, users' interests in places, and bring us opportunities to understand the correlation between users and locations. In this article, we move towards this direction and report on a personalized friend and location recommender for the geographical information systems (GIS) on the Web. First, in this recommender system, a particular individual's visits to a geospatial region in the real world are used as their implicit ratings on that region. Second, we measure the similarity between users in terms of their location histories and recommend to each user a group of potential friends in a GIS community. Third, we estimate an individual's interests in a set of unvisited regions by involving his her location history and those of other users. Some unvisited locations that might match their tastes can be recommended to the individual. A framework, referred to as a hierarchical-graph-based similarity measurement (HGSM), is proposed to uniformly model each individual's location history, and effectively measure the similarity among users. In this framework, we take into account three factors: 1) the sequence property of people's outdoor movements, 2) the visited popularity of a geospatial region, and 3) the hierarchical property of geographic spaces. Further, we incorporated a content-based method into a user-based collaborative filtering algorithm, which uses HGSM as the user similarity measure, to estimate the rating of a user on an item. We evaluated this recommender system based on the GPS data collected by 75 subjects over a period of 1 year in the real world. As a result, HGSM outperforms related similarity measures, namely similarity-by-count, cosine similarity, and Pearson similarity measures. Moreover, beyond the item-based CF method and random recommendations, our system provides users with more attractive locations and better user experiences of recommendation. We study the problem of representation learning in heterogeneous networks. Its unique challenges come from the existence of multiple types of nodes and links, which limit the feasibility of the conventional network embedding techniques. We develop two scalable representation learning models, namely metapath2vec and metapath2vec++. The metapath2vec model formalizes meta-path-based random walks to construct the heterogeneous neighborhood of a node and then leverages a heterogeneous skip-gram model to perform node embeddings. The metapath2vec++ model further enables the simultaneous modeling of structural and semantic correlations in heterogeneous networks. Extensive experiments show that metapath2vec and metapath2vec++ are able to not only outperform state-of-the-art embedding models in various heterogeneous network mining tasks, such as node classification, clustering, and similarity search, but also discern the structural and semantic correlations between diverse network objects. 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).