Intrinsic dimension of a dataset: what properties does one expect?

We propose an axiomatic approach to the concept of an intrinsic dimension of a dataset, based on a viewpoint of geometry of high-dimensional structures. Our first axiom postulates that high values of dimension be indicative of the presence of the curse of dimensionality (in a certain precise mathematical sense). The second axiom requires the dimension to depend smoothly on a distance between datasets (so that the dimension of a dataset and that of an approximating principal manifold would be close to each other). The third axiom is a normalization condition: the dimension of the Euclidean $n$-sphere \s^n is $\Theta(n)$. We give an example of a dimension function satisfying our axioms, even though it is in general computationally unfeasible, and discuss a computationally cheap function satisfying most but not all of our axioms (the ``intrinsic dimensionality'' of Ch\'avez et al.)Comment: 6 pages, 6 figures, 1 table, latex with IEEE macros, final submission to Proceedings of the 22nd IJCNN (Orlando, FL, August 12-17, 2007

Indexing Metric Spaces for Exact Similarity Search

With the continued digitalization of societal processes, we are seeing an explosion in available data. This is referred to as big data. In a research setting, three aspects of the data are often viewed as the main sources of challenges when attempting to enable value creation from big data: volume, velocity and variety. Many studies address volume or velocity, while much fewer studies concern the variety. Metric space is ideal for addressing variety because it can accommodate any type of data as long as its associated distance notion satisfies the triangle inequality. To accelerate search in metric space, a collection of indexing techniques for metric data have been proposed. However, existing surveys each offers only a narrow coverage, and no comprehensive empirical study of those techniques exists. We offer a survey of all the existing metric indexes that can support exact similarity search, by i) summarizing all the existing partitioning, pruning and validation techniques used for metric indexes, ii) providing the time and storage complexity analysis on the index construction, and iii) report on a comprehensive empirical comparison of their similarity query processing performance. Here, empirical comparisons are used to evaluate the index performance during search as it is hard to see the complexity analysis differences on the similarity query processing and the query performance depends on the pruning and validation abilities related to the data distribution. This article aims at revealing different strengths and weaknesses of different indexing techniques in order to offer guidance on selecting an appropriate indexing technique for a given setting, and directing the future research for metric indexes

HD-Index: Pushing the Scalability-Accuracy Boundary for Approximate kNN Search in High-Dimensional Spaces

Nearest neighbor searching of large databases in high-dimensional spaces is inherently difficult due to the curse of dimensionality. A flavor of approximation is, therefore, necessary to practically solve the problem of nearest neighbor search. In this paper, we propose a novel yet simple indexing scheme, HD-Index, to solve the problem of approximate k-nearest neighbor queries in massive high-dimensional databases. HD-Index consists of a set of novel hierarchical structures called RDB-trees built on Hilbert keys of database objects. The leaves of the RDB-trees store distances of database objects to reference objects, thereby allowing efficient pruning using distance filters. In addition to triangular inequality, we also use Ptolemaic inequality to produce better lower bounds. Experiments on massive (up to billion scale) high-dimensional (up to 1000+) datasets show that HD-Index is effective, efficient, and scalable.Comment: PVLDB 11(8):906-919, 201

QDR-Tree: An Efcient Index Scheme for Complex Spatial Keyword Query

With the popularity of mobile devices and the development of geo-positioning technology, location-based services (LBS) attract much attention and top-k spatial keyword queries become increasingly complex. It is common to see that clients issue a query to find a restaurant serving pizza and steak, low in price and noise level particularly. However, most of prior works focused only on the spatial keyword while ignoring these independent numerical attributes. In this paper we demonstrate, for the first time, the Attributes-Aware Spatial Keyword Query (ASKQ), and devise a two-layer hybrid index structure called Quad-cluster Dual-filtering R-Tree (QDR-Tree). In the keyword cluster layer, a Quad-Cluster Tree (QC-Tree) is built based on the hierarchical clustering algorithm using kernel k-means to classify keywords. In the spatial layer, for each leaf node of the QC-Tree, we attach a Dual-Filtering R-Tree (DR-Tree) with two filtering algorithms, namely, keyword bitmap-based and attributes skyline-based filtering. Accordingly, efficient query processing algorithms are proposed. Through theoretical analysis, we have verified the optimization both in processing time and space consumption. Finally, massive experiments with real-data demonstrate the efficiency and effectiveness of QDR-Tree

Efficient Joinable Table Discovery in Data Lakes: A High-Dimensional Similarity-Based Approach

Finding joinable tables in data lakes is key procedure in many applications such as data integration, data augmentation, data analysis, and data market. Traditional approaches that find equi-joinable tables are unable to deal with misspellings and different formats, nor do they capture any semantic joins. In this paper, we propose PEXESO, a framework for joinable table discovery in data lakes. We embed textual values as high-dimensional vectors and join columns under similarity predicates on high-dimensional vectors, hence to address the limitations of equi-join approaches and identify more meaningful results. To efficiently find joinable tables with similarity, we propose a block-and-verify method that utilizes pivot-based filtering. A partitioning technique is developed to cope with the case when the data lake is large and the index cannot fit in main memory. An experimental evaluation on real datasets shows that our solution identifies substantially more tables than equi-joins and outperforms other similarity-based options, and the join results are useful in data enrichment for machine learning tasks. The experiments also demonstrate the efficiency of the proposed method.Comment: Full version of paper in ICDE 202