Efficient similarity search in high-dimensional data spaces

Similarity search in high-dimensional data spaces is a popular paradigm for many modern database applications, such as content based image retrieval, time series analysis in financial and marketing databases, and data mining. Objects are represented as high-dimensional points or vectors based on their important features. Object similarity is then measured by the distance between feature vectors and similarity search is implemented via range queries or k-Nearest Neighbor (k-NN) queries. Implementing k-NN queries via a sequential scan of large tables of feature vectors is computationally expensive. Building multi-dimensional indexes on the feature vectors for k-NN search also tends to be unsatisfactory when the dimensionality is high. This is due to the poor index performance caused by the dimensionality curse. Dimensionality reduction using the Singular Value Decomposition method is the approach adopted in this study to deal with high-dimensional data. Noting that for many real-world datasets, data distribution tends to be heterogeneous, dimensionality reduction on the entire dataset may cause a significant loss of information. More efficient representation is sought by clustering the data into homogeneous subsets of points, and applying dimensionality reduction to each cluster respectively, i.e., utilizing local rather than global dimensionality reduction. The thesis deals with the improvement of the efficiency of query processing associated with local dimensionality reduction methods, such as the Clustering and Singular Value Decomposition (CSVD) and the Local Dimensionality Reduction (LDR) methods. Variations in the implementation of CSVD are considered and the two methods are compared from the viewpoint of the compression ratio, CPU time, and retrieval efficiency. An exact k-NN algorithm is presented for local dimensionality reduction methods by extending an existing multi-step k-NN search algorithm, which is designed for global dimensionality reduction. Experimental results show that the new method requires less CPU time than the approximate method proposed original for CSVD at a comparable level of accuracy. Optimal subspace dimensionality reduction has the intent of minimizing total query cost. The problem is complicated in that each cluster can retain a different number of dimensions. A hybrid method is presented, combining the best features of the CSVD and LDR methods, to find optimal subspace dimensionalities for clusters generated by local dimensionality reduction methods. The experiments show that the proposed method works well for both real-world datasets and synthetic datasets

Discrete Elastic Inner Vector Spaces with Application in Time Series and Sequence Mining

This paper proposes a framework dedicated to the construction of what we call discrete elastic inner product allowing one to embed sets of non-uniformly sampled multivariate time series or sequences of varying lengths into inner product space structures. This framework is based on a recursive definition that covers the case of multiple embedded time elastic dimensions. We prove that such inner products exist in our general framework and show how a simple instance of this inner product class operates on some prospective applications, while generalizing the Euclidean inner product. Classification experimentations on time series and symbolic sequences datasets demonstrate the benefits that we can expect by embedding time series or sequences into elastic inner spaces rather than into classical Euclidean spaces. These experiments show good accuracy when compared to the euclidean distance or even dynamic programming algorithms while maintaining a linear algorithmic complexity at exploitation stage, although a quadratic indexing phase beforehand is required.Comment: arXiv admin note: substantial text overlap with arXiv:1101.431

Ptolemaic Indexing

This paper discusses a new family of bounds for use in similarity search, related to those used in metric indexing, but based on Ptolemy's inequality, rather than the metric axioms. Ptolemy's inequality holds for the well-known Euclidean distance, but is also shown here to hold for quadratic form metrics in general, with Mahalanobis distance as an important special case. The inequality is examined empirically on both synthetic and real-world data sets and is also found to hold approximately, with a very low degree of error, for important distances such as the angular pseudometric and several Lp norms. Indexing experiments demonstrate a highly increased filtering power compared to existing, triangular methods. It is also shown that combining the Ptolemaic and triangular filtering can lead to better results than using either approach on its own

From Frequency to Meaning: Vector Space Models of Semantics

Computers understand very little of the meaning of human language. This profoundly limits our ability to give instructions to computers, the ability of computers to explain their actions to us, and the ability of computers to analyse and process text. Vector space models (VSMs) of semantics are beginning to address these limits. This paper surveys the use of VSMs for semantic processing of text. We organize the literature on VSMs according to the structure of the matrix in a VSM. There are currently three broad classes of VSMs, based on term-document, word-context, and pair-pattern matrices, yielding three classes of applications. We survey a broad range of applications in these three categories and we take a detailed look at a specific open source project in each category. Our goal in this survey is to show the breadth of applications of VSMs for semantics, to provide a new perspective on VSMs for those who are already familiar with the area, and to provide pointers into the literature for those who are less familiar with the field