High-dimensional indexing methods utilizing clustering and dimensionality reduction

The emergence of novel database applications has resulted in the prevalence of a new paradigm for similarity search. These applications include multimedia databases, medical imaging databases, time series databases, DNA and protein sequence databases, and many others. Features of data objects are extracted and transformed into high-dimensional data points. Searching for objects becomes a search on points in the high-dimensional feature space. The dissimilarity between two objects is determined by the distance between two feature vectors. Similarity search is usually implemented as nearest neighbor search in feature vector spaces. The cost of processing k-nearest neighbor (k-NN) queries via a sequential scan increases as the number of objects and the number of features increase. A variety of multi-dimensional index structures have been proposed to improve the efficiency of k-NN query processing, which work well in low-dimensional space but lose their efficiency in high-dimensional space due to the curse of dimensionality. This inefficiency is dealt in this study by Clustering and Singular Value Decomposition - CSVD with indexing, Persistent Main Memory - PMM index, and Stepwise Dimensionality Increasing - SDI-tree index. CSVD is an approximate nearest neighbor search method. The performance of CSVD with indexing is studied and the approximation to the distance in original space is investigated. For a given Normalized Mean Square Error - NMSE, the higher the degree of clustering, the higher the recall. However, more clusters require more disk page accesses. Certain number of clusters can be obtained to achieve a higher recall while maintaining a relatively lower query processing cost. Clustering and Indexing using Persistent Main Memory - CIPMM framework is motivated by the following consideration: (a) a significant fraction of index pages are accessed randomly, incurring a high positioning time for each access; (b) disk transfer rate is improving 40% annually, while the improvement in positioning time is only 8%; (c) query processing incurs less CPU time for main memory resident than disk resident indices. CIPMM aims at reducing the elapsed time for query processing by utilizing sequential, rather than random disk accesses. A specific instance of the CIPMM framework CIPOP, indexing using Persistent Ordered Partition - OP-tree, is elaborated and compared with clustering and indexing using the SR-tree, CISR. The results show that CIPOP outperforms CISR, and the higher the dimensionality, the higher the performance gains. The SDI-tree index is motivated by fanouts decrease with dimensionality increasing and shorter vectors reduce cache misses. The index is built by using feature vectors transformed via principal component analysis, resulting in a structure with fewer dimensions at higher levels and increasing the number of dimensions from one level to the other. Dimensions are retained in nonincreasing order of their variance according to a parameter p, which specifies the incremental fraction of variance at each level of the index. Experiments on three datasets have shown that SDL-trees with carefully tuned parameters access fewer disk accesses than SR-trees and VAMSR-trees and incur less CPU time than VA-Files in addition

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

Dimensionality Reduced Clustered Data and Order Partition and Stepwise Dimensionality Increasing Indices

One of the goals of NASA funded project at IBM T. J. Watson Research Center was to build an index for similarity searching satellite images, which were characterized by high-dimensional feature image texture vectors. Reviewed is our effort on data clustering, dimensionality reduction via Singular Value Decomposition - SVD and indexing to build a smaller index and more efficient k-Nearest Neighbor - k-NN query processing for similarity search. k-NN queries based on scanning of the feature vectors of all images is obviously too costly for ever-increasing number of images. The ubiquitous multidimensional R-tree index and its extensions were not an option given their limited scalability dimension-wise. The cost of processing k-NN queries was further reduced by building memory resident Ordered Partition indices on dimensionality reduced clusters. Further research in a university setting included the following: (1) Clustered SVD was extended to yield exact k-NN queries by issuing appropriate less costly range queries, (2) Stepwise Dimensionality Increasing - SDI index outperformed other known indices, (3) selection of optimal number of dimensions to reduce query processing cost, (4) two methods to make the OP-trees persistent and loadable as a single file access

k-Nearest Neighbor Search for Large Scale High Dimensional data

Tohoku University徳山

Incremental dimension reduction of tensors with random index

We present an incremental, scalable and efficient dimension reduction technique for tensors that is based on sparse random linear coding. Data is stored in a compactified representation with fixed size, which makes memory requirements low and predictable. Component encoding and decoding are performed on-line without computationally expensive re-analysis of the data set. The range of tensor indices can be extended dynamically without modifying the component representation. This idea originates from a mathematical model of semantic memory and a method known as random indexing in natural language processing. We generalize the random-indexing algorithm to tensors and present signal-to-noise-ratio simulations for representations of vectors and matrices. We present also a mathematical analysis of the approximate orthogonality of high-dimensional ternary vectors, which is a property that underpins this and other similar random-coding approaches to dimension reduction. To further demonstrate the properties of random indexing we present results of a synonym identification task. The method presented here has some similarities with random projection and Tucker decomposition, but it performs well at high dimensionality only (n>10^3). Random indexing is useful for a range of complex practical problems, e.g., in natural language processing, data mining, pattern recognition, event detection, graph searching and search engines. Prototype software is provided. It supports encoding and decoding of tensors of order >= 1 in a unified framework, i.e., vectors, matrices and higher order tensors.Comment: 36 pages, 9 figure

DROP: Dimensionality Reduction Optimization for Time Series

Dimensionality reduction is a critical step in scaling machine learning pipelines. Principal component analysis (PCA) is a standard tool for dimensionality reduction, but performing PCA over a full dataset can be prohibitively expensive. As a result, theoretical work has studied the effectiveness of iterative, stochastic PCA methods that operate over data samples. However, termination conditions for stochastic PCA either execute for a predetermined number of iterations, or until convergence of the solution, frequently sampling too many or too few datapoints for end-to-end runtime improvements. We show how accounting for downstream analytics operations during DR via PCA allows stochastic methods to efficiently terminate after operating over small (e.g., 1%) subsamples of input data, reducing whole workload runtime. Leveraging this, we propose DROP, a DR optimizer that enables speedups of up to 5x over Singular-Value-Decomposition-based PCA techniques, and exceeds conventional approaches like FFT and PAA by up to 16x in end-to-end workloads

Sampling from large matrices: an approach through geometric functional analysis

We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(r log r) with a small error in the spectral norm, where r = ||A||_F^2 / ||A||_2^2 is the numerical rank of A. The numerical rank is always bounded by, and is a stable relaxation of, the rank of A. This yields an asymptotically optimal guarantee in an algorithm for computing low-rank approximations of A. We also prove asymptotically optimal estimates on the spectral norm and the cut-norm of random submatrices of A. The result for the cut-norm yields a slight improvement on the best known sample complexity for an approximation algorithm for MAX-2CSP problems. We use methods of Probability in Banach spaces, in particular the law of large numbers for operator-valued random variables.Comment: Our initial claim about Max-2-CSP problems is corrected. We put an exponential failure probability for the algorithm for low-rank approximations. Proofs are a little more explaine