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

A randomized approach to speed up the analysis of large-scale read-count data in the application of CNV detection

Abstract Background The application of high-throughput sequencing in a broad range of quantitative genomic assays (e.g., DNA-seq, ChIP-seq) has created a high demand for the analysis of large-scale read-count data. Typically, the genome is divided into tiling windows and windowed read-count data is generated for the entire genome from which genomic signals are detected (e.g. copy number changes in DNA-seq, enrichment peaks in ChIP-seq). For accurate analysis of read-count data, many state-of-the-art statistical methods use generalized linear models (GLM) coupled with the negative-binomial (NB) distribution by leveraging its ability for simultaneous bias correction and signal detection. However, although statistically powerful, the GLM+NB method has a quadratic computational complexity and therefore suffers from slow running time when applied to large-scale windowed read-count data. In this study, we aimed to speed up substantially the GLM+NB method by using a randomized algorithm and we demonstrate here the utility of our approach in the application of detecting copy number variants (CNVs) using a real example. Results We propose an efficient estimator, the randomized GLM+NB coefficients estimator (RGE), for speeding up the GLM+NB method. RGE samples the read-count data and solves the estimation problem on a smaller scale. We first theoretically validated the consistency and the variance properties of RGE. We then applied RGE to GENSENG, a GLM+NB based method for detecting CNVs. We named the resulting method as “R-GENSENG". Based on extensive evaluation using both simulated and empirical data, we concluded that R-GENSENG is ten times faster than the original GENSENG while maintaining GENSENG’s accuracy in CNV detection. Conclusions Our results suggest that RGE strategy developed here could be applied to other GLM+NB based read-count analyses, i.e. ChIP-seq data analysis, to substantially improve their computational efficiency while preserving the analytic power