There has been considerable interest in random projections, an approximate algorithm for estimating distances between pairs of points in a high-dimensional vector space. Let A in Rn x D be our n points in D dimensions. The method multiplies A by a random matrix R in RD x k, reducing the D dimensions down to just k for speeding up the computation. R typically consists of entries of standard normal N(0,1). It is well known that random projections preserve pairwise distances (in the expectation). Achlioptas proposed sparse random projections by replacing the N(0,1) entries in R with entries in -1,0,1 with probabilities 1/6, 2/3, 1/6, achieving a threefold speedup in processing time.We recommend using R of entries in -1,0,1 with probabilities 1/2√D, 1-1√D, 1/2√D for achieving a significant √D-fold speedup, with little loss in accuracy.

/pdf/very-sparse-random-projections-omb9acpovm.pdf

Very sparse random projections

Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis, and this work was performed by individuals from many different research communities. This monograph will provide a detailed overview of recent work on the theory of randomized matrix algorithms as well as the application of those ideas to the solution of practical problems in large-scale data analysis. An emphasis will be placed on a few simple core ideas that underlie not only recent theoretical advances but also the usefulness of these tools in large-scale data applications. Crucial in this context is the connection with the concept of statistical leverage. This concept has long been used in statistical regression diagnostics to identify outliers; and it has recently proved crucial in the development of improved worst-case matrix algorithms that are also amenable to high-quality numerical implementation and that are useful to domain scientists. Randomized methods solve problems such as the linear least-squares problem and the low-rank matrix approximation problem by constructing and operating on a randomized sketch of the input matrix. Depending on the specifics of the situation, when compared with the best previously-existing deterministic algorithms, the resulting randomized algorithms have worst-case running time that is asymptotically faster; their numerical implementations are faster in terms of clock-time; or they can be implemented in parallel computing environments where existing numerical algorithms fail to run at all. Numerous examples illustrating these observations will be described in detail.

Randomized algorithms for matrices and data

Randomized algorithms for very large matrix problems have received a great deal of attention in recent years. Much of this work was motivated by problems in large-scale data analysis, largely since matrices are popular structures with which to model data drawn from a wide range of application domains, and this work was performed by individuals from many different research communities. While the most obvious benefit of randomization is that it can lead to faster algorithms, either in worst-case asymptotic theory and/or numerical implementation, there are numerous other benefits that are at least as important. For example, the use of randomization can lead to simpler algorithms that are easier to analyze or reason about when applied in counterintuitive settings; it can lead to algorithms with more interpretable output, which is of interest in applications where analyst time rather than just computational time is of interest; it can lead implicitly to regularization and more robust output; and randomized algorithms can often be organized to exploit modern computational architectures better than classical numerical methods.

This monograph will provide a detailed overview of recent work on the theory of randomized matrix algorithms as well as the application of those ideas to the solution of practical problems in large-scale data analysis. Throughout this review, an emphasis will be placed on a few simple core ideas that underlie not only recent theoretical advances but also the usefulness of these tools in large-scale data applications. Crucial in this context is the connection with the concept of statistical leverage. This concept has long been used in statistical regression diagnostics to identify outliers; and it has recently proved crucial in the development of improved worst-case matrix algorithms that are also amenable to high-quality numerical implementation and that are useful to domain scientists. This connection arises naturally when one explicitly decouples the effect of randomization in these matrix algorithms from the underlying linear algebraic structure. This decoupling also permits much finer control in the application of randomization, as well as the easier exploitation of domain knowledge.

Most of the review will focus on random sampling algorithms and random projection algorithms for versions of the linear least-squares problem and the low-rank matrix approximation problem. These two problems are fundamental in theory and ubiquitous in practice. Randomized methods solve these problems by constructing and operating on a randomized sketch of the input matrix A — for random sampling methods, the sketch consists of a small number of carefully-sampled and rescaled columns/rows of A, while for random projection methods, the sketch consists of a small number of linear combinations of the columns/rows of A. Depending on the specifics of the situation, when compared with the best previously-existing deterministic algorithms, the resulting randomized algorithms have worst-case running time that is asymptotically faster; their numerical implementations are faster in terms of clock-time; or they can be implemented in parallel computing environments where existing numerical algorithms fail to run at all. Numerous examples illustrating these observations will be described in detail.

/pdf/randomized-algorithms-for-matrices-and-data-48ivrhw4vw.pdf

Randomized Algorithms for Matrices and Data

Inspired by theories of sparse representation and compressed sensing, this paper presents a simple, novel, yet very powerful approach for texture classification based on random projection, suitable for large texture database applications. At the feature extraction stage, a small set of random features is extracted from local image patches. The random features are embedded into a bag--of-words model to perform texture classification; thus, learning and classification are carried out in a compressed domain. The proposed unconventional random feature extraction is simple, yet by leveraging the sparse nature of texture images, our approach outperforms traditional feature extraction methods which involve careful design and complex steps. We have conducted extensive experiments on each of the CUReT, the Brodatz, and the MSRC databases, comparing the proposed approach to four state-of-the-art texture classification methods: Patch, Patch-MRF, MR8, and LBP. We show that our approach leads to significant improvements in classification accuracy and reductions in feature dimensionality.

Texture Classification from Random Features

Sparse representation has shown an attractive performance in a number of applications. However, the available sparse representation methods still suffer from some problems, and it is necessary to design more efficient methods. Particularly, to design a computationally inexpensive, easily solvable, and robust sparse representation method is a significant task. In this paper, we explore the issue of designing the simple, robust, and powerfully efficient sparse representation methods for image classification. The contributions of this paper are as follows. First, a novel discriminative sparse representation method is proposed and its noticeable performance in image classification is demonstrated by the experimental results. More importantly, the proposed method outperforms the existing state-of-the-art sparse representation methods. Second, the proposed method is not only very computationally efficient but also has an intuitive and easily understandable idea. It exploits a simple algorithm to obtain a closed-form solution and discriminative representation of the test sample. Third, the feasibility, computational efficiency, and remarkable classification accuracy of the proposed $l_{2}$ regularization-based representation are comprehensively shown by extensive experiments and analysis. The code of the proposed method is available at http://www.yongxu.org/lunwen.html .

A New Discriminative Sparse Representation Method for Robust Face Recognition via $l_{2}$ Regularization

There has been a strong trend lately in face processing research away from geometric models towards appearance models. Appearance-based methods employ dimensionality reduction to represent faces more compactly in a low-dimensional subspace which is found by optimizing certain criteria. The most popular appearance-based method is the method of eigenfaces that uses Principal Component Analysis (PCA) to represent faces in a low-dimensional subspace spanned by the eigenvectors of the covariance matrix of the data corresponding to the largest eigenvalues (i.e., directions of maximum variance). Recently, Random Projection (RP) has emerged as a powerful method for
dimensionality reduction. It represents a computationally simple and efficient method that preserves the structure of the data without introducing significant distortion. Despite its simplicity, RP has promising theoretical properties that make it an attractive tool for dimensionality reduction. Our focus in this paper is on investigating the feasibility of RP for face recognition. In this context, we have performed a large number of experiments using three popular face databases and comparisons using PCA. Our experimental results illustrate that although RP represents faces in a random, low-dimensional subspace, its overall performance is comparable to that of PCA while having lower computational requirements and being data independent.

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Navin Goel

Papers

Face recognition experiments with random projection