Randomized methods for matrix computations

The purpose of this text is to provide an accessible introduction to a set of recently developed algorithms for factorizing matrices. These new algorithms attain high practical speed by reducing the dimensionality of intermediate computations using randomized projections. The algorithms are particularly powerful for computing low-rank approximations to very large matrices, but they can also be used to accelerate algorithms for computing full factorizations of matrices. A key competitive advantage of the algorithms described is that they require less communication than traditional deterministic methods

Motion detection using randomized methods

The detection and recogni6on of a moving object in a sequence of time varying images proves to be a very important task in machine intelligence in general and computer vision in particular. Recently, parametric domain techniques have been successfully used with a number of variants. In such melhods, the image is transfonned into some parameter space and the motion detection process is applied in that space. A recent parametric domain is the Randomized Hough Transform (RHT) that uses random sampling mechanism in the image space, score accumulation in the parameter space, and bridge between them using a converging mapping. The use of such method for motion detection is called the Motion Detection Randomized Hough Transform (MDRHT). Since random sampling is used, the process of establishing correspondence between sets of points belonging to the same object in successive motion frames proves to be the most important problem in this methodology. Improving the accuracy of correspondence rules will improve the performance of the algorithm. In the present work, motion detection was considered through the analysis of a sequence of time varying gay level image stream using the RHT algorithm that provides an efficient simple non-model based methodology using edge pixels as features. The objective of our work was to construct a set of correspondence rules that would maximize the ability of the methodology to detect motion parameters for both pure translational and pure rotational motions restricted to 2-D rigid objects. Analysis of accuracy and efficiency of correspondence were restricted to the cases of two points and three point pairs to select rules maximizing the performance. For that purpose, five different correspondence rules were investigated. The first three were 2-point rules that were used in previous researches. They measure correspondence through 2-point x-and y-differences, City Block distances and Euclidean distances, respectively. The present work introduced the two remaining rules for the first time. These are 3-point rules that measure correspondences through 3-point City Dlock distances and triangular areas, respectively. We have developed a mathematical analysis of the invariance of the five rules given for both pure translational and pure rotational motions. The analysis proved that the five selected rules are invariant under pure translation while only rule (3) and rule (5) were invariant under pure rotation. In order to compare the performances of our randomized motion detection methodology for the different rules, a perfonnance parameter was introduced to measure the capability of peak detection in the RHT space. For translational motion, different simulation experiments were conducted with varying sizes of random samples. The results obtained for the translational motion indicated that our 3-point algorithms are in general superior to the previous 2-point algorithms. In particular, algorithm (5) that uses equal triangle areas gave the hjghest performance, outperforming the next in performance (2-point City Block distance) by a factor of almost 3 times. In order to study the affect of noise on the algorithms\u27 performance, a salt and pepper noise with different levels was added to the frames images. The results for translational motion showed that algorithm (5) again has the performance which is three times better than other algorithms and proved to be robust against noisy conditions. Different simulation experiments were also conducted for the case of pure rotational motion. The two algorithms that proved to be invariant under rotation (using correspondence rules 3 and 5) have been tested for different rotation angles using various numbers of trials NT. Also in this case, our 3-point algorithm proved to outperform the 2- point algorithm by a factor of almost 7 times. Similar conclusions are obtained for the robustness of algorithm (5) under varying angles of rotation, RHT spatial resolution, and correspondence tolerance

Faster Randomized Methods for Orthogonality Constrained Problems

Recent literature has advocated the use of randomized methods for accelerating the solution of various matrix problems arising throughout data science and computational science. One popular strategy for leveraging randomization is to use it as a way to reduce problem size. However, methods based on this strategy lack sufficient accuracy for some applications. Randomized preconditioning is another approach for leveraging randomization, which provides higher accuracy. The main challenge in using randomized preconditioning is the need for an underlying iterative method, thus randomized preconditioning so far have been applied almost exclusively to solving regression problems and linear systems. In this article, we show how to expand the application of randomized preconditioning to another important set of problems prevalent across data science: optimization problems with (generalized) orthogonality constraints. We demonstrate our approach, which is based on the framework of Riemannian optimization and Riemannian preconditioning, on the problem of computing the dominant canonical correlations and on the Fisher linear discriminant analysis problem. For both problems, we evaluate the effect of preconditioning on the computational costs and asymptotic convergence, and demonstrate empirically the utility of our approach