The Numbers Σ J. Are Called The Singular Values Of Z

The numbers ÏÆ'j are called the singular values of Z. They are organised in weakly decreasing order: ÏÆ'1 ≠¥ ÏÆ'2 ≠¥ †¢ †¢ †¢ ≠¥ ÏÆ'k ≠¥ 0. The columns of X and Y are called left singular vectors and right singular vectors, respectively. Singular values are connected with the approximability of matrices. For each j, the number ÏÆ'j+1 equals the spectral-norm [5] discrepancy between Z and an optimal rank-j approximation. That is, ÏÆ'j+1 = min{kZ − Bk : B has rank j}. 3.2 Randomized Singular Value Decomposition (RSVD): As previously described the singular value decomposition (SVD) of a matrix Z∈RmÃâ€"n is defined as , Z=X ∑YT,†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦(1) Where X and Y are orthonormal, and ∑ is a rectangular diagonal matrix whose diagonal entries are the singular values signified as ÏÆ'i. The column vectors of X and Y are left and right singular vectors, respectively, symbolized as xi and yi. Define the truncated SVD(TSVD) approximation [7] of Z as a matrix Zk such that †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.(2) The randomized SVD (rSVD) of A can be defined as †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.(3) Where and are both orthonormal and is diagonal with diagonal entries symbolized as . Designate the column vectors [8] of and as and and correspondingly. Elucidate the residual matrix of a TSVD approximation as follows Rk = Z - Zk†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.(4) and the residual matrix of a rSVD approximation as follows: †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.(5) Elucidate the random projection of a matrix as follows: †¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦.(6) where is a random matrix with independent and identically distributed [9] entries.Show MoreRelatedMonte Carlo Simulation218872 Words   |  876 Pages1 summarizes the theoretical foundations of derivatives pricing and Monte Carlo. It explains the principles by which a pricing problem can be formulated as an integration problem to which Monte Carlo is then applicable. Chapter 2 discusses random number generation and methods for sampling from nonuniform distributions, tools fundamental to every application of Monte Carlo. Chapter 3 provides an overview of some of the most important models used in ï ¬ nancial engineering and discusses their implementationRead MoreQwertyui17452 Words   |  70 PagesKKT-system formed from the QP Hessian and the working-set constraint gradients. It is shown that, under certain circumstances, the solution of this KKT-system may be updated using a simple recurrence relation, thereby giving a signiï ¬ cant reduction in the number of KKT systems that need to be solved. Furthermore, the nonbinding-direction framework is applied to QP problems with constraints in standard form, and to the dual of a convex QP. The second part of the paper focuses on implementation issues. FirstRead More_x000C_Introduction to Statistics and Data Analysis355457 Words   |  1422 PagesMacintosh are registered trademarks of Apple Computer, Inc. Used herein under license. Library of Congress Control Number: 2006933904 Student Edition: ISBN-13: 978-0-495-11873-2 ISBN-10: 0-495-11873-7 ââ€"   To my nephews, Jesse and Luke Smidt, who bet I wouldn’t put their names in this book. R. P. ââ€"   To my wife, Sally, and my daughter, Anna C. O. ââ€"   To Carol, Allie, and Teri. J. D. ââ€"   About the Authors puter Teacher of the Year award in 1988 and received the Siemens Award for AdvancedRead MoreData Mining16277 Words   |  66 Pageswe have little or no control over the data gathering process, with data often being collected for some entirely different purpose. For example, customer transaction logs may be maintained from an auditing perspective and data mining would then be called upon to analyse the logs for estimating customer buying patterns. The second major difference (between temporal data mining and classical time series analysis) lies in the kind of information that we want to estimate or unearth from the data. TheRead MoreGame Theory and Economic Analyst83847 Words   |  336 Pageswho was himself co-author of a treatise on bridge. Nothing about this singular and rather marginal branch of mathematics would at this time have suggested its later encounter with economics.1 The analogy between economic activity and what goes on in casinos was only suggested much later, in a far diï ¬â‚¬erent economic environment than that which these two mathematicians would have been able to observe. One could say that J. Von Neumann was the person who both conferred a sense of scientiï ¬ c legitimacy