The Decompositional Approach to Matrix Computation

Total Page:16

File Type:pdf, Size:1020Kb

The Decompositional Approach to Matrix Computation the Top T HEME ARTICLE THE DECOMPOSITIONAL APPROACH TO MATRIX COMPUTATION The introduction of matrix decomposition into numerical linear algebra revolutionized matrix computations. This article outlines the decompositional approach, comments on its history, and surveys the six most widely used decompositions. n 1951, Paul S. Dwyer published Linear used scalar equations whereas Householder used Computations, perhaps the first book de- partitioned matrices. But a deeper difference is voted entirely to numerical linear algebra.1 that while Dwyer started from a system of equa- Digital computing was in its infancy, and tions, Householder worked with a (block) LU IDwyer focused on computation with mechani- decomposition—the factorization of a matrix cal calculators. Nonetheless, the book was state into the product of lower and upper triangular of the art. Figure 1 reproduces a page of the matrices. book dealing with Gaussian elimination. In Generally speaking, a decomposition is a fac- 1954, Alston S. Householder published Princi- torization of a matrix into simpler factors. The ples of Numerical Analysis,2 one of the first mod- underlying principle of the decompositional ap- ern treatments of high-speed digital computa- proach to matrix computation is that it is not the tion. Figure 2 reproduces a page from this book, business of the matrix algorithmists to solve par- also dealing with Gaussian elimination. ticular problems but to construct computational The contrast between these two pages is strik- platforms from which a variety of problems can ing. The most obvious difference is that Dwyer be solved. This approach, which was in full swing by the mid-1960s, has revolutionized ma- trix computation. To illustrate the nature of the decompositional 1521-9615/00/$10.00 © 2000 IEEE approach and its consequences, I begin with a discussion of the Cholesky decomposition and G.W. STEWART the solution of linear systems—essentially the University of Maryland decomposition that Gauss computed in his elim- 50 COMPUTING IN SCIENCE & ENGINEERING Figure 1. This page from Linear Computations Figure 2. On this page from Principles of Numerical shows that Paul Dwyer’s approach begins with a Analysis, Alston Householder uses partitioned system of scalar equations. Courtesy of John Wiley matrices and LU decomposition. Courtesy of Mc- & Sons. Graw-Hill. ination algorithm. This article also provides a A can be factored in the form tour of the five other major matrix decomposi- tions, including the pivoted LU decomposition, A = RTR (2) the QR decomposition, the spectral decompo- sition, the Schur decomposition, and the singu- where R is an upper triangular matrix. The fac- lar value decomposition. torization is called the Cholesky decomposition of A disclaimer is in order. This article deals pri- A. marily with dense matrix computations. Al- The factorization in Equation 2 can be used though the decompositional approach has greatly to solve linear systems as follows. If we write the − influenced iterative and direct methods for sparse system in the form RTRx = b and set y = R Tb, matrices, the ways in which it has affected them then x is the solution of the triangular system are different from what I describe here. Rx = y. However, by definition y is the solution of the system RTy = b. Consequently, we have re- duced the problem to the solution of two trian- The Cholesky decomposition and gular systems, as illustrated in the following al- linear systems gorithm: We can use the decompositional approach to solve the system 1. Solve the system RTy = b. 2. Solve the system Rx = y. (3) Ax = b (1) Because triangular systems are easy to solve, the where A is positive definite. It is well known that introduction of the Cholesky decomposition has JANUARY/FEBRUARY 2000 51 used to solve a system at one point in a compu- tation, you can reuse the decomposition to do the same thing later without having to recom- pute it. Historically, Gaussian elimination and its variants (including Cholesky’s algorithm) have solved the system in Equation 1 by reducing it to an equivalent triangular system. This mixes the computation of the decomposition with the solution of the first triangular system in Equa- tion 3, and it is not obvious how to reuse the elimination when a new right-hand side presents itself. A naive programmer is in danger of per- forming the reduction from the beginning, thus repeating the lion’s share of the work. On the other hand, a program that knows a decomposi- tion is in the background can reuse it as needed. (By the way, the problem of recomputing de- compositions has not gone away. Some matrix packages hide the fact that they repeatedly com- pute a decomposition by providing drivers to solve linear systems with a call to a single rou- tine. If the program calls the routine again with the same matrix, it recomputes the decomposi- tion—unnecessarily. Interpretive matrix systems such as Matlab and Mathematica have the same problemthey hide decompositions behind op- Figure 3. These varieties of Gaussian elimination are all numerically erators and function calls. Such are the conse- equivalent. quences of not stressing the decompositional ap- proach to the consumers of matrix algorithms.) Another advantage of working with decompo- transformed our problem into one for which the sitions is unity. There are different ways of or- solution can be readily computed. ganizing the operations involved in solving lin- We can use such decompositions to solve ear systems by Gaussian elimination in general more than one problem. For example, the fol- and Cholesky’s algorithm in particular. Figure 3 lowing algorithm solves the system ATx = b: illustrates some of these arrangements: a white area contains elements from the original matrix, 1. Solve the system Ry = b. a dark area contains the factors, a light gray 2. Solve the system RTx = y. (4) area contains partially processed elements, and the boundary strips contain elements about to Again, in many statistical applications we want be processed. Most of these variants were origi- − to compute the quantity ρ = xTA 1x. Because nally presented in scalar form as new algorithms. Once you recognize that a decomposition is in- − − − − xTA 1x = xT(RTR) 1x = (R Tx)T(R Tx) (5) volved, it is easy to see the essential unity of the various algorithms. we can compute ρ as follows: All the variants in Figure 3 are numerically equivalent. This means that one rounding-error 1. Solve the system RTy = x. analysis serves all. For example, the Cholesky al- 2. ρ = yTy. (6) gorithm, in whatever guise, is backward stable: the computed factor R satisfies The decompositional approach can also save computation. For example, the Cholesky de- (A + E) = RTR (7) composition requires O(n3) operations to com- pute, whereas the solution of triangular systems where E is of the size of the rounding unit rela- requires only O(n2) operations. Thus, if you rec- tive to A. Establishing this backward is usually ognize that a Cholesky decomposition is being the most difficult part of an analysis of the use 52 COMPUTING IN SCIENCE & ENGINEERING of a decomposition to solve a problem. For ex- ample, once Equation 7 has been established, the Benefits of the decompositional rounding errors involved in the solutions of the approach triangular systems in Equation 3 can be incor- porated in E with relative ease. Thus, another • A matrix decomposition solves not one but many problems. advantage of the decompositional approach is • A matrix decomposition, which is generally expensive to that it concentrates the most difficult aspects of compute, can be reused to solve new problems involving the rounding-error analysis in one place. original matrix. In general, if you change the elements of a • The decompositional approach often shows that apparently positive definite matrix, you must recompute its different algorithms are actually computing the same object. Cholesky decomposition from scratch. However, • The decompositional approach facilitates rounding-error if the change is structured, it may be possible to analysis. compute the new decomposition directly from • Many matrix decompositions can be updated, sometimes the old—a process known as updating. For ex- with great savings in computation. ample, you can compute the Cholesky decom- • By focusing on a few decompositions instead of a host of position of A + xxT from that of A in O(n2) op- specific problems, software developers have been able to erations, an enormous savings over the ab initio produce highly effective matrix packages. computation of the decomposition. Finally, the decompositional approach has greatly affected the development of software for matrix computation. Instead of wasting energy the reduction of a quadratic form ϕ(x) = xTAx developing code for a variety of specific applica- (I am simplifying a little here). In terms of the tions, the producers of a matrix package can con- Cholesky factorization A = RTR, Gauss wrote centrate on the decompositions themselves, per- ϕ(x) in the form haps with a few auxiliary routines to handle the 2 2 2 most important applications. This approach has ϕ = T + T + + T (x) (r1 x) (r2 x) K (rn x) informed the major public-domain packages: the ≡ ρ2 + ρ2 + + ρ2 Handbook series,3 Eispack,4 Linpack,5 and La- 1 (x) 2(x) K n(x) (8) pack.6 A consequence of this emphasis on de- T compositions is that software developers have where ri is the ith row of R. Thus Gauss re- found that most algorithms have broad compu- duced ϕ(x) to a sum of squares of linear functions ρ tational features in common—features than can i. Because R is upper triangular, the function ρ be relegated to basic linear-algebra subprograms i(x) depends only on the components xi,…xn (such as Blas), which can then be optimized for of x.
Recommended publications
  • Conjugate Gradient Method (Part 4)  Pre-Conditioning  Nonlinear Conjugate Gradient Method
    18-660: Numerical Methods for Engineering Design and Optimization Xin Li Department of ECE Carnegie Mellon University Pittsburgh, PA 15213 Slide 1 Overview Conjugate Gradient Method (Part 4) Pre-conditioning Nonlinear conjugate gradient method Slide 2 Conjugate Gradient Method Step 1: start from an initial guess X(0), and set k = 0 Step 2: calculate ( ) ( ) ( ) D 0 = R 0 = B − AX 0 Step 3: update solution D(k )T R(k ) X (k +1) = X (k ) + µ (k )D(k ) where µ (k ) = D(k )T AD(k ) Step 4: calculate residual ( + ) ( ) ( ) ( ) R k 1 = R k − µ k AD k Step 5: determine search direction R(k +1)T R(k +1) (k +1) = (k +1) + β (k ) β = D R k +1,k D where k +1,k (k )T (k ) D R Step 6: set k = k + 1 and go to Step 3 Slide 3 Convergence Rate k ( + ) κ(A) −1 ( ) X k 1 − X ≤ ⋅ X 0 − X κ(A) +1 Conjugate gradient method has slow convergence if κ(A) is large I.e., AX = B is ill-conditioned In this case, we want to improve convergence rate by pre- conditioning Slide 4 Pre-Conditioning Key idea Convert AX = B to another equivalent equation ÃX̃ = B̃ Solve ÃX̃ = B̃ by conjugate gradient method Important constraints to construct ÃX̃ = B̃ à is symmetric and positive definite – so that we can solve it by conjugate gradient method à has a small condition number – so that we can achieve fast convergence Slide 5 Pre-Conditioning AX = B −1 −1 L A⋅ X = L B L−1 AL−T ⋅ LT X = L−1B A X B ̃ ̃ ̃ L−1AL−T is symmetric and positive definite, if A is symmetric and positive definite T (L−1 AL−T ) = L−1 AL−T T X T L−1 AL−T X = (L−T X ) ⋅ A⋅(L−T X )> 0 Slide
    [Show full text]
  • Hybrid Algorithms for Efficient Cholesky Decomposition And
    Hybrid Algorithms for Efficient Cholesky Decomposition and Matrix Inverse using Multicore CPUs with GPU Accelerators Gary Macindoe A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of UCL. 2013 ii I, Gary Macindoe, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Signature : Abstract The use of linear algebra routines is fundamental to many areas of computational science, yet their implementation in software still forms the main computational bottleneck in many widely used algorithms. In machine learning and computational statistics, for example, the use of Gaussian distributions is ubiquitous, and routines for calculating the Cholesky decomposition, matrix inverse and matrix determinant must often be called many thousands of times for com- mon algorithms, such as Markov chain Monte Carlo. These linear algebra routines consume most of the total computational time of a wide range of statistical methods, and any improve- ments in this area will therefore greatly increase the overall efficiency of algorithms used in many scientific application areas. The importance of linear algebra algorithms is clear from the substantial effort that has been invested over the last 25 years in producing low-level software libraries such as LAPACK, which generally optimise these linear algebra routines by breaking up a large problem into smaller problems that may be computed independently. The performance of such libraries is however strongly dependent on the specific hardware available. LAPACK was originally de- veloped for single core processors with a memory hierarchy, whereas modern day computers often consist of mixed architectures, with large numbers of parallel cores and graphics process- ing units (GPU) being used alongside traditional CPUs.
    [Show full text]
  • Geometric Polarimetry − Part I: Spinors and Wave States
    1 Geometric Polarimetry Part I: Spinors and − Wave States David Bebbington, Laura Carrea, and Ernst Krogager, Member, IEEE Abstract A new approach to polarization algebra is introduced. It exploits the geometric properties of spinors in order to represent wave states consistently in arbitrary directions in three dimensional space. In this first expository paper of an intended series the basic derivation of the spinorial wave state is seen to be geometrically related to the electromagnetic field tensor in the spatio-temporal Fourier domain. Extracting the polarization state from the electromagnetic field requires the introduction of a new element, which enters linearly into the defining relation. We call this element the phase flag and it is this that keeps track of the polarization reference when the coordinate system is changed and provides a phase origin for both wave components. In this way we are able to identify the sphere of three dimensional unit wave vectors with the Poincar´esphere. Index Terms state of polarization, geometry, covariant and contravariant spinors and tensors, bivectors, phase flag, Poincar´esphere. I. INTRODUCTION arXiv:0804.0745v1 [physics.optics] 4 Apr 2008 The development of applications in radar polarimetry has been vigorous over the past decade, and is anticipated to continue with ambitious new spaceborne remote sensing missions (for example TerraSAR-X [1] and TanDEM-X [2]). As technical capabilities increase, and new application areas open up, innovations in data analysis often result. Because polarization data are This work was supported by the Marie Curie Research Training Network AMPER (Contract number HPRN-CT-2002-00205). D. Bebbington and L.
    [Show full text]
  • Matrix Inversion Using Cholesky Decomposition
    Matrix Inversion Using Cholesky Decomposition Aravindh Krishnamoorthy, Deepak Menon ST-Ericsson India Private Limited, Bangalore [email protected], [email protected] Abstract—In this paper we present a method for matrix inversion Upper triangular elements, i.e. : based on Cholesky decomposition with reduced number of operations by avoiding computation of intermediate results; further, we use fixed point simulations to compare the numerical ( ∑ ) accuracy of the method. … (4) Keywords-matrix, inversion, Cholesky, LDL. Note that since older values of aii aren’t required for computing newer elements, they may be overwritten by the I. INTRODUCTION value of rii, hence, the algorithm may be performed in-place Matrix inversion techniques based on Cholesky using the same memory for matrices A and R. decomposition and the related LDL decomposition are efficient Cholesky decomposition is of order and requires techniques widely used for inversion of positive- operations. Matrix inversion based on Cholesky definite/symmetric matrices across multiple fields. decomposition is numerically stable for well conditioned Existing matrix inversion algorithms based on Cholesky matrices. decomposition use either equation solving [3] or triangular matrix operations [4] with most efficient implementation If , with is the linear system with requiring operations. variables, and satisfies the requirement for Cholesky decomposition, we can rewrite the linear system as In this paper we propose an inversion algorithm which … (5) reduces the number of operations by 16-17% compared to the existing algorithms by avoiding computation of some known By letting , we have intermediate results. … (6) In section 2 of this paper we review the Cholesky and LDL decomposition techniques, and discuss solutions to linear and systems based on them.
    [Show full text]
  • AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 3: Positive-Definite Systems; Cholesky Factorization
    AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 3: Positive-Definite Systems; Cholesky Factorization Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 11 Symmetric Positive-Definite Matrices n×n Symmetric matrix A 2 R is symmetric positive definite (SPD) if T n x Ax > 0 for x 2 R nf0g n×n Hermitian matrix A 2 C is Hermitian positive definite (HPD) if ∗ n x Ax > 0 for x 2 C nf0g SPD matrices have positive real eigenvalues and orthogonal eigenvectors Note: Most textbooks only talk about SPD or HPD matrices, but a positive-definite matrix does not need to be symmetric or Hermitian! A real matrix A is positive definite iff A + AT is SPD T n If x Ax ≥ 0 for x 2 R nf0g, then A is said to be positive semidefinite Xiangmin Jiao Numerical Analysis I 2 / 11 Properties of Symmetric Positive-Definite Matrices SPD matrix often arises as Hessian matrix of some convex functional I E.g., least squares problems; partial differential equations If A is SPD, then A is nonsingular Let X be any n × m matrix with full rank and n ≥ m. Then T I X X is symmetric positive definite, and T I XX is symmetric positive semidefinite n×m If A is n × n SPD and X 2 R has full rank and n ≥ m, then X T AX is SPD Any principal submatrix (picking some rows and corresponding columns) of A is SPD and aii > 0 Xiangmin Jiao Numerical Analysis I 3 / 11 Cholesky Factorization If A is symmetric positive definite, then there is factorization of A A = RT R where R is upper triangular, and all its diagonal entries are positive Note: Textbook calls is “Cholesky
    [Show full text]
  • A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices
    Appl. Math. Inf. Sci. 9, No. 2L, 319-324 (2015) 319 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/092L06 A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices Waldemar Hołubowski1, Dariusz Kurzyk1,∗ and Tomasz Trawi´nski2 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, Gliwice 44–100, Poland 2 Mechatronics Division, Silesian University of Technology, Akademicka 10a, Gliwice 44–100, Poland Received: 6 Jul. 2014, Revised: 7 Oct. 2014, Accepted: 8 Oct. 2014 Published online: 1 Apr. 2015 Abstract: We propose an effective method to find the inverse of symmetric block arrowhead matrices which often appear in areas of applied science and engineering such as head-positioning systems of hard disk drives or kinematic chains of industrial robots. Block arrowhead matrices can be considered as generalisation of arrowhead matrices occurring in physical problems and engineering. The proposed method is based on LDLT decomposition and we show that the inversion of the large block arrowhead matrices can be more effective when one uses our method. Numerical results are presented in the examples. Keywords: matrix inversion, block arrowhead matrices, LDLT decomposition, mechatronic systems 1 Introduction thermal and many others. Considered subsystems are highly differentiated, hence formulation of uniform and simple mathematical model describing their static and A square matrix which has entries equal zero except for dynamic states becomes problematic. The process of its main diagonal, a one row and a column, is called the preparing a proper mathematical model is often based on arrowhead matrix. Wide area of applications causes that the formulation of the equations associated with this type of matrices is popular subject of research related Lagrangian formalism [9], which is a convenient way to with mathematics, physics or engineering, such as describe the equations of mechanical, electromechanical computing spectral decomposition [1], solving inverse and other components.
    [Show full text]
  • 1. Positive Definite Matrices a Matrix a Is Positive Definite If X>Ax > 0 for All Nonzero X
    CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8 GENE H. GOLUB 1. Positive Definite Matrices A matrix A is positive definite if x>Ax > 0 for all nonzero x. A positive definite matrix has real and positive eigenvalues, and its leading principal submatrices all have positive determinants. From the definition, it is easy to see that all diagonal elements are positive. To solve the system Ax = b where A is positive definite, we can compute the Cholesky decom- position A = F >F where F is upper triangular. This decomposition exists if and only if A is symmetric and positive definite. In fact, attempting to compute the Cholesky decomposition of A is an efficient method for checking whether A is symmetric positive definite. It is important to distinguish the Cholesky decomposition from the square root factorization.A square root of a matrix A is defined as a matrix S such that S2 = SS = A. Note that the matrix F in A = F >F is not the square root of A, since it does not hold that F 2 = A unless A is a diagonal matrix. The square root of a symmetric positive definite A can be computed by using the fact that A has an eigendecomposition A = UΛU > where Λ is a diagonal matrix whose diagonal elements are the positive eigenvalues of A and U is an orthogonal matrix whose columns are the eigenvectors of A. It follows that A = UΛU > = (UΛ1/2U >)(UΛ1/2U >) = SS and so S = UΛ1/2U > is a square root of A. 2. The Cholesky Decomposition The Cholesky decomposition can be computed directly from the matrix equation A = F >F .
    [Show full text]
  • Cholesky Decomposition 1 Cholesky Decomposition
    Cholesky decomposition 1 Cholesky decomposition In linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a symmetric, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices and is an example of a square root of a matrix. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.[1] Statement If A has real entries and is symmetric (or more generally, is Hermitian) and positive definite, then A can be decomposed as where L is a lower triangular matrix with strictly positive diagonal entries, and L* denotes the conjugate transpose of L. This is the Cholesky decomposition. The Cholesky decomposition is unique: given a Hermitian, positive-definite matrix A, there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL*. The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite. The requirement that L have strictly positive diagonal entries can be dropped to extend the factorization to the positive-semidefinite case. The statement then reads: a square matrix A has a Cholesky decomposition if and only if A is Hermitian and positive semi-definite. Cholesky factorizations for positive-semidefinite matrices are not unique in general. In the special case that A is a symmetric positive-definite matrix with real entries, L can be assumed to have real entries as well.
    [Show full text]
  • Structured Factorizations in Scalar Product Spaces
    STRUCTURED FACTORIZATIONS IN SCALAR PRODUCT SPACES D. STEVEN MACKEY¤, NILOUFER MACKEYy , AND FRANC»OISE TISSEURz Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors and singular values that persists across a wide range of scalar products. A key feature of our analysis is the identi¯cation of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of di®erent characterizations of these scalar product classes is given. Key words. automorphism group, Lie group, Lie algebra, Jordan algebra, bilinear form, sesquilinear form, scalar product, inde¯nite inner product, orthosymmetric, adjoint, factorization, symplectic, Hamiltonian, pseudo-orthogonal, polar decomposition, matrix sign function, matrix square root, generalized polar decomposition, eigenvalues, eigenvectors, singular values, structure preservation. AMS subject classi¯cations. 15A18, 15A21, 15A23, 15A57,
    [Show full text]
  • The CMA Evolution Strategy: a Tutorial
    The CMA Evolution Strategy: A Tutorial Nikolaus Hansen June 28, 2011 Contents Nomenclature3 0 Preliminaries4 0.1 Eigendecomposition of a Positive Definite Matrix...............5 0.2 The Multivariate Normal Distribution.....................6 0.3 Randomized Black Box Optimization.....................7 0.4 Hessian and Covariance Matrices........................8 1 Basic Equation: Sampling8 2 Selection and Recombination: Moving the Mean9 3 Adapting the Covariance Matrix 10 3.1 Estimating the Covariance Matrix From Scratch................ 10 3.2 Rank-µ-Update................................. 12 3.3 Rank-One-Update................................ 14 3.3.1 A Different Viewpoint......................... 14 3.3.2 Cumulation: Utilizing the Evolution Path............... 16 3.4 Combining Rank-µ-Update and Cumulation.................. 17 4 Step-Size Control 17 5 Discussion 21 A Algorithm Summary: The CMA-ES 25 B Implementational Concerns 27 B.1 Multivariate normal distribution........................ 28 B.2 Strategy internal numerical effort........................ 28 B.3 Termination criteria............................... 28 B.4 Flat fitness.................................... 29 B.5 Boundaries and Constraints........................... 29 C MATLAB Source Code 31 1 D Reformulation of Learning Parameter ccov 33 2 Nomenclature We adopt the usual vector notation, where bold letters, v, are column vectors, capital bold letters, A, are matrices, and a transpose is denoted by vT. A list of used abbreviations and symbols is given in alphabetical order. Abbreviations CMA Covariance Matrix Adaptation EMNA Estimation of Multivariate Normal Algorithm ES Evolution Strategy (µ/µfI;Wg; λ)-ES, Evolution Strategy with µ parents, with recombination of all µ parents, either Intermediate or Weighted, and λ offspring. RHS Right Hand Side. Greek symbols λ ≥ 2, population size, sample size, number of offspring, see (5).
    [Show full text]
  • Linear Algebra - Part II Projection, Eigendecomposition, SVD
    Linear Algebra - Part II Projection, Eigendecomposition, SVD (Adapted from Punit Shah's slides) 2019 Linear Algebra, Part II 2019 1 / 22 Brief Review from Part 1 Matrix Multiplication is a linear tranformation. Symmetric Matrix: A = AT Orthogonal Matrix: AT A = AAT = I and A−1 = AT L2 Norm: s X 2 jjxjj2 = xi i Linear Algebra, Part II 2019 2 / 22 Angle Between Vectors Dot product of two vectors can be written in terms of their L2 norms and the angle θ between them. T a b = jjajj2jjbjj2 cos(θ) Linear Algebra, Part II 2019 3 / 22 Cosine Similarity Cosine between two vectors is a measure of their similarity: a · b cos(θ) = jjajj jjbjj Orthogonal Vectors: Two vectors a and b are orthogonal to each other if a · b = 0. Linear Algebra, Part II 2019 4 / 22 Vector Projection ^ b Given two vectors a and b, let b = jjbjj be the unit vector in the direction of b. ^ Then a1 = a1 · b is the orthogonal projection of a onto a straight line parallel to b, where b a = jjajj cos(θ) = a · b^ = a · 1 jjbjj Image taken from wikipedia. Linear Algebra, Part II 2019 5 / 22 Diagonal Matrix Diagonal matrix has mostly zeros with non-zero entries only in the diagonal, e.g. identity matrix. A square diagonal matrix with diagonal elements given by entries of vector v is denoted: diag(v) Multiplying vector x by a diagonal matrix is efficient: diag(v)x = v x is the entrywise product. Inverting a square diagonal matrix is efficient: 1 1 diag(v)−1 = diag [ ;:::; ]T v1 vn Linear Algebra, Part II 2019 6 / 22 Determinant Determinant of a square matrix is a mapping to a scalar.
    [Show full text]
  • Reducing Dimensionality in Text Mining Using Conjugate Gradients and Hybrid Cholesky Decomposition
    (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 8, No. 7, 2017 Reducing Dimensionality in Text Mining using Conjugate Gradients and Hybrid Cholesky Decomposition Jasem M. Alostad The Public Authority for Applied Education and Training (PAAET), College of Basic Education P.O.Box 23167, Safat 13092, Kuwait Abstract—Generally, data mining in larger datasets consists The data reduction is the process of reducing the size or of certain limitations in identifying the relevant datasets for the dimensionality of the data, however, the representation of the given queries. The limitations include: lack of interaction in the data should be retained. Selection of instance is one better way required objective space, inability to handle the data sets or to reduce the data by reducing the total number of instances. In discrete variables in datasets, especially in the presence of spite of many efforts to deal with such instances, data mining missing variables and inability to classify the records as per the algorithm, however undergoes severe challenges due to non- given query, and finally poor generation of explicit knowledge for applicability of datasets with large instances. Hence, the a query increases the dimensionality of the data. Hence, this computational complexity of the system increases with larger paper aims at resolving the problems with increasing data instances [3], [4] and leads to problems in scaling, increased dimensionality in datasets using modified non-integer matrix storage requirements and clustering accuracy. The other factorization (NMF). Further, the increased dimensionality arising due to non-orthogonally of NMF is resolved with problems associated with larger data instances include, Cholesky decomposition (cdNMF).
    [Show full text]