Extrinsic Antimean and Bootstrap
|
|
- Byron Bishop
- 5 years ago
- Views:
Transcription
1 Extrinsic Antimean and Bootstrap Yunfan Wang Department of Statistics Florida State University April 12, 2018 Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
2 Outline 1 Introduction Distance Fréchet Function Extrinsic Mean and Antimean 2 Kendall Shape Spaces Extrinsic antimean on Complex Space 3 Application Computational Example Results Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
3 Outline 1 Introduction Distance Fréchet Function Extrinsic Mean and Antimean 2 Kendall Shape Spaces Extrinsic antimean on Complex Space 3 Application Computational Example Results Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
4 Two Types of Distance In statistics one mainly considers two types of distances on a manifold M: 1 A geodesic distance (arc distance), that is the Riemmannian distance ρ g associated with a Riemannian structure g on M. 2 A chord distance, is the distance ρ j induced by the Euclidean distance on R N via an embedding j : M R N, that is given by ρ j (p, q) = j(q) j(p) 2 0 An intrinsic data analysis on a manifold is a statistical analysis of a probability measure, using a geodesic distance based statistics. An extrinsic data analysis is a statistical analysis based on a chord distance based statistics. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
5 Embedding There are infinitely many embeddings of RP m in an Euclidean space, when consider sample mean and sample antimean, it is preferred to use an embedding j that is compatible with two transitive group actions of SO(m + 1) on RP m, respectively on j(rp m ), that is j(t [x]) = T j([x]), T SO(m + 1), [x] RP m (1) where T [x] = [Tx] Such an embedding is said to be equivariant. The equivariant embedding of RP m that was used so far in the axial data analysis literature is the Veronese Whitney (VW) embedding j : RP m Sym + (m + 1, R), that associates to an axis the matrix of the orthogonal projection on this axis: j([x]) = xx T, x = 1 (2) Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
6 Fréchet Function Consider object spaces provided with a chord distance associated with the embedding of an object space into a numerical space, and the statistical analysis performed relative to a chord distance is termed extrinsic data analysis. The expected square distance from the random object X to an arbitrary point p defines what we call the Fréchet function associated with X : F(p) = E(ρ 2 (p, X)), (3) and its minimizers form the Fréchet mean set. In case when ρ is the chord distance on M induced by the Euclidean distance in R N via an embedding j : M R N, the Fréchet function becomes F(p) = j(x) j(p) 2 0Q(dx), (4) M where Q = P X is the probability measure on M, associated with X. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
7 Extrinsic Mean and Antimean In this case the Fréchet mean set is called the extrinsic mean set, and if we have a unique point in the extrinsic mean set of X, this point is the extrinsic mean of X, and is labeled µ E (X) or simply µ E. Also, given X 1,..., X n i.i.d.r.v. s from Q, their extrinsic sample mean (set) is the extrinsic mean (set) of the empirical distribution ˆQ n = 1 n ni=1 δ Xi. Introducing a new location parameter for X. Definition The set of maximizers of the Fréchet function, is called the extrinsic antimean set. In case the extrinsic antimean set has one point only, that point is called extrinsic antimean of X, and is labeled αµ j,e (Q), or simply αµ E, when j is known. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
8 Focal and non-focal Remark Let (M, ρ 0 ) be a compact metric space, where ρ 0 is the chord distance via the embedding j : M R N, that is ρ 0 (p 1, p 2 ) = j(p 1 ) j(p 2 ) = d 0 (j(p 1 ), j(p 2 )), (p 1, p 2 ) M 2 where d 0 is the Euclidean distance in R N. Recall that a point y R N for which there is a unique point p M satisfying the equality, d 0 (y, j(m)) = inf x M y j(x) 0 = d 0(y, j(p)) is called j-nonfocal. A point which is not j-nonfocal is said to be j-focal. And if y is a j-nonfocal point, its projection on j(m) is the unique point j(p) = P j (y) j(m) with d o (y, j(m)) = d 0 (y, j(p)). Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
9 Focal and non-focal cont Definition (a)a point y R N for which there is unique point p M satisfying the equality, sup y j(x) 0 = d 0 (y, j(p)) x M is called αj-nonfocal. A point which is not αj-nonfocal is said to be αj-focal. (b)if y is an αj-nonfocal point, its projection on j(m) is the unique point z = P F,J (y) j(m) with sup x M y j(x) 0 = d 0 (y, j(p)). Definition A probability distribution Q on M is said to be αj-nonfocal if the mean µ of j(q) is αjnonfocal. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
10 Extrinsic Mean The extrinsic mean µ j,e (Q) of a j-nonfocal probability measure Q on M w.r.t. an embedding j, when it exists,is given by µ j,e (Q) = j 1 (P F,j (µ)) where µ is the mean of j(q). For the VW embedding, F is the set of matrices in S + (m + 1, R), whose largest eigenvalues are of multiplicity at least 2. The projection P j assigns to each non negative definite symmetric matrix A with highest eigenvalue of multiplicity 1, the matrix vv T, where v is a unit eigenvector of A corresponding to its largest eigenvalue. Furthermore, the VW mean of a random object [X] RP m, [X T X] = 1, is given by µ j,e (Q) = [γ(m + 1)] and (λ(a), γ(a)), a = 1,.., m + 1 are eigenvalues and unit eigenvectors pairs. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
11 Extrinsic Antimean PROPOSITION (i) The set of αvw -nonfocal points in Sym + (m + 1, R), is the set of matrices in Sym + (m + 1, R) whose smallest eigenvalue has multiplicity 1. (ii) The projection P F,j : (αf ) c j(rp m ) assigns to each nonnegative definite symmetric matrix A, of rank 1, with a smallest eigenvalue of multiplicity 1, the matrix j([ν]), where ν = 1 and ν is an eigenvector of A corresponding to that eigenvalue. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
12 Extrinsic Antimean - con t PROPOSITION Let Q be a distribution on RP m. The VW-antimean set of a random object [X], X T X = 1 on RP m, is the set of points p = [v] V 1, where V 1 is the eigenspace corresponding to the smallest eigenvalue λ(1) of E(XX T ). If in addition Q = P [X] is αvw -nonfocal, then µ j,e (Q) = j 1 (P F,j (µ)) = γ(1) where (λ(a), γ(a)), a = 1,.., m + 1 are eigenvalues in increasing order and the corresponding unit eigenvectors of µ = E(XX T ). Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
13 Extrinsic Antimean - con t PROPOSITION Let Q be a distribution on RP m. Let x 1,..., x n be random observations from a distribution Q on RP m, such that j(x) is αvw-nonfocal. Then the VW sample antimean of x 1,..., x n is given by; ax j,e = j 1 (P F,j (j(x))) = g(1) where (d(a), g(a)) are the eigenvalues in increasing order and the corresponding unit eigenvectors of J = 1 n x i xi T. n i=1 Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
14 Outline 1 Introduction Distance Fréchet Function Extrinsic Mean and Antimean 2 Kendall Shape Spaces Extrinsic antimean on Complex Space 3 Application Computational Example Results Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
15 Kendall Shape Spaces We are concerned with a landmark based nonparametric analysis of similarity shape data. For landmark based shape data, one considers a k-ad x = (x 1,..., x k (R m ) k, which consists of k labeled points in R m that represents coordinates of landmarks. G is the group direct similarities of R m. A similarity is a function f : R m R m, that uniformly dilates distance, that is, for which there is K > 0, such that f (x) f (y) = k x y, x, y R m. A direct similarity is given by f (x) = Ax + b, A T A = ci m, c > 0, where A has a positive determinant. Direct similarities form under composition, the group of direct similarities. The sample spaces considered here are called them shape spaces of k-ads in R m. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
16 Kendall Shape Spaces - cont Direct Similarity (Kendall) Shape: Geometrical information that remains when location, scale and rotational effects are filtered out from a k-ads. Two k-ads (z 1, z 2,..., z k ) and (z 1, z 2,..., z k ) are said to have the same shape if there is a direct similarity T in the plane, that is, a composition of a rotation, a translation and a homothety such that T (z j ) = z j for j = 1,..., k. Having the same shape is an equivalence relationship in the spaces of planar k-ads, and the set of all equivalence classes of k-ads is called the planar shape spaces of k-ads, or the space Σ k 2. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
17 Kendall Shape Spaces - cont Without loss of generality one may assume that two k-ads that have the same shape also have the same center of mass, that is, Σz j = Σz j = 0, and they have the same shape if there is a composition of a transformation T which keeps the origin fixed, and is a rotation followed by a homothety such that T (z j ) = z j for j = 1,..., k 1. Such a transformation T is determined by a nonzero complex number, that is to say, the two k-ads with center of mass 0 have the same shape if there is a z C\{0} such that zz j = z j for j = 1,..., k 1. Thus the shape equivalence class of a planar k-ad is uniquely determined by a point in CP k 2, that is to say, Σ k 2 is identified with CP k 2. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
18 Embedding The Procrustean distance, in our terminology, is the distance induced by the Euclidean distance on CP k 2 via a quadratic Veronese-Whitney embedding into a unit sphere of the linear space S(k 1, C of selfadjoint complex matrices of order k 1. In order to define j : CP k 2 S(k 1, C) it is useful to note that CP k 2 = S 2k 3 /S 1, where S 2k 3 is the space of complex vectors C k 1 of norm 1, and the equivalence relation on S 2k 3 is by multiplication with scalars in S 1 (complex numbers of modulus 1). If z = (z 1, z 2,..., z k 1 ) is in S 2k 3, we will denote by [z] the equivalence class of z in CP k 2. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
19 Embedding - cont The Veronese-Whitney (or simply Veronese) map is in this case j([z]) = zz where, if z is considered as a column vector, z is the adjoint of z, that is, the conjugate of the transpose of z. The Euclidean distance in the space of Hermitian matrices S(k 1, C) is d 2 0 (A, B) = Tr((A B) (A B) ) = Tr((A B) 2 ). Definition Let L k = {(Z 1,..., Z k ) C k Z Z k = 0} Theorem The Kendall planar shape space Σ k 2 can be identified with the complex projective space P(L k ). Moreover, since L k has complex dimension k 1, P(L k ) P(C k 1 ) = CP k 2, therefore the Kendall planar shape analysis is data analysis on the complex projective space CP k 2. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
20 Extrinsic Mean on the Complex Projective Space Theorem Let Q be a probability distribution on CP k 2 and let {[Z r ], Z r = 1 r=1,...,n } be a random sample from Q. (a) Q is nonfocal iff λ the largest eigenvalue of E[Z 1 Z1 ] is simple and in this case µ E Q = [m], where m is an eigenvector of E[Z 1 Z1 ] corresponding to λ, with m = 1. (b) The extrinsic sample mean X E = [m], where m is an eigenvector of norm 1 of 1 ni=1 n Z i Zi, Z i = 1, i = 1,..., n, corresponding to the largest eigenvalue. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
21 Extrinsic Antimean on the Complex Projective Space Theorem Let Q be a probability distribution on CP k 2 and let {[Z r ], Z r = 1 r=1,...,n } be a random sample from Q. (a) Q is nonfocal iff λ the smallest eigenvalue of E[Z 1 Z1 ] is simple and in this case µ E Q = [m], where m is an eigenvector of E[Z 1 Z1 ] corresponding to λ, with m = 1. (b) The extrinsic sample antimean αx E = [m], where m is an eigenvector of norm 1 of 1 ni=1 n Z i Zi, Z i = 1, i = 1,..., n, corresponding to the smallest eigenvalue. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
22 Outline 1 Introduction Distance Fréchet Function Extrinsic Mean and Antimean 2 Kendall Shape Spaces Extrinsic antimean on Complex Space 3 Application Computational Example Results Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
23 Computational Example Use the VW-embedding of the complex projective space (Kendall shape space) to compare VW means and VW antimeans for a configuration of landmarks on midface in a population of normal children, based on a study on growth measured from X-rays at 8 and 14 years of age. Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
24 Coordinates of skull image Figure: The coordinates for children skull Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
25 Extrinsic Sample Mean Figure: Extrinsic Sample Mean Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
26 Bootstraps of extrinsic Sample Mean Figure: Bootstraps of Extrinsic Sample Mean Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
27 Extrinsic Sample Antimean Figure: Extrinsic Sample Antimean Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
28 Bootstraps of extrinsic Sample antimean Figure: Bootstraps for Extrinsic Sample Antimean Yunfan Wang (Florida State University) Extrinsic Antimean and Bootstrap April 12, / 28
Extrinsic Means and Antimeans
Extrinsic Means and Antimeans Vic Patrangenaru 1, K. David Yao 2, Ruite Guo 1 Florida State University 1 Department of Statistics, 2 Department of Mathematics December 28, 2015 1 Introduction Fréchet (1948)
More informationMeans and Antimeans. Vic Patrangenaru 1,K. David Yao 2, Ruite Guo 1 Florida State University 1 Department of Statistics, 2 Department of Mathematics
Means and Antimeans Vic Patrangenaru 1,K. David Yao 2, Ruite Guo 1 Florida State University 1 Department of Statistics, 2 Department of Mathematics May 10, 2015 1 Introduction Fréchet (1948) noticed that
More informationGEOMETRY AND DISTRIBUTIONS OF SHAPES Rabi Bhattacharya, Univ. of Arizona, USA (With V. Patrangenaru & A. Bhattacharya)
GEOMETRY AND DISTRIBUTIONS OF SHAPES Rabi Bhattacharya, Univ. of Arizona, USA (With V. Patrangenaru & A. Bhattacharya) CONTENTS 1. PROBAB. MEASURES ON MANIFOLDS (a) FRÉCHET MEAN (b) EXTRINSIC MEAN (c)
More informationSTATISTICS ON SHAPE MANIFOLDS: THEORY AND APPLICATIONS Rabi Bhattacharya, Univ. of Arizona, Tucson (With V. Patrangenaru & A.
STATISTICS ON SHAPE MANIFOLDS: THEORY AND APPLICATIONS Rabi Bhattacharya, Univ. of Arizona, Tucson (With V. Patrangenaru & A. Bhattacharya) CONTENTS 1. INTRODUCTION - EXAMPLES 2. PROBAB. MEASURES ON MANIFOLDS
More informationINTRINSIC MEAN ON MANIFOLDS. Abhishek Bhattacharya Project Advisor: Dr.Rabi Bhattacharya
INTRINSIC MEAN ON MANIFOLDS Abhishek Bhattacharya Project Advisor: Dr.Rabi Bhattacharya 1 Overview Properties of Intrinsic mean on Riemannian manifolds have been presented. The results have been applied
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationThe Singular Value Decomposition
The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors week -2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n
More informationExample: Filter output power. maximization. Definition. Eigenvalues, eigenvectors and similarity. Example: Stability of linear systems.
Lecture 2: Eigenvalues, eigenvectors and similarity The single most important concept in matrix theory. German word eigen means proper or characteristic. KTH Signal Processing 1 Magnus Jansson/Emil Björnson
More informationII. DIFFERENTIABLE MANIFOLDS. Washington Mio CENTER FOR APPLIED VISION AND IMAGING SCIENCES
II. DIFFERENTIABLE MANIFOLDS Washington Mio Anuj Srivastava and Xiuwen Liu (Illustrations by D. Badlyans) CENTER FOR APPLIED VISION AND IMAGING SCIENCES Florida State University WHY MANIFOLDS? Non-linearity
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationBasic Elements of Linear Algebra
A Basic Review of Linear Algebra Nick West nickwest@stanfordedu September 16, 2010 Part I Basic Elements of Linear Algebra Although the subject of linear algebra is much broader than just vectors and matrices,
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationSection 3.9. Matrix Norm
3.9. Matrix Norm 1 Section 3.9. Matrix Norm Note. We define several matrix norms, some similar to vector norms and some reflecting how multiplication by a matrix affects the norm of a vector. We use matrix
More informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 2
EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 2 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 5, 2012 Andre Tkacenko
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationNormed & Inner Product Vector Spaces
Normed & Inner Product Vector Spaces ECE 174 Introduction to Linear & Nonlinear Optimization Ken Kreutz-Delgado ECE Department, UC San Diego Ken Kreutz-Delgado (UC San Diego) ECE 174 Fall 2016 1 / 27 Normed
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More informationNONPARAMETRIC BAYESIAN INFERENCE ON PLANAR SHAPES
NONPARAMETRIC BAYESIAN INFERENCE ON PLANAR SHAPES Author: Abhishek Bhattacharya Coauthor: David Dunson Department of Statistical Science, Duke University 7 th Workshop on Bayesian Nonparametrics Collegio
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationStatistics on Placenta Shapes
IMS Lecture Notes Monograph Series c Institute of Mathematical Statistics, Statistics on Placenta Shapes Abhishek Bhattacharya University of Arizona Abstract: This report presents certain recent methodologies
More informationStatistics on Manifolds and Landmarks Based Image Analysis: A Nonparametric Theory with Applications
Statistics on Manifolds and Landmarks Based Image Analysis: A Nonparametric Theory with Applications Rabi Bhattacharya, Department of Mathematics, The University of Arizona, Tucson, AZ, 85721, USA and
More informationChapter 5 Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n
More informationComputational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Recap: A N N matrix A has an eigenvector x (non-zero) with corresponding
More informationExercise Sheet 1.
Exercise Sheet 1 You can download my lecture and exercise sheets at the address http://sami.hust.edu.vn/giang-vien/?name=huynt 1) Let A, B be sets. What does the statement "A is not a subset of B " mean?
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More information6 Inner Product Spaces
Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space
More informationCHAPTER 3. Matrix Eigenvalue Problems
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationPrincipal Components Theory Notes
Principal Components Theory Notes Charles J. Geyer August 29, 2007 1 Introduction These are class notes for Stat 5601 (nonparametrics) taught at the University of Minnesota, Spring 2006. This not a theory
More informationLecture 10 - Eigenvalues problem
Lecture 10 - Eigenvalues problem Department of Computer Science University of Houston February 28, 2008 1 Lecture 10 - Eigenvalues problem Introduction Eigenvalue problems form an important class of problems
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p
More informationRIGIDITY OF MINIMAL ISOMETRIC IMMERSIONS OF SPHERES INTO SPHERES. Christine M. Escher Oregon State University. September 10, 1997
RIGIDITY OF MINIMAL ISOMETRIC IMMERSIONS OF SPHERES INTO SPHERES Christine M. Escher Oregon State University September, 1997 Abstract. We show two specific uniqueness properties of a fixed minimal isometric
More informationRegularization and Inverse Problems
Regularization and Inverse Problems Caroline Sieger Host Institution: Universität Bremen Home Institution: Clemson University August 5, 2009 Caroline Sieger (Bremen and Clemson) Regularization and Inverse
More information7. Symmetric Matrices and Quadratic Forms
Linear Algebra 7. Symmetric Matrices and Quadratic Forms CSIE NCU 1 7. Symmetric Matrices and Quadratic Forms 7.1 Diagonalization of symmetric matrices 2 7.2 Quadratic forms.. 9 7.4 The singular value
More informationSymmetric and anti symmetric matrices
Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal
More informationStatistical Analysis on Manifolds: A Nonparametric Approach for Inference on Shape Spaces
Statistical Analysis on Manifolds: A Nonparametric Approach for Inference on Shape Spaces Abhishek Bhattacharya Department of Statistical Science, Duke University Abstract. This article concerns nonparametric
More informationAppendix A: Matrices
Appendix A: Matrices A matrix is a rectangular array of numbers Such arrays have rows and columns The numbers of rows and columns are referred to as the dimensions of a matrix A matrix with, say, 5 rows
More informationThe goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T
1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Eigenvalue Problems; Similarity Transformations Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 18 Eigenvalue
More informationLearning Eigenfunctions: Links with Spectral Clustering and Kernel PCA
Learning Eigenfunctions: Links with Spectral Clustering and Kernel PCA Yoshua Bengio Pascal Vincent Jean-François Paiement University of Montreal April 2, Snowbird Learning 2003 Learning Modal Structures
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationEconomics 204 Summer/Fall 2010 Lecture 10 Friday August 6, 2010
Economics 204 Summer/Fall 2010 Lecture 10 Friday August 6, 2010 Diagonalization of Symmetric Real Matrices (from Handout Definition 1 Let δ ij = { 1 if i = j 0 if i j A basis V = {v 1,..., v n } of R n
More information2. Matrix Algebra and Random Vectors
2. Matrix Algebra and Random Vectors 2.1 Introduction Multivariate data can be conveniently display as array of numbers. In general, a rectangular array of numbers with, for instance, n rows and p columns
More informationPart IA. Vectors and Matrices. Year
Part IA Vectors and Matrices Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2018 Paper 1, Section I 1C Vectors and Matrices For z, w C define the principal value of z w. State de Moivre s
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationFFTs in Graphics and Vision. The Laplace Operator
FFTs in Graphics and Vision The Laplace Operator 1 Outline Math Stuff Symmetric/Hermitian Matrices Lagrange Multipliers Diagonalizing Symmetric Matrices The Laplacian Operator 2 Linear Operators Definition:
More informationLinear algebra and applications to graphs Part 1
Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces
More informationChapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors
Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there
More informationMath 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
More informationSpectral Properties of Elliptic Operators In previous work we have replaced the strong version of an elliptic boundary value problem
Spectral Properties of Elliptic Operators In previous work we have replaced the strong version of an elliptic boundary value problem L u x f x BC u x g x with the weak problem find u V such that B u,v
More informationMATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003
MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space
More informationTRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE. Luis Guijarro and Gerard Walschap
TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE Luis Guijarro and Gerard Walschap Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold
More informationElliptic Regularity. Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n.
Elliptic Regularity Throughout we assume all vector bundles are smooth bundles with metrics over a Riemannian manifold X n. 1 Review of Hodge Theory In this note I outline the proof of the following Fundamental
More informationTutorials in Optimization. Richard Socher
Tutorials in Optimization Richard Socher July 20, 2008 CONTENTS 1 Contents 1 Linear Algebra: Bilinear Form - A Simple Optimization Problem 2 1.1 Definitions........................................ 2 1.2
More informationUniversity of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm
University of Colorado at Denver Mathematics Department Applied Linear Algebra Preliminary Exam With Solutions 16 January 2009, 10:00 am 2:00 pm Name: The proctor will let you read the following conditions
More informationGeometry of symmetric R-spaces
Geometry of symmetric R-spaces Makiko Sumi Tanaka Geometry and Analysis on Manifolds A Memorial Symposium for Professor Shoshichi Kobayashi The University of Tokyo May 22 25, 2013 1 Contents 1. Introduction
More informationLECTURE 9: THE WHITNEY EMBEDDING THEOREM
LECTURE 9: THE WHITNEY EMBEDDING THEOREM Historically, the word manifold (Mannigfaltigkeit in German) first appeared in Riemann s doctoral thesis in 1851. At the early times, manifolds are defined extrinsically:
More informationLaplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation
Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation Author: Raif M. Rustamov Presenter: Dan Abretske Johns Hopkins 2007 Outline Motivation and Background Laplace-Beltrami Operator
More informationEcon 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms. 1 Diagonalization and Change of Basis
Econ 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms De La Fuente notes that, if an n n matrix has n distinct eigenvalues, it can be diagonalized. In this supplement, we will provide
More informationON THE SINGULAR DECOMPOSITION OF MATRICES
An. Şt. Univ. Ovidius Constanţa Vol. 8, 00, 55 6 ON THE SINGULAR DECOMPOSITION OF MATRICES Alina PETRESCU-NIŢǍ Abstract This paper is an original presentation of the algorithm of the singular decomposition
More informationCHAPTER 3. Gauss map. In this chapter we will study the Gauss map of surfaces in R 3.
CHAPTER 3 Gauss map In this chapter we will study the Gauss map of surfaces in R 3. 3.1. Surfaces in R 3 Let S R 3 be a submanifold of dimension 2. Let {U i, ϕ i } be a DS on S. For any p U i we have a
More informationMath 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
More information1 Hermitian symmetric spaces: examples and basic properties
Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................
More informationOptimization Theory. Linear Operators and Adjoints
Optimization Theory Linear Operators and Adjoints A transformation T. : X Y y Linear Operators y T( x), x X, yy is the image of x under T The domain of T on which T can be defined : D X The range of T
More information1 Linear Algebra Problems
Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and
More informationAs always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).
An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted
More informationLecture 1: Review of linear algebra
Lecture 1: Review of linear algebra Linear functions and linearization Inverse matrix, least-squares and least-norm solutions Subspaces, basis, and dimension Change of basis and similarity transformations
More informationCourse Summary Math 211
Course Summary Math 211 table of contents I. Functions of several variables. II. R n. III. Derivatives. IV. Taylor s Theorem. V. Differential Geometry. VI. Applications. 1. Best affine approximations.
More informationAlgebra I Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary
More informationStatistics on Manifolds with Applications to Shape Spaces
Statistics on Manifolds with Applications to Shape Spaces Rabi Bhattacharya and Abhishek Bhattacharya Abstract. This article provides an exposition of recent developments on the analysis of landmark based
More informationp 2 p 3 p y p z It will not be considered in the present context; the interested reader can find more details in [05].
1. Geometrical vectors A geometrical vector p represents a point P in space. The point P is an abstraction that often, but not always, requires a representation. Vector representations are given wrt a
More informationLecture: Linear algebra. 4. Solutions of linear equation systems The fundamental theorem of linear algebra
Lecture: Linear algebra. 1. Subspaces. 2. Orthogonal complement. 3. The four fundamental subspaces 4. Solutions of linear equation systems The fundamental theorem of linear algebra 5. Determining the fundamental
More informationj=1 [We will show that the triangle inequality holds for each p-norm in Chapter 3 Section 6.] The 1-norm is A F = tr(a H A).
Math 344 Lecture #19 3.5 Normed Linear Spaces Definition 3.5.1. A seminorm on a vector space V over F is a map : V R that for all x, y V and for all α F satisfies (i) x 0 (positivity), (ii) αx = α x (scale
More informationLinear Algebra Review
January 29, 2013 Table of contents Metrics Metric Given a space X, then d : X X R + 0 and z in X if: d(x, y) = 0 is equivalent to x = y d(x, y) = d(y, x) d(x, y) d(x, z) + d(z, y) is a metric is for all
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 207/8 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationEigenvalue (mis)behavior on manifolds
Bucknell University Lehigh University October 20, 2010 Outline 1 Isoperimetric inequalities 2 3 4 A little history Rayleigh quotients The Original Isoperimetric Inequality The Problem of Queen Dido: maximize
More informationFunctional Analysis Review
Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all
More information1. Foundations of Numerics from Advanced Mathematics. Linear Algebra
Foundations of Numerics from Advanced Mathematics Linear Algebra Linear Algebra, October 23, 22 Linear Algebra Mathematical Structures a mathematical structure consists of one or several sets and one or
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationX-RAY TRANSFORM ON DAMEK-RICCI SPACES. (Communicated by Jan Boman)
Volume X, No. 0X, 00X, X XX Web site: http://www.aimsciences.org X-RAY TRANSFORM ON DAMEK-RICCI SPACES To Jan Boman on his seventy-fifth birthday. François Rouvière Laboratoire J.A. Dieudonné Université
More informationMobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti
Mobile Robotics 1 A Compact Course on Linear Algebra Giorgio Grisetti SA-1 Vectors Arrays of numbers They represent a point in a n dimensional space 2 Vectors: Scalar Product Scalar-Vector Product Changes
More informationTangent bundles, vector fields
Location Department of Mathematical Sciences,, G5-109. Main Reference: [Lee]: J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer-Verlag, 2002. Homepage for the book,
More informationIntroduction to Teichmüller Spaces
Introduction to Teichmüller Spaces Jing Tao Notes by Serena Yuan 1. Riemann Surfaces Definition 1.1. A conformal structure is an atlas on a manifold such that the differentials of the transition maps lie
More informationEigenvalue and Eigenvector Problems
Eigenvalue and Eigenvector Problems An attempt to introduce eigenproblems Radu Trîmbiţaş Babeş-Bolyai University April 8, 2009 Radu Trîmbiţaş ( Babeş-Bolyai University) Eigenvalue and Eigenvector Problems
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661 - Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
More informationNonparametric Inference on Manifolds
Nonparametric Inference on Manifolds This book introduces in a systematic manner a general nonparametric theory of statistics on manifolds, with emphasis on manifolds of shapes. The theory has important
More informationarxiv: v1 [math.dg] 25 Dec 2018 SANTIAGO R. SIMANCA
CANONICAL ISOMETRIC EMBEDDINGS OF PROJECTIVE SPACES INTO SPHERES arxiv:82.073v [math.dg] 25 Dec 208 SANTIAGO R. SIMANCA Abstract. We define inductively isometric embeddings of and P n (C) (with their canonical
More informationThroughout these notes we assume V, W are finite dimensional inner product spaces over C.
Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More information21 Symmetric and skew-symmetric matrices
21 Symmetric and skew-symmetric matrices 21.1 Decomposition of a square matrix into symmetric and skewsymmetric matrices Let C n n be a square matrix. We can write C = (1/2)(C + C t ) + (1/2)(C C t ) =
More informationGeneral Relativity by Robert M. Wald Chapter 2: Manifolds and Tensor Fields
General Relativity by Robert M. Wald Chapter 2: Manifolds and Tensor Fields 2.1. Manifolds Note. An event is a point in spacetime. In prerelativity physics and in special relativity, the space of all events
More information