The line, the circle, and the ray. R + x r. Science is linear, is nt? But behaviors take place in nonlinear spaces. The line The circle The ray
|
|
- Blaise Whitehead
- 6 years ago
- Views:
Transcription
1 Science is linear, is nt The line, the circle, and the ray Nonlinear spaces with efficient linearizations R. Sepulchre -- University of Cambridge Francqui Chair UCL, 05 Page rank algorithm Consensus algorithms Power method all share the same linear iteration + = A But behaviors take place in nonlinear spaces The line The circle The ray The power algorithm is an iteration on the projective space (orthogonality constraint, i.e. the circle) Perron-Frobenius theorem is a fied point in the projective space (positivity constraints, i.e. the ray) R S R + r
2 Part I: Homogeneity is essential to nonlinear behaviors The line : linear spaces Homogeneity is the net best thing to linearity It is necessary to account for the nonlinear nature of data y + y R Linear combinations: the basis of calculus It is sufficient to make local analysis efficient R, R n, R n n, C, C n, Sym(n), Skew(n),... The circle : phase and rotation spaces The ray : intensity spaces e i Embedding: Projection: S! C :! e i C! S : z! arg(z) Embedding: R +! R : r! log r S, S n, SO(n), SU(n),... R +, + (n), GL + (n),... phase, rotation, attitude, orthogonal matrices, unitary matrices,... Representation: linear spaces + orthogonality constraints radius, intensity, concentration, probability, density, Representation: linear spaces + positivity constraints
3 Polar coordinates r z A fundamental result of linear matri analysis A = QR Linear transformations (matrices) have intensities and orientation A = U V T z = re i C Any linear transformation can be decomposed as two rotations and one diagonal scaling. Linear objects (vectors) have intensities and orientation Nonlinear data Eamples of nonlinear measurements Most sensors have a preference for phase or intensity. For good reasons. concentration signals Our ears favor amplitude. Our eyes favor phase. Our nose favors concentrations. phase & intensity signals intensity signals
4 Behaviors (Willems) Linear behaviors (a mature theory) (V, B) V T! W vector space The Universum : space in which we observe/measure/collect the data The Behavior : mathematical law that govern the data B linear subspace E.g. Dynamical systems: V signal space, i.e. T! W :(t, )! w(t, ) B local law in (t, ) : F (w, ẇ,..., w (m) )=0 Linearization = local (and efficient) calculus Linearized behaviors Filtering Interpolating Optimizing (least squares, grad) Averaging,... (lecture 4) Linearization principle (Newton): linear behaviors capture local behaviors near a nominal solution w B w w + w w B(w ) linearized behavior around w From Taylor s epansion: DF(w ) w =0 Note: this requires W to be a linear space...
5 Linearized phase behavior Local analysis of nonlinear behaviors Embed the space W in a linear space and make the embedding part of the behavior W = S, B : F ( ) =0 B(w ) B(w ) B(w 3) is equivalent to z W = C, B : F ( ) =0 z = e i B Patching linearized behaviors: intractable Note: conceptually, the embedding trick works for arbitrary differentiable manifolds. What is special about the phase constraint Invariance and nonlinear data We like to think of local laws over to be invariant with respect to data phases intensities rotation scaling Homogeneity: linearization should look the same everywhere... Moon phase measured in Tokyo or Paris T measured in C or F...
6 Behaviors in invariant spaces LT(S)I behaviors are maimally invariant Laws independent from the locality of our data are invariant to specific transformations of the space The law is the same everywhere and everytime W = R W = S W = R + Invariance to translation of data Invariance to rotation of data Invariance to scaling of data shift-invariant in T translation invariant in W Note: those are the eact assumptions under which behavioral theory is mature and efficient. The key property The geometry of scaling and rotating Phase and rotation spaces are homogeneous spaces. In those spaces, linearization (and hence local calculus) can be made the same everywhere. The line, the circle, and the ray share a common mathematical structure. + z e i.e iz e.e z Transitive group action. Reaching any point from any point.
7 Lie groups Matri Lie groups From Wikipedia: three major themes in 9th century mathematics were combined by Lie in creating his new theory: the idea of symmetry, as eemplified by Galois through the algebraic notion of a group; geometric theory and the eplicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi; and the new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject. R n n SO(n) matri translation matri rotation Pioneers in engineering: e.g. A.S. Willsky, Dynamical Systems Defined on Groups: Structural Properties and Estimation, Ph.D. Thesis, MIT Dept. of Aeronautics and Astronautics, May 973. Thesis Advisors: R. W. Brockett, Wallace E. Vander Velde. GL(n) matri scaling Homogeneity is essential to nonlinear modeling The local description of the law can be made independent from the locality of the data window only if W is homogeneous. Homogeneity is key to tractability: local coordinates can be made the same everywhere... Part II: Eamples of behaviors on homogeneous spaces A homogeneous space is a space with a transitive group action by a Lie group. (The sphere is not a Lie group but it is a homogenous space)
8 Homogeneous spaces A homogeneous space M is a space with a transitive group action by a Lie group G. Notation: M G/H H is the stabilizer: Two (important) eamples S + (n) Symmetric positive definite matrices (Behaviors in homogeneous spaces; positivity constraints) H = {g G g.e = e} Gr(p, n) Sphere: S O(3)/O() Behaviors on the space of positive definite matrices the set of p-dimensional subspaces of Rn (Homogeneous behaviors in linear spaces; orthogonality and rank constraints) Diffusion Tensor Imaging voel data = local measure of diffusion of water molecules Phase Intensity Homogeneous data processing (Filtering, interpolating, registering,...)
9 Quadratic forms on linear data Zero-mean gaussian distributions are characterized by covariance matrices A linear transformation of the data points results in the group action GL(n) X = E( T )! A X! AXA T S + (n) The group acts transitively on by congruence. S + (n) GL(n)/O(n) Other eamples of quadratic forms: kernels, distance matrices,... Engineering impact of affine-invariant geometry of the cone Statistical engineering S. T. Smith, Covariance, subspace, and intrinsic Cramer-Rao bounds, IEEE Trans. Signal Process., 53 (005), pp Conve optimization Yu. E. Nesterov and M. J. Todd, On the Riemannian geometry defined for selfconcordant barriers and interior point methods, Found. Comput. Math., (00), pp DTI imaging X. Pennec, P. Fillard, and N. Ayache, A Riemannian framework for tensor computing, International Journal of Computer Vision, 66 (006), pp Behaviors on quadratic forms: gaussian processes, kernel optimization,... Different ways to make a space homogeneous Diffusion Tensor Imaging : filtering and interpolating smarties X = AA T = U U T = ep(z) Affine-invariant geometry (intrinsic) X GL(n)/O(n) Group embedding (etrinsic) (U, ) O(n) + linear embedding (etrinsic) Z Sym(n) Issues: computation, singularities, invariance properties (PhD Anne Collard, 03: anisotropy preserving midpoints)
10 Matri completion: a popular benchmark Big data behaviors ~ 07 known ratings (0.0% - 0.%) A recurrent theme: Scarcity of data points in huge dimensional spaces make behaviors ill-posed. Remedy: rank and orthogonality constraints ~ 05 items ~ 06 users Matri completion with a low-rank prior Statistics with scarce data Make the search space dimension consistent with the number of data points spots are gene epression levels each row is an eperiment (~0) each column is a gene (~04) DNA. mrna Protein
11 Statistics with a low-rank /sparsity prior The Grassmann manifold Gr(p, n) the set of p-dimensional subspaces of Rn A subspace is determined by the first p columns of an orthogonal matri Gr(p, n) O(n)/stab e epression of a component for all eperiments test correlation with clinical data gene signature of a component test overlap with pathways, regulatory modules Rank constraints M (p, m n) Transitive group action : A key homogeneous space of behavioral theory Miing rank and positivity constraints Space of m by n matrices of rank p (A, B)! AXB T M (p, m n) Gl(n) Gl(m)/stab ep The set of positive semidefinite matrices of size n and rank p S+ (p, n) = {X Rn n X = X T 0} p Transitive group action: A! AXAT S+ (p, n) GL(n)/stabe 44
12 The line, the circle, and the ray Nonlinear spaces with efficient linearizations Nonlinear data is a source of nonlinear behaviors. Phase and intensities spaces are homogeneous. Rank, orthogonality, and positivity constraints are homogeneous. Behaviors on homogeneous spaces can be made independent from the locality of data. Calculus on homogeneous spaces can be made invariant (lecture 3). Even if they are not invariant, behaviors on homogeneous spaces might have invariant properties (see lecture 6). Behaviors defined on non homogeneous spaces are ill-posed Behaviors with invariant properties are tractable
Lyapunov functions on nonlinear spaces S 1 R R + V =(1 cos ) Constructing Lyapunov functions: a personal journey
Constructing Lyapunov functions: a personal journey Lyapunov functions on nonlinear spaces R. Sepulchre -- University of Liege, Belgium Reykjavik - July 2013 Lyap functions in linear spaces (1994-1997)
More informationDifferential Geometry and Lie Groups with Applications to Medical Imaging, Computer Vision and Geometric Modeling CIS610, Spring 2008
Differential Geometry and Lie Groups with Applications to Medical Imaging, Computer Vision and Geometric Modeling CIS610, Spring 2008 Jean Gallier Department of Computer and Information Science University
More informationdistances between objects of different dimensions
distances between objects of different dimensions Lek-Heng Lim University of Chicago joint work with: Ke Ye (CAS) and Rodolphe Sepulchre (Cambridge) thanks: DARPA D15AP00109, NSF DMS-1209136, NSF IIS-1546413,
More informationGEOMETRIC DISTANCE BETWEEN POSITIVE DEFINITE MATRICES OF DIFFERENT DIMENSIONS
GEOMETRIC DISTANCE BETWEEN POSITIVE DEFINITE MATRICES OF DIFFERENT DIMENSIONS LEK-HENG LIM, RODOLPHE SEPULCHRE, AND KE YE Abstract. We show how the Riemannian distance on S n ++, the cone of n n real symmetric
More informationThe nonsmooth Newton method on Riemannian manifolds
The nonsmooth Newton method on Riemannian manifolds C. Lageman, U. Helmke, J.H. Manton 1 Introduction Solving nonlinear equations in Euclidean space is a frequently occurring problem in optimization and
More information1 Kernel methods & optimization
Machine Learning Class Notes 9-26-13 Prof. David Sontag 1 Kernel methods & optimization One eample of a kernel that is frequently used in practice and which allows for highly non-linear discriminant functions
More informationFocus was on solving matrix inversion problems Now we look at other properties of matrices Useful when A represents a transformations.
Previously Focus was on solving matrix inversion problems Now we look at other properties of matrices Useful when A represents a transformations y = Ax Or A simply represents data Notion of eigenvectors,
More information6. Linear Transformations.
6. Linear Transformations 6.1. Matrices as Transformations A Review of Functions domain codomain range x y preimage image http://en.wikipedia.org/wiki/codomain 6.1. Matrices as Transformations A Review
More informationProjective geometry and spacetime structure. David Delphenich Bethany College Lindsborg, KS USA
Projective geometry and spacetime structure David Delphenich Bethany College Lindsborg, KS USA delphenichd@bethanylb.edu Affine geometry In affine geometry the basic objects are points in a space A n on
More informationStatistical Geometry Processing Winter Semester 2011/2012
Statistical Geometry Processing Winter Semester 2011/2012 Linear Algebra, Function Spaces & Inverse Problems Vector and Function Spaces 3 Vectors vectors are arrows in space classically: 2 or 3 dim. Euclidian
More informationSparse Optimization Lecture: Basic Sparse Optimization Models
Sparse Optimization Lecture: Basic Sparse Optimization Models Instructor: Wotao Yin July 2013 online discussions on piazza.com Those who complete this lecture will know basic l 1, l 2,1, and nuclear-norm
More informationSpectral Processing. Misha Kazhdan
Spectral Processing Misha Kazhdan [Taubin, 1995] A Signal Processing Approach to Fair Surface Design [Desbrun, et al., 1999] Implicit Fairing of Arbitrary Meshes [Vallet and Levy, 2008] Spectral Geometry
More informationStochastic gradient descent on Riemannian manifolds
Stochastic gradient descent on Riemannian manifolds Silvère Bonnabel 1 Centre de Robotique - Mathématiques et systèmes Mines ParisTech SMILE Seminar Mines ParisTech Novembre 14th, 2013 1 silvere.bonnabel@mines-paristech
More informationIT is well-known that the cone of real symmetric positive
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 1 Geometric distance between positive definite matrices of different dimensions Lek-Heng Lim, Rodolphe Sepulchre, Fellow, IEEE, and Ke Ye Abstract We
More informationA Riemannian Framework for Denoising Diffusion Tensor Images
A Riemannian Framework for Denoising Diffusion Tensor Images Manasi Datar No Institute Given Abstract. Diffusion Tensor Imaging (DTI) is a relatively new imaging modality that has been extensively used
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 informationCS168: The Modern Algorithmic Toolbox Lecture #8: How PCA Works
CS68: The Modern Algorithmic Toolbox Lecture #8: How PCA Works Tim Roughgarden & Gregory Valiant April 20, 206 Introduction Last lecture introduced the idea of principal components analysis (PCA). The
More informationMath Subject GRE Questions
Math Subject GRE Questions Calculus and Differential Equations 1. If f() = e e, then [f ()] 2 [f()] 2 equals (a) 4 (b) 4e 2 (c) 2e (d) 2 (e) 2e 2. An integrating factor for the ordinary differential equation
More informationStochastic gradient descent on Riemannian manifolds
Stochastic gradient descent on Riemannian manifolds Silvère Bonnabel 1 Robotics lab - Mathématiques et systèmes Mines ParisTech Gipsa-lab, Grenoble June 20th, 2013 1 silvere.bonnabel@mines-paristech Introduction
More informationDistances between spectral densities. V x. V y. Genesis of this talk. The key point : the value of chordal distances. Spectrum approximation problem
Genesis of this talk Distances between spectral densities The shortest distance between two points is always under construction. (R. McClanahan) R. Sepulchre -- University of Cambridge Celebrating Anders
More informationMATH 311: COMPLEX ANALYSIS CONFORMAL MAPPINGS LECTURE
MATH 311: COMPLEX ANALYSIS CONFORMAL MAPPINGS LECTURE 1. Introduction Let D denote the unit disk and let D denote its boundary circle. Consider a piecewise continuous function on the boundary circle, {
More informationNumerical Integration (Quadrature) Another application for our interpolation tools!
Numerical Integration (Quadrature) Another application for our interpolation tools! Integration: Area under a curve Curve = data or function Integrating data Finite number of data points spacing specified
More informationLecture 2: Linear Algebra Review
EE 227A: Convex Optimization and Applications January 19 Lecture 2: Linear Algebra Review Lecturer: Mert Pilanci Reading assignment: Appendix C of BV. Sections 2-6 of the web textbook 1 2.1 Vectors 2.1.1
More informationC/CS/Phys C191 Grover s Quantum Search Algorithm 11/06/07 Fall 2007 Lecture 21
C/CS/Phys C191 Grover s Quantum Search Algorithm 11/06/07 Fall 2007 Lecture 21 1 Readings Benenti et al, Ch 310 Stolze and Suter, Quantum Computing, Ch 84 ielsen and Chuang, Quantum Computation and Quantum
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 informationRanking from Crowdsourced Pairwise Comparisons via Matrix Manifold Optimization
Ranking from Crowdsourced Pairwise Comparisons via Matrix Manifold Optimization Jialin Dong ShanghaiTech University 1 Outline Introduction FourVignettes: System Model and Problem Formulation Problem Analysis
More informationManifolds, Lie Groups, Lie Algebras, with Applications. Kurt W.A.J.H.Y. Reillag (alias Jean Gallier) CIS610, Spring 2005
Manifolds, Lie Groups, Lie Algebras, with Applications Kurt W.A.J.H.Y. Reillag (alias Jean Gallier) CIS610, Spring 2005 1 Motivations and Goals 1. Motivations Observation: Often, the set of all objects
More informationGeneralized Shifted Inverse Iterations on Grassmann Manifolds 1
Proceedings of the Sixteenth International Symposium on Mathematical Networks and Systems (MTNS 2004), Leuven, Belgium Generalized Shifted Inverse Iterations on Grassmann Manifolds 1 J. Jordan α, P.-A.
More informationLecture 4.2 Finite Difference Approximation
Lecture 4. Finite Difference Approimation 1 Discretization As stated in Lecture 1.0, there are three steps in numerically solving the differential equations. They are: 1. Discretization of the domain by
More information18-660: Numerical Methods for Engineering Design and Optimization
8-66: Numerical Methods or Engineering Design and Optimization Xin Li Department o ECE Carnegie Mellon University Pittsburgh, PA 53 Slide Overview Linear Regression Ordinary least-squares regression Minima
More informationABSTRACT ALGEBRA WITH APPLICATIONS
ABSTRACT ALGEBRA WITH APPLICATIONS IN TWO VOLUMES VOLUME I VECTOR SPACES AND GROUPS KARLHEINZ SPINDLER Darmstadt, Germany Marcel Dekker, Inc. New York Basel Hong Kong Contents f Volume I Preface v VECTOR
More informationVisual SLAM Tutorial: Bundle Adjustment
Visual SLAM Tutorial: Bundle Adjustment Frank Dellaert June 27, 2014 1 Minimizing Re-projection Error in Two Views In a two-view setting, we are interested in finding the most likely camera poses T1 w
More informationMathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More information33A Linear Algebra and Applications: Practice Final Exam - Solutions
33A Linear Algebra and Applications: Practice Final Eam - Solutions Question Consider a plane V in R 3 with a basis given by v = and v =. Suppose, y are both in V. (a) [3 points] If [ ] B =, find. (b)
More informationSummary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture)
Building Geometric Structures: Summary of Prof. Yau s lecture, Monday, April 2 [with additional references and remarks] (for people who missed the lecture) A geometric structure on a manifold is a cover
More informationUsing Hankel structured low-rank approximation for sparse signal recovery
Using Hankel structured low-rank approximation for sparse signal recovery Ivan Markovsky 1 and Pier Luigi Dragotti 2 Department ELEC Vrije Universiteit Brussel (VUB) Pleinlaan 2, Building K, B-1050 Brussels,
More informationThe Erlangen Program and General Relativity
The Erlangen Program and General Relativity Derek K. Wise University of Erlangen Department of Mathematics & Institute for Quantum Gravity Colloquium, Utah State University January 2014 What is geometry?
More informationVector and Affine Math
Vector and Affine Math Computer Science Department The Universit of Teas at Austin Vectors A vector is a direction and a magnitude Does NOT include a point of reference Usuall thought of as an arrow in
More informationM E M O R A N D U M. Faculty Senate approved November 1, 2018
M E M O R A N D U M Faculty Senate approved November 1, 2018 TO: FROM: Deans and Chairs Becky Bitter, Sr. Assistant Registrar DATE: October 23, 2018 SUBJECT: Minor Change Bulletin No. 5 The courses listed
More informationGiven the vectors u, v, w and real numbers α, β, γ. Calculate vector a, which is equal to the linear combination α u + β v + γ w.
Selected problems from the tetbook J. Neustupa, S. Kračmar: Sbírka příkladů z Matematiky I Problems in Mathematics I I. LINEAR ALGEBRA I.. Vectors, vector spaces Given the vectors u, v, w and real numbers
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
More informationRiemannian Metric Learning for Symmetric Positive Definite Matrices
CMSC 88J: Linear Subspaces and Manifolds for Computer Vision and Machine Learning Riemannian Metric Learning for Symmetric Positive Definite Matrices Raviteja Vemulapalli Guide: Professor David W. Jacobs
More informationGeometric Modeling Summer Semester 2010 Mathematical Tools (1)
Geometric Modeling Summer Semester 2010 Mathematical Tools (1) Recap: Linear Algebra Today... Topics: Mathematical Background Linear algebra Analysis & differential geometry Numerical techniques Geometric
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 informationInterval solutions for interval algebraic equations
Mathematics and Computers in Simulation 66 (2004) 207 217 Interval solutions for interval algebraic equations B.T. Polyak, S.A. Nazin Institute of Control Sciences, Russian Academy of Sciences, 65 Profsoyuznaya
More informationNPTEL
NPTEL Syllabus Selected Topics in Mathematical Physics - Video course COURSE OUTLINE Analytic functions of a complex variable. Calculus of residues, Linear response; dispersion relations. Analytic continuation
More informationCLASS NOTES Computational Methods for Engineering Applications I Spring 2015
CLASS NOTES Computational Methods for Engineering Applications I Spring 2015 Petros Koumoutsakos Gerardo Tauriello (Last update: July 2, 2015) IMPORTANT DISCLAIMERS 1. REFERENCES: Much of the material
More informationNumerical Methods I Singular Value Decomposition
Numerical Methods I Singular Value Decomposition Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 9th, 2014 A. Donev (Courant Institute)
More informationA Padé approximation to the scalar wavefield extrapolator for inhomogeneous media
A Padé approimation A Padé approimation to the scalar wavefield etrapolator for inhomogeneous media Yanpeng Mi, Zhengsheng Yao, and Gary F. Margrave ABSTRACT A seismic wavefield at depth z can be obtained
More informationPopulation Games and Evolutionary Dynamics
Population Games and Evolutionary Dynamics William H. Sandholm The MIT Press Cambridge, Massachusetts London, England in Brief Series Foreword Preface xvii xix 1 Introduction 1 1 Population Games 2 Population
More informationLecture: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab Lecture 11-1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 19: More on Arnoldi Iteration; Lanczos Iteration Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical Analysis I 1 / 17 Outline 1
More informationExact and Approximate Numbers:
Eact and Approimate Numbers: The numbers that arise in technical applications are better described as eact numbers because there is not the sort of uncertainty in their values that was described above.
More informationSubgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold.
Recollections from finite group theory. The notion of a group acting on a set is extremely useful. Indeed, the whole of group theory arose through this route. As an example of the abstract power of this
More informationPaul Heckbert. Computer Science Department Carnegie Mellon University. 26 Sept B - Introduction to Scientific Computing 1
Paul Heckbert Computer Science Department Carnegie Mellon University 26 Sept. 2 5-859B - Introduction to Scientific Computing aerospace: simulate subsonic & supersonic air flow around full aircraft, no
More informationDIFFERENTIAL GEOMETRY HW 12
DIFFERENTIAL GEOMETRY HW 1 CLAY SHONKWILER 3 Find the Lie algebra so(n) of the special orthogonal group SO(n), and the explicit formula for the Lie bracket there. Proof. Since SO(n) is a subgroup of GL(n),
More informationSTA414/2104 Statistical Methods for Machine Learning II
STA414/2104 Statistical Methods for Machine Learning II Murat A. Erdogdu & David Duvenaud Department of Computer Science Department of Statistical Sciences Lecture 3 Slide credits: Russ Salakhutdinov Announcements
More informationREGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES
REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES MAKIKO SUMI TANAKA 1. Introduction This article is based on the collaboration with Tadashi Nagano. In the first part of this article we briefly review basic
More information2. On integer geometry (22 March 2011)
2. On integer geometry (22 March 2011) 2.1. asic notions and definitions. notion of geometry in general can be interpreted in many different ways. In our course we think of geometry as of a set of objects
More informationThe Karcher Mean of Points on SO n
The Karcher Mean of Points on SO n Knut Hüper joint work with Jonathan Manton (Univ. Melbourne) Knut.Hueper@nicta.com.au National ICT Australia Ltd. CESAME LLN, 15/7/04 p.1/25 Contents Introduction CESAME
More information1 Computing with constraints
Notes for 2017-04-26 1 Computing with constraints Recall that our basic problem is minimize φ(x) s.t. x Ω where the feasible set Ω is defined by equality and inequality conditions Ω = {x R n : c i (x)
More informationLecture 13 The Fundamental Forms of a Surface
Lecture 13 The Fundamental Forms of a Surface In the following we denote by F : O R 3 a parametric surface in R 3, F(u, v) = (x(u, v), y(u, v), z(u, v)). We denote partial derivatives with respect to the
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 informationInduced Representations and Frobenius Reciprocity. 1 Generalities about Induced Representations
Induced Representations Frobenius Reciprocity Math G4344, Spring 2012 1 Generalities about Induced Representations For any group G subgroup H, we get a restriction functor Res G H : Rep(G) Rep(H) that
More informationCS 4300 Computer Graphics. Prof. Harriet Fell Fall 2011 Lecture 11 September 29, 2011
CS 4300 Computer Graphics Prof. Harriet Fell Fall 2011 Lecture 11 September 29, 2011 October 8, 2011 College of Computer and Information Science, Northeastern Universit 1 Toda s Topics Linear Algebra Review
More information9-12 Mathematics Vertical Alignment ( )
Algebra I Algebra II Geometry Pre- Calculus U1: translate between words and algebra -add and subtract real numbers -multiply and divide real numbers -evaluate containing exponents -evaluate containing
More information(MTH5109) GEOMETRY II: KNOTS AND SURFACES LECTURE NOTES. 1. Introduction to Curves and Surfaces
(MTH509) GEOMETRY II: KNOTS AND SURFACES LECTURE NOTES DR. ARICK SHAO. Introduction to Curves and Surfaces In this module, we are interested in studing the geometr of objects. According to our favourite
More informationWe wish the reader success in future encounters with the concepts of linear algebra.
Afterword Our path through linear algebra has emphasized spaces of vectors in dimension 2, 3, and 4 as a means of introducing concepts which go forward to IRn for arbitrary n. But linear algebra does not
More informationRobustness of Principal Components
PCA for Clustering An objective of principal components analysis is to identify linear combinations of the original variables that are useful in accounting for the variation in those original variables.
More informationFrom Wikipedia, the free encyclopedia
1 of 8 27/03/2013 12:41 Quadratic form From Wikipedia, the free encyclopedia In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic
More informationL26: Advanced dimensionality reduction
L26: Advanced dimensionality reduction The snapshot CA approach Oriented rincipal Components Analysis Non-linear dimensionality reduction (manifold learning) ISOMA Locally Linear Embedding CSCE 666 attern
More informationMLCC 2015 Dimensionality Reduction and PCA
MLCC 2015 Dimensionality Reduction and PCA Lorenzo Rosasco UNIGE-MIT-IIT June 25, 2015 Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component
More informationLAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM
LAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM ORIGINATION DATE: 8/2/99 APPROVAL DATE: 3/22/12 LAST MODIFICATION DATE: 3/28/12 EFFECTIVE TERM/YEAR: FALL/ 12 COURSE ID: COURSE TITLE: MATH2800 Linear Algebra
More informationSlice Sampling with Adaptive Multivariate Steps: The Shrinking-Rank Method
Slice Sampling with Adaptive Multivariate Steps: The Shrinking-Rank Method Madeleine B. Thompson Radford M. Neal Abstract The shrinking rank method is a variation of slice sampling that is efficient at
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 informationReview of Linear Algebra
Review of Linear Algebra Dr Gerhard Roth COMP 40A Winter 05 Version Linear algebra Is an important area of mathematics It is the basis of computer vision Is very widely taught, and there are many resources
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationThe structure tensor in projective spaces
The structure tensor in projective spaces Klas Nordberg Computer Vision Laboratory Department of Electrical Engineering Linköping University Sweden Abstract The structure tensor has been used mainly for
More informationS.F. Xu (Department of Mathematics, Peking University, Beijing)
Journal of Computational Mathematics, Vol.14, No.1, 1996, 23 31. A SMALLEST SINGULAR VALUE METHOD FOR SOLVING INVERSE EIGENVALUE PROBLEMS 1) S.F. Xu (Department of Mathematics, Peking University, Beijing)
More information0.1. Linear transformations
Suggestions for midterm review #3 The repetitoria are usually not complete; I am merely bringing up the points that many people didn t now on the recitations Linear transformations The following mostly
More informationGeometry and Kinematics with Uncertain Data
Geometry and Kinematics with Uncertain Data Christian Perwass, Christian Gebken, and Gerald Sommer Institut für Informatik, CAU Kiel Christian-Albrechts-Platz 4, 24118 Kiel, Germany {chp,chg,gs}@ks.informatik.uni-kiel.de
More informationChapter 3 Numerical Methods
Chapter 3 Numerical Methods Part 2 3.2 Systems of Equations 3.3 Nonlinear and Constrained Optimization 1 Outline 3.2 Systems of Equations 3.3 Nonlinear and Constrained Optimization Summary 2 Outline 3.2
More informationFitting Linear Statistical Models to Data by Least Squares I: Introduction
Fitting Linear Statistical Models to Data by Least Squares I: Introduction Brian R. Hunt and C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling February 5, 2014 version
More informationSome preconditioners for systems of linear inequalities
Some preconditioners for systems of linear inequalities Javier Peña Vera oshchina Negar Soheili June 0, 03 Abstract We show that a combination of two simple preprocessing steps would generally improve
More informationNumerical Methods in Matrix Computations
Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices
More informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationDiscriminative Direction for Kernel Classifiers
Discriminative Direction for Kernel Classifiers Polina Golland Artificial Intelligence Lab Massachusetts Institute of Technology Cambridge, MA 02139 polina@ai.mit.edu Abstract In many scientific and engineering
More informationCovariance Tracking Algorithm on Bilateral Filtering under Lie Group Structure Yinghong Xie 1,2,a Chengdong Wu 1,b
Applied Mechanics and Materials Online: 014-0-06 ISSN: 166-748, Vols. 519-50, pp 684-688 doi:10.408/www.scientific.net/amm.519-50.684 014 Trans Tech Publications, Switzerland Covariance Tracking Algorithm
More informationHow curvature shapes space
How curvature shapes space Richard Schoen University of California, Irvine - Hopf Lecture, ETH, Zürich - October 30, 2017 The lecture will have three parts: Part 1: Heinz Hopf and Riemannian geometry Part
More informationLecture: Face Recognition and Feature Reduction
Lecture: Face Recognition and Feature Reduction Juan Carlos Niebles and Ranjay Krishna Stanford Vision and Learning Lab 1 Recap - Curse of dimensionality Assume 5000 points uniformly distributed in the
More informationVector-valued quadratic forms in control theory
MTNS 2002, To appear Chapter 1 Vector-valued quadratic forms in control theory Francesco Bullo Coordinated Science Laboratory University of Illinois Urbana-Champaign, IL 61801 United States bullo@uiuc.edu
More informationLecture 7: Weak Duality
EE 227A: Conve Optimization and Applications February 7, 2012 Lecture 7: Weak Duality Lecturer: Laurent El Ghaoui 7.1 Lagrange Dual problem 7.1.1 Primal problem In this section, we consider a possibly
More informationSYMMETRIES OF SECTIONAL CURVATURE ON 3 MANIFOLDS. May 27, 1992
SYMMETRIES OF SECTIONAL CURVATURE ON 3 MANIFOLDS Luis A. Cordero 1 Phillip E. Parker,3 Dept. Xeometría e Topoloxía Facultade de Matemáticas Universidade de Santiago 15706 Santiago de Compostela Spain cordero@zmat.usc.es
More informationComputational Methods CMSC/AMSC/MAPL 460. EigenValue decomposition Singular Value Decomposition. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 EigenValue decomposition Singular Value Decomposition Ramani Duraiswami, Dept. of Computer Science Hermitian Matrices A square matrix for which A = A H is said
More informationApplied Linear Algebra
Applied Linear Algebra Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN 55455 olver@math.umn.edu http://www.math.umn.edu/ olver Chehrzad Shakiban Department of Mathematics University
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
More informationLecture 3: Latent Variables Models and Learning with the EM Algorithm. Sam Roweis. Tuesday July25, 2006 Machine Learning Summer School, Taiwan
Lecture 3: Latent Variables Models and Learning with the EM Algorithm Sam Roweis Tuesday July25, 2006 Machine Learning Summer School, Taiwan Latent Variable Models What to do when a variable z is always
More information