DOCSLIB.ORG

Generating Functions from Symmetric Functions Anthony Mendes And

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generating Functions from Symmetric Functions Anthony Mendes And Generating functions from symmetric functions Anthony Mendes and Jeffrey Remmel Abstract. This monograph introduces a method of finding and refining gen- erating functions. By manipulating combinatorial objects known as brick tabloids, we will show how many well known generating functions may be found and subsequently generalized. New results are given as well. The techniques described in this monograph originate from a thorough understanding of a connection between symmetric functions and the permu- tation enumeration of the symmetric group. Define a homomorphism ξ on the ring of symmetric functions by defining it on the elementary symmetric n−1 function en such that ξ(en) = (1 − x) /n!. Brenti showed that applying ξ to the homogeneous symmetric function gave a generating function for the Eulerian polynomials [14, 13]. Beck and Remmel reproved the results of Brenti combinatorially [6]. A handful of authors have tinkered with their proof to discover results about the permutation enumeration for signed permutations and multiples of permuta- tions [4, 5, 51, 52, 53, 58, 70, 71]. However, this monograph records the true power and adaptability of this relationship between symmetric functions and permutation enumeration. We will give versatile methods unifying a large number of results in the theory of permutation enumeration for the symmet- ric group, subsets of the symmetric group, and assorted Coxeter groups, and many other objects. Contents Chapter 1. Brick tabloids in permutation enumeration 1 1.1. The ring of formal power series 1 1.2. The ring of symmetric functions 7 1.3. Brenti’s homomorphism 21 1.4. Published uses of brick tabloids in permutation enumeration 30 1.5. First extensions of Brenti’s homomorphism 34 Chapter 2. Generating functions for permutations 51 2.1. Alternating permutations 51 2.2. Consecutive descents 56 2.3. Consecutive patterns 67 2.4. Descents, major indices, and inversions 91 Chapter 3. Generating functions for other objects 97 3.1. Wreath product statistics 97 3.2. Words 102 3.3. Fibonacci numbers 113 3.4. The exponential formula 113 Chapter 4. Conclusions 125 Appendix A. Permutation statistics 133 Appendix. Bibliography 135 iii CHAPTER 1 Brick tabloids in permutation enumeration We will begin with a discussion on the ring of formal power series in order to establish basic definitions and concepts needed understand the rest of this work. Generating functions and the benefits of their use are introduced here. The objec- tives of this monograph and a brief introduction to permutation statistics are also included in Section 1.1. Symmetric function theory will be used heavily throughout this work. In Sec- tion 1.2, the ring of symmetric functions is described completely from scratch. Our approach is not like that of any other published work in that we attempt to give combinatorial proofs for every basic symmetric function identity. It is in Section 1.2 where we introduce the notion of brick tabloids, a combinatorial object used extensively in the rest of this monograph. Section 1.3 contains the ideas at the heart of this monograph. Symmetric func- tion theory is used to create generating functions here. The rest of this monograph is devoted to further developing the methods in Section 1.3—this section is essential reading for those wanting a complete understanding of our building of generating functions. We systematically compare all previous published works relating brick tabloids and permutation statistics with the results in this monograph in Section 1.4. This should help to clarify the advances we will make in this work. Many proofs in the first three sections have been revamped and elements of the combinatorial approach to basic symmetric function identities included in this section have not been previously published, but this content is well known and none of it should be considered brand new. This is not the case for Section 1.5, however. Although some of the generating functions in this section are known, the methodology is original. It is in this section where we make our first significant advances in our development when we discuss ways to extend the homomorphism introduced in Section 1.3 to find and refine more results about descents in the symmetric group. 1.1. The ring of formal power series Let a0,a1,... be a sequence in a ring R and t an indeterminate. The formal power series for the sequence a0,a1,... is the expression a t0 + a t1 + a t2 + . 0 1 2 · · · n Such an object may be denoted n∞=0 ant . At face value, although we are using the plus symbol and summation notation, we are not performing the operation of P addition. We are simply presenting the sequence a0,a1,... in a specific way, using plus symbols to separate terms and using powers of t as placeholders. The set of 1 2 1. BRICK TABLOIDS IN PERMUTATION ENUMERATION all formal power series in t representing sequences in the ring R will be denoted by R[[t]]. For instance, consider the sequence in the ring of rational numbers defined by n(n 1)/2 an =2 − /n!. This sequence recorded as a formal power series is 8 64 ∞ 2n(n 1)/2 (1.1) 1+ t + t2 + t3 + t4 + = − tn. 6 24 · · · n! n=0 X If interpreted as a complex-valued function in the variable t, the above series would have a radius of convergence of 0 and thus most techniques from analysis would not apply. The object in (1.1) does, however, define a perfectly valid formal power series. In fact, this formal power series will be used in Section 3.4 to count the number of graphs on n nodes. If a0,a1,... is a sequence in R such that a0 = a1 = = aj 1 = 0 for some n n · · · − j 0, then we may denote ∞ a t as ∞ a t to reflect this fact. Along ≥ n=0 n n=j n j n similar lines, we write n=0 aPnt provided aPn = 0 for n > j. There are a handful of operations on R[[t]] which we now define. First, for each P nonnegative integer j, define a map j from R[[t]] to R such that ·|t ∞ n ant = aj . n=0 tj X j The element in R found by an application of j is called the coefficient of t . Two ·|t elements in R[[t]] are equal provided the coefficients of tj in each formal power series are equal for all j 0. Define the sum≥ of two formal power series by the rule ∞ n ∞ n ∞ n ant + bnt = (an + bn)t n=0 ! n=0 ! n=0 X X X where the plus symbol on the right hand side of the equation denotes the sum of two elements in the ring R. Define the product of two formal power series by the rule ∞ ∞ ∞ n n n ant bnt = (a0bn + a1bn 1 + + an 1b1 + anb0) t − · · · − n=0 ! n=0 ! n=0 X X X where the plus symbols and the adjacent elements on the right hand side of the above equation denote the sum and product of two elements in R, respectively. With these definitions, it is not difficult to show that R[[t]] is a ring; naturally, R[[t]] is called the ring of formal power series. Notice that R is commutative if and only if R[[t]] is commutative. For greater simplicity in our development of the ring of power series, we make the assumption that R is a commutative ring with unity from now on. If ∞ n ∞ n ant bnt =1 n=0 ! n=0 ! X X 1.1. THE RING OF FORMAL POWER SERIES 3 2 n where 1 represents 1 + 0t +0t + , then we say n∞=0 bnt is the reciprocal of n · · · ∞ ant and write n=0 P 1 P ∞ ∞ − n n 1 bnt = ant = . ∞ a tn n=0 n=0 ! n=0 n X X For example, by our definition of the product of twoP formal power series, (1 t)(1+ 2 2 1 − t + t + ) = 1 and therefore (1+ t + t + )=(1 t)− =1/(1 t). Formally, this is the· · · familiar formula for the sum of a· geometric · · − series. − n m Define the composition of n∞=0 ant and m∞=1 bmt as the formal power series P Pn ∞ ∞ m an bmt . n=0 m=1 ! X X A potential problem in this definition arises if any coefficient in the above formal power series is an infinite sum of elements in R. However, notice that n ∞ ∞ ∞ n a b tm = a b t1 + + b tj n m n 1 · · · j n=0 m=1 ! j n=0 j X X t X t j 1 j n = an b1t + + bj t · · · n=0 tj X j where in the last expression we are selecting the coefficient of t in a finite sum. j This shows that the stipulation b0 = 0 forces the coefficient of t in the composition as defined above to be a finite sum of elements in R for j 0. Thus, there are no problems with our definition of composition. ≥ The derivative of a formal power series is a map d/dt( ) from R[[t]] to R[[t]] defined by · d ∞ ∞ a tn = (n + 1)a tn dt n n+1 n=0 ! n=0 X X where n + 1 is the element 1 + + 1 (n + 1 times) in R. Similarly, the integral of a formal power series is a map · · · dt from R[[t]] to R[[t]] defined by · R ∞ n ∞ an 1 n a t dt = − t n n n=0 ! n=1 Z X X provided the multiplicative inverse of n exists in R for n 1.
Load more

Recommended publications
  • Covariances of Symmetric Statistics
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector JOURNAL OF MULTIVARIATE ANALYSIS 41, 14-26 (1992) Covariances of Symmetric Statistics RICHARD A. VITALE Universily of Connecticut Communicared by C. R. Rao We examine the second-order structure arising when a symmetric function is evaluated over intersecting subsets of random variables. The original work of Hoeffding is updated with reference to later results. New representations and inequalities are presented for covariances and applied to U-statistics. 0 1992 Academic Press. Inc. 1. INTRODUCTION The importance of the symmetric use of sample observations was apparently first observed by Halmos [6] and, following the pioneering work of Hoeffding [7], has been reflected in the active study of U-statistics (e.g., Hoeffding [8], Rubin and Vitale [ 111, Dynkin and Mandelbaum [2], Mandelbaum and Taqqu [lo] and, for a lucid survey of the classical theory Serfling [12]). An important feature is the ANOVA-type expansion introduced by Hoeffding and its application to finite-sample inequalities, Efron and Stein [3], Karlin and Rinnott [9], Bhargava Cl], Takemura [ 143, Vitale [16], Steele [ 131). The purpose here is to survey work on this latter topic and to present new results. After some preliminaries, Section 3 organizes for comparison three approaches to the ANOVA-type expansion. The next section takes up characterizations and representations of covariances. These are then used to tighten an important inequality of Hoeffding and, in the last section, to study the structure of U-statistics. 2. NOTATION AND PRELIMINARIES The general setup is a supply X,, X,, .
    [Show full text]
  • Math 263A Notes: Algebraic Combinatorics and Symmetric Functions
    MATH 263A NOTES: ALGEBRAIC COMBINATORICS AND SYMMETRIC FUNCTIONS AARON LANDESMAN CONTENTS 1. Introduction 4 2. 10/26/16 5 2.1. Logistics 5 2.2. Overview 5 2.3. Down to Math 5 2.4. Partitions 6 2.5. Partial Orders 7 2.6. Monomial Symmetric Functions 7 2.7. Elementary symmetric functions 8 2.8. Course Outline 8 3. 9/28/16 9 3.1. Elementary symmetric functions eλ 9 3.2. Homogeneous symmetric functions, hλ 10 3.3. Power sums pλ 12 4. 9/30/16 14 5. 10/3/16 20 5.1. Expected Number of Fixed Points 20 5.2. Random Matrix Groups 22 5.3. Schur Functions 23 6. 10/5/16 24 6.1. Review 24 6.2. Schur Basis 24 6.3. Hall Inner product 27 7. 10/7/16 29 7.1. Basic properties of the Cauchy product 29 7.2. Discussion of the Cauchy product and related formulas 30 8. 10/10/16 32 8.1. Finishing up last class 32 8.2. Skew-Schur Functions 33 8.3. Jacobi-Trudi 36 9. 10/12/16 37 1 2 AARON LANDESMAN 9.1. Eigenvalues of unitary matrices 37 9.2. Application 39 9.3. Strong Szego limit theorem 40 10. 10/14/16 41 10.1. Background on Tableau 43 10.2. KOSKA Numbers 44 11. 10/17/16 45 11.1. Relations of skew-Schur functions to other fields 45 11.2. Characters of the symmetric group 46 12. 10/19/16 49 13. 10/21/16 55 13.1.
    [Show full text]
  • Some New Symmetric Function Tools and Their Applications
    Some New Symmetric Function Tools and their Applications A. Garsia ∗Department of Mathematics University of California, La Jolla, CA 92093 [email protected] J. Haglund †Department of Mathematics University of Pennsylvania, Philadelphia, PA 19104-6395 [email protected] M. Romero ‡Department of Mathematics University of California, La Jolla, CA 92093 [email protected] June 30, 2018 MR Subject Classifications: Primary: 05A15; Secondary: 05E05, 05A19 Keywords: Symmetric functions, modified Macdonald polynomials, Nabla operator, Delta operators, Frobenius characteristics, Sn modules, Narayana numbers Abstract Symmetric Function Theory is a powerful computational device with applications to several branches of mathematics. The Frobenius map and its extensions provides a bridge translating Representation Theory problems to Symmetric Function problems and ultimately to combinatorial problems. Our main contributions here are certain new symmetric func- tion tools for proving a variety of identities, some of which have already had significant applications. One of the areas which has been nearly untouched by present research is the construction of bigraded Sn modules whose Frobenius characteristics has been conjectured from both sides. The flagrant example being the Delta Conjecture whose symmetric function side was conjectured to be Schur positive since the early 90’s and there are various unproved recent ways to construct the combinatorial side. One of the most surprising applications of our tools reveals that the only conjectured bigraded Sn modules are remarkably nested by Sn the restriction ↓Sn−1 . 1 Introduction Frobenius constructed a map Fn from class functions of Sn to degree n homogeneous symmetric poly- λ nomials and proved the identity Fnχ = sλ[X], for all λ ⊢ n.
    [Show full text]
  • Symmetric Group Characters As Symmetric Functions, Arxiv:1605.06672
    SYMMETRIC GROUP CHARACTERS AS SYMMETRIC FUNCTIONS (EXTENDED ABSTRACT) ROSA ORELLANA AND MIKE ZABROCKI Abstract. The irreducible characters of the symmetric group are a symmetric polynomial in the eigenvalues of a permutation matrix. They can therefore be realized as a symmetric function that can be evaluated at a set of variables and form a basis of the symmetric functions. This basis of the symmetric functions is of non-homogeneous degree and the (outer) product structure coefficients are the stable Kronecker coefficients. We introduce the irreducible character basis by defining it in terms of the induced trivial char- acters of the symmetric group which also form a basis of the symmetric functions. The irreducible character basis is closely related to character polynomials and we obtain some of the change of basis coefficients by making this connection explicit. Other change of basis coefficients come from a repre- sentation theoretic connection with the partition algebra, and still others are derived by developing combinatorial expressions. This document is an extended abstract which can be used as a review reference so that this basis can be implemented in Sage. A more full version of the results in this abstract can be found in arXiv:1605.06672. 1. Introduction We begin with a very basic question in representation theory. Can the irreducible characters of the symmetric group be realized as functions of the eigenvalues of the permutation matrices? In general, characters of matrix groups are symmetric functions in the set of eigenvalues of the matrix and (as we will show) the symmetric group characters define a basis of inhomogeneous degree for the space of symmetric functions.
    [Show full text]
  • An Introduction to Symmetric Functions
    An Introduction to Symmetric Functions Ira M. Gessel Department of Mathematics Brandeis University Summer School on Algebraic Combinatorics Korea Institute for Advanced Study Seoul, Korea June 13, 2016 They are formal power series in the infinitely many variables x1; x2;::: that are invariant under permutation of the subscripts. In other words, if i1;:::; im are distinct positive integers and α1; : : : ; αm are arbitrary nonnegative integers then the coefficient of xα1 ··· xαm in a symmetric function is the same as i1 im α1 αm the coefficient of x1 ··· xm . Examples: 2 2 I x1 + x2 + ::: P I i≤j xi xj P 2 But not i≤j xi xj What are symmetric functions? Symmetric functions are not functions. In other words, if i1;:::; im are distinct positive integers and α1; : : : ; αm are arbitrary nonnegative integers then the coefficient of xα1 ··· xαm in a symmetric function is the same as i1 im α1 αm the coefficient of x1 ··· xm . Examples: 2 2 I x1 + x2 + ::: P I i≤j xi xj P 2 But not i≤j xi xj What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x1; x2;::: that are invariant under permutation of the subscripts. Examples: 2 2 I x1 + x2 + ::: P I i≤j xi xj P 2 But not i≤j xi xj What are symmetric functions? Symmetric functions are not functions. They are formal power series in the infinitely many variables x1; x2;::: that are invariant under permutation of the subscripts. In other words, if i1;:::; im are distinct positive integers and α1; : : : ; αm are arbitrary nonnegative integers then the coefficient of xα1 ··· xαm in a symmetric function is the same as i1 im α1 αm the coefficient of x1 ··· xm .
    [Show full text]
  • Introduction to Symmetric Functions Chapter 3 Mike Zabrocki
    Introduction to Symmetric Functions Chapter 3 Mike Zabrocki Abstract. A development of the symmetric functions using the plethystic notation. CHAPTER 2 Symmetric polynomials Our presentation of the ring of symmetric functions has so far been non-standard and re- visionist in the sense that the motivation for defining the ring Λ was historically to study the ring of polynomials which are invariant under the permutation of the variables. In this chapter we consider the relationship between Λ and this ring. In this section we wish to consider polynomials f(x1, x2, . , xn) ∈ Q[x1, x2, . , xn] such that f(xσ1 , xσ2 , ··· , xσn ) = f(x1, x2, . , xn) for all σ ∈ Symn. These polynomials form a ring since clearly they are closed under multiplication and contain the element 1 as a unit. We will denote this ring Xn Λ = {f ∈ Q[x1, x2, . , xn]: f(x1, x2, . , xn) =(2.1) f(xσ1 , xσ2 , . , xσn ) for all σ ∈ Symn} Xn Pn k Now there is a relationship between Λ and Λ by setting pk[Xn] := k=1 xi and define a map Λ −→ ΛXn by the linear homomorphism (2.2) pλ 7→ pλ1 [Xn]pλ1 [Xn] ··· pλ`(λ) [Xn] with the natural extension to linear combinations of the pλ. In a more general setting we will take the elements Λ to be a set of functors on polynomials k pk[xi] = xi and pk[cE + dF ] = cpk[E] + dpk[F ] for E, F ∈ Q[x1, x2, . , xn] and coefficients c, d ∈ Q then pλ[E] := pλ1 [E]pλ2 [E] ··· pλ`(λ) [E]. This means that f ∈ Λ will also be a Xn function from Q[x1, x2, .
    [Show full text]
  • Symmetric Functions
    Symmetric functions Contents 1 Preliminaries 1 2 Monomial symmetric functions 2 2.1 Multiplying Monomials . 2 2.2 Dominance order . 3 3 Elementary symmetric functions 3 4 Homogeneous symmetric functions: hλ 3 4.1 Generating Functions . 4 5 Power symmetric functions: pλ 4 5.1 Generating Function: . 4 6 Schur functions 5 6.1 Skew Schur functions . 5 6.2 Symmetric group action . 5 6.3 Schur functions . 6 6.4 The Pieri Rule . 6 7 Littlewood-Richardson coefficients 6 7.1 The bi-alternant formula . 7 7.2 Littlewood-Richardson coefficients . 8 7.3 RSK-Correspondence and the Cauchy identity . 8 8 Jacobi-Trudi Determinants 9 1 Preliminaries Let R be a commutative ring and x1, . , xn a set of indeterminants. Let α = (α1, . , αn) be a vector of non-negative integer coefficients. The set of all α form a monoid under addition which is isomorphic to the monoid of all monomials xα under multiplication: α β α+β α α αn x x = x with x = x1 ··· xn . The corresponding algebra is the ring of polynomials, denoted R[x1, . , xn], consisting of all polyno- mials in n variables with coefficients in R. α We write |α| = deg x = α1 + ··· + αn. An element of R[x1, . , xn] of the form X α f(x1, . , xn) = cαx |α|=d is called a homogeneous polynomial of degree d. With set of all such polynomials denoted by d R[x1, . , xn] , R[x1, . , xn] is a graded ring: M d R[x1, . , xn] = R[x1, . , xn] d≥0 1 There is a natural degree-preserving Sn action on polynomials, where elements of the symmetric group act by permuting the variables.
    [Show full text]
  • On Representing (Anti)Symmetric Functions
    On Representing (Anti)Symmetric Functions Marcus Hutter DeepMind & ANU http://www.hutter1.net/ 12 Jun 2020 Abstract Permutation-invariant, -equivariant, and -covariant functions and anti- symmetric functions are important in quantum physics, computer vision, and other disciplines. Applications often require most or all of the following properties: (a) a large class of such functions can be approximated, e.g. all continuous function, (b) only the (anti)symmetric functions can be represented, (c) a fast algorithm for computing the approximation, (d) the representation itself is continuous or differentiable, (e) the architecture is suitable for learning the function from data. (Anti)symmetric neural networks have recently been developed and applied with great success. A few theoretical approximation results have been proven, but many questions are still open, especially for particles in more than one dimension and the anti-symmetric case, which this work focusses on. More concretely, we derive nat- ural polynomial approximations in the symmetric case, and approximations based on a single generalized Slater determinant in the anti-symmetric case. Unlike some previous super-exponential and discontinuous approximations, these seem a more promising basis for future tighter bounds. We provide a complete and explicit uni- versality proof of the Equivariant MultiLayer Perceptron, which implies universality of symmetric MLPs and the FermiNet. Contents 1 Introduction 2 2 Related Work 3 arXiv:2007.15298v1 [cs.NE] 30 Jul 2020 3 One-Dimensional Symmetry 4 4 One-Dimensional AntiSymmetry 8 5 d-dimensional (Anti)Symmetry 11 6 Universality of Neural Networks 13 7 Universal Equivariant Networks 15 8 (Anti)Symmetric Networks 18 9 Discussion 19 A List of Notation 22 Keywords Neural network, approximation, universality, Slater determinant, Vandermonde matrix, equivariance, symmetry, anti-symmetry, symmetric polynomials, polarized basis, multilayer perceptron, continuity, smoothness, ..
    [Show full text]
  • Combinatorics of Symmetric Functions
    Combinatorics of Symmetric Functions Kevin Anderson Supervised by Dr. Hadi Salmasian August 28, 2019 Contents Introduction 3 1 Preliminaries 4 1.1 Partitions . .4 2 Symmetric Polynomials 7 2.1 Elementary Symmetric Polynomials . .7 2.2 Power Sums and Newton's Identities . .9 2.3 Classical Application: The Discriminant . 10 2.4 Fundamental Theorem of Symmetric Polynomials . 12 2.5 Monomial Symmetric Polynomials . 13 2.6 Complete Homogeneous Symmetric Polynomials . 15 2.7 Ring of Symmetric Functions . 18 3 Schur Polynomials 23 3.1 Alternating Polynomials . 23 3.2 Schur Polynomials . 24 3.3 Schur Polynomials and Other Symmetric Polynomial Bases . 28 3.4 Cauchy Formulas . 30 4 Combinatorics of Young Tableaux 33 4.1 Young Tableaux . 33 4.2 Schensted Row-Insertion Algorithm, Knuth-Equivalence . 35 4.3 Sliding, Rectification of Skew Tableaux . 40 4.4 Products of Tableaux . 42 4.5 Robinson-Schensted-Knuth Correspondence . 45 1 Combinatorics of Symmetric Functions Kevin Anderson 5 Combinatorics of Schur Polynomials 50 5.1 Hall Inner Product . 50 5.2 Skew Schur Polynomials . 52 5.3 Schur Polynomials as Sums of Tableau Monomials . 57 5.4 Littlewood-Richardson Rule . 60 6 Generalizations of Schur Polynomials 66 6.1 Generalized Hall Inner Product . 66 6.2 Macdonald Polynomials . 71 6.3 Pieri Formulas for the Macdonald Polynomials . 75 6.4 Skew Macdonald Polynomials . 82 References 87 2 Combinatorics of Symmetric Functions Kevin Anderson Introduction Symmetric polynomials are an important object of study in many fields of modern mathematics. The Schur polynomials (and their generalizations) are a particularly pervasive family of symmetric polynomials which arise naturally in such diverse settings as representation theory, algebraic geom- etry, and mathematical physics.
    [Show full text]
  • Compressed Principal Component Analysis of Non-Gaussian Vectors Marc Mignolet, Christian Soize
    Compressed principal component analysis of non-Gaussian vectors Marc Mignolet, Christian Soize To cite this version: Marc Mignolet, Christian Soize. Compressed principal component analysis of non-Gaussian vectors. SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2020, 8 (4), pp.1261-1286. 10.1137/20M1322029. hal-02966143 HAL Id: hal-02966143 https://hal-upec-upem.archives-ouvertes.fr/hal-02966143 Submitted on 13 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. COMPRESSED PRINCIPAL COMPONENT ANALYSIS OF NON-GAUSSIAN VECTORS MARC MIGNOLET∗ AND CHRISTIAN SOIZEy Abstract. A novel approximate representation of non-Gaussian random vectors is introduced and validated, which can be viewed as a Compressed Principal Component Analysis (CPCA). This representation relies on the eigenvectors of the covariance matrix obtained as in a Principal Compo- nent Analysis (PCA) but expresses the random vector as a linear combination of a random sample of N of these eigenvectors. In this model, the indices of these eigenvectors are independent discrete random variables with probabilities proportional to the corresponding eigenvalues. Moreover, the coefficients of the linear combination are zero mean unit variance random variables. Under these conditions, it is first shown that the covariance matrix of this CPCA matches exactly its PCA coun- terpart independently of the value of N.
    [Show full text]
  • Improved Principal Component Analysis and Linear Regression Classification for Face Recognition
    Accepted Manuscript Improved Principal Component Analysis and Linear regression classification for face recognition Yani Zhu , Chaoyang Zhu , Xiaoxin Li PII: S0165-1684(17)30413-9 DOI: 10.1016/j.sigpro.2017.11.018 Reference: SIGPRO 6669 To appear in: Signal Processing Received date: 15 April 2017 Revised date: 24 November 2017 Accepted date: 25 November 2017 Please cite this article as: Yani Zhu , Chaoyang Zhu , Xiaoxin Li , Improved Principal Component Analysis and Linear regression classification for face recognition, Signal Processing (2017), doi: 10.1016/j.sigpro.2017.11.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Highlights IPCA is used to extract the useful information from the original face images through reducing the dimension of feature vectors to better recognize images. Linear regression classification (LRC) is employed to deal with the problem of face recognition as a linear regression problem. LRC uses the least-square method to decide the class with the minimum reconstruction error. ACCEPTED MANUSCRIPT Improved Principal Component Analysis and Linear regression classification for face recognition Yani Zhua,b*, Chaoyang Zhub, Xiaoxin Lia a Zhejiang University of Technology , Hangzhou, 310014, China b Hangzhou Dianzi University, Hangzhou 310018,China *Corresponding author.
    [Show full text]
  • Combinatorics of Symmetric Functions
    Combinatorics of Symmetric Functions Supervisor: Eric Egge Terms: Winter and Spring of 2018-19 Prerequisite: Previous experience with permutations, partitions, and generating functions. Math 333 (Combinatorial Theory), which is offered in the fall, will be sufficient. However, you could also gain this experience by taking certain courses in Budapest, in an REU, or by doing some supervised independent reading. This project will also involve some occasional computer work, so comfort learning how to use a program like Mathematica or SAGE will be important. However, you do not need to have any previous programming experience. Short Description: We will read a book I've written on the combinatorics of symmetric functions, and you will teach me the subject. If there's time and interest, then we might also work on some open problems. Longer Description: A symmetric polynomial is a polynomial in variables x1; : : : ; xn which is unchanged under every permutation of x1; : : : ; xn. For example, if we swap the variables x2 and x3 in the polynomial x1x2 +x2x3 then we get x1x3 +x2x3. Since this is a different polynomial, x1x2 +x2x3 is not a symmetric polynomial. As another example, consider the polynomial x1x3 + 3x1x2x3. If we swap the variables x1 and x3 then we get the same polynomial. But if we swap x2 and x3 we get a different polynomial, so x1x3 + 3x1x2x3 is also not a symmetric polynomial. On the other hand, if our variables are x1; x2; x3, then 2x1 + 2x2 + 2x3 + 5x1x2 + 5x2x3 + 5x1x3 is a symmetric polynomial, because it is unchanged by every permutation of x1; x2; x3.
    [Show full text]