DOCSLIB.ORG

Majorization and the Schur-Horn Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Majorization and the Schur-Horn Theorem MAJORIZATION AND THE SCHUR-HORN THEOREM A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master of Science In Mathematics University of Regina By Maram Albayyadhi Regina, Saskatchewan January 2013 c Copyright 2013: Maram Albayyadhi UNIVERSITY OF REGINA FACULTY OF GRADUATE STUDIES AND RESEARCH SUPERVISORY AND EXAMINING COMMITTEE Maram Albayyadhi, candidate for the degree of Master of Science in Mathematics, has presented a thesis titled, Majorization and the Schur-Horn Theorem, in an oral examination held on December 18, 2012. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Daryl Hepting, Department of Computer Science Supervisor: Dr. Martin Argerami, Department of Mathematics and Statistics Committee Member: Dr. Douglas Farenick, Department of Mathematics and Statistics Chair of Defense: Dr. Sandra Zilles, Department of Computer Science Abstract We study majorization in Rn and some of its properties. The concept of ma- jorization plays an important role in matrix analysis by producing several useful re- lationships. We find out that there is a strong relationship between majorization and doubly stochastic matrices; this relation has been perfectly described in Birkhoff's Theorem. On the other hand, majorization characterizes the connection between the eigenvalues and the diagonal elements of self adjoint matrices. This relation is summarized in the Schur-Horn Theorem. Using this theorem, we prove versions of Kadison's Carpenter's Theorem. We discuss A. Neumann's extension of the concept of majorization to infinite dimension to that provides a Schur-Horn Theorem in this context. Finally, we detail the work of W. Arveson and R.V. Kadison in proving a strict Schur-Horn Theorem for positive trace-class operators. i Acknowledgments Throughout my studying, I could have not done my project without the support of my professors whom I insist on thanking even though my words cannot adequately express my gratitude. Dr. Martin Argerami, I would like to thank you from the bottom of my heart for all the support and the guidance that you provided for me. Dr. Shaun Fallat and Dr. Remus Floricel, if it were not for your classes, I would not have learned as much about my field. I'm also honored to thank all my amazing colleagues in the math department, especially my friend Angshuman Bhattacharya. To my father Ibrahim Albayyadhi and my Mother Suad Bakkari, words cannot express my love for you. Your prayers, belief in me and encouragement are the main reasons for my success. If I kept on thanking you all of my life, I could not pay you back. To my husband, Dr. Hadi Mufti, you always make it easier for me whenever I face obstacles; you have always been the wind beneath my wings. Finally, I would like to thank the one who kept wiping my tears on the hard days and saying, \Mom ::: don't give up" { my son Yazan. ii Contents Abstract i Acknowledgments ii Table of Contents iii 1 Preliminaries 1 1.1 Majorization . 1 1.2 Doubly Stochastic Matrices . 4 1.3 Doubly Stochastic Matrices and Majorization . 6 2 The Schur-Horn Theorem in the Finite Dimensional Case 14 2.1 The Pythagorean Theorem in Finite Dimension . 14 2.2 The Schur-Horn Theorem in the Finite Dimensional Case . 20 2.3 A Pythagorean Theorem for Finite Doubly Stochastic Matrices . 24 3 The Carpenter Theorem in the Infinite Dimensional Case 27 3.1 The Subspaces K and K? both have Infinite Dimension . 28 3.2 One of the Subspaces K; K? has Finite Dimension . 45 3.3 A Pythagorean Theorem for Infinite Doubly Stochastic Matrices . 47 iii 4 A Schur-Horn Theorem in Infinite Dimensional Case 52 4.1 Majorization in Infinite Dimension . 52 4.2 Neumann's Schur-Horn Theorem . 55 4.3 A Strict Schur-Horn Theorem for Positive Trace-Class Operators . 57 5 Conclusion and Future Work 66 iv Chapter 1 Preliminaries In this chapter we provide some basic information about majorization and some of it properties, which we will use later. The material in this chapter is basic and can be found in many matrix analysis books [4, 7]. 1.1 Majorization n " " " # # # Let x = (x1; ··· ; xn) 2 R . Let x = (x1; ··· ; xn) and x = (x1; ··· ; xn) denote the vector x with it's coordinates rearranged in increasing and decreasing orders respectively. Then " # xi = xn−i+1; 1 ≤ i ≤ n: Definition 1.1. Given x; y in Rn, we say x is majorized by y, denoted by x ≺ y, 1 if k k X # X # xi ≤ yi ; 1 ≤ k ≤ n (1.1) i=1 i=1 and n n X # X # xi = yi : (1.2) i=1 i=1 Pn Example 1.2. If xi 2 [0; 1], and i=1 xi = 1, then we have 1 1 ( ; ··· : ) ≺ (x ; ··· ; x ) ≺ (1; 0; ··· ; 0): n n 1 n Proposition 1.3. If x ≺ y and y ≺ x, then there exists a permutation matrix P such that y = Px. Proof. Assume first that x; y are rearranged in decreasing order,i.e. x1 ≥ x2 ≥ · · · ≥ xn, y1 ≥ y2 · · · ≥ yn. We proceed by induction on k. For k = 1 we get x1 ≤ y1, and y1 ≤ x1 from the majorization equation (1.1), which implies x1 = y1: By induction hypothesis we assume that xi = yi, for i = 1; ··· ; k and we will show that it works up to k + 1. From ( 1.1) x1 + x2 + ··· + xk + xk+1 ≤ y1 + y2 + ··· + yk + yk+1, and by our assumption x1 + x2 + ··· + xk = y1 + y2 + ··· + yk, so if we cancel them we get xk+1 ≤ yk+1, as the roles of x and y can be reversed yk+1 ≤ xk+1, and this implies xk+1 = yk+1: In general, if we write x# and y# for the non-increasing rearrangements, there exist # # permutations P1 and P2 such that x = P1x, y = P2y. By the first part of the proof, # # −1 x = y , i.e. x = P1 P2y. Although majorization is defined through non-increasing rearrangements, it can 2 also be done by non-decreasing ones: Proposition 1.4. If x; y 2 Rn, then x ≺ y if and only if k k X " X " xi ≥ yi ; 1 ≤ k ≤ n (1.3) i=1 i=1 and n n X " X " xi = yi (1.4) i=1 i=1 Pn # Pn # Pn " Proof. For equation(1.4), we have i=1 xi = i=1 yi , since x ≺ y; but i=1 xi = Pn # Pn " Pn # Pn # Pn " Pn " Pn " i=1 xi ; so i=1 xi = i=1 xi = i=1 yi = i=1 yi ; and i=1 xi = i=1 yi : And for equation (1.3), we know that " # xi = xn−i+1; 1 ≤ i ≤ n: So k k X " X # xi = xn−i+1: i=1 i=1 Then k n n n−k X " X # X X # xi = xl = xl − xl i=1 l=n−k+1 l=1 l=1 n−k X # = tr (x) − xl l=1 n−k X # ≥ tr (y) − yl l=1 n n−k n k X X # X # X " = yl − yl = yl = yi : l=1 l=1 l=n−k+1 i=1 3 1.2 Doubly Stochastic Matrices There is a deep relation between majorization and doubly stochastic matrices; we will discuss that relation in this section. Definition 1.5. Let B = (bij) be a square matrix. We say B is doubly stochastic if : • bij ≥ 0 8i; j, Pn • i=1 bij = 1 8j, Pn • j=1 bij = 1 8i. Proposition 1.6. The set of square doubly stochastic matrices is a convex set and it is closed under multiplication and the adjoint operation. But it is not a group. Pr Proof. If t1; ··· ; tr 2 [0; 1] with j=1 tj = 1 and P1; ··· ;Pr are permutations, let Pr A = j=1 tjPj. Clearly every entry of A is non-negative. Also, n n r X X X Akl = tj(Pj)kl k=1 k=1 j=1 r n X X = tj (Pj)kl j=1 k=1 r X = tj = 1; 8l: j=1 Pn A similar computation shows that l=1 Akl = 1 for all k. If A; B are doubly stochastic matrices then AB is also doubly stochastic. Indeed, 4 the sum over the rows is n n n X X X (AB)ij = AikBkj j=1 j=1 k=1 n n X X = Aik Bkj k=1 j=1 n X = Aik = 1: k=1 And the same thing can be done for the columns, which shows that AB is also doubly stochastic. Also it is clear that if A doubly stochastic, then so is its adjoint A∗: The class of doubly stochastic matrices is not a group since not every doubly stochastic matrix is invertible; for example if we take the 2 × 2 matrix, with all its 1 entries equal to 2 , this matrix is doubly stochastic and its determinant is zero. Proposition 1.7. Every permutation matrix is doubly stochastic and is an extreme point of the convex set of all doubly stochastic matrices. Proof. A permutation matrix has exactly one entry +1 in each row and in each column and all other entries are zero. So it is doubly stochastic. Now let A = α1B + α2C, with A a permutation matrix such that α1; α2 2 (0; 1), α1 + α2 = 1, and B; C are doubly stochastic matrices. Then every entry of B; C that corresponds to the zero element aij = 0 of A must be zero; indeed, 0 = aij = α1bij + α2cij.
Load more

Recommended publications
  • Arxiv:1306.4805V3 [Math.OC] 6 Feb 2015 Used Greedy Techniques to Reorder Matrices
    CONVEX RELAXATIONS FOR PERMUTATION PROBLEMS FAJWEL FOGEL, RODOLPHE JENATTON, FRANCIS BACH, AND ALEXANDRE D’ASPREMONT ABSTRACT. Seriation seeks to reconstruct a linear order between variables using unsorted, pairwise similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We write seri- ation as an optimization problem by proving the equivalence between the seriation and combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic minimization problem over permutations). The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we derive several convex relax- ations for 2-SUM to improve the robustness of seriation solutions in noisy settings. These convex relaxations also allow us to impose structural constraints on the solution, hence solve semi-supervised seriation problems. We derive new approximation bounds for some of these relaxations and present numerical experiments on archeological data, Markov chains and DNA assembly from shotgun gene sequencing data. 1. INTRODUCTION We study optimization problems written over the set of permutations. While the relaxation techniques discussed in what follows are applicable to a much more general setting, most of the paper is centered on the seriation problem: we are given a similarity matrix between a set of n variables and assume that the variables can be ordered along a chain, where the similarity between variables decreases with their distance within this chain. The seriation problem seeks to reconstruct this linear ordering based on unsorted, possibly noisy, pairwise similarity information. This problem has its roots in archeology [Robinson, 1951] and also has direct applications in e.g.
    [Show full text]
  • Doubly Stochastic Matrices Whose Powers Eventually Stop
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 330 (2001) 25–30 www.elsevier.com/locate/laa Doubly stochastic matrices whose powers eventually stopୋ Suk-Geun Hwang a,∗, Sung-Soo Pyo b aDepartment of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea bCombinatorial and Computational Mathematics Center, Pohang University of Science and Technology, Pohang, South Korea Received 22 June 2000; accepted 14 November 2000 Submitted by R.A. Brualdi Abstract In this note we characterize doubly stochastic matrices A whose powers A, A2,A3,... + eventually stop, i.e., Ap = Ap 1 =···for some positive integer p. The characterization en- ables us to determine the set of all such matrices. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A51 Keywords: Doubly stochastic matrix; J-potent 1. Introduction Let R denote the real field. For positive integers m, n,letRm×n denote the set of all m × n matrices with real entries. As usual let Rn denote the set Rn×1.We call the members of Rn the n-vectors. The n-vector of 1’s is denoted by e,andthe identity matrix of order n is denoted by In. For two matrices A, B of the same size, let A B denote that all the entries of A − B are nonnegative. A matrix A is called nonnegative if A O. A nonnegative square matrix is called a doubly stochastic matrix if all of its row sums and column sums equal 1.
    [Show full text]
  • Notes on Birkhoff-Von Neumann Decomposition of Doubly Stochastic Matrices Fanny Dufossé, Bora Uçar
    Notes on Birkhoff-von Neumann decomposition of doubly stochastic matrices Fanny Dufossé, Bora Uçar To cite this version: Fanny Dufossé, Bora Uçar. Notes on Birkhoff-von Neumann decomposition of doubly stochastic matri- ces. Linear Algebra and its Applications, Elsevier, 2016, 497, pp.108–115. 10.1016/j.laa.2016.02.023. hal-01270331v6 HAL Id: hal-01270331 https://hal.inria.fr/hal-01270331v6 Submitted on 23 Apr 2016 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. Notes on Birkhoff-von Neumann decomposition of doubly stochastic matrices Fanny Dufoss´ea, Bora U¸carb,∗ aInria Lille, Nord Europe, 59650, Villeneuve d'Ascq, France bLIP, UMR5668 (CNRS - ENS Lyon - UCBL - Universit´ede Lyon - INRIA), Lyon, France Abstract Birkhoff-von Neumann (BvN) decomposition of doubly stochastic matrices ex- presses a double stochastic matrix as a convex combination of a number of permutation matrices. There are known upper and lower bounds for the num- ber of permutation matrices that take part in the BvN decomposition of a given doubly stochastic matrix. We investigate the problem of computing a decom- position with the minimum number of permutation matrices and show that the associated decision problem is strongly NP-complete.
    [Show full text]
  • On Matrices with a Doubly Stochastic Pattern
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICALANALYSISAND APPLICATIONS 34,648-652(1971) On Matrices with a Doubly Stochastic Pattern DAVID LONDON Technion-Israel Institute of Technology, Haifa Submitted by Ky Fan 1. INTRODUCTION In [7] Sinkhorn proved that if A is a positive square matrix, then there exist two diagonal matrices D, = {@r,..., d$) and D, = {dj2),..., di2r) with positive entries such that D,AD, is doubly stochastic. This problem was studied also by Marcus and Newman [3], Maxfield and Mint [4] and Menon [5]. Later Sinkhorn and Knopp [8] considered the same problem for A non- negative. Using a limit process of alternately normalizing the rows and columns sums of A, they obtained a necessary and sufficient condition for the existence of D, and D, such that D,AD, is doubly stochastic. Brualdi, Parter and Schneider [l] obtained the same theorem by a quite different method using spectral properties of some nonlinear operators. In this note we give a new proof of the same theorem. We introduce an extremal problem, and from the existence of a solution to this problem we derive the existence of D, and D, . This method yields also a variational characterization for JJTC1(din dj2’), which can be applied to obtain bounds for this quantity. We note that bounds for I7y-r (d,!‘) dj”)) may be of interest in connection with inequalities for the permanent of doubly stochastic matrices [3]. 2. PRELIMINARIES Let A = (aij) be an 12x n nonnegative matrix; that is, aij > 0 for i,j = 1,***, n.
    [Show full text]
  • An Inequality for Doubly Stochastic Matrices*
    JOURNAL OF RESEARCH of the National Bureau of Standards-B. Mathematical Sciences Vol. 80B, No.4, October-December 1976 An Inequality for Doubly Stochastic Matrices* Charles R. Johnson** and R. Bruce Kellogg** Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 (June 3D, 1976) Interrelated inequalities involving doubly stochastic matrices are presented. For example, if B is an n by n doubly stochasti c matrix, x any nonnegative vector and y = Bx, the n XIX,· •• ,x" :0:::; YIY" •• y ... Also, if A is an n by n nonnegotive matrix and D and E are positive diagonal matrices such that B = DAE is doubly stochasti c, the n det DE ;:::: p(A) ... , where p (A) is the Perron· Frobenius eigenvalue of A. The relationship between these two inequalities is exhibited. Key words: Diagonal scaling; doubly stochasti c matrix; P erron·Frobenius eigenvalue. n An n by n entry·wise nonnegative matrix B = (b i;) is called row (column) stochastic if l bi ; = 1 ;= 1 for all i = 1,. ',n (~l bij = 1 for all j = 1,' . ',n ). If B is simultaneously row and column stochastic then B is said to be doubly stochastic. We shall denote the Perron·Frobenius (maximal) eigenvalue of an arbitrary n by n entry·wise nonnegative matrix A by p (A). Of course, if A is stochastic, p (A) = 1. It is known precisely which n by n nonnegative matrices may be diagonally scaled by positive diagonal matrices D, E so that (1) B=DAE is doubly stochastic. If there is such a pair D, E, we shall say that A has property (").
    [Show full text]
  • Doubly Stochastic Normalization for Spectral Clustering
    Doubly Stochastic Normalization for Spectral Clustering Ron Zass and Amnon Shashua ∗ Abstract In this paper we focus on the issue of normalization of the affinity matrix in spec- tral clustering. We show that the difference between N-cuts and Ratio-cuts is in the error measure being used (relative-entropy versus L1 norm) in finding the closest doubly-stochastic matrix to the input affinity matrix. We then develop a scheme for finding the optimal, under Frobenius norm, doubly-stochastic approxi- mation using Von-Neumann’s successive projections lemma. The new normaliza- tion scheme is simple and efficient and provides superior clustering performance over many of the standardized tests. 1 Introduction The problem of partitioning data points into a number of distinct sets, known as the clustering problem, is central in data analysis and machine learning. Typically, a graph-theoretic approach to clustering starts with a measure of pairwise affinity Kij measuring the degree of similarity between points xi, xj, followed by a normalization step, followed by the extraction of the leading eigenvectors which form an embedded coordinate system from which the partitioning is readily available. In this domain there are three principle dimensions which make a successful clustering: (i) the affinity measure, (ii) the normalization of the affinity matrix, and (iii) the particular clustering algorithm. Common practice indicates that the former two are largely responsible for the performance whereas the particulars of the clustering process itself have a relatively smaller impact on the performance. In this paper we focus on the normalization of the affinity matrix. We first show that the existing popular methods Ratio-cut (cf.
    [Show full text]
  • Majorization Results for Zeros of Orthogonal Polynomials
    Majorization results for zeros of orthogonal polynomials ∗ Walter Van Assche KU Leuven September 4, 2018 Abstract We show that the zeros of consecutive orthogonal polynomials pn and pn−1 are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for the zeros of pn (1) and the associated polynomial pn−1 and for the zeros of the polynomial obtained by deleting the kth row and column (1 ≤ k ≤ n) in the corresponding Jacobi matrix. 1 Introduction 1.1 Majorization Recently S.M. Malamud [3, 4] and R. Pereira [6] have given a nice extension of the Gauss- Lucas theorem that the zeros of the derivative p′ of a polynomial p lie in the convex hull of the roots of p. They both used the theory of majorization of sequences to show that if x =(x1, x2,...,xn) are the zeros of a polynomial p of degree n, and y =(y1,...,yn−1) are the zeros of its derivative p′, then there exists a doubly stochastic (n − 1) × n matrix S such that y = Sx ([4, Thm. 4.7], [6, Thm. 5.4]). A rectangular (n − 1) × n matrix S =(si,j) is said to be doubly stochastic if si,j ≥ 0 and arXiv:1607.03251v1 [math.CA] 12 Jul 2016 n n− 1 n − 1 s =1, s = . i,j i,j n j=1 i=1 X X In this paper we will use the notion of majorization to rephrase some known results for the zeros of orthogonal polynomials on the real line.
    [Show full text]
  • The Symbiotic Relationship of Combinatorics and Matrix Theory
    The Symbiotic Relationship of Combinatorics and Matrix Theory Richard A. Brualdi* Department of Mathematics University of Wisconsin Madison. Wisconsin 53 706 Submitted by F. Uhlig ABSTRACT This article demonstrates the mutually beneficial relationship that exists between combinatorics and matrix theory. 3 1. INTRODUCTION According to The Random House College Dictionary (revised edition, 1984) the word symbiosis is defined as follows: symbiosis: the living together of two dissimilar organisms, esp. when this association is mutually beneficial. In applying, as I am, this definition to the relationship between combinatorics and matrix theory, I would have preferred that the qualifying word “dissimilar” was omitted. Are combinatorics and matrix theory really dissimilar subjects? On the other hand I very much like the inclusion of the phrase “when the association is mutually beneficial” because I believe the development of each of these subjects shows that combinatorics and matrix theory have a mutually beneficial relationship. The fruits of this mutually beneficial relationship is the subject that I call combinatorial matrix theory or, for brevity here, CMT. Viewed broadly, as I do, CMT is an amazingly rich and diverse subject. If it is * Research partially supported by National Science Foundation grant DMS-8901445 and National Security Agency grant MDA904-89 H-2060. LZNEAR ALGEBRA AND ITS APPLZCATZONS 162-164:65-105 (1992) 65 0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010 0024.3795/92/$5.00 66 RICHARD A. BRUALDI combinatorial and uses matrix theory r in its formulation or proof, it’s CMT. If it is matrix theory and has a combinatorial component to it, it’s CMT.
    [Show full text]
  • RANDOM DOUBLY STOCHASTIC MATRICES: the CIRCULAR LAW 3 Easily Be Bounded by a Polynomial in N
    The Annals of Probability 2014, Vol. 42, No. 3, 1161–1196 DOI: 10.1214/13-AOP877 c Institute of Mathematical Statistics, 2014 RANDOM DOUBLY STOCHASTIC MATRICES: THE CIRCULAR LAW By Hoi H. Nguyen1 Ohio State University Let X be a matrix sampled uniformly from the set of doubly stochastic matrices of size n n. We show that the empirical spectral × distribution of the normalized matrix √n(X EX) converges almost − surely to the circular law. This confirms a conjecture of Chatterjee, Diaconis and Sly. 1. Introduction. Let M be a matrix of size n n, and let λ , . , λ be × 1 n the eigenvalues of M. The empirical spectral distribution (ESD) µM of M is defined as 1 µ := δ . M n λi i n X≤ We also define µcir as the uniform distribution over the unit disk 1 µ (s,t) := mes( z 1; (z) s, (z) t). cir π | |≤ ℜ ≤ ℑ ≤ Given a random n n matrix M, an important problem in random matrix theory is to study the× limiting distribution of the empirical spectral distri- bution as n tends to infinity. We consider one of the simplest random matrix ensembles, when the entries of M are i.i.d. copies of the random variable ξ. When ξ is a standard complex Gaussian random variable, M can be viewed as a random matrix drawn from the probability distribution P(dM)= 1 tr(MM ∗) 2 e− dM on the set of complex n n matrices. This is known as arXiv:1205.0843v3 [math.CO] 27 Mar 2014 πn × the complex Ginibre ensemble.
    [Show full text]
  • On Constructing Orthogonal Generalized Doubly Stochastic
    On Constructing Orthogonal Generalized Doubly Stochastic Matrices Gianluca Oderda∗, Alicja Smoktunowicz†and Ryszard Kozera‡ September 21, 2018 Abstract A real quadratic matrix is generalized doubly stochastic (g.d.s.) if all of its row sums and column sums equal one. We propose numeri- cally stable methods for generating such matrices having possibly or- thogonality property or/and satisfying Yang-Baxter equation (YBE). Additionally, an inverse eigenvalue problem for finding orthogonal generalized doubly stochastic matrices with prescribed eigenvalues is solved here. The tests performed in MATLAB illustrate our pro- posed algorithms and demonstrate their useful numerical properties. AMS Subj. Classification: 15B10, 15B51, 65F25, 65F15. Keywords: stochastic matrix, orthogonal matrix, Householder QR de- composition, eigenvalues, condition number. 1 Introduction We propose efficient algorithms for constructing generalized doubly stochas- n n arXiv:1809.07618v1 [cs.NA] 20 Sep 2018 tic matrix A R × . Recall that A is a generalized doubly stochastic matrix ∈ (g.d.s.) if all of its row sums and column sums equal one. Let In denote the ∗Ersel Asset Management SGR S.p.A., Piazza Solferino, 11, 10121 Torino, Italy, e- mail: [email protected] †Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland, e-mail: [email protected] ‡Faculty of Applied Informatics and Mathematics, Warsaw University of Life Sciences - SGGW, Nowoursynowska str. 159, 02-776 Warsaw, Poland and Department of Com- puter Science and Software Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Perth, Australia, e-mail: [email protected] 1 T n n n n n identity matrix and e = (1, 1,..., 1) = ei R , where ei × i=1 ∈ { }i=1 forms a canonical basis in Rn.
    [Show full text]
  • Package 'Spbsampling'
    Package ‘Spbsampling’ August 24, 2020 Title Spatially Balanced Sampling Version 1.3.4 Description Selection of spatially balanced samples. In particular, the implemented sampling de- signs allow to select probability samples well spread over the population of interest, in any di- mension and using any distance function (e.g. Euclidean distance, Manhattan dis- tance). For more details, Benedetti R and Piersi- moni F (2017) <doi:10.1002/bimj.201600194> and Benedetti R and Piersi- moni F (2017) <arXiv:1710.09116>. The implementa- tion has been done in C++ through the use of 'Rcpp' and 'RcppArmadillo'. Depends R (>= 3.1) License GPL-3 Encoding UTF-8 LazyData true LinkingTo Rcpp, RcppArmadillo Imports Rcpp RoxygenNote 7.1.1 NeedsCompilation yes Author Francesco Pantalone [aut, cre], Roberto Benedetti [aut], Federica Piersimoni [aut] Maintainer Francesco Pantalone <[email protected]> Repository CRAN Date/Publication 2020-08-24 14:00:02 UTC R topics documented: hpwd ............................................2 income_emilia . .3 lucas_abruzzo . .4 pwd .............................................5 sbi..............................................6 1 2 hpwd simul1 . .7 simul2 . .8 simul3 . .9 Spbsampling . 10 stprod . 10 stsum . 11 swd ............................................. 12 Index 15 hpwd Heuristic Product Within Distance (Spatially Balanced Sampling De- sign) Description Selects spatially balanced samples through the use of Heuristic Product Within Distance design (HPWD). To have constant inclusion probabilities πi = n=N, where n is sample size and N is population size, the distance matrix has to be standardized with function stprod. Usage hpwd(dis, n, beta = 10, nrepl = 1L) Arguments dis A distance matrix NxN that specifies how far all the pairs of units in the popu- lation are.
    [Show full text]
  • Learning Permutations with Exponential Weights
    Learning permutations with exponential weights David P. Helmbold and Manfred K. Warmuth Computer Science Department University of California, Santa Cruz {dph | manfred}@cse.ucsc.edu Abstract. We give an algorithm for learning a permutation on-line. The algorithm maintains its uncertainty about the target permutation as a doubly stochastic matrix. This matrix is updated by multiplying the current matrix entries by exponential factors. These factors destroy the doubly stochastic property of the matrix and an iterative procedure is needed to re-normalize the rows and columns. Even though the result of the normalization procedure does not have a closed form, we can still bound the additional loss of our algorithm over the loss of the best permutation chosen in hindsight. 1 Introduction Finding a good permutation is a key aspect of many problems such as the rank- ing of search results or matching workers to tasks. In this paper we present an efficient and effective algorithm for learning permutations in the on-line setting called PermELearn. In each trial, the algorithm probabilistically chooses a per- mutation and incurs a loss based on how appropriate the permutation was for that trial. The goal of the algorithm is to have low total expected loss compared to the best permutation chosen in hindsight for the whole sequence of trials. Since there are n! permutations on n elements, it is infeasible to simply treat each permutation as an expert and apply Weighted Majority or another expert algorithm. Our solution is to implicitly represent the weights of the n!permu- tations with a more concise representation.
    [Show full text]