DOCSLIB.ORG

Nonnegative Matrix Factorization Nicolas Gillis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Nonnegative Matrix Factorization Nicolas Gillis Université catholique de Louvain École polytechnique de Louvain Département d’ingénierie mathématique Nonnegative Matrix Factorization Complexity, Algorithms and Applications Nicolas Gillis Thesis submitted in partial fulfillment of the requirements for the degree of Docteur en Sciences de l’Ingénieur Dissertation committee: Prof. Philippe Lefèvre (UCL, President) Prof. François Glineur (UCL, Advisor) Prof. Pierre-Antoine Absil (UCL) Prof. Paul Van Dooren (UCL) Prof. Samuel Fiorini (Université libre de Bruxelles, Belgium) Prof. Didier Henrion (LAAS-CNRS, University of Toulouse, France) Prof. Sabine Van Huffel (Katholieke Universiteit Leuven, Belgium) Prof. Stephen Vavasis (University of Waterloo, Canada) February 2011 Acknowledgments I would like to thank my advisor, François Glineur. I have enjoyed working with him since my master’s thesis, and he has constantly supported and encouraged me for my different projects. I also thank Robert Plemmons who warmly greeted me at Wake Forest Uni- versity during the Fall semester 2009. It was a pleasure working with him, Paùl Pauca and their colleagues. I am grateful to my support committee, Didier Henrion and Paul Van Dooren, with whom I had the chance to discuss my problems, and benefit from their experience. I also thank the members of my defense committee Pierre-Antoine Absil, Samuel Fiorini, Sabine Van Huffel, Stephen Vavasis, and the chair Philippe Lefèvre. I am also grateful to the other people who have helped me in various ways, and have interacted with this work: Michael Berry, Mathieu Van Vyve, Raphaël Jungers, Chia-Tche Chang, Quentin Rentmeesters, Santanu Dey, Lau- rence Wolsey, and others. This experience could not have been what it has without the warm am- biance at CORE. I thank my office mates, Gauthier, Joachim, Salome, Grégory, Stéphane, Laurent, Alexis, Vladislav and Julie ; my colleagues Robert, Olivier, Rafael, Sylvette, Joël, Peter, Maia, Margherita, Olivier’, Jean-Sébastien, Clau- dia, Mikel and many others. I also thank the precious help of the CORE administratives. As a F.R.S.-FNRS research fellow, I thank the Fonds de la recherche scien- tifique (F.R.S.-FNRS) for their administrative and financial support. Last but not least, I thank my family and my friends. I dedicate this thesis to my grandparents. i ii Abstract Linear dimensionality reduction techniques such as principal component anal- ysis are powerful tools for the analysis of high-dimensional data. In this thesis, we explore a closely related problem, namely nonnegative matrix factorization (NMF), a low-rank matrix approximation problem with nonnegativity con- straints. More precisely, we seek to approximate a given nonnegative matrix with the product of two low-rank nonnegative matrices. These nonnegative fac- tors can be interpreted in the same way as the data, e.g., as images (described by pixel intensities) or texts (represented by vectors of word counts), and lead to an additive and sparse representation. However, they render the problem much more difficult to solve (i.e., NP-hard). A first goal of this thesis is to study theoretical issues related to NMF. In particular, we make connections with well-known problems in graph theory, combinatorial optimization and computational geometry. We also study com- putational complexity issues and show, using reductions from the maximum- edge biclique problem, NP-hardness of several low-rank matrix approximation problems, including the rank-one subproblems arising in NMF, a problem in- volving underapproximation constraints (NMU) and the unconstrained version of the factorization problem where some data is missing or unknown. Our second goal is to improve existing techniques and develop new tools for the analysis of nonnegative data. We propose accelerated variants of sev- eral NMF algorithms based on a careful analysis of their computational cost. We also introduce a multilevel approach to speed up their initial convergence. Finally, a new greedy heuristic based on NMU is presented and used for the analysis of hyperspectral images, in which each pixel is measured along hun- dreds of wavelengths, which allows for example spectroscopy of satellite images. iii iv Table of contents Acknowledgments i Abstract iii Table of Contents vi Notation vii 1 Introduction 1 Thesis outline and related publications . 5 2 Preliminaries 9 2.1 Optimization ............................ 9 2.2 Low-RankMatrixApproximation. 12 2.3 ComputationalComplexity . 16 3 Nonnegative Rank 21 3.1 ComputationalComplexity . 22 3.2 ExtendedFormulations . 25 3.3 RestrictedNonnegativeRank . 26 3.4 Lower Bounds for the Nonnegative Rank . 38 3.5 Applications: Slack Matrices and Linear EDM’s . 45 3.6 Improvements using the Matrix Transpose . 57 4 Algorithms for NMF 63 4.1 Three Existing Algorithms: MU, ANLS and HALS . 65 4.2 AcceleratedMUandHALSAlgorithms . 73 4.3 AMultilevelApproach. 90 5 Nonnegative Factorization 109 5.1 Rank-oneNonnegativeFactorization . 110 5.2 StationaryPoints. .. .. .. .. .. .. .. .. .. .. .. 114 v TABLE OF CONTENTS 5.3 BicliqueFindingAlgorithm . 119 5.4 MUforNonnegativeFactorization . 126 6 Nonnegative Matrix Underapproximation 135 6.1 Sparsity ............................... 137 6.2 RelatedWork............................ 138 6.3 ComputationalComplexity . 139 6.4 ConvexFormulations. 141 6.5 Lagrangian Relaxation based Algorithm for NMU . 148 6.6 NMUvssparseNMF........................ 153 7 Hyperspectral Data Analysis using NMU 167 7.1 TheIdealCase ........................... 169 7.2 TheNon-IdealCase ........................ 175 7.3 ℓ0-Pseudo-Norm Minimization and ℓ1-NormRelaxation . 176 7.4 NumericalExperiments . 180 8 Weights and Missing Data 199 8.1 PreviousResults .......................... 201 8.2 WeightedLow-RankApproximation . 204 8.3 Low-Rank Approximation with Missing Data . 208 Conclusion 215 Bibliography 220 Appendix 232 A Active set methods for NNLS 233 vi Notation Scalars, Vectors, Matrices R set of real numbers R0 set of nonzero real numbers R+ set of nonnegative real numbers Rn set of real column vectors of dimension n m n R × set of real matrices of dimension m-by-n n R+ set of nonnegative real column vectors of dimension n m n R+ × set of nonnegative real matrices of dimension m-by-n Norms n n . 1 ℓ1-norm, x 1 = i=1 xi , x R || || || || | | n∈ 2 n . 2 vector ℓ2-norm, x 2 = i=1 xi , x R || || ||P|| ∈ m n n matrix ℓ2-norm, A 2 = maxx R , x 2=1 Ax 2, A R × || || pP ∈ || || || ||n ∈ . vector ℓ -norm, x = max1 i n xi , x R || ||∞ ∞ || ||∞ ≤ ≤ | | ∈ . ℓ -‘norm’, x = i x =0 , x Rm || ||0 0 || ||0 { | i 6 } ∈ m n 2 m n . F Frobenius norm, A F = i=1 j=1 Aij , A R × || || || || ∈ qPvii P TABLE OF CONTENTS Functions on Matrices m n m n ., . scalar product, A, B = A B ,A,B R × h i h i i=1 j=1 ij ij ∈ σ (.) ith singular values of a matrix, in non-decreasing order i P P rank(.) rank of a matrix rank+(.) nonnegative rank of a nonnegative matrix rank+∗ (.) restricted nonnegative rank of a nonnegative matrix conv(.) convex hull of the columns of a matrix col(.) column space of a matrix th Ai: or A(i, :) i row of A th A:j or A(:, j) j column of A Aij or A(i, j) entry at position (i, j) of A A(I, J) submatrix of A with row (resp. column) indices in I (resp. J) component-wise multiplication, (A B) = A B ◦ ◦ ij ij ij [ . ] [A] Aij [ . ] component-wise division, [B] = B ij ij T T (.) transpose of a matrix, (A )ij = Aji Miscellaneous 1m vector of all ones of dimension m 1m n matrix of all ones of dimension m-by-n × 0m n matrix of zeros of dimension m-by-n I × identity matrix of dimension n .n = is equal by definition a:b set a,a +1,...,b 1,b (for a and b integers with a b) f It is{ the the gradient− of} the function f ≤ ∇2f It is the hessian of the function f ∇ . x is the smallest integer greater or equal to x R ⌈ ⌉ ⌈ ⌉ ∈ . x is the largest integer smaller or equal to x R ⌊ ⌋ ⌊substraction⌋ of two sets, R S is the set of elements∈ in R not in S \. cardinality of a set, S is the\ number of elements in S ¯.| | complement of a set,| for| S R, S¯ = R S ⊂ \ n supp(.) support (set of non-zero entries), supp(x)= 1 i n xi =0 , x R supp(.) sparsity pattern (set of zero entries, complement{ ≤ of≤ the| support)6 } ∈ viii TABLE OF CONTENTS Acronyms PCA principal component analysis (p. 1) SPCA sparse principal component analysis (p. 1) WLRA weighted low-rank approximation (p. 1 and p. 200) ICA independent component analysis (p. 2) NMF nonnegative matrix factorization (p. 3) LP linear programming (p. 10) QP quadratic programming (p. 10) SOCP second order cone programming (p. 10) SDP semidefinite positive (p. 10) GP geometric programming (p. 10 and p. 143) SVD singular value decomposition (p. 13) LRA low-rank matrix approximation (p. 12) LRA-1 rank-one matrix approximation (p. 13) PSV principal singular vectors (p. 14) PC principal component (p. 14) Exact NMF exact nonnegative matrix factorization (p. 22) IS intermediate simplex (p. 22) RNR restricted nonnegative rank (p. 26) NPP nested polytopes problem (p. 27) EDM Euclidean distance matrix (p. 48) NNLS nonnegative least squares (p. 64) MU multiplicative updates (p. 65) ANLS alternating nonnegative least squares (p. 68) HALS hierarchical alternating least squares (p. 70) R1NF rank-one nonnegative factorization (p. 110) NF nonnegative factorization (p. 127) NMU nonnegative matrix underapproximation (p. 135) NMU-1 rank-one nonnegative matrix underapproximation (p. 139) PCAMD PCA with missing data (p. 199) SFM structure from motion (p. 203) LRAMD low-rank matrix approximation with missing data (p. 208) ix TABLE OF CONTENTS x Chapter 1 Introduction Approximating a matrix with one of lower rank is a key problem in data anal- ysis, and is widely used for linear dimensionality reduction. Typically, we are m n given a matrix M R × , where each of the n columns represents an ele- ment of a dataset in∈ an m-dimensional space.
Load more

Recommended publications
  • Parametrizations of K-Nonnegative Matrices
    Parametrizations of k-Nonnegative Matrices Anna Brosowsky, Neeraja Kulkarni, Alex Mason, Joe Suk, Ewin Tang∗ October 2, 2017 Abstract Totally nonnegative (positive) matrices are matrices whose minors are all nonnegative (positive). We generalize the notion of total nonnegativity, as follows. A k-nonnegative (resp. k-positive) matrix has all minors of size k or less nonnegative (resp. positive). We give a generating set for the semigroup of k-nonnegative matrices, as well as relations for certain special cases, i.e. the k = n − 1 and k = n − 2 unitriangular cases. In the above two cases, we find that the set of k-nonnegative matrices can be partitioned into cells, analogous to the Bruhat cells of totally nonnegative matrices, based on their factorizations into generators. We will show that these cells, like the Bruhat cells, are homeomorphic to open balls, and we prove some results about the topological structure of the closure of these cells, and in fact, in the latter case, the cells form a Bruhat-like CW complex. We also give a family of minimal k-positivity tests which form sub-cluster algebras of the total positivity test cluster algebra. We describe ways to jump between these tests, and give an alternate description of some tests as double wiring diagrams. 1 Introduction A totally nonnegative (respectively totally positive) matrix is a matrix whose minors are all nonnegative (respectively positive). Total positivity and nonnegativity are well-studied phenomena and arise in areas such as planar networks, combinatorics, dynamics, statistics and probability. The study of total positivity and total nonnegativity admit many varied applications, some of which are explored in “Totally Nonnegative Matrices” by Fallat and Johnson [5].
    [Show full text]
  • Crystals and Total Positivity on Orientable Surfaces 3
    CRYSTALS AND TOTAL POSITIVITY ON ORIENTABLE SURFACES THOMAS LAM AND PAVLO PYLYAVSKYY Abstract. We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated. We describe in terms of our model the crystal structure and R-matrix of the affine geometric crystal of products of symmetric and dual symmetric powers of type A. Local realizations of the R-matrix and crystal actions are used to construct a double affine geometric crystal on a torus, generalizing the commutation result of Kajiwara-Noumi- Yamada [KNY] and an observation of Berenstein-Kazhdan [BK07b]. We show that our model on a cylinder gives a decomposition and parametrization of the totally nonnegative part of the rational unipotent loop group of GLn. Contents 1. Introduction 3 1.1. Networks on orientable surfaces 3 1.2. Factorizations and parametrizations of totally positive matrices 4 1.3. Crystals and networks 5 1.4. Measurements and moves 6 1.5. Symmetric functions and loop symmetric functions 7 1.6. Comparison of examples 7 Part 1. Boundary measurements on oriented surfaces 9 2. Networks and measurements 9 2.1. Oriented networks on surfaces 9 2.2. Polygon representation of oriented surfaces 9 2.3. Highway paths and cycles 10 arXiv:1008.1949v1 [math.CO] 11 Aug 2010 2.4. Boundary and cycle measurements 10 2.5. Torus with one vertex 11 2.6. Flows and intersection products in homology 12 2.7. Polynomiality 14 2.8. Rationality 15 2.9.
    [Show full text]
  • A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality
    Electrical and Computer Engineering Publications Electrical and Computer Engineering 6-15-2017 A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality Songtao Lu Iowa State University, [email protected] Mingyi Hong Iowa State University Zhengdao Wang Iowa State University, [email protected] Follow this and additional works at: https://lib.dr.iastate.edu/ece_pubs Part of the Industrial Technology Commons, and the Signal Processing Commons The complete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ ece_pubs/162. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Article is brought to you for free and open access by the Electrical and Computer Engineering at Iowa State University Digital Repository. It has been accepted for inclusion in Electrical and Computer Engineering Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality Abstract Symmetric nonnegative matrix factorization (SymNMF) has important applications in data analytics problems such as document clustering, community detection, and image segmentation. In this paper, we propose a novel nonconvex variable splitting method for solving SymNMF. The proposed algorithm is guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points of the nonconvex SymNMF problem. Furthermore, it achieves a global sublinear convergence rate. We also show that the algorithm can be efficiently implemented in parallel. Further, sufficient conditions ear provided that guarantee the global and local optimality of the obtained solutions.
    [Show full text]
  • Minimal Rank Decoupling of Full-Lattice CMV Operators With
    MINIMAL RANK DECOUPLING OF FULL-LATTICE CMV OPERATORS WITH SCALAR- AND MATRIX-VALUED VERBLUNSKY COEFFICIENTS STEPHEN CLARK, FRITZ GESZTESY, AND MAXIM ZINCHENKO Abstract. Relations between half- and full-lattice CMV operators with scalar- and matrix-valued Verblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV oper- ators into a direct sum of two half-lattice CMV operators by a perturbation of minimal rank is studied. Contrary to the Jacobi case, decoupling a full-lattice CMV matrix by changing one of the Verblunsky coefficients results in a perturbation of twice the minimal rank. The explicit form for the minimal rank perturbation and the resulting two half-lattice CMV matrices are obtained. In addition, formulas relating the Weyl–Titchmarsh m-functions (resp., matrices) associated with the involved CMV operators and their Green’s functions (resp., matrices) are derived. 1. Introduction CMV operators are a special class of unitary semi-infinite or doubly-infinite five-diagonal matrices which received enormous attention in recent years. We refer to (2.8) and (3.18) for the explicit form of doubly infinite CMV operators on Z in the case of scalar, respectively, matrix-valued Verblunsky coefficients. For the corresponding half-lattice CMV operators we refer to (2.16) and (3.26). The actual history of CMV operators (with scalar Verblunsky coefficients) is somewhat intrigu- ing: The corresponding unitary semi-infinite five-diagonal matrices were first introduced in 1991 by Bunse–Gerstner and Elsner [15], and subsequently discussed in detail by Watkins [82] in 1993 (cf. the discussion in Simon [73]). They were subsequently rediscovered by Cantero, Moral, and Vel´azquez (CMV) in [17].
    [Show full text]
  • Random Matrices, Magic Squares and Matching Polynomials
    Random Matrices, Magic Squares and Matching Polynomials Persi Diaconis Alex Gamburd∗ Departments of Mathematics and Statistics Department of Mathematics Stanford University, Stanford, CA 94305 Stanford University, Stanford, CA 94305 [email protected] [email protected] Submitted: Jul 22, 2003; Accepted: Dec 23, 2003; Published: Jun 3, 2004 MR Subject Classifications: 05A15, 05E05, 05E10, 05E35, 11M06, 15A52, 60B11, 60B15 Dedicated to Richard Stanley on the occasion of his 60th birthday Abstract Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zeta- function, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices (magic squares). Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed. 1 Introduction Two noteworthy developments took place recently in Random Matrix Theory. One is the discovery and exploitation of the connections between eigenvalue statistics and the longest- increasing subsequence problem in enumerative combinatorics [1, 4, 5, 47, 59]; another is the outburst of interest in characteristic polynomials of Random Matrices and associated global statistics, particularly in connection with the moments of the Riemann zeta function and other L-functions [41, 14, 35, 36, 15, 16]. The purpose of this paper is to point out some connections between the distribution of the coefficients of characteristic polynomials of random matrices and some classical problems in enumerative combinatorics.
    [Show full text]
  • Z Matrices, Linear Transformations, and Tensors
    Z matrices, linear transformations, and tensors M. Seetharama Gowda Department of Mathematics and Statistics University of Maryland, Baltimore County Baltimore, Maryland, USA [email protected] *************** International Conference on Tensors, Matrices, and their Applications Tianjin, China May 21-24, 2016 Z matrices, linear transformations, and tensors – p. 1/35 This is an expository talk on Z matrices, transformations on proper cones, and tensors. The objective is to show that these have very similar properties. Z matrices, linear transformations, and tensors – p. 2/35 Outline • The Z-property • M and strong (nonsingular) M-properties • The P -property • Complementarity problems • Zero-sum games • Dynamical systems Z matrices, linear transformations, and tensors – p. 3/35 Some notation • Rn : The Euclidean n-space of column vectors. n n • R+: Nonnegative orthant, x ∈ R+ ⇔ x ≥ 0. n n n • R++ : The interior of R+, x ∈++⇔ x > 0. • hx,yi: Usual inner product between x and y. • Rn×n: The space of all n × n real matrices. • σ(A): The set of all eigenvalues of A ∈ Rn×n. Z matrices, linear transformations, and tensors – p. 4/35 The Z-property A =[aij] is an n × n real matrix • A is a Z-matrix if aij ≤ 0 for all i =6 j. (In economics literature, −A is a Metzler matrix.) • We can write A = rI − B, where r ∈ R and B ≥ 0. Let ρ(B) denote the spectral radius of B. • A is an M-matrix if r ≥ ρ(B), • nonsingular (strong) M-matrix if r > ρ(B). Z matrices, linear transformations, and tensors – p. 5/35 The P -property • A is a P -matrix if all its principal minors are positive.
    [Show full text]
  • Totally Positive Toeplitz Matrices and Quantum Cohomology of Partial Flag Varieties
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 2, Pages 363{392 S 0894-0347(02)00412-5 Article electronically published on November 29, 2002 TOTALLY POSITIVE TOEPLITZ MATRICES AND QUANTUM COHOMOLOGY OF PARTIAL FLAG VARIETIES KONSTANZE RIETSCH 1. Introduction A matrix is called totally nonnegative if all of its minors are nonnegative. Totally nonnegative infinite Toeplitz matrices were studied first in the 1950's. They are characterized in the following theorem conjectured by Schoenberg and proved by Edrei. Theorem 1.1 ([10]). The Toeplitz matrix ∞×1 1 a1 1 0 1 a2 a1 1 B . .. C B . a2 a1 . C B C A = B .. .. .. C B ad . C B C B .. .. C Bad+1 . a1 1 C B C B . C B . .. .. a a .. C B 2 1 C B . C B .. .. .. .. ..C B C is totally nonnegative@ precisely if its generating function is of theA form, 2 (1 + βit) 1+a1t + a2t + =exp(tα) ; ··· (1 γit) i Y2N − where α R 0 and β1 β2 0,γ1 γ2 0 with βi + γi < . 2 ≥ ≥ ≥···≥ ≥ ≥···≥ 1 This beautiful result has been reproved many times; see [32]P for anP overview. It may be thought of as giving a parameterization of the totally nonnegative Toeplitz matrices by ~ N N ~ (α;(βi)i; (~γi)i) R 0 R 0 R 0 i(βi +~γi) < ; f 2 ≥ × ≥ × ≥ j 1g i X2N where β~i = βi βi+1 andγ ~i = γi γi+1. − − Received by the editors December 10, 2001 and, in revised form, September 14, 2002. 2000 Mathematics Subject Classification. Primary 20G20, 15A48, 14N35, 14N15.
    [Show full text]
  • Tropical Totally Positive Matrices 3
    TROPICAL TOTALLY POSITIVE MATRICES STEPHANE´ GAUBERT AND ADI NIV Abstract. We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally nonnegative matrices, and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields. Keywords: Total positivity; total nonnegativity; tropical geometry; compound matrix; permanent; Monge matrices; Grassmannian; Pl¨ucker coordinates. AMSC: 15A15 (Primary), 15A09, 15A18, 15A24, 15A29, 15A75, 15A80, 15B99. 1. Introduction 1.1. Motivation and background. A real matrix is said to be totally positive (resp. totally nonneg- ative) if all its minors are positive (resp. nonnegative). These matrices arise in several classical fields, such as oscillatory matrices (see e.g. [And87, §4]), or approximation theory (see e.g. [GM96]); they have appeared more recently in the theory of canonical bases for quantum groups [BFZ96]. We refer the reader to the monograph of Fallat and Johnson in [FJ11] or to the survey of Fomin and Zelevin- sky [FZ00] for more information. Totally positive/nonnegative matrices can be defined over any real closed field, and in particular, over nonarchimedean fields, like the field of Puiseux series with real coefficients.
    [Show full text]
  • [Math.QA] 3 Dec 2001 Minors)
    QUANTUM COHOMOLOGY RINGS OF GRASSMANNIANS AND TOTAL POSITIVITY KONSTANZE RIETSCH Abstract. We give a proof of a result of D. Peterson’s identifying the quan- tum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of GLn. The totally positive part of this subvariety is then constructed and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of Vafa and Intriligator and Bertram for the structure constants (Gromov–Witten invariants). Finally, we use the positiv- ity of these Gromov–Witten invariants to prove certain inequalities for Schur polynomials at roots of unity. 1. Introduction + The group U of unipotent upper–triangular matrices in GLn have on their coordinate ring a nice basis with positive structure constants. Namely one has the dual of the classical limit of Lusztig’s geometrically constructed canonical basis of the quantized enveloping algebra. The existence of this basis has been closely tied [10] to the ‘totally positive part’ of U + (the matrices with only nonnegative minors). + In this paper we study certain remarkable subvarieties Vd,n of U which come up in the stabilizer of a particular standard principal nilpotent element e as clo- sures of the 1–dimensional components under Bruhat decomposition. By a theorem of Dale Peterson’s [16] the quantum cohomology rings of Grassmannians may be identified with the coordinate rings of these varieties. Therefore like U + itself these varieties have coordinate rings with ‘canonical’ bases on them (this time coming from Schubert bases), and with positive structure constants.
    [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]
  • Convex and Semi-Nonnegative Matrix Factorizations: Ding, Li and Jordan 1
    CONVEX AND SEMI-NONNEGATIVE MATRIX FACTORIZATIONS: DING, LI AND JORDAN 1 Convex and Semi-Nonnegative Matrix Factorizations Chris Ding, Tao Li, and Michael I. Jordan Chris Ding is with the Department of Computer Science and Engineering, University of Texas, Arlington, TX 76019. Tao Li is with the School of Computer Science at Florida International University, Miami, FL 33199, USA. Michael I. Jordan is with the Department of Electrical Engineering and Computer Science and the Department of Statistics at the University of California at Berkeley, CA 94720, USA. October 24, 2008 DRAFT CONVEX AND SEMI-NONNEGATIVE MATRIX FACTORIZATIONS: DING, LI AND JORDAN 2 Abstract We present several new variations on the theme of nonnegative matrix factorization (NMF). Con- sidering factorizations of the form X = F GT , we focus on algorithms in which G is restricted to contain nonnegative entries, but allow the data matrix X to have mixed signs, thus extending the applicable range of NMF methods. We also consider algorithms in which the basis vectors of F are constrained to be convex combinations of the data points. This is used for a kernel extension of NMF. We provide algorithms for computing these new factorizations and we provide supporting theoretical analysis. We also analyze the relationships between our algorithms and clustering algorithms, and consider the implications for sparseness of solutions. Finally, we present experimental results that explore the properties of these new methods. Index Terms Nonnegative Matrix Factorization, Singular Value Decomposition, Clustering I. INTRODUCTION Matrix factorization is a unifying theme in numerical linear algebra. A wide variety of matrix factorization algorithms have been developed over many decades, providing a numerical platform for matrix operations such as solving linear systems, spectral decomposition, and subspace identification.
    [Show full text]
  • Contents September 1
    Contents September 1 : Examples 2 September 3 : The Gessel-Lindstr¨om-ViennotLemma 3 September 15 : Parametrizing totally positive unipotent matrices { Part 1 6 September 17 : Parametrizing totally positive unipotent matrices { part 2 7 September 22 : Introduction to the 0-Hecke monoid 9 September 24 : The 0-Hecke monoid continued 11 September 29 : The product has the correct ranks 12 October 1 : Bruhat decomposition 14 October 6 : A unique Bruhat decomposition 17 October 8: B−wB− \ N − + as a manifold 18 October 13: Grassmannians 21 October 15: The flag manifold, and starting to invert the unipotent product 23 October 20: Continuing to invert the unipotent product 28 October 22 { Continuing to invert the unipotent product 30 October 27 { Finishing the inversion of the unipotent product 32 October 29 { Chevalley generators in GLn(R) 34 November 5 { LDU decomposition, double Bruhat cells, and total positivity for GLn(R) 36 November 10 { Kasteleyn labelings 39 November 12 { Kasteleyn labellings continued 42 November 17 { Using graphs in a disc to parametrize the totally nonnegative Grassmannian 45 November 19 { cyclic rank matrices 45 December 1 { More on cyclic rank matrices 48 December 3 { Bridges and Chevalley generators 51 December 8 { finishing the proof 55 1 2 September 1 : Examples We write [m] = f1; 2; : : : ; mg. Given an m × n matrix M, and subsets I =⊆ [m] and I J ⊆ [n] with #(I) = #(J), we define ∆J (M) = det(Mij)i2I; j2J , where we keep the elements of I and the elements of J in the same order. For example, 13 M12 M15 ∆25(M) = det : M32 M35 I The ∆J (M) are called the minors of M.
    [Show full text]