DOCSLIB.ORG

Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem UWB Mathematics Society

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem UWB Mathematics Society Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem UWB Mathematics Society Pietro Paparella Division of Engineering & Mathematics School of Science, Technology, Engineering, & Mathematics 14 April 2016 B&N PFT NIEP SC/STFW Overview 1 Background & Notation 2 The Perron-Frobenius Theorem for Nonnegative Matrices 3 Nonnegative Inverse Eigenvalue Problem 4 Perron Spectracones & Spectratopes 5 Future Work P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. P. Paparella Perron Spectratopes A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. P. Paparella Perron Spectratopes B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). P. Paparella Perron Spectratopes the scalar λ 2 C is is called an eigenvalue of A if there is a nonzero vector x such that Ax = λx. The vector x is called an eigenvector of A associated with λ. σ (A) denotes the spectrum of a A, i.e., σ (A) := fλ 2 C : det(A − λI ) = 0g = fλ1; : : : ; λng: ρ (A) denotes the spectral radius of A, i.e., ρ (A) := maxi2hni jλi j. aij denotes the (i; j)-entry of A (i.e., the entry in row i, column j). A> denotes the transpose of A, i.e., A> is the n-by-m matrix whose (i; j)-entry is aji . If A is an n-by-n matrix, then n For x 2 F , denote by Dx the diagonal matrix whose (i; i)-entry is xi . th n Denote by ei the i standard basis vector of R . Let e := [1 ··· 1]> 2 Rn. B&N PFT NIEP SC/STFW For any m-by-n matrix A: P. Paparella Perron Spectratopes the scalar λ 2 C is is called an eigenvalue of A if there is a nonzero vector x such that Ax = λx. The vector x is called an eigenvector of A associated with λ. σ (A) denotes the spectrum of a A, i.e., σ (A) := fλ 2 C : det(A − λI ) = 0g = fλ1; : : : ; λng: ρ (A) denotes the spectral radius of A, i.e., ρ (A) := maxi2hni jλi j. A> denotes the transpose of A, i.e., A> is the n-by-m matrix whose (i; j)-entry is aji . If A is an n-by-n matrix, then n For x 2 F , denote by Dx the diagonal matrix whose (i; i)-entry is xi . th n Denote by ei the i standard basis vector of R . Let e := [1 ··· 1]> 2 Rn. B&N PFT NIEP SC/STFW For any m-by-n matrix A: aij denotes the (i; j)-entry of A (i.e., the entry in row i, column j). P. Paparella Perron Spectratopes the scalar λ 2 C is is called an eigenvalue of A if there is a nonzero vector x such that Ax = λx. The vector x is called an eigenvector of A associated with λ. σ (A) denotes the spectrum of a A, i.e., σ (A) := fλ 2 C : det(A − λI ) = 0g = fλ1; : : : ; λng: ρ (A) denotes the spectral radius of A, i.e., ρ (A) := maxi2hni jλi j.
Load more

Recommended publications
  • Binary Polynomial Transforms for Nonlinear Signal Processing
    BINARY POLYNOMIAL TRANSFORMS FOR NONLINEAR SIGNAL PROCESSING Jaakko Astola, Karen Egiazarian, Rugen oktern SOS Agaian Signal Processing Laboratory Electrical Engineering Department Tampere University of Technology University of Texas at San Antonio Tampere, FINLAND San Antonio, Texas, USA ABSTRACT ing computers or systems suitable for simultaneous arith- metical or logical operations. In this paper we introduce parametric binary Rademacher functions of two types. Based on them using different kinds 2. BINARY POLYNOMIAL FUNCTIONS AND of arithmetical and logical operations we generate a set of MATRICES binary polynomial transforms (BPT). Some applications of BPT in nonlinear filtering and in compression of binary 2.1. Rademacher Functions and Matrices images are presented. Let a and b be arbitrary integers. We form the classes r,(t,a, b) and s,(t,a, b) of real periodic functions in the 1. INTRODUCTION following way. Let Spectral analysis is a powerful tool in communication, sig- ro(t.a,b) E so(t,a, b) E 1, (1) nal/ image processing and applied mathematics and there are many reasons behind the usefulness of spectral analysis. First, in many applications it is convenient to transform a a, fortEU~~ol[$,fk++i), problem into another, easier problem; the spectral domain rn+l (t, a, b) = is essentially a steady-state viewpoint; this not only gives b, for t E U~~ol[ $t + 5&l *).2 insight for analysis but also for design. Second, and more (2) important: the spectral approach allows us to treat entire and classes of signals which have similar properties in the spec- tral domain; it has excellent properties such as fast algo- a: for t E LO,&I, n = 0, 1,2, .
    [Show full text]
  • LU Decomposition - Wikipedia, the Free Encyclopedia
    LU decomposition - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/LU_decomposition From Wikipedia, the free encyclopedia In linear algebra, LU decomposition (also called LU factorization) is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear equations or calculate the determinant of a matrix. LU decomposition can be viewed as a matrix form of Gaussian elimination. LU decomposition was introduced by mathematician Alan Turing [1] 1 Definitions 2 Existence and uniqueness 3 Positive definite matrices 4 Explicit formulation 5 Algorithms 5.1 Doolittle algorithm 5.2 Crout and LUP algorithms 5.3 Theoretical complexity 6 Small example 7 Sparse matrix decomposition 8 Applications 8.1 Solving linear equations 8.2 Inverse matrix 8.3 Determinant 9 See also 10 References 11 External links Let A be a square matrix. An LU decomposition is a decomposition of the form where L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal. For a matrix, this becomes: An LDU decomposition is a decomposition of the form where D is a diagonal matrix and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U LDU decomposition of a Walsh matrix 1 of 7 1/11/2012 5:26 PM LU decomposition - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/LU_decomposition are one.
    [Show full text]
  • Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem
    Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem a b, Charles R. Johnson , Pietro Paparella ∗ aDepartment of Mathematics, College of William & Mary, Williamsburg, VA 23187-8795, USA bDivision of Engineering and Mathematics, University of Washington Bothell, Bothell, WA 98011-8246, USA Abstract Call an n-by-n invertible matrix S a Perron similarity if there is a real non- 1 scalar diagonal matrix D such that SDS− is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra (S) := n 1 C x R : SDxS− 0;Dx := diag (x) and (S) := x (S): x1 = 1 , whichf 2 we call the Perron≥ spectracone andgPerronP spectratopef 2, respectively. C Theg set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes, so that enumerating them is of interest. The Perron spectracone and spectratope of Hadamard matrices are of par- ticular interest and tend to have large volume. For the canonical Hadamard matrix (as well as other matrices), the Perron spectratope coincides with the convex hull of its rows. In addition, we provide a constructive version of a result due to Fiedler ([9, Theorem 2.4]) for Hadamard orders, and a constructive version of [2, Theorem 5.1] for Sule˘ımanova spectra. Keywords: Perron spectracone, Perron spectratope, real nonnegative inverse eigenvalue problem, Hadamard matrix, association scheme, relative gain array 2010 MSC: 15A18, 15B48, 15A29, 05B20, 05E30 1. Introduction The real nonnegative inverse eigenvalue problem (RNIEP) is to determine arXiv:1508.07400v2 [math.RA] 19 Nov 2015 which sets of n real numbers occur as the spectrum of an n-by-n nonnegative matrix.
    [Show full text]
  • Fast Random Projections Using Lean Walsh Transforms
    Fast Random Projections using Lean Walsh Transforms Abstract We present a random projection matrix that is applicable to vectors x 2 Rd in O(d) operations if 0 d ¸ k2+± for arbitrary small ±0 and k = O(log(n)="2). The projection succeeds with probability 1 ¡ 1=n ¡(1+±) and it preserves lengths up to distortion " for all vectors such that kxk1 · kxk2k for arbitrary small ±. Previous results are either not applicable in linear time or require a bound on kxk1 that depends on d. 1 Introduction We are interested in producing a random projection procedure such that applying it to a vector in Rd requires O(d) operation. The idea is to ¯rst project an incoming vector x 2 Rd into an intermediate dimension d~ = d® (0 < ® < 1) and then use a known construction to project it further down to the minimal Johnson 2 Lindenstrauss (JL) dimension k = O(log(n)=" )). The mapping is composed of three matrices x 7! RADsx. ~ The ¯rst, Ds, is a random §1 diagonal matrix. The second, A, is a dense matrix of size d £ d termed the Lean Walsh Matrix. The third, R, is a k £ d~ Fast Johnson Lindenstrauss matrix. As R, we use the fast JL construction by Ailon and Liberty [2]. Their matrix, R, can be applied ~ ~ 2+±00 00 (2+±00)=® to ADsx in O(d log(k)) operations as long as d ¸ k for arbitrary small ± . If d > k this 1 condition is met . Therefore, for such d, applying R to ADsx is length preserving w.p and accomplishable in O(d~log(k)) = O(d® log(k)) = O(d) operations.
    [Show full text]
  • A Novel 4-Point Discrete Fourier Transforms Circuit Based on Product of Rademacher Functions
    The 5th International Conference on Electrical Engineering and Informatics 2015 August 10-11, 2015, Bali, Indonesia A Novel 4-Point Discrete Fourier Transforms Circuit based on Product of Rademacher Functions Zulfikar1, Hubbul Walidainy2 Department of Electrical Engineering Syiah Kuala University Banda Aceh 23111, Indonesia. E-mail: [email protected], [email protected] Abstract—This paper presents a new circuit design for algorithms have been developed for certain purposes and each implementing 4-point DFT algorithm based on product of of them comes with advantages and drawbacks. Rademacher functions. The circuit has been derived from the similarity of how Fourier transforms and Walsh transforms are Walsh transforms algorithm is simpler than Fourier implemented. Walsh matrices contain numbers either +1 or -1 transform, but this transformation model is rarely used in except for first row. Similarly, the 4-point DFT matrix contain application and almost forgotten. Walsh transforms has a few numbers either positive or negative except for first row. This similarity to the Fourier transforms [2]-[5]. Based on this, some similarity has been taken into the case of how to implement the researchers have adopted this algorithm for developing the DFT circuit based on how Walsh transforms is generated. Since more efficient Fourier transforms [6]-[8]. An algorithm for Walsh transforms is derived based on product of Rademacher calculating DFT using Walsh transforms is developed through functions, the proposed 4-point DFT circuit is designed according the factorization of intermediate transform T [6]. Monir T et al to product Rademacher functions. The circuit consist of negative proposed an efficient combination of Walsh and DFT circuit, multiplexers, accumulator (real and imaginary), buffers, calculation.
    [Show full text]
  • Accelerated Dense Random Projections
    Accelerated Dense Random Projections A Dissertation Presented to the faculty of the Graduate School of Yale University in Candidacy for the degree of Doctor of Philosophy by Edo Liberty Dissertation Director: Steven Zucker May 2009 Accelerated Dense Random Projections Edo Liberty May 2009 ABSTRACT In dimensionality reduction, a set of points in Rd is mapped into Rk, with the target dimension k smaller than the original dimension d, while distances between all pairs of points are approximately preserved. Currently popular methods for achieving this involve random projection, or choosing a linear mapping (a k £ d matrix) from a distribution that is independent of the input points. Applying the mapping (chosen according to this distribution) is shown to give the desired property with at least constant probability. The contributions in this thesis are twofold. First, we provide a framework for designing such distributions. Second, we derive e±cient random projection algorithms using this framework. Our results achieve performance exceeding other existing approaches. When the target dimension is signi¯cantly smaller than the original dimension we gain signi¯cant improvement by designing e±cient algorithms for applying certain linear algebraic transforms. To demonstrate practicality, we supplement the theoretical derivations with experimental results. Acknowledgments: Special thanks to Steven Zucker, Nir Ailon, Daniel Spielman, Ronald Coifman, Amit Singer and Mark Tygert. Their work, ideas and guidance can be seen throughout this manuscript. 1. LIST OF COMMON NOTATIONS d . .input vectors' original dimension Rd ...............................................................................real d dimensional space d x or xi . .input vector(s) in R n . number of input vectors or a constant polynomial in that number k .
    [Show full text]
  • New Perspectives on Non-Negative Matrix Factorization for Grouped Topic Models
    New Perspectives on Non-negative Matrix Factorization for Grouped Topic Models by Gabriel C. Phelan B.Sc., Rochester Institute of Technology, 2017 Project Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Statistics and Actuarial Science Faculty of Science c Gabriel C. Phelan 2020 SIMON FRASER UNIVERSITY Summer 2020 Copyright in this work rests with the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Approval Name: Gabriel C. Phelan Degree: Master of Science (Statistics) Title: New Perspectives on Non-negative Matrix Factorization for Grouped Topic Models Examining Committee: Chair: Jinko Graham Professor David Campbell Senior Supervisor Professor School of Mathematics and Statistics Carleton University Lloyd T. Elliott Supervisor Assistant Professor Thomas M. Loughin Internal Examiner Professor Date Defended: August 18, 2020 ii Abstract Probabilistic topic models (PTM’s) have become a ubiquitous approach for finding a set of latent themes (“topics”) in collections of unstructured text. A simpler, linear algebraic technique for the same problem is non-negative matrix factorization: we are given a matrix with non-negative entries and asked to find a pair of low-rank matrices, also non-negative, whose product is approximately the original matrix. A drawback of NMF is the non-convex nature of the optimization problem it poses. Recent work by the theoretical computer sci- ence community addresses this issue, utilizing NMF’s inherent structure to find conditions under which the objective function admits convexity. With convexity comes tractability, and the central theme of this thesis is the exploitation of this tractability to ally NMF with resampling-based nonparametrics.
    [Show full text]
  • Human Identification Using Compressed ECG Signals
    This is a postprint version of the following published document: C. Camara, P. Peris-Lopez, and J. E. Tapiador (2015). Human Identification Using Compressed ECG Signals. Journal of Medical Systems, 39: 148. Avalaible in https://doi.org/10.1007/s10916-015-0323-2 © Springer Science+Business Media New York 2015 Human ldentification Using Compressed ECG Signals Carmen Camara1 • Pedro Peris-Lopez1 • Juan E. Tapiador1 Abstract As a result of the increased demand for improved Introduction life styles and the increment of senior citizens over the age of 65, new home care services are demanded. Simul­ According to [3], the medica! sector is the area that has taneously, the medica! sector is increasingly becoming the suffered the major number of hacking incidents over the new target of cybercrirninals due the potential value of last two year�3 % of the data breaches in US. Medica! users' medica! information. Toe use of biometrics seems companies and hospitals have begun to introduce biometric an effective tool as a deterrent for many of such attacks. solutions to rnitigate attacks and reduce costs. Further­ In this paper, we propase the use of electrocardiograms more, the proper identification of patients when they walk (ECGs) for the identification of individuals. For instance, through the door is a major issue nowadays for all the for a telecare service, a user could be authenticated using hospitals around the world. Errors in medica} records, or the information extracted from her ECG signa!. Toe major­ even incorrect treatrnents, are very costly for the medica! ity of ECG-based biometrics systems extract information centres and harmful for the patient.
    [Show full text]
  • Complex Hadamard Matrices and Applications Teo Banica
    Complex Hadamard matrices and applications Teo Banica To cite this version: Teo Banica. Complex Hadamard matrices and applications. 2021. hal-02317067v2 HAL Id: hal-02317067 https://hal.archives-ouvertes.fr/hal-02317067v2 Preprint submitted on 27 Jul 2021 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. COMPLEX HADAMARD MATRICES AND APPLICATIONS TEO BANICA Abstract. A complex Hadamard matrix is a square matrix H 2 MN (C) whose entries are on the unit circle, jHijj = 1, and whose rows and pairwise orthogonal. The main ij 2πi=N example is the Fourier matrix, FN = (w ) with w = e . We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. Contents Introduction2 1. Hadamard matrices 7 2. Analytic aspects 25 3. Norm maximizers 43 4. Partial matrices 61 5. Complex matrices 79 6. Roots of unity 97 7. Geometry, defect 115 8. Special matrices 133 9. Circulant matrices 151 10. Bistochastic form 169 11. Glow computations 187 12. Local estimates 205 13. Quantum groups 223 14. Hadamard models 241 15. Generalizations 259 16. Fourier models 277 References 295 2010 Mathematics Subject Classification. 15B10.
    [Show full text]
  • A Demonstration of the Use of Walsh Functions for Multiplexed Imaging
    Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1990-12 A demonstration of the use of Walsh functions for multiplexed imaging McKenzie, Robert Hugh, III Monterey, California: Naval Postgraduate School http://hdl.handle.net/10945/27636 NAVAL POSTGRADUATE SCHOOL " Monterey, California AD-A246 138 ft LECTE i R ADt' FEB 2 0, 199 2 U THESIS A DEMONSTRATION OF THE USE OF WALSH FUNCTIONS FOR MULTIPLEXED IMAGING by Robert Hugh McKenzie III DECEMBER 1990 Thesis Advisor: David S. Davis Approved for public release: Distribution is unlimited 92-03974 92 2 NCLASSIFIED CURITY CLASS!FICATION OF THIS PAGE i i i iiN I Form Approved REPORT DOCUMENTATION PAGE OMB No 0704-0188 a REPORT SECURITY CLASSIFiCATION lb RESTRICT:VE MARK!NGS Unclassified a SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION ,AVAI.ABILITY OF EPORT Approved for pubi c release: b DECLASSIFICATION! DOWNGRADING SCHEDULE Distribution is unlimited PERFORMING ORGANIZATION REPORT NUMBER(S) S MONITORING ORGANIZATION REPORT NUMBER(S) a NAME OF PERFORMING ORGANIZATION r6b OFFICE SYMBOL 7a NAME OF MONITORING ORGANiZATION 4aval Postgraduate School I6Dv(If applicable) Naval Postgraduate School ,c.ADDRESS (City, State, and ZIPCode) 7b ADDRESS (City, State, and ZIP Code) Monterey, CA 93943-5000 Monterey, CA 93943-5000 Ia NAME OF FUNDING/SPONSORING r8b OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NU.VBER ORGANIZATIONj (ifapplicable) c. ADDRESS (City, State, and ZIP Code) 10 SOURCE OF FUNDING NUMBERS PROGRAM IPROjECT ITASK I VORo, UNIT ELEMENT NO NO NO ACCESSiO% NO 1 1 TITLE (Include Security Classification) A DEMONSTRATION OF THE USE OF W 7 SH FUNCTIONS FOR MULTIPLEXED IMAGING "2 P - MCKENZIE III 13a TYPE OF REPORT 13b TIME COVERED 14 D REPORT ~onth.0Day) :5 PAGE CO AJT Master's Thesis FROM _ TO _____ 131 16 SUPPLEMENTARY NOTATION The iews expressed in this thesis are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S.
    [Show full text]
  • Spectral Methods for Boolean and Multiple-Valued Input Logic Functions
    Portland State University PDXScholar Dissertations and Theses Dissertations and Theses 1991 Spectral Methods for Boolean and Multiple-Valued Input Logic Functions Bogdan Jaroslaw Falkowski Portland State University Let us know how access to this document benefits ouy . Follow this and additional works at: http://pdxscholar.library.pdx.edu/open_access_etds Recommended Citation Falkowski, Bogdan Jaroslaw, "Spectral Methods for Boolean and Multiple-Valued Input Logic Functions" (1991). Dissertations and Theses. Paper 1152. 10.15760/etd.1151 This Dissertation is brought to you for free and open access. It has been accepted for inclusion in Dissertations and Theses by an authorized administrator of PDXScholar. For more information, please contact [email protected]. SPECTRAL METHODS FOR BOOLEAN AND MULTIPLE-VALUED INPUT LOGIC F&-ICTIONS by BOGDAN JAROSLAW FALKOWSKI A disseitation submitted in partial fulfillment ofthe requirements for the degree of DOcrOR OF PHILOSOPHY in ELECTRICAL AND COMPUTER ENGINEERING Portland State University 1991 AN ABSTRACT OF THE DISSERTATION OF Bogdan Jaroslaw Falkowski for the Doctor of Philosophy in Electrical and Computer Engineering presented May 2, 1991. Title: Spectra! Methods for Boolean and Multiple-Valued Input Logic Functions. Jack12. Riley ,/ ; .~ / E. Balogh Spectral techniques in digital logic design have been known for more than thirty years. They have been used for Boolean function classification, disjoint decomposition, parallel and serial linear decomposition, spectral translation synthesis (extraction of 2 linear pre- and post-filters), multiplexer synthesis, prime implicant extraction by spectral summation, threshold logic synthesis, estimation of logic complexity, testing, and state assignment. This dissertation resolves many important issues concerning the efficient applica­ tion of spectral methods used in the computer-aided design of digital circuits.
    [Show full text]
  • Accelerated Dense Random Projections CONTENTS
    Edo Liberty Accelerated Dense Random Projections CONTENTS 1. List of common notations ::::::::::::::::::::::::::::::::::::::::: 5 2. Introduction to random projections :::::::::::::::::::::::::::::::::::: 7 2.1 Linear embedding and the JL property . 7 2.2 Classic results, review of known JL distributions . 9 2.3 Motivation for accelerating random projections . 10 2.4 Sparse Projective matrices . 11 2.5 Our contributions . 15 2.5.1 Fast linear transforms . 15 2.5.2 A two stage framework . 16 2.5.3 Dense fast projective matrices . 17 2.5.4 Results summary . 18 3. Revised Fast Johnson Lindenstrauss transform ::::::::::::::::::::::::::::: 20 3.1 Review of the FJLT algorithm . 20 3.2 Trimmed Walsh Hadamard transform . 21 3.3 Trimmed Discrete Fourier Transform . 23 3.4 FJLTr conclusion . 24 4. Two stage projection process ::::::::::::::::::::::::::::::::::::::: 26 4.1 Concentration result . 26 4.2 Â(A) the probabilistic Image . 28 4.3 `p bounds on A-norms . 29 4.4 Conclusion . 30 5. Four-wise independence and random projections :::::::::::::::::::::::::::: 31 5.1 Tools from Error Correcting Codes . 32 T 5.2 Bounding kB k2!4 Using BCH Codes . 32 1=2¡± 5.3 Controlling kxk4 for k < d ................................... 33 5.4 Reducing to Manhattan Space for k < d1=2¡± ........................... 35 5.5 Reducing the running time to O(d log k) for small k ....................... 37 5.6 JL concatenation . 40 5.7 Results summary . 42 6. Towards linear time dimensionality reduction :::::::::::::::::::::::::::::: 43 6.1 Lean Walsh transforms . 44 6.2 An `p bound on k ¢ kA ........................................ 46 6.2.1 Controlling ® and choosing R ................................ 47 6.3 Comparison to sparse projections .
    [Show full text]