Correlation Clustering with Noisy Input
|
|
- Berniece McGee
- 5 years ago
- Views:
Transcription
1 Correlaton Clsterng wth Nosy Inpt Clare Mathe Warren Schdy Brown Unversty SODA 2010
2 Nosy Correlaton Clsterng Model Unknown base clsterng B of n obects Nose: each edge s controlled by an adversary wth probablty p and tells the trth otherwse Problem: reconstrct B from the edge labels I n p t B = = = =
3 One of or reslts Theorem: assme p 1/3. If all clsters have sze at least 1 then the natral sem-defnte program (SDP) recovers B exactly wth hgh probablty. Prevos best: 2 n log n [Bansal, Blm, Chawla 04, Shamr and Tsr 07], combnatoral. See paper for other reslts (ncldng approxmaton algorthms) n
4 Plan The sem-defnte program Its dal Usng the dal
5 Clsterngs Clsterngs are represented by 0/1 matrces: =1: and n same clster In general a clsterng satsfes: E.g. v k k v T k for some 0/1 orthogonal vectors v, v2,..., v one per clster v cat v dog : : 1 m ,
6 Relaxaton Clsterng Relaxaton of clsterngs =1 for all 0 for all, The followng are eqvalent ( symmetrc): T vkvk for some 0/1 vectors v1, v2,..., v k T vkvk for some vectors v1, v2,..., v k s postve sem-defnte (p.s.d.) m m
7 Obectve Maxmze nmber of agreements: max f 1 f Drop the constant = I.e. max E where E 1 f = -1 f
8 Smmary of SDP max p.s.d. E s.t. Ths SDP was prevosly sed by: [Charkar, Grswam, Wrth 05] [Swamy 04] 1 0 E = = = =
9 Dscsson Algorthm: Solve SDP If ntegral, otpt t. Otherwse fal. Thm: assme p 1/3. If all clsters have sze at least 1 n then the SDP recovers B exactly wth hgh probablty. An example matrx from solver n [Elsner and Schdy 09]. That solver scales to a few thosand obects.
10 Plan The sem-defnte program Its dal Usng the dal
11 Translate SDP nto LP The followng are eqvalent ( symmetrc): postve sem-defnte SDP agan: max T 1 0 p.s.d. 0 for all E vectors s.t. max LP form: T Lnear n for fxed E s.t for all vectors
12 SDP Dal Prmal: Dal: 0, a for all ) 1( s.t. mn h E h d a d E T 0 for all, 0 for all 1for all s.t. max a h d
13 Translate dal LP nto SDP The followng are eqvalent ( symmetrc): a wth a 0 T k k k postve sem-defnte Dal agan: mn a d s.t. d 1( ) h a, h E 0 Matrx form: mn Trace ( D) E D H a D H s.t. dagonal 0 0 Arbtrary postve semdefnte matrx a T
14 The Dal SDP mn Trace( D) s.t. E D H postve sem- defnte D dagonal H 0
15 Plan The sem-defnte program Its dal Usng the dal Ths proof s nspred by a smlar reslt for the planted clqe problem [Fege and Krathgamer 00].
16 Usng the dal - overvew Prove optmalty of the base clsterng by presentng dal solton (D,H) whose vale matches vale of base clsterng B (see paper) Dffclt part: provng that E D H s p.s.d. The followng are eqvalent (Y symmetrc): Y postve sem-defnte All egenvales of Y are 0 We present b egenvectors wth egenvale 0 (see paper), where b s the nmber of clsters n B We prove that the b+1 th smallest egenvale, denoted, s postve (sketched next) b1 E D H Hence all egenvales of E D H are 0
17 Egenvale analyss E D H M M M M b 1 (mn clster sze) (see paper) 1 ( n) (next) We apply the followng: Theorem [Weyl]: If M and N are symmetrc matrces then M N) ( M) ( ) b1( 1 b1 N Hence for sffcently large mn clster sze b E D H. 1 0
18 Random matrces Theorem [Füred and Komlós 81]: Let M be a random symmetrc matrx wth ndependent entres of mean zero. Then wth hgh probablty Applcaton: 1 M E Expectaton E n 2 M O n for all. 1 To analyze M 3 we developed a generalzaton of ths theorem that handles lmted dependence between the entres.
19 Recap Theorem: assme p 1/3. If all clsters have sze at least then the SDP recovers B exactly wth hgh probablty. Proof: We wrote a dal solton matrx as a sm of 4 random matrces, sed Füred-Komlós varants to bond ther egenvales, sed Weyl to nfer bond on egenvales of the matrx, hence p.s.d., hence solton s feasble. That solton has vale eqal to the vale of B, hence by dalty B s prmal optmal B s the nqe prmal optmm (see paper), hence SDP wll exactly retrn B Hence algorthm reconstrcts B exactly when all clsters have sze at least n. 1 1 n
20 Sppose some clsters are sze c 3 n and others are sze 1. Can the SDP be sed to reconstrct the large clsters? Open Qeston 1 Software: [Elsner and Schdy 09].
21 Open Qeston 2 Planted clqe problem = correlaton clsterng wth only one non-sngleton and no corrpton of wthn-clster edges Exst polynomal-tme algorthm when clqe sze = c 1 n O(log n) Exsts n -tme algorthm when clqe sze = c1 log n Can polynomal-tme algorthms beat the n barrer? c 1
22 Clsterng References Nr Alon, Moses Charkar, and Alantha Newman. Aggregatng nconsstent nformaton: rankng and clsterng. In STOC 05, pages , Nkhl Bansal, Avrm Blm, and Shch Chawla. Correlaton clsterng. Mach. Learn., 56(1-3):89 113, Moses Charkar, Venkatesan Grswam, and Anthony Wrth. Clsterng wth qaltatve nformaton. J. Compt. Syst. Sc., 71(3): , M. Elsner and W. Schdy. Bondng and Comparng Methods for Correlaton Clsterng Beyond ILP. In ILP-NLP 09: Proc. NAACL/HLT 2009 Workshop on Integer Lnear Programmng for Natral Langage Processng, pages 19 27, Ron Shamr and Dekel Tsr. Improved algorthms for the random clster graph model. Random Strctres and Algorthms, 31(4): , 2007.
23 Other References F. Alzadeh. Interor pont methods n semdefnte programmng wth applcatons to combnatoral optmzaton. SIAM Jornal on Optmzaton, 5(1):13 51, 1995 Urel Fege and Robert Krathgamer. Fndng and certfyng a large hdden clqe n a semrandom graph. Random Strct. Algorthms, 16(2): , 2000 Zoltán Füred and János Komlós. The egenvales of random symmetrc matrces. Combnatorca, 1(3): , 1981
U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationCS 3750 Machine Learning Lecture 6. Monte Carlo methods. CS 3750 Advanced Machine Learning. Markov chain Monte Carlo
CS 3750 Machne Learnng Lectre 6 Monte Carlo methods Mlos Haskrecht mlos@cs.ptt.ed 5329 Sennott Sqare Markov chan Monte Carlo Importance samplng: samples are generated accordng to Q and every sample from
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpenCourseWare http://ocw.mt.edu 6.854J / 18.415J Advanced Algorthms Fall 2008 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 18.415/6.854 Advanced Algorthms
More informationEigenvalues of Random Graphs
Spectral Graph Theory Lecture 2 Egenvalues of Random Graphs Danel A. Spelman November 4, 202 2. Introducton In ths lecture, we consder a random graph on n vertces n whch each edge s chosen to be n the
More informationThe Prncpal Component Transform The Prncpal Component Transform s also called Karhunen-Loeve Transform (KLT, Hotellng Transform, oregenvector Transfor
Prncpal Component Transform Multvarate Random Sgnals A real tme sgnal x(t can be consdered as a random process and ts samples x m (m =0; ;N, 1 a random vector: The mean vector of X s X =[x0; ;x N,1] T
More informationCOS 521: Advanced Algorithms Game Theory and Linear Programming
COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton
More informationFinding Dense Subgraphs in G(n, 1/2)
Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng
More informationLecture 20: Lift and Project, SDP Duality. Today we will study the Lift and Project method. Then we will prove the SDP duality theorem.
prnceton u. sp 02 cos 598B: algorthms and complexty Lecture 20: Lft and Project, SDP Dualty Lecturer: Sanjeev Arora Scrbe:Yury Makarychev Today we wll study the Lft and Project method. Then we wll prove
More informationBAR & TRUSS FINITE ELEMENT. Direct Stiffness Method
BAR & TRUSS FINITE ELEMENT Drect Stness Method FINITE ELEMENT ANALYSIS AND APPLICATIONS INTRODUCTION TO FINITE ELEMENT METHOD What s the nte element method (FEM)? A technqe or obtanng approxmate soltons
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
More informationStanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011
Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationVARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES
VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue
More informationA 2D Bounded Linear Program (H,c) 2D Linear Programming
A 2D Bounded Lnear Program (H,c) h 3 v h 8 h 5 c h 4 h h 6 h 7 h 2 2D Lnear Programmng C s a polygonal regon, the ntersecton of n halfplanes. (H, c) s nfeasble, as C s empty. Feasble regon C s unbounded
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationProbability Theory (revisited)
Probablty Theory (revsted) Summary Probablty v.s. plausblty Random varables Smulaton of Random Experments Challenge The alarm of a shop rang. Soon afterwards, a man was seen runnng n the street, persecuted
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationAE/ME 339. K. M. Isaac. 8/31/2004 topic4: Implicit method, Stability, ADI method. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Comptatonal Fld Dynamcs (CFD) Comptatonal Fld Dynamcs (AE/ME 339) Implct form of dfference eqaton In the prevos explct method, the solton at tme level n,,n, depended only on the known vales of,
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationSolutions HW #2. minimize. Ax = b. Give the dual problem, and make the implicit equality constraints explicit. Solution.
Solutons HW #2 Dual of general LP. Fnd the dual functon of the LP mnmze subject to c T x Gx h Ax = b. Gve the dual problem, and make the mplct equalty constrants explct. Soluton. 1. The Lagrangan s L(x,
More informationρ some λ THE INVERSE POWER METHOD (or INVERSE ITERATION) , for , or (more usually) to
THE INVERSE POWER METHOD (or INVERSE ITERATION) -- applcaton of the Power method to A some fxed constant ρ (whch s called a shft), x λ ρ If the egenpars of A are { ( λ, x ) } ( ), or (more usually) to,
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationOn the Multicriteria Integer Network Flow Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More informationLecture 3: Dual problems and Kernels
Lecture 3: Dual problems and Kernels C4B Machne Learnng Hlary 211 A. Zsserman Prmal and dual forms Lnear separablty revsted Feature mappng Kernels for SVMs Kernel trck requrements radal bass functons SVM
More informationChat eld, C. and A.J.Collins, Introduction to multivariate analysis. Chapman & Hall, 1980
MT07: Multvarate Statstcal Methods Mke Tso: emal mke.tso@manchester.ac.uk Webpage for notes: http://www.maths.manchester.ac.uk/~mkt/new_teachng.htm. Introducton to multvarate data. Books Chat eld, C. and
More informationMATH Homework #2
MATH609-601 Homework #2 September 27, 2012 1. Problems Ths contans a set of possble solutons to all problems of HW-2. Be vglant snce typos are possble (and nevtable). (1) Problem 1 (20 pts) For a matrx
More informationClustering gene expression data & the EM algorithm
CG, Fall 2011-12 Clusterng gene expresson data & the EM algorthm CG 08 Ron Shamr 1 How Gene Expresson Data Looks Entres of the Raw Data matrx: Rato values Absolute values Row = gene s expresson pattern
More informationFeature Selection: Part 1
CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?
More informationConvex Optimization. Optimality conditions. (EE227BT: UC Berkeley) Lecture 9 (Optimality; Conic duality) 9/25/14. Laurent El Ghaoui.
Convex Optmzaton (EE227BT: UC Berkeley) Lecture 9 (Optmalty; Conc dualty) 9/25/14 Laurent El Ghaou Organsatonal Mdterm: 10/7/14 (1.5 hours, n class, double-sded cheat sheet allowed) Project: Intal proposal
More informationCombining Constraint Programming and Integer Programming
Combnng Constrant Programmng and Integer Programmng GLOBAL CONSTRAINT OPTIMIZATION COMPONENT Specal Purpose Algorthm mn c T x +(x- 0 ) x( + ()) =1 x( - ()) =1 FILTERING ALGORITHM COST-BASED FILTERING ALGORITHM
More informationLecture 17: Lee-Sidford Barrier
CSE 599: Interplay between Convex Optmzaton and Geometry Wnter 2018 Lecturer: Yn Tat Lee Lecture 17: Lee-Sdford Barrer Dsclamer: Please tell me any mstake you notced. In ths lecture, we talk about the
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Handout 6 Luca Trevisan September 12, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 6 Luca Trevsan September, 07 Scrbed by Theo McKenze Lecture 6 In whch we study the spectrum of random graphs. Overvew When attemptng to fnd n polynomal
More informationApproximate D-optimal designs of experiments on the convex hull of a finite set of information matrices
Approxmate D-optmal desgns of experments on the convex hull of a fnte set of nformaton matrces Radoslav Harman, Mára Trnovská Department of Appled Mathematcs and Statstcs Faculty of Mathematcs, Physcs
More informationAn introduction to chaining, and applications to sublinear algorithms
An ntroducton to channg, and applcatons to sublnear algorthms Jelan Nelson Harvard August 28, 2015 What s ths talk about? What s ths talk about? Gven a collecton of random varables X 1, X 2,...,, we would
More informationCS : Algorithms and Uncertainty Lecture 17 Date: October 26, 2016
CS 29-128: Algorthms and Uncertanty Lecture 17 Date: October 26, 2016 Instructor: Nkhl Bansal Scrbe: Mchael Denns 1 Introducton In ths lecture we wll be lookng nto the secretary problem, and an nterestng
More informationHOMOGENEOUS LEAST SQUARES PROBLEM
the Photogrammetrc Jornal of nl, Vol. 9, No., 005 HOMOGENEOU LE QURE PROLEM Kejo Inklä Helsnk Unversty of echnology Laboratory of Photogrammetry Remote ensng P.O.ox 00, IN-005 KK kejo.nkla@tkk.f RC he
More informationEnsemble Methods: Boosting
Ensemble Methods: Boostng Ncholas Ruozz Unversty of Texas at Dallas Based on the sldes of Vbhav Gogate and Rob Schapre Last Tme Varance reducton va baggng Generate new tranng data sets by samplng wth replacement
More informationLagrange Multipliers Kernel Trick
Lagrange Multplers Kernel Trck Ncholas Ruozz Unversty of Texas at Dallas Based roughly on the sldes of Davd Sontag General Optmzaton A mathematcal detour, we ll come back to SVMs soon! subject to: f x
More informationNonlinear Programming Formulations for On-line Applications
Nonlnear Programmng Formlatons for On-lne Applcatons L. T. Begler Carnege Mellon Unversty Janary 2007 Jont work wth Vctor Zavala and Carl Lard NLP for On-lne Process Control Nonlnear Model Predctve Control
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationLecture 10: May 6, 2013
TTIC/CMSC 31150 Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander,
More informationAssortment Optimization under the Paired Combinatorial Logit Model
Assortment Optmzaton under the Pared Combnatoral Logt Model Heng Zhang, Paat Rusmevchentong Marshall School of Busness, Unversty of Southern Calforna, Los Angeles, CA 90089 hengz@usc.edu, rusmevc@marshall.usc.edu
More informationOPTIMAL COMBINATION OF FOURTH ORDER STATISTICS FOR NON-CIRCULAR SOURCE SEPARATION. Christophe De Luigi and Eric Moreau
OPTIMAL COMBINATION OF FOURTH ORDER STATISTICS FOR NON-CIRCULAR SOURCE SEPARATION Chrstophe De Lug and Erc Moreau Unversty of Toulon LSEET UMR CNRS 607 av. G. Pompdou BP56 F-8362 La Valette du Var Cedex
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More informationLecture 17. Solving LPs/SDPs using Multiplicative Weights Multiplicative Weights
Lecture 7 Solvng LPs/SDPs usng Multplcatve Weghts In the last lecture we saw the Multplcatve Weghts (MW) algorthm and how t could be used to effectvely solve the experts problem n whch we have many experts
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationLecture 3 January 31, 2017
CS 224: Advanced Algorthms Sprng 207 Prof. Jelan Nelson Lecture 3 January 3, 207 Scrbe: Saketh Rama Overvew In the last lecture we covered Y-fast tres and Fuson Trees. In ths lecture we start our dscusson
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationFisher Linear Discriminant Analysis
Fsher Lnear Dscrmnant Analyss Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan Fsher lnear
More information18.1 Introduction and Recap
CS787: Advanced Algorthms Scrbe: Pryananda Shenoy and Shjn Kong Lecturer: Shuch Chawla Topc: Streamng Algorthmscontnued) Date: 0/26/2007 We contnue talng about streamng algorthms n ths lecture, ncludng
More information1 The Mistake Bound Model
5-850: Advanced Algorthms CMU, Sprng 07 Lecture #: Onlne Learnng and Multplcatve Weghts February 7, 07 Lecturer: Anupam Gupta Scrbe: Bryan Lee,Albert Gu, Eugene Cho he Mstake Bound Model Suppose there
More informationStanford University Graph Partitioning and Expanders Handout 3 Luca Trevisan May 8, 2013
Stanford Unversty Graph Parttonng and Expanders Handout 3 Luca Trevsan May 8, 03 Lecture 3 In whch we analyze the power method to approxmate egenvalues and egenvectors, and we descrbe some more algorthmc
More informationIntroduction. - The Second Lyapunov Method. - The First Lyapunov Method
Stablty Analyss A. Khak Sedgh Control Systems Group Faculty of Electrcal and Computer Engneerng K. N. Toos Unversty of Technology February 2009 1 Introducton Stablty s the most promnent characterstc of
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationA Modeling System to Combine Optimization and Constraint. Programming. INFORMS, November Carnegie Mellon University.
A Modelng Sstem to Combne Optmzaton and Constrant Programmng John Hooker Carnege Mellon Unverst INFORMS November 000 Based on ont work wth Ignaco Grossmann Hak-Jn Km Mara Axlo Osoro Greger Ottosson Erlendr
More informationLecture 5 September 17, 2015
CS 229r: Algorthms for Bg Data Fall 205 Prof. Jelan Nelson Lecture 5 September 7, 205 Scrbe: Yakr Reshef Recap and overvew Last tme we dscussed the problem of norm estmaton for p-norms wth p > 2. We had
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationLOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin
Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, M-H Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence
More information= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.
Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationFinding Primitive Roots Pseudo-Deterministically
Electronc Colloquum on Computatonal Complexty, Report No 207 (205) Fndng Prmtve Roots Pseudo-Determnstcally Ofer Grossman December 22, 205 Abstract Pseudo-determnstc algorthms are randomzed search algorthms
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationA FAST HEURISTIC FOR TASKS ASSIGNMENT IN MANYCORE SYSTEMS WITH VOLTAGE-FREQUENCY ISLANDS
Shervn Haamn A FAST HEURISTIC FOR TASKS ASSIGNMENT IN MANYCORE SYSTEMS WITH VOLTAGE-FREQUENCY ISLANDS INTRODUCTION Increasng computatons n applcatons has led to faster processng. o Use more cores n a chp
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More informationSupport Vector Machines. Vibhav Gogate The University of Texas at dallas
Support Vector Machnes Vbhav Gogate he Unversty of exas at dallas What We have Learned So Far? 1. Decson rees. Naïve Bayes 3. Lnear Regresson 4. Logstc Regresson 5. Perceptron 6. Neural networks 7. K-Nearest
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 3: Large deviations bounds and applications Lecturer: Sanjeev Arora
prnceton unv. F 13 cos 521: Advanced Algorthm Desgn Lecture 3: Large devatons bounds and applcatons Lecturer: Sanjeev Arora Scrbe: Today s topc s devaton bounds: what s the probablty that a random varable
More informationSpectral Clustering. Shannon Quinn
Spectral Clusterng Shannon Qunn (wth thanks to Wllam Cohen of Carnege Mellon Unverst, and J. Leskovec, A. Raaraman, and J. Ullman of Stanford Unverst) Graph Parttonng Undrected graph B- parttonng task:
More informationReading Assignment. Panel Data Cross-Sectional Time-Series Data. Chapter 16. Kennedy: Chapter 18. AREC-ECON 535 Lec H 1
Readng Assgnment Panel Data Cross-Sectonal me-seres Data Chapter 6 Kennedy: Chapter 8 AREC-ECO 535 Lec H Generally, a mxtre of cross-sectonal and tme seres data y t = β + β x t + β x t + + β k x kt + e
More informationAPPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
More informationLow correlation tensor decomposition via entropy maximization
CS369H: Herarches of Integer Programmng Relaxatons Sprng 2016-2017 Low correlaton tensor decomposton va entropy maxmzaton Lecture and notes by Sam Hopkns Scrbes: James Hong Overvew CS369H). These notes
More informationProseminar Optimierung II. Victor A. Kovtunenko SS 2012/2013: LV
Prosemnar Optmerung II Vctor A. Kovtunenko Insttute for Mathematcs and Scentfc Computng, Karl-Franzens Unversty of Graz, Henrchstr. 36, 8010 Graz, Austra; Lavrent ev Insttute of Hydrodynamcs, Sberan Dvson
More informationLecture Randomized Load Balancing strategies and their analysis. Probability concepts include, counting, the union bound, and Chernoff bounds.
U.C. Berkeley CS273: Parallel and Dstrbuted Theory Lecture 1 Professor Satsh Rao August 26, 2010 Lecturer: Satsh Rao Last revsed September 2, 2010 Lecture 1 1 Course Outlne We wll cover a samplng of the
More informationOutline and Reading. Dynamic Programming. Dynamic Programming revealed. Computing Fibonacci. The General Dynamic Programming Technique
Outlne and Readng Dynamc Programmng The General Technque ( 5.3.2) -1 Knapsac Problem ( 5.3.3) Matrx Chan-Product ( 5.3.1) Dynamc Programmng verson 1.4 1 Dynamc Programmng verson 1.4 2 Dynamc Programmng
More informationLecture 2: Gram-Schmidt Vectors and the LLL Algorithm
NYU, Fall 2016 Lattces Mn Course Lecture 2: Gram-Schmdt Vectors and the LLL Algorthm Lecturer: Noah Stephens-Davdowtz 2.1 The Shortest Vector Problem In our last lecture, we consdered short solutons to
More informationCS 331 DESIGN AND ANALYSIS OF ALGORITHMS DYNAMIC PROGRAMMING. Dr. Daisy Tang
CS DESIGN ND NLYSIS OF LGORITHMS DYNMIC PROGRMMING Dr. Dasy Tang Dynamc Programmng Idea: Problems can be dvded nto stages Soluton s a sequence o decsons and the decson at the current stage s based on the
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationMEM Chapter 4b. LMI Lab commands
1 MEM8-7 Chapter 4b LMI Lab commands setlms lmvar lmterm getlms lmedt lmnbr matnbr lmnfo feasp dec2mat evallm showlm setmvar mat2dec mncx dellm delmvar gevp 2 Intalzng the LMI System he descrpton of an
More informationU.C. Berkeley CS278: Computational Complexity Professor Luca Trevisan 2/21/2008. Notes for Lecture 8
U.C. Berkeley CS278: Computatonal Complexty Handout N8 Professor Luca Trevsan 2/21/2008 Notes for Lecture 8 1 Undrected Connectvty In the undrected s t connectvty problem (abbrevated ST-UCONN) we are gven
More informationThe Second Anti-Mathima on Game Theory
The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player
More informationCalculation of time complexity (3%)
Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add
More informationComputing Correlated Equilibria in Multi-Player Games
Computng Correlated Equlbra n Mult-Player Games Chrstos H. Papadmtrou Presented by Zhanxang Huang December 7th, 2005 1 The Author Dr. Chrstos H. Papadmtrou CS professor at UC Berkley (taught at Harvard,
More informationShort running title: A generating function approach A GENERATING FUNCTION APPROACH TO COUNTING THEOREMS FOR SQUARE-FREE POLYNOMIALS AND MAXIMAL TORI
Short runnng ttle: A generatng functon approach A GENERATING FUNCTION APPROACH TO COUNTING THEOREMS FOR SQUARE-FREE POLYNOMIALS AND MAXIMAL TORI JASON FULMAN Abstract. A recent paper of Church, Ellenberg,
More informationCSCI B609: Foundations of Data Science
CSCI B609: Foundatons of Data Scence Lecture 13/14: Gradent Descent, Boostng and Learnng from Experts Sldes at http://grgory.us/data-scence-class.html Grgory Yaroslavtsev http://grgory.us Constraned Convex
More informationCommunication Complexity 16:198: February Lecture 4. x ij y ij
Communcaton Complexty 16:198:671 09 February 2010 Lecture 4 Lecturer: Troy Lee Scrbe: Rajat Mttal 1 Homework problem : Trbes We wll solve the thrd queston n the homework. The goal s to show that the nondetermnstc
More informationLecture 4: Constant Time SVD Approximation
Spectral Algorthms and Representatons eb. 17, Mar. 3 and 8, 005 Lecture 4: Constant Tme SVD Approxmaton Lecturer: Santosh Vempala Scrbe: Jangzhuo Chen Ths topc conssts of three lectures 0/17, 03/03, 03/08),
More informationMATHEMATICAL ENGINEERING TECHNICAL REPORTS. Successive Lagrangian Relaxation Algorithm for Nonconvex Quadratic Optimization
MATHEMATICAL ENGINEERING TECHNICAL REPORTS Successve Lagrangan Relaxaton Algorthm for Nonconvex Quadratc Optmzaton Shnj YAMADA and Akko TAKEDA METR 2017 08 March 2017 DEPARTMENT OF MATHEMATICAL INFORMATICS
More informationErratum: A Generalized Path Integral Control Approach to Reinforcement Learning
Journal of Machne Learnng Research 00-9 Submtted /0; Publshed 7/ Erratum: A Generalzed Path Integral Control Approach to Renforcement Learnng Evangelos ATheodorou Jonas Buchl Stefan Schaal Department of
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,
More informationDecentralized Computation for Robust Stability Analysis of Large State-Space Systems Using Polya s Theorem
Decentralzed Computaton for Robust Stablty Analyss of Large State-Space Systems Usng Polya s Theorem Reza Kamyar Matthew M. Peet Abstract In ths paper, we propose a parallel algorthm to solve large robust
More informationFaster and Simpler Width-Independent Parallel Algorithms for Positive Semidefinite Programming
Faster and Smpler Wdth-Independent Parallel Algorthms for Postve Semdefnte Programmng Rchard Peng Kanat Tangwongsan Carnege Mellon Unversty {yangp, ktangwon}@cs.cmu.edu ASTRACT Ths paper studes the problem
More informationNorms, Condition Numbers, Eigenvalues and Eigenvectors
Norms, Condton Numbers, Egenvalues and Egenvectors 1 Norms A norm s a measure of the sze of a matrx or a vector For vectors the common norms are: N a 2 = ( x 2 1/2 the Eucldean Norm (1a b 1 = =1 N x (1b
More informationSingle-Facility Scheduling over Long Time Horizons by Logic-based Benders Decomposition
Sngle-Faclty Schedulng over Long Tme Horzons by Logc-based Benders Decomposton Elvn Coban and J. N. Hooker Tepper School of Busness, Carnege Mellon Unversty ecoban@andrew.cmu.edu, john@hooker.tepper.cmu.edu
More informationDepartment of Chemical and Biological Engineering LECTURE NOTE II. Chapter 3. Function of Several Variables
LECURE NOE II Chapter 3 Functon of Several Varables Unconstraned multvarable mnmzaton problem: mn f ( x), x R x N where x s a vector of desgn varables of dmenson N, and f s a scalar obectve functon - Gradent
More informationFeb 14: Spatial analysis of data fields
Feb 4: Spatal analyss of data felds Mappng rregularly sampled data onto a regular grd Many analyss technques for geophyscal data requre the data be located at regular ntervals n space and/or tme. hs s
More information