Machine Learning. Support Vector Machines. Le Song. CSE6740/CS7641/ISYE6740, Fall Lecture 8, Sept. 13, 2012 Based on slides from Eric Xing, CMU
|
|
|
- Rudolph Payne
- 6 years ago
- Views:
Transcription
1 Mchne Lernng CSE6740/CS764/ISYE6740 Fll 0 Support Vector Mchnes Le Song Lecture 8 Sept. 3 0 Bsed on sldes fro Erc Xng CMU Redng: Chp. 6&7 C.B ook
2 Outlne Mu rgn clssfcton Constrned optzton Lgrngn dult Kernel trck Non-seprle cses
3 Wht s good Decson Boundr? Consder nr clssfcton tsk th = ± lels not 0/ s efore. When the trnng eples re lnerl seprle e cn set the preters of lner clssfer so tht ll the trnng eples re clssfed correctl Mn decson oundres! Genertve clssfers Logstc regressons Are ll decson oundres equll good? Clss Clss
4 Not All Decson Boundres Are Equl! Wh e hve such oundres? Irregulr dstruton Ilnced trnng szes outlners
5 Clssfcton nd Mrgn Preterzng decson oundr Let denote vector orthogonl to the decson oundr nd denote sclr "offset" ter then e cn rte the decson oundr s: 0 Clss Clss d - d +
6 Clssfcton nd Mrgn Preterzng decson oundr Let denote vector orthogonl to the decson oundr nd denote sclr "offset" ter then e cn rte the decson oundr s: 0 Mrgn Clss Clss d - d + + > +c for ll n clss + < -c for ll n clss Or ore copctl: + >c he rgn eteen to ponts = d - + d + =
7 Mu Mrgn Clssfcton he rgn s: Here s our Mu Mrgn Clssfcton prole: c - * * c c s.t
8 Mu Mrgn Clssfcton con'd. he optzton prole: s.t But note tht the gntude of c erel scles nd nd does not chnge the clssfcton oundr t ll! h? So e nsted ork on ths clener prole: s.t c he soluton to ths leds to the fous Support Vector Mchnes - -- eleved n to e the est "off-the-shelf" supervsed lernng lgorth c
9 Support vector chne A conve qudrtc progrng prole th lner constrns: s.t he ttned rgn s no gven Onl fe of the clssfcton constrnts re relevnt support vectors Constrned optzton We cn drectl solve ths usng coercl qudrtc progrng QP code But e nt to tke ore creful nvestgton of Lgrnge dult nd the soluton of the ove s ts dul for. deeper nsght: support vectors kernels ore effcent lgorth d - d +
10 Lgrngn Dult he Prl Prole Prl: he generlzed Lgrngn: the 's 0 nd 's re clled the Lgrngn ultplers Le: A re-rtten Prl: l h k g f s.t. n 0 0 l k h g f L o/ s constrnt prl stsfes f f L 0 n L 0
11 Lgrngn Dult cont. Recll the Prl Prole: n L 0 he Dul Prole: n L 0 heore ek dult: d * 0 n L n 0 L p * heore strong dult: Iff there est sddle pont of * d L p * e hve
12 he KK condtons If there ests soe sddle pont of L then the sddle pont stsfes the follong "Krush-Kuhn-ucker" KK condtons: heore: If * * nd * stsf the KK condton then t s lso soluton to the prl nd the dul proles. k k g k g α l n L L
13 Solvng optl rgn clssfer Recll our opt prole: hs s equvlent to Wrte the Lgrngn: Recll tht * cn e reforulted s No e solve ts dul prole: s.t - s.t n L * n L 0 n L 0
14 *** he Dul Prole We nze L th respect to nd frst: Note tht * ples: Plus *** ck to L nd usng ** e hve: n L 0-0 L 0 L * - L **
15 he Dul prole cont. No e hve the follong dul opt prole: hs s gn qudrtc progrng prole. A glol u of cn ls e found. But ht's the g del?? Note to thngs:. cn e recovered. he "kernel" s.t. J 0 - k 0. See net More lter
16 Support vectors Note the KK condton --- onl fe 's cn e nonzero!! α g 0 k 5 =0 Clss 8 =0.6 0 =0 7 =0 =0 Cll the trnng dt ponts hose 's re nonzero the support vectors SV 4 =0 9 =0 Clss 3 =0 6 =.4 =0.8
17 Support vector chnes Once e hve the Lgrnge ultplers { } e cn reconstruct the preter vector s eghted conton of the trnng eples: SV For testng th ne dt z Copute z SV z nd clssf z s clss f the su s postve nd clss otherse Note: need not e fored eplctl
18 Interpretton of support vector chnes he optl s lner conton of sll nuer of dt ponts. hs sprse representton cn e veed s dt copresson s n the constructon of knn clssfer o copute the eghts { } nd to use support vector chnes e need to specf onl the nner products or kernel eteen the eples We ke decsons coprng ech ne eple z th onl the support vectors: * sgn SV z
19 Non-lnerl Seprle Proles Clss Clss We llo error n clssfcton; t s sed on the output of the dscrnnt functon + pprotes the nuer of sclssfed sples
20 Soft Mrgn Hperplne No e hve slghtl dfferent opt prole: re slck vrles n optzton Note tht =0 f there s no error for s n upper ound of the nuer of errors C : trdeoff preter eteen error nd rgn s.t - 0 C n
21 he Optzton Prole he dul of ths ne constrned optzton prole s hs s ver slr to the optzton prole n the lner seprle cse ecept tht there s n upper ound C on no Once gn QP solver cn e used to fnd - J. 0 s.t. k C 0
22 Etenson to Non-lner Decson Boundr So fr e hve onl consdered lrge-rgn clssfer th lner decson oundr Ho to generlze t to ecoe nonlner? Ke de: trnsfor to hgher densonl spce to ke lfe eser Input spce: the spce the pont re locted Feture spce: the spce of f fter trnsforton Wh trnsfor? Lner operton n the feture spce s equvlent to non-lner operton n nput spce Clssfcton cn ecoe eser th proper trnsforton. In the XOR prole for eple ddng ne feture of ke the prole lnerl seprle hoeork
23 rnsforng the Dt Input spce f. Coputton n the feture spce cn e costl ecuse t s hgh densonl he feture spce s tpcll nfnte-densonl! he kernel trck coes to rescue f f f f f f f f f f f f f f f f f f Feture spce Note: feture spce s of hgher denson thn the nput spce n prctce
24 he Kernel rck Recll the SVM optzton prole he dt ponts onl pper s nner product As long s e cn clculte the nner product n the feture spce e do not need the ppng eplctl Mn coon geoetrc opertons ngles dstnces cn e epressed nner products Defne the kernel functon K - J. 0 s.t. k C 0 K f f
25 An Eple for feture ppng nd kernels Consder n nput =[ ] Suppose f. s gven s follos An nner product n the feture spce s So f e defne the kernel functon s follos there s no need to crr out f. eplctl f ' ' ' ' ' ' ' ' f f ' ' K
26 More eples of kernel functons Lner kernel e've seen t K ' ' Polnol kernel e ust s n eple K here p = 3 o get the feture vectors e conctente ll pth order polnol ters of the coponents of eghted pproprtel ' ' p Rdl ss kernel K ' ep - - ' In ths cse the feture spce conssts of functons nd results n nonpretrc clssfer.
27 SVM eples
28 Eples for Non Lner SVMs Gussn Kernel
29 Cross-vldton error he leve-one-out cross-vldton error does not depend on the densonlt of the feture spce ut onl on the # of support vectors! Leve - one - out CV error # support ve ctors # of trnng eples
30 Sur Support Vector Mchne: M-rgn clssfer Constrned conve optzton prole Dult Support vectors Kernels
Machine Learning. Support Vector Machines. Eric Xing , Fall Lecture 9, October 6, 2015
Machne Learnng 0-70 Fall 205 Support Vector Machnes Erc Xng Lecture 9 Octoer 6 205 Readng: Chap. 6&7 C.B ook and lsted papers Erc Xng @ CMU 2006-205 What s a good Decson Boundar? Consder a nar classfcaton
Machine Learning. What is a good Decision Boundary? Support Vector Machines
Machne Learnng 0-70/5 70/5-78 78 Sprng 200 Support Vector Machnes Erc Xng Lecture 7 March 5 200 Readng: Chap. 6&7 C.B book and lsted papers Erc Xng @ CMU 2006-200 What s a good Decson Boundar? Consder
Machine Learning. Support Vector Machines. Eric Xing , Fall Lecture 9, October 8, 2015
Machne Learnng 0-70 Fall 205 Support Vector Machnes Erc Xng Lecture 9 Octoer 8 205 Readng: Chap. 6&7 C.B ook and lsted papers Erc Xng @ CMU 2006-205 What s a good Decson Boundar? Consder a nar classfcaton
Machine Learning. Support Vector Machines. Eric Xing. Lecture 4, August 12, Reading: Eric CMU,
Machne Learnng Support Vector Machnes Erc Xng Lecture 4 August 2 200 Readng: Erc Xng @ CMU 2006-200 Erc Xng @ CMU 2006-200 2 What s a good Decson Boundar? Wh e a have such boundares? Irregular dstrbuton
Recap: the SVM problem
Machne Learnng 0-70/5-78 78 Fall 0 Advanced topcs n Ma-Margn Margn Learnng Erc Xng Lecture 0 Noveber 0 Erc Xng @ CMU 006-00 Recap: the SVM proble We solve the follong constraned opt proble: a s.t. J 0
SVMs for regression Non-parametric/instance based classification method
S 75 Mchne ernng ecture Mos Huskrecht mos@cs.ptt.edu 539 Sennott Squre SVMs for regresson Non-prmetrc/nstnce sed cssfcton method S 75 Mchne ernng Soft-mrgn SVM Aos some fet on crossng the seprtng hperpne
Support vector machines for regression
S 75 Mchne ernng ecture 5 Support vector mchnes for regresson Mos Huskrecht mos@cs.ptt.edu 539 Sennott Squre S 75 Mchne ernng he decson oundr: ˆ he decson: Support vector mchnes ˆ α SV ˆ sgn αˆ SV!!: Decson
Machine Learning Support Vector Machines SVM
Mchne Lernng Support Vector Mchnes SVM Lesson 6 Dt Clssfcton problem rnng set:, D,,, : nput dt smple {,, K}: clss or lbel of nput rget: Construct functon f : X Y f, D Predcton of clss for n unknon nput
SVMs for regression Multilayer neural networks
Lecture SVMs for regresson Muter neur netors Mos Husrecht mos@cs.ptt.edu 539 Sennott Squre Support vector mchne SVM SVM mmze the mrgn round the seprtng hperpne. he decson functon s fu specfed suset of
An Ising model on 2-D image
School o Coputer Scence Approte Inerence: Loopy Bele Propgton nd vrnts Prolstc Grphcl Models 0-708 Lecture 4, ov 7, 007 Receptor A Knse C Gene G Receptor B Knse D Knse E 3 4 5 TF F 6 Gene H 7 8 Hetunndn
An Introduction to Support Vector Machines
An Introducton to Support Vector Mchnes Wht s good Decson Boundry? Consder two-clss, lnerly seprble clssfcton problem Clss How to fnd the lne (or hyperplne n n-dmensons, n>)? Any de? Clss Per Lug Mrtell
CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 9
CS434/541: Pttern Recognton Prof. Olg Veksler Lecture 9 Announcements Fnl project proposl due Nov. 1 1-2 prgrph descrpton Lte Penlt: s 1 pont off for ech d lte Assgnment 3 due November 10 Dt for fnl project
18.7 Artificial Neural Networks
310 18.7 Artfcl Neurl Networks Neuroscence hs hypotheszed tht mentl ctvty conssts prmrly of electrochemcl ctvty n networks of brn cells clled neurons Ths led McCulloch nd Ptts to devse ther mthemtcl model
Chapter 2 Introduction to Algebra. Dr. Chih-Peng Li ( 李 )
Chpter Introducton to Algebr Dr. Chh-Peng L 李 Outlne Groups Felds Bnry Feld Arthetc Constructon of Glos Feld Bsc Propertes of Glos Feld Coputtons Usng Glos Feld Arthetc Vector Spces Groups 3 Let G be set
Partially Observable Systems. 1 Partially Observable Markov Decision Process (POMDP) Formalism
CS294-40 Lernng for Rootcs nd Control Lecture 10-9/30/2008 Lecturer: Peter Aeel Prtlly Oservle Systems Scre: Dvd Nchum Lecture outlne POMDP formlsm Pont-sed vlue terton Glol methods: polytree, enumerton,
UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II
Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )
Lecture 3 Camera Models 2 & Camera Calibration. Professor Silvio Savarese Computational Vision and Geometry Lab
Lecture Cer Models Cer Clbrton rofessor Slvo Svrese Coputtonl Vson nd Geoetry Lb Slvo Svrese Lecture - Jn 7 th, 8 Lecture Cer Models Cer Clbrton Recp of cer odels Cer clbrton proble Cer clbrton wth rdl
Least squares. Václav Hlaváč. Czech Technical University in Prague
Lest squres Václv Hlváč Czech echncl Unversty n Prgue hlvc@fel.cvut.cz http://cmp.felk.cvut.cz/~hlvc Courtesy: Fred Pghn nd J.P. Lews, SIGGRAPH 2007 Course; Outlne 2 Lner regresson Geometry of lest-squres
Rank One Update And the Google Matrix by Al Bernstein Signal Science, LLC
Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses
Review of linear algebra. Nuno Vasconcelos UCSD
Revew of lner lgebr Nuno Vsconcelos UCSD Vector spces Defnton: vector spce s set H where ddton nd sclr multplcton re defned nd stsf: ) +( + ) (+ )+ 5) λ H 2) + + H 6) 3) H, + 7) λ(λ ) (λλ ) 4) H, - + 8)
Support Vector Machines
Spport Vector Mchnes Note to other techers nd sers of these sldes. Andrew wold e delghted f yo fond ths sorce terl sefl n gvng yor own lectres. Feel free to se these sldes vert, or to odfy the to ft yor
Geometric Correction or Georeferencing
Geoetrc Correcton or Georeferencng GEOREFERENCING: fro ge to p Coordntes on erth: (λ, φ) ge: (, ) p: (, ) rel nteger Trnsfortons (nvolvng deforton): erth-to-ge: χ erth-to-p: ψ (crtogrphc proecton) ge-to-p:
SVMs: Duality and Kernel Trick. SVMs as quadratic programs
/8/9 SVMs: Dualt and Kernel rck Machne Learnng - 6 Geoff Gordon MroslavDudík [[[partl ased on sldes of Zv-Bar Joseph] http://.cs.cmu.edu/~ggordon/6/ Novemer 8 9 SVMs as quadratc programs o optmzaton prolems:
DCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
Constructing Free Energy Approximations and GBP Algorithms
3710 Advnced Topcs n A ecture 15 Brnslv Kveton kveton@cs.ptt.edu 5802 ennott qure onstructng Free Energy Approxtons nd BP Algorths ontent Why? Belef propgton (BP) Fctor grphs egon-sed free energy pproxtons
mywbut.com Lesson 13 Representation of Sinusoidal Signal by a Phasor and Solution of Current in R-L-C Series Circuits
wut.co Lesson 3 Representtion of Sinusoil Signl Phsor n Solution of Current in R-L-C Series Circuits wut.co In the lst lesson, two points were escrie:. How sinusoil voltge wvefor (c) is generte?. How the
SVMs: Duality and Kernel Trick. SVMs as quadratic programs
11/17/9 SVMs: Dualt and Kernel rck Machne Learnng - 161 Geoff Gordon MroslavDudík [[[partl ased on sldes of Zv-Bar Joseph] http://.cs.cmu.edu/~ggordon/161/ Novemer 18 9 SVMs as quadratc programs o optmzaton
Support Vector Machines
Spport Vector Mchnes Modfed fro the sldes y Dr. Andrew W. Moore http://www.cs.c.ed/~w/ttorls Copyrght 00, 003, Andrew W. Moore Nov 3rd, 00 Mterl Sectons 5.3 nd 6.4 n We, Copsey: Sttstcl Pttern ecognton,
( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
CISE 301: Numerical Methods Lecture 5, Topic 4 Least Squares, Curve Fitting
CISE 3: umercl Methods Lecture 5 Topc 4 Lest Squres Curve Fttng Dr. Amr Khouh Term Red Chpter 7 of the tetoo c Khouh CISE3_Topc4_Lest Squre Motvton Gven set of epermentl dt 3 5. 5.9 6.3 The reltonshp etween
Unit 2 Exponents Study Guide
Unit Eponents Stud Guide 7. Integer Eponents Prt : Zero Eponents Algeric Definition: 0 where cn e n non-zero vlue 0 ecuse 0 rised to n power less thn or equl to zero is n undefined vlue. Eple: 0 If ou
Applied Statistics Qualifier Examination
Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng
8 factors of x. For our second example, let s raise a power to a power:
CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,
Neural Network Introduction. Hung-yi Lee
Neu Neto Intoducton Hung- ee Reve: Supevsed enng Mode Hpothess Functon Set f, f : : (e) Tnng: Pc the est Functon f * Best Functon f * Testng: f Tnng Dt : functon nput : functon output, ˆ,, ˆ, Neu Neto
Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. with respect to λ. 1. χ λ χ λ ( ) λ, and thus:
More on χ nd errors : uppose tht we re fttng for sngle -prmeter, mnmzng: If we epnd The vlue χ ( ( ( ; ( wth respect to. χ n Tlor seres n the vcnt of ts mnmum vlue χ ( mn χ χ χ χ + + + mn mnmzes χ, nd
Formulae For. Standard Formulae Of Integrals: x dx k, n 1. log. a dx a k. cosec x.cot xdx cosec. e dx e k. sec. ax dx ax k. 1 1 a x.
Forule For Stndrd Forule Of Integrls: u Integrl Clculus By OP Gupt [Indir Awrd Winner, +9-965 35 48] A B C D n n k, n n log k k log e e k k E sin cos k F cos sin G tn log sec k OR log cos k H cot log sin
Principle Component Analysis
Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors
6 Roots of Equations: Open Methods
HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 6 Roots of Equtons: Open Methods Smple Fed-Pont Iterton Newton-Rphson Secnt Methods MATLAB Functon: fzero Polynomls Cse Study: Ppe Frcton Brcketng
Support 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
r = cos θ + 1. dt ) dt. (1)
MTHE 7 Proble Set 5 Solutions (A Crdioid). Let C be the closed curve in R whose polr coordintes (r, θ) stisfy () Sketch the curve C. r = cos θ +. (b) Find pretriztion t (r(t), θ(t)), t [, b], of C in polr
Exponents and Powers
EXPONENTS AND POWERS 9 Exponents nd Powers CHAPTER. Introduction Do you know? Mss of erth is 5,970,000,000,000, 000, 000, 000, 000 kg. We hve lredy lernt in erlier clss how to write such lrge nubers ore
Recap: the SVM problem
Machne Learnng 0-70/5-78, 78, Fall 20 Advanced topcs n Max-Margn Margn Learnng Erc Xng Lecture 20, Noveber 2, 20 Erc Xng @ CMU, 2006-200 Recap: the SVM proble We solve the follong constraned opt proble:
Lecture 8: Camera Calibration
Lecture 8: Cer Clbrton rofessor Fe-Fe L Stnford Vson Lb Fe-Fe L 9-Oct- Wht we wll lern tody? Revew cer preters Affne cer odel (roble Set (Q4)) Cer clbrton Vnshng ponts nd lnes (roble Set (Q)) Redng: [F]
Chapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
VECTORS VECTORS VECTORS VECTORS. 2. Vector Representation. 1. Definition. 3. Types of Vectors. 5. Vector Operations I. 4. Equal and Opposite Vectors
1. Defnton A vetor s n entt tht m represent phsl quntt tht hs mgntude nd dreton s opposed to slr tht ls dreton.. Vetor Representton A vetor n e represented grphll n rrow. The length of the rrow s the mgntude
September 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
Lecture 9: LTL and Büchi Automata
Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled
PHYS 601 HW3 Solution
3.1 Norl force using Lgrnge ultiplier Using the center of the hoop s origin, we will describe the position of the prticle with conventionl polr coordintes. The Lgrngin is therefore L = 1 2 ṙ2 + 1 2 r2
OXFORD H i g h e r E d u c a t i o n Oxford University Press, All rights reserved.
Renshw: Mths for Econoics nswers to dditionl exercises Exercise.. Given: nd B 5 Find: () + B + B 7 8 (b) (c) (d) (e) B B B + B T B (where 8 B 6 B 6 8 B + B T denotes the trnspose of ) T 8 B 5 (f) (g) B
Work and Energy (Work Done by a Varying Force)
Lecture 1 Chpter 7 Physcs I 3.5.14 ork nd Energy (ork Done y Vryng Force) Course weste: http://fculty.uml.edu/andry_dnylov/techng/physcsi Lecture Cpture: http://echo36.uml.edu/dnylov13/physcs1fll.html
Partial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
5.1 Estimating with Finite Sums Calculus
5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during
Haddow s Experiment:
schemtc drwng of Hddow's expermentl set-up movng pston non-contctng moton sensor bems of sprng steel poston vres to djust frequences blocks of sold steel shker Hddow s Experment: terr frm Theoretcl nd
Supporting information How to concatenate the local attractors of subnetworks in the HPFP
n Effcen lgorh for Idenfyng Prry Phenoype rcors of Lrge-Scle Boolen Newor Sng-Mo Choo nd Kwng-Hyun Cho Depren of Mhecs Unversy of Ulsn Ulsn 446 Republc of Kore Depren of Bo nd Brn Engneerng Kore dvnced
? plate in A G in
Proble (0 ponts): The plstc block shon s bonded to rgd support nd to vertcl plte to hch 0 kp lod P s ppled. Knong tht for the plstc used G = 50 ks, deterne the deflecton of the plte. Gven: G 50 ks, P 0
Ideal Gas behaviour: summary
Lecture 4 Rel Gses Idel Gs ehviour: sury We recll the conditions under which the idel gs eqution of stte Pn is vlid: olue of individul gs olecules is neglected No interctions (either ttrctive or repulsive)
ESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus
ESI 34 tmospherc Dnmcs I Lesson 1 Vectors nd Vector lculus Reference: Schum s Outlne Seres: Mthemtcl Hndbook of Formuls nd Tbles Suggested Redng: Mrtn Secton 1 OORDINTE SYSTEMS n orthonorml coordnte sstem
Support Vector Machines CS434
Support Vector Machnes CS434 Lnear Separators Many lnear separators exst that perfectly classfy all tranng examples Whch of the lnear separators s the best? Intuton of Margn Consder ponts A, B, and C We
Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
Discussion Question 1A P212, Week 1 P211 Review: 2-D Motion with Uniform Force
Discussion Question 1A P1, Week 1 P11 Review: -D otion with Unifor Force The thetics nd phsics of the proble below re siilr to probles ou will encounter in P1, where the force is due to the ction of n
Second degree generalized gauss-seidel iteration method for solving linear system of equations. ABSTRACT
Ethiop. J. Sci. & Technol. 7( 5-, 0 5 Second degree generlized guss-seidel itertion ethod for solving liner syste of equtions Tesfye Keede Bhir Dr University, College of Science, Deprtent of Mthetics tk_ke@yhoo.co
Introduction to Electrical & Electronic Engineering ENGG1203
Introduction to Electricl & Electronic Engineering ENGG23 2 nd Semester, 27-8 Dr. Hden Kwok-H So Deprtment of Electricl nd Electronic Engineering Astrction DIGITAL LOGIC 2 Digitl Astrction n Astrct ll
Elastic Collisions. Definition: two point masses on which no external forces act collide without losing any energy.
Elastc Collsons Defnton: to pont asses on hch no external forces act collde thout losng any energy v Prerequstes: θ θ collsons n one denson conservaton of oentu and energy occurs frequently n everyday
Support Vector Machines
CS 2750: Machne Learnng Support Vector Machnes Prof. Adrana Kovashka Unversty of Pttsburgh February 17, 2016 Announcement Homework 2 deadlne s now 2/29 We ll have covered everythng you need today or at
The Schur-Cohn Algorithm
Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for
Things to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
Introduction to Numerical Integration Part II
Introducton to umercl Integrton Prt II CS 75/Mth 75 Brn T. Smth, UM, CS Dept. Sprng, 998 4/9/998 qud_ Intro to Gussn Qudrture s eore, the generl tretment chnges the ntegrton prolem to ndng the ntegrl w
Linear and Nonlinear Optimization
Lner nd Nonlner Optmzton Ynyu Ye Deprtment of Mngement Scence nd Engneerng Stnford Unversty Stnford, CA 9430, U.S.A. http://www.stnford.edu/~yyye http://www.stnford.edu/clss/msnde/ Ynyu Ye, Stnford, MS&E
Lecture 4: Piecewise Cubic Interpolation
Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml
Numerical Solution of Freholm-Volterra Integral Equations by Using Scaling Function Interpolation Method
Aled Mthetcs 3 4 4-9 htt://ddoorg/436/34a3 Pulshed Onlne Jnury 3 (htt://wwwscrorg/ournl/) uercl Soluton of Frehol-Volterr Integrl Equtons y Usng Sclng Functon Interolton Method Yousef Al-Jrrh En-Bng Ln
Thomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
Advanced Machine Learning. An Ising model on 2-D image
Advnced Mchne Lernng Vrtonl Inference Erc ng Lecture 12, August 12, 2009 Redng: Erc ng Erc ng @ CMU, 2006-2009 1 An Isng model on 2-D mge odes encode hdden nformton ptchdentty. They receve locl nformton
The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
Multiple view geometry
EECS 442 Computer vson Multple vew geometry Perspectve Structure from Moton - Perspectve structure from moton prolem - mgutes - lgerc methods - Fctorzton methods - Bundle djustment - Self-clrton Redng:
Lecture 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
Computer Graphics (CS 543) Lecture 3 (Part 1): Linear Algebra for Graphics (Points, Scalars, Vectors)
Coputer Grphics (CS 543) Lecture 3 (Prt ): Liner Alger for Grphics (Points, Sclrs, Vectors) Prof Enuel Agu Coputer Science Dept. Worcester Poltechnic Institute (WPI) Points, Sclrs nd Vectors Points, vectors
Outline. Review Quadrilateral Equation. Review Linear φ i Quadrilateral. Review x and y Derivatives. Review φ Derivatives
E 5 Engneerng nlss ore on Fnte Eleents n ore on Fnte Eleents n Two Densons Two Densons Lrr Cretto echncl Engneerng 5 Senr n Engneerng nlss prl 7-9 9 Otlne Revew lst lectre Qrtc ss nctons n two ensons orer
Introduction to Algebra - Part 2
Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite
Creative Practicing. By Jimmy Wyble edited by David Oakes
Cretve Prctcng By Jy Wyble edted by Dvd Okes Edtors Note: Ths terl s n excert fro Jy s lecture tht he resented t Muscns Insttute on Arl, 008. Ths s the thrd eek of ten-eek qurter. In the revous eeks, he
INTRODUCTION TO COMPLEX NUMBERS
INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced
Kernels in Support Vector Machines. Based on lectures of Martin Law, University of Michigan
Kernels n Support Vector Machnes Based on lectures of Martn Law, Unversty of Mchgan Non Lnear separable problems AND OR NOT() The XOR problem cannot be solved wth a perceptron. XOR Per Lug Martell - Systems
LINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
International Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sc. Technol., () (), pp. 44-49 Interntonl Journl of Pure nd Appled Scences nd Technolog ISSN 9-67 Avlle onlne t www.jopst.n Reserch Pper Numercl Soluton for Non-Lner Fredholm Integrl
MATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
The Number of Rows which Equal Certain Row
Interntonl Journl of Algebr, Vol 5, 011, no 30, 1481-1488 he Number of Rows whch Equl Certn Row Ahmd Hbl Deprtment of mthemtcs Fcult of Scences Dmscus unverst Dmscus, Sr hblhmd1@gmlcom Abstrct Let be X
M344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
Advanced Introduction to Machine Learning
Advanced Introducton to Machne Learnng 10715, Fall 2014 The Kernel Trck, Reproducng Kernel Hlbert Space, and the Representer Theorem Erc Xng Lecture 6, September 24, 2014 Readng: Erc Xng @ CMU, 2014 1
II The Z Transform. Topics to be covered. 1. Introduction. 2. The Z transform. 3. Z transforms of elementary functions
II The Z Trnsfor Tocs o e covered. Inroducon. The Z rnsfor 3. Z rnsfors of eleenry funcons 4. Proeres nd Theory of rnsfor 5. The nverse rnsfor 6. Z rnsfor for solvng dfference equons II. Inroducon The
Lecture 36. Finite Element Methods
CE 60: Numercl Methods Lecture 36 Fnte Element Methods Course Coordntor: Dr. Suresh A. Krth, Assocte Professor, Deprtment of Cvl Engneerng, IIT Guwht. In the lst clss, we dscussed on the ppromte methods
Image classification. Given the bag-of-features representations of images from different classes, how do we learn a model for distinguishing i them?
Image classfcaton Gven te bag-of-features representatons of mages from dfferent classes ow do we learn a model for dstngusng tem? Classfers Learn a decson rule assgnng bag-offeatures representatons of
Abhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no
hlsh Clsses Clss- XII Dte: 0- - SOLUTION Chp - 9,0, MM 50 Mo no-996 If nd re poston vets of nd B respetvel, fnd the poston vet of pont C n B produed suh tht C B vet r C B = where = hs length nd dreton
Section 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
PHY 5246: Theoretical Dynamics, Fall Assignment # 5, Solutions. θ = l mr 2 = l
PHY 546: Theoreticl Dynics, Fll 15 Assignent # 5, Solutions 1 Grded Probles Proble 1 (1.) Using the eqution of the orbit or force lw d ( 1 dθ r)+ 1 r = r F(r), (1) l with r(θ) = ke αθ one finds fro which
Name: SID: Discussion Session:
Nme: SID: Dscusson Sesson: hemcl Engneerng hermodynmcs -- Fll 008 uesdy, Octoer, 008 Merm I - 70 mnutes 00 onts otl losed Book nd Notes (5 ponts). onsder n del gs wth constnt het cpctes. Indcte whether
Linear Classification, SVMs and Nearest Neighbors
1 CSE 473 Lecture 25 (Chapter 18) Lnear Classfcaton, SVMs and Nearest Neghbors CSE AI faculty + Chrs Bshop, Dan Klen, Stuart Russell, Andrew Moore Motvaton: Face Detecton How do we buld a classfer to dstngush
Variable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia
Vrble tme mpltude mplfcton nd quntum lgorthms for lner lgebr Andrs Ambns Unversty of Ltv Tlk outlne. ew verson of mpltude mplfcton;. Quntum lgorthm for testng f A s sngulr; 3. Quntum lgorthm for solvng
Chapter I Vector Analysis
. Chpte I Vecto nlss . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) ( (v) he lw
Nondeterminism and Nodeterministic Automata
Nondeterminism nd Nodeterministic Automt 61 Nondeterminism nd Nondeterministic Automt The computtionl mchine models tht we lerned in the clss re deterministic in the sense tht the next move is uniquely
Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
The Properties of Stars
10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product