Anouncements. Conjugate Gradients. Steepest Descent. Outline. Steepest Descent. Steepest Descent

Transcription

1 oucms Couga Gas Mchal Kazha (6.657) Ifomao abou h Sma (6.757) hav b pos ol: hp:// Tch Spcs: o M o Tusay afoo. o Two paps scuss ach w. o Vos fo w s caa paps u by Thusay vg. Oul Rvw of Sps Dsc Couga Gas Sps Dsc Rvw: Th a bh hs appoach s o p h soluo of h quao b as h mmzao of h fuco: b 2 Sps Dsc Rvw: Th a bh hs appoach s o p h soluo of h quao b as h mmzao of h fuco: b 2 Gv a guss fo h soluo h guss + s ga by ag a sp h co oppos o h co whch F cass: ) + Sps Dsc Rvw: Sc h ga of F a s h sual: F ( ) b : hs gvs h upa ul: + wh

2 Sps Dsc Sps Dsc Eampl: Eampl: b b 8 Fo hs ma a hs vco b h plo of h so-coous of h fuco s show o h gh. Sag wh a al guss f w a hough h sps sc algohm w ma h sps: Shwchu 994 Shwchu 994 Sps Dsc Eampl: b 8 Oul Rvw of Sps Dsc Couga Gas Sag wh a al guss f w a hough h sps sc algohm w ma h W sps: of up -vsg sach cos w ha alay Shwchu 994 Couga Gas Goal: To f a av appoach ha:. Gs us clos a clos o h soluo. 2. Esus w o o vs h sam co wc. Couga Gas To o hs w wll h of wog wh h squc of os { } ah ha h squc of gusss { }: 2

3 Couga Gas To o hs w wll h of wog wh h squc of os { } ah ha h squc of gusss { }: Tha s ah ha yg o ga a squc of gusss wh: lm Couga Gas To o hs w wll h of wog wh h squc of os { } ah ha h squc of gusss { }: Tha s ah ha yg o ga a squc of gusss wh: lm W y o ga a squc of os wh: lm Couga Gas No: If w h of a upa ul as ag som vco ε o o gv us + : + + ε Couga Gas No: If w h of a upa ul as ag som vco ε o o gv us + : + + ε Ths s quval o subacg h vco ε fom o gv us + : + ε Couga Gas (Fs Pass) ppoach: Suppos ha w hav a al guss a w hav a s of ohoomal cos { - }. Couga Gas (Fs Pass) ppoach: Suppos ha w hav a al guss a w hav a s of ohoomal cos { - }. W woul l o sg a algohm ha fs h (+)-s o by movg h compo of h o lyg alog h co. + 3

4 Couga Gas (Fs Pass) Clam: Ths mho s guaa o g h gh asw af aos. Couga Gas (Fs Pass) Poof: Sc h { - } a ohoomal w ca w h o h al guss as: Couga Gas (Fs Pass) Poof: Sc h { - } a ohoomal w ca w h o h al guss as: f h fs ao w hav: Couga Gas (Fs Pass) Poof: Sc h { - } a ohoomal w ca w h o h al guss as: f h fs ao w hav: Couga Gas (Fs Pass) Poof: Sc h { - } a ohoomal w ca w h o h al guss as: f h -h ao w hav: Couga Gas (Fs Pass) Poof: Sc h { - } a ohoomal w ca w h o h al guss as: af h -h ao w hav: 4

5 Couga Gas (Fs Pass) Poblm: W o ow h coc soluo Couga Gas (Fs Pass) Poblm: W o ow h coc soluo W o ow h valu of - Couga Gas (Fs Pass) Poblm: W o ow h coc soluo W o ow h valu of - W ca fgu ou wha h compo of h o co s:? Couga Gas Soluo: To ass hs poblm w wll chag ou oo of sac so ha w ca compu h compo of h o co whou v owg h valu of. Couga Gas Obsvao: If w hav a symmc posv f ma w ca h of h ma as fg a w -pouc: u v u v Couga Gas Obsvao: If w hav a symmc posv f ma w ca h of h ma as fg a w -pouc: u v u v Ths w pouc has h sam pops ha h aoal pouc has:. Symmy: u v v u 2. Posvy: u u 3. Dfss: u u u 5

6 Couga Gas Ky Ia: lhough w cao compu h o-pouc: usg h aoal -pouc Couga Gas Ky Ia: lhough w cao compu h o-pouc: usg h aoal -pouc W ca compu usg h -pouc f by : ( ) b Couga Gas Ky Ia: lhough w cao compu h o-pouc: usg h aoal -pouc W ca compu usg h -pouc f by : ( ) b Couga Gas Ky Ia: lhough w cao compu h o-pouc: usg h aoal -pouc W ca compu usg h -pouc f by : ( ) b Couga Gas ppoach: If h vcos { - } a -ohoomal: δ Couga Gas ppoach: If h vcos { - } a -ohoomal: δ W ca f a aalogous algohm sag wh a al o w ga h os by succssvly movg h o compo co : + 6

7 Couga Gas ppoach: If h vcos { - } a -ohoomal: δ W ca f a aalogous algohm sag wh a al o w ga h os by succssvly movg h o compo co : + s bfo hs mho s guaa o gv h coc asw af aos. Couga Gas ppoach: If h vcos { - } a -ohoomal: δ W ca f a aalogous algohm sag wh a al o w ga h os by succssvly movg h o compo co : + s bfo hs mho s guaa o gv Howv h coc os asw o qu af aos. owg h vco avac oly b. Couga Gas Cocpually: Sc w o ow h soluo w cao ally al abou upag h o. Couga Gas Cocpually: Sc w o ow h soluo w cao ally al abou upag h o. Howv w ca al abou upag h sual: b Couga Gas Cocpually: I hs co h upa sp bcoms: + Couga Gas Cocpually: I hs co h upa sp bcoms: + + ( ) 7

8 Couga Gas Cocpually: I hs co h upa sp bcoms: + + ( ) Couga Gas Quso: How o w ga a goo s of sach cos { - }? o Th cos a -ohoomal. o Th cos hav h popy ha mos of h covgc happs aly o (so w o hav o u a full aos). + Couga Gas b 2 Couga Gas b 2 Choosg Dcos: ) b Choosg h fs co s asy. Choosg Dcos: ) b Choosg h fs co s asy. Gv h guss w wa o choos a co o upa o o mmz. Couga Gas b 2 Couga Gas b 2 Choosg Dcos: Choosg h fs co s asy. Gv h guss w wa o choos a co o upa o o mmz. Usg h fac ha h ga a s: ) ) b Choosg Dcos: Choosg h fs co s asy. Gv h guss w wa o choos a co o upa o o mmz. Usg h fac ha h ga a s: ) ) b hs gvs: 8

9 Couga Gas b 2 Couga Gas b 2 Choosg Dcos: ) b To choos h co w sa wh h ga co: a upa so ha { } a -ohoomal: ) Choosg Dcos: ) b To choos h co w sa wh h Th ga poblm co: wh hs appoach s ha smacs of Gam-Schm F ohogoalzao. ( ) a upa so ha { } a -ohoomal: Couga Gas b 2 Couga Gas b 2 Choosg Dcos: ) b To choos h co w sa wh h Th ga poblm co: wh hs appoach s ha smacs of Gam-Schm F ohogoalzao. ( ) Gag h vco qus compug a upa h o-pouc so ha {wh } all a -ohoomal: wh <. Choosg Dcos: ) b To choos h co w sa wh h Th ga poblm co: wh hs appoach s ha smacs of Gam-Schm F ohogoalzao. ( ) Gag h vco qus compug a upa h o-pouc so ha {wh } all a -ohoomal: Th comply of compug wh h <. fs cos s O( 2 ). Couga Gas b 2 Couga Gas b 2 Choosg Dcos: Tus ou ha lf s o so ba. ) b Choosg Dcos: ) b Tus ou ha lf s o so ba. Fo ay < h sual + sasfs h popy: + 9

10 Couga Gas Choosg Dcos: Tus ou ha lf s o so ba. b 2 ) b Fo ay < h sual + sasfs h popy: + Thus pfomg h Gam-Schm ohogoalzao oly qus wo o-poucs Couga Gas Poof: To show hs w wll us wo facs:. Th -h sual s ohogoal ( h aoal ss) o all cos wh <. 2. Th vco ca b pss as h la sum of h vcos { + }. Couga Gas Poof: To show hs w wll us wo facs:. Th -h sual s ohogoal ( h aoal ss) o all cos wh <. 2. Th vco ca b pss as h la sum of h vcos { + }. ssum Tu: Th fo ay < w hav: α + Couga Gas Poof: To show hs w wll us wo facs:. Th -h sual s ohogoal ( h aoal ss) o all cos wh <. 2. Th vco ca b pss as h la sum of h vcos { + }. ssum Tu: Th fo ay < w hav: α + Couga Gas Poof: To show hs w wll us wo facs:. Th -h sual s ohogoal ( h aoal ss) o all cos wh <. 2. Th vco ca b pss as h la sum of h vcos { + }. ssum Tu: Th fo ay < w hav: α + Couga Gas Clam : Th -h sual s ohogoal ( h aoal ss) o all cos wh <.

11 Couga Gas Poof: Sc w hav: Couga Gas Poof: Sc w hav: W ow ha fo <: Couga Gas Poof: Sc w hav: W ow ha fo <: Couga Gas Poof: Sc w hav: W ow ha fo <: Couga Gas Poof: Sc w hav: W ow ha fo <: Couga Gas Clam 2: Th vco ca b pss as h la sum of h vcos { + }.

12 Couga Gas Clam 2: Th vco ca b pss as h la sum of h vcos { + }. Poof: L us o by D h vco sub-spac: D Spa{ K } Couga Gas Clam 2: Th vco ca b pss as h la sum of h vcos { + }. Poof: L us o by D h vco sub-spac: D Spa{ K } W woul l o show ha D +. Couga Gas Poof: D Spa{ K } Sc s oba by compug h compo of ohogoal o { - } w hav: D Spa{ D } Couga Gas Poof: D Spa{ K } Sc s oba by compug h compo of ohogoal o { - } w hav: D Spa{ D } Coug a cusv fasho w ow ha: D Spa{ K } Couga Gas Poof: Bu w also ow ha: + Couga Gas Poof: Bu w also ow ha: + + D D So ha f w mus hav D +. 2

13 Couga Gas Poof: Bu w also ow ha: + + D D So ha f w mus hav D +. (If hs mpls ha h -h sual s zo a w hav ach h soluo a sp.) 3