STATISTICAL MECHANICS OF THE INVERSE ISING MODEL

Transcription

1 STATISTICAL MECHANICS OF THE INVESE ISING MODEL Muro Cro Supervsors: rof. Mchele Cselle rof. ccrdo Zecchn uly 2009

2 INTODUCTION SUMMAY OF THE ESENTATION Defnton of the drect nd nverse prole Approton ethods of the drect prole: vrtonl pproches Bethe Overvew of drect nd nverse lgorths Sultons GOALS OF THE THESIS Use of lgorths whch generlze Bethe pproton Gp n order to solve the nverse prole Upgrde of the Gp lgorth to get ore ccurte clculton of the correltons nd pplcton to the nverse prole

3 ISING MODEL We cn represent the usul Boltznn prolty dstruton wth fctor grph: > < h H e e e } { } { β β β > < h H } {

4 ISING MODEL H { } < > h h > < > < Inverse role!

5 EXAMLES OF ALICATIONS Neuron networks reconstructon [E.Schnedn M..Berry.Segev W.Blek 2006] Genes networks reconstructon [A.Brusten A.gn M.Wegt.Zecchn 2008] roten networks recontructons [G.Tkck 2007] Why Isng odel? Mu entropy prncple

6 VAIATIONAL AOACHES Men feld pprotons G[ ] < E > T S G[ ] ~ F Gs Free Energy Boltznn Dstr. Heloltz Free Energy Men feld pproton Mnzton of G In sudon We chose for for nd we nze the functonl G

7 Locl consstency: BELIEFS: 1 1 VAIATIONAL AOACHES Bethe pproton [ ] > < q p 1 } { [ ] [ ] log 1 log E q E G > < Mnu equtons Constrnts equtons 1 Couplngs f 2 Couplngs f Messges equtons t the f pont

8 B Drect Isng B equtons t t \ 1 η ν... k k t k t \ \ ν ψ η..k..

9 B Drect Isng * η t t s s \ * \ * η η ψ

10 GB [.S.Yedd W.T. Freen Y.Wess 2001] Drect Isng Let s defne: c c U E S Log : vrles close to functon node c 1 c U U A Generlzton of regons {} : ng[{}] Bethe pproton New for of G GB equtons

11 GB Drect Isng \ \ D I I N I I F ψ

12 GB Drect Isng \ \ D I I N I I F ψ Ψ \ ' ' D D E D D D A

13 GB [.S.Yedd W.T. Freen Y.Wess 2001] Drect Isng Is G vld? depends on regons c Condton: c I c I 1 I 1 f We wnt to count every node only once

14 INVESE ISING Solvng the nverse prole through n tertve ethod Coplete Grph χ Itertve process Updte rules for h Grph wth: < σ > < σσ > χ

15 B/GB Inverse Isng Self-consstent equtons n the essges Self consstent equtons n the nputs χ f M h χ g M h Messges fed h f 1 M χ g M 1 χ

16 B/GB Inverse Isng χ Otnng eperentl vlues of Intlze coplete grph wth rndo h For t 1T Itertons Bp/Gp sve essges Functon nodes updte n the grph h f M 1 g M 1 χ χ End

17 B Inverse Isng EXAMLE ~ h h Tnh h ATnh h ~ ν ν ν ν ~ ~ ~ ~ h h h h h h h h e e e e

18 B Inverse Isng EXAMLE χ e ν ν 2 1 ν 2 1 ν 1 Tnh Tnh χ ATnh χ χ 1

19 Sus.rop. [M.Mezrd T.Mor 2008] Drect nd Inverse Isng B lttons χ only f < > Fluctuton-esponse theore < σ σ > < σ >< σ > h h h wth η We hve to know dervtves of essges! B equtons ν η B dervtves ν η ν η χ f h Invertng f New updte rule for

20 GS Drect Isng A: prove ccurcy n χ n the GB schee B Sus.rop. Flutt.- esp. T. GB GS Fnd equtons for dervtves of essges n GB Gsp equtons Do the dervtves of the 1-elefs equtons Correltons

21 GS Isng Inverso Etrctng couplngs fro GS wth rtrry regons? Contrnt: t lest ll the 1 nd 2 regons 1-regons see Bethe ppro. rtl reuse of Sus.rop. forls

22 Sll correltons epnson [.Monsson V.Sessk 2008] Lkelhood zton Let s suppose to hve n copes of the syste Let s ze the prolty to hve these copes gven the theory: 1 [ Log { σ} n h ] n n where : S Log Z h ns h c Let s solve the prole for c 0 c β c Anltcl epressons for e h

23 Sultons eples 2 Coplete 2 2 N N 1 < Grph N 20 Bp Gp Nve M.F. Sus.rop. Gsp S.C.E.

24 Sultons eples N N 1 < Tree N 20 Bp Gp Nve M.F. Sus.rop. Gsp S.C.E.

25 Conclusons nd future B Sus.rop. GB GS OBLEMS Bg correltons Glssy phse Loopy B Sll Correlton Epnson Necessty to redefne the prole Iterton ethods n g correltons rege Cvty ethods for GB n 1SB [T.zzo et l. 2009] New pprotons for the free energy

26 Acronys & Notton B GB Sus.rop. GS S.C.E. Nve M.F. GB 3 GS 3 Belef ropgton Generlzed Belef ropgton Susceptlty ropgton Generlzed Susceptlty ropgton Sll Correlton Epnson Men Feld Theory Fluctuton esponse Theore GB wth ll regons wth 1 or 2 or 3 spns GS wth ll regons wth 1 or 2 or 3 spns \ Identty true ecept for norlzton fctor Set of nodes whch re connected to wth lnk Set wthout the node node representng vrle leled y k node representng functon leled y c