A brief introduction to the inverse Ising problem and some algorithms to solve it

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1 A brief introduction to the inverse Ising problem and some algorithms to solve it Federico Ricci-Tersenghi Physics Department Sapienza University, Roma original results in collaboration with Jack Raymond and Aurelien Decelle

2 Simplifications for this talk Ising variables s i = ±1 At most pairwise (two body) interactions. The most general Hamiltonian is H(s J, h) = X J ij s i s j + X i h i s i with corresponding measure P (s 1,...,s N )= 1 Z(J, h) exp i=j J ij s i s j + i h i s i

3 The direct problem (main pb. in stat. mech.) Given the Hamiltonian, compute the free energy F (J, h) = log Z(J, h) = log X exp X J ij s i s j + X {s i } i h i s i! and average values hoi = X s O(s)P (s) The sum is over exponentially many terms

4 The inverse Ising problem Given data generated according to which may be P (s 1,...,s N )= 1 Z(J, h) exp i=j J ij s i s j + i h i s i either M configurations of spins either magnetizations and correlations m i = hs i i C ij = hs i s j i m i m j GOAL: estimate couplings and fields (J, h)

5 The exact solution Maximize the log-likelihood = X i L(J, h s) = 1 M log M Y h i hs i i + X ij k=1 P (s (k) J, h) = J ij hs i s j i log Z(J, h) = X i h i m i + X ij J ij (C ij + m i m j )+F (J, h) Taking the derivatives m i hi F (J, h) =0 =) m i (DATA) = m i (J, h) C ij + m i m j Jij F (J, h) =0 =) C ij (DATA) = C ij (J, h)

6 Input data: Magnetizations and Correlations Less information than having configurations If only m i = hs i i C ij = hs i s j i m i m j are given then maximum entropy principle implies Hamiltonian contains only single-body and two-body interactions H(s J, h) = X J ij s i s j + X i h i s i

7 Brute force solution by Monte Carlo Monte Carlo -> unbiased solution...but it is slow! Initial guess for J(0),h(0) Given J(t),h(t) MC compute m(t),c(t) Return J(t),h(t) Yes No dm<eps? dc<eps? Update J(t+1),h(t+1) according to dm(t)=m(t)-m dc(t)=c(t)-c

8 Mean field approximations (MFA) Log-likelihood L(J, h s) = 1 M log M Y P (s (k) J, h) = = X i h i m i + X ij k=1 J ij (C ij + m i m j ) log Z(J, h) F MFA (J, h) m MFA i hi F MFA (J, h) m MFA i (J, h) =m i (DATA) Cij MFA (J, h) =C ij (DATA)

9 MFA to the free-energy naive mean-field (nmf) F nmf = i 1+mi 1 H + H 2 P (s) = Q i P i(s i ) mi 2 + i h i m i + i=j J ij m i m j H(x) x ln(x) F nmf m i = j J ij m j + h i atanh(m i )=0 m i = tanh h i + j J ij m j

10 MFA to the free-energy nmf + Onsager reaction term (TAP) F TAP = 1+mi 1 mi H + H i + h i m i + J ij m i m j J ij(1 2 m 2 i )(1 m 2 j) i i=j m i = tanh h i + J ij m j J ij (1 m 2 j)m i j reaction term

11 MFA to the free-energy Plefka expansion in small J F nmf = i H 1+mi + H 2 1 mi 2 + i h i m i + J ij m i m j i=j F TAP = i + i H 1+mi + H 2 h i m i + i=j 1 mi 2 + J ij m i m j J 2 ij(1 m 2 i )(1 m 2 j)

12 MFA to the free-energy Bethe approximation (BA) tries to include any correlations between n.n. spins in principle no small J required (but beware to phase transitions) factorization over links exact on trees P (s) = Q i P i(s i ) Q ij P ij (s i,s j ) P i (s i )P j (s j )

13 MFA to the free-energy Bethe approximation (BA) F BA = (1 + mi )(1 + m j )+c ij (1 mi )(1 m j )+c ij H + H i=j (1 + mi )(1 m j ) c ij (1 mi )(1 + m j ) c ij + H + H mi 1 mi (1 d i ) H + H + h i m i + J ij (c ij + m i m j ), 2 2 i i i=j t ij = tanh(j ij ) F BA / c ij = 0 + c ij (m i,m j,t ij ) = 1 1+t 2 ij (1 t 2 ij 2t )2 4t ij (m i t ij m j )(m j t ij m i ) ij m i m j

14 MFA to the free-energy Bethe approximation (BA) and cavity method m i = m(j) i 1+m (j) i m j = t ij m (j) i 1+m (j) i + t ij m (i) j t ij m (i) j + m (i) j t ij m (i) j m (j) i i : magnetization of i in absence of j t ij j m (j) i = f(m i,m j,t ij ) m (i) j = f(m j,m i,t ij ) f(m 1,m 2,t)= 1 t2 (1 t 2 ) 2 4t(m 1 m 2 t)(m 2 m 1 t) 2t(m 2 m 1 t)

15 MFA to the free-energy Bethe approximation (BA) and cavity method f(m 1,m 2,t)= 1 t2 (1 t 2 ) 2 4t(m 1 m 2 t)(m 2 m 1 t) 2t(m 2 m 1 t) m i = tanh h i + j atanh t ij f(m j,m i,t ij ) Small J expansion gives nmf, TAP,... h i + j atanh t ij f(m j,m i,t ij ) h i + J ij m j Jij(1 2 m 2 j)m i +... j

16 Computing correlations by linear response Correlations are trivial in MFA C ij =0 in nmf, TAP and BA (between distant spins) Non trivial correlations can be obtained by using the linear response (Kappen Rodriguez, 1998) ij j ( 1 ) ij j ( (CnMF 1 ) ij = ( 1 1 TAP ) ij (C 1 TAP ) ij = ij 1 m 2 i 1 1 m 2 i J ij, + k J 2 ik (1 m2 k ) ij J ij +2J 2 ijm i m j

17 Computing correlations by linear response in BA Analytic expression for the linear responses in BA (C 1 BA ) ij = ( BA ) ij 1 1 m 2 i Coincide with the fixed point of Susceptibility Propagation No need to run any algorithm! k t ik f 2 (m k,m i,t ik ) 1 t 2 ik f(m k,m i,t ik ) 2 ij t ij f 1 (m j,m i,t ij ) 1 t 2 ij f(m j,m i,t ij ) 2

18 Solving the inverse problem by MFA Match measured magnetizations and correlations with MF approximated magnetizations and linear responses m MFA i (J, h) =m i (DATA) MFA ij (J, h) =C ij (DATA) Under the Bethe approx. one could use either c BA ij or BA ij for n.n. correlations. Which one is better?

19 Zero field case is simpler If all field are zero, then magnetizations are null by symmetry, and expressions simplify to naive MF (C 1 nmf ) ij = ij J ij, " # TAP (C 1 TAP ) ij = Bethe (C 1 BA ) ij = " 1+ X k 1+ X k J 2 ik t 2 ik 1 t 2 ik ij J ij, # ij t ij, 1 t 2 ij

20 Exactly solvable case for the inverse Ising problem? Curie-Weiss model, fully connected nmf approximation J ij = /(N 1) ij with i 6= j N=10,20,40 exact correlations ii

21 Exactly solvable case for the inverse Ising problem? Curie-Weiss model, fully connected more MFA (N=20) ij with i 6= j ii J ij = /(N 1) TAP ~ BA nmf 3rd th

22 Exactly solvable case for the inverse Ising problem? Bethe approximation on trees is ok Bethe approximation on random graph is ok only far from the critical point (as nmf for the Curie-Weiss model) How much the paramagnetic properties of a model on a finite size random graph are different from those of the same model defined on a tree? see recent works on finite size corrections for models defined on random graphs (Lucibello and Morone)

23 Numerical results on estimating correlations 1e+02 C 1 N 2 i,j (C ij C ij ) 2 1 1e+00 1 C 1e-02 1e-04 1e-06 2D ferromagnet diluted (p=0.7) nmf TAP 3 rd 4 th BA 1e

24 Matching data and MF predictions 0 1 C ij... 1? C A = 0 11 ij... 1 C A C ij 1 ij NN Usually only off-diagonal elements are used J ij = (C 1 ) ij and diagonal elements are ignored...

25 MFA for the inverse Ising problem J BA ij = atanh (C 1 nmf ) ij = (C 1 TAP ) ij = J TAP ij = + ij =) J nmf 1 m 2 J ij, i 1 1 m 2 + Jik 2 (1 m2 k ) i k 1 8mi m j (C 1 ) ij 1 4m i m j ij = (C 1 ) ij 1 2(C (1 m 2 i ) )(1 m2 j )(C 1 ) 2 ij m i m j ij ij J ij +2J 2 ijm i m j 1 2(C 1 ) ij 1 + 4(1 m 2 i )(1 m2 j )(C 1 ) 2 ij 2m i m j (C 1 ) ij 2 4(C 1 ) 2 ij equal for m=0

26 More MFA for the inverse Ising problem Independent pair (IP) approximation J IP ij = 1 4 ln (1 + m i)(1 + m j )+C ij (1 + m i )(1 m j ) C ij (1 m i )(1 m j )+C ij (1 m i )(1 + m j ) C ij Sessak-Monasson (SM) small correlation expansion Jij SM = (C 1 ) ij + Jij IP C ij (1 m 2 i )(1 m2 j ) (C ij) 2

27 Numerical results for the inverse Ising problem 10 2D ferromagnet N=7 2 diluted (p=0.7) J IP TAP SM BA BA norm

28 Improving correlations by a normalization trick In ferromagnetic models with loops, linear response correlations in BA are too strong because of loops, which are unexpected in SuscProp 2D lattice Leading to ii > 1 which is unphysical Trick: enforce ii =1 by a normalization b ij = p ij ii jj

29 Make MFA & LR consistent Consistency is more important than truth (S. Ting) Add Lagrange multipliers to your referred MF free-energy F MFA ({m i }, {C ij },...) to enforce consistency with linear response estimates ii =1 m 2 i ij = C ij free energy minimum curvature free energy minimum location

30 General framework (MFA + LR) F = F MFA ({m i }, {C ij },...)+ X i im 2 i + X i<j ijc ij Your preferred MFA can be set to zero to recover known approx. or used to satisfy ii =1 m 2 i ij = C ij

31 Nearest-neighbor correlation (2D square lattice) 0.8 NN correlations Bethe approx. LR ME exact ME+LR

32 Off-diagonal constraint only 0.01 NMF/TAP (λ=0) Bethe (λ=0) Bethe (λ=0) [KR] Bethe Plaquette J β Bethe + off-diagonal constraints = SM

33 Random field Ising model 2D square lattice D RFIM <h> = 0.0 σ h = 0.2 β = Bethe, no constraint Bethe, diag. constraint Bethe, both constraints m infer - m exact m exact

34 Random field Ising model 2D square lattice 1e-02 2D RFIM <h> = 0.0 σ h = 0.2 β = e-03 m infer - m exact 1e-04 1e-05 Bethe, no constraint Bethe, diag. constraint Bethe, both constraints 1e m exact

35 Random field Ising model 2D square lattice 1e-01 2D RFIM <h> = 0.0 σ h = 0.2 β = e-02 self correlations 1e-03 C infer - C exact 1e-04 1e-05 1e-06 Bethe, no constraint (LR) Bethe, no constraint (ME) Bethe, diag. constraint (LR) Bethe, both constraints (ME=LR) 1e C exact

36 Random field Ising model 2D square lattice 1e-01 2D RFIM <h> = 0.0 σ h = 0.2 β = 0.25 NNN correlations NN correlations 1e-02 C infer - C exact 1e-03 1e-04 1e-05 Bethe, no constraint (LR) Bethe, diag. constraint (LR) Bethe, both constraints (ME=LR) C exact

37 Input data: Configurations More information than knowing only m i = hs i i C ij = hs i s j i m i m j In principle one can access to all higher order correlations (but these are much more noisy) Many different inference algorithms. Among these: Adaptive cluster expansion cluster configurations & apply MFA within any state

38 Pseudo-likelihood method (PLM) For each variable define a conditional probability P i (s i s \i )= exp[s i(h i + P j J ijs j )] 2 cosh(h i + P j J ijs j ) Maximize the local log-likelihood L i = hlog P i (s i s \i )i = J ij (C ij + m i m j ) hlog 2 cosh(h i + X j J ij s j )i h i m i + X j to estimate h i and J ij Note that for each coupling J ij PLM returns 2 estimates Better maximizing PL(h, J) = X i L i

39 PLM vs. MFA 10 2D ferromagnet N=7 2 =49 diluted (p=0.7) 1 M=5000 samples J IP TAP SM BA PLM BA norm

40 Inferring topology in sparse models For simplicity let s assume J ij 2 {0, } non-zero couplings are sparse Maximize L1-regularized pseudo-likelihood PL (h, J) = X i L1-regularization gives a bias to the estimates! L i X J ij ij

41 Couplings inferred by PLM β=0.5 β=0.9 β=0.9 L1 regularized 1e-4 1e-2 1 1e-4 1e-2 1 log(j inferred) 1e-12 1e-8 1e-4 1 2D Ising model (30% dilution) M=4500

42 Decimation procedure No L1-regularizer -> no bias Maximize PL(h, J) Set to zero a constant fraction of couplings (those inferred to be the smallest) Maximize again PL(h, J) to zero (this is impossible within a MFA) only on couplings still not set Iterate until maximum of sensibly PL(h, J) starts decreasing

43 PLM + decimation tplf Error tplf Error Number of non-decimated couplings

44 PLM + decimation True Positive Rate PLM+L1 δ=1e-8 PLM+L1 δ=1e-6 PLM+L1 δ=1e-4 PLM+L1 δ=1e-2 PLM+L1 δ=1e-1 PLM+Decimation STOP point True Negative Rate

45 Some conclusions Mean field approximations Inverse problem harder than direct problem Requires (at least) improvement in the direct problem Fundamental problem of going beyond Bethe and trees... Pseudo-likelihood method Better performances in general Specially well suited for inferring topology in sparse models via L1-regularization, thresholding or decimation