Some recent results on the inverse Ising problem. Federico Ricci-Tersenghi Physics Department Sapienza University, Roma
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1 Some recent results on the inverse Ising problem Federico Ricci-Tersenghi Physics Department Sapienza University, Roma
2 Outline of the talk Bethe approx. for inverse Ising problem Comparison among several mean-field approximations (IP, nmf, TAP, SM, Bethe...) Correlations normalization trick Major limitation in using TAP and Bethe in frustrated models with a field New method: pseudo-likelihood + decimation (Aurelien DECELLE)
3 Inverse Ising problem M configurations of N Ising variables ( s i = ±1) extracted from P (s 1,...,s N )= 1 Z(J, h) exp [alternatively only magnetizations correlations i=j C ij = s i s j m i m j J ij s i s j + i and are given] m i = s i h i s i GOAL: estimate couplings and fields (J, h)
4 Solving inverse Ising pb. 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
5 Solving inverse Ising pb. 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
6 Solving inverse Ising pb. Mean field approximations (MFA) m MFA i (J, h) =m i Cij MFA (J, h) =C ij
7 Solving inverse Ising pb. Mean field approximations (MFA) m MFA i (J, h) =m i Cij MFA (J, h) =C ij
8 Solving inverse Ising pb. Mean field approximations (MFA) m MFA i (J, h) =m i Cij MFA (J, h) =C ij Log-likelihood L(J, h s) = 1 M log M P (s (k) J, h) = = i h i m i + ij k=1 J ij (C ij + m i m j ) log Z(J, h)
9 Solving inverse Ising pb. Mean field approximations (MFA) m MFA i (J, h) =m i Cij MFA (J, h) =C ij Log-likelihood L(J, h s) = 1 M log M P (s (k) J, h) = = i h i m i + ij k=1 J ij (C ij + m i m j ) log Z(J, h) F MFA (J, h)
10 Solving inverse Ising pb. Mean field approximations (MFA) m MFA i (J, h) =m i Cij MFA (J, h) =C ij Log-likelihood L(J, h s) = 1 M log M = i h i m i + ij k=1 P (s (k) J, h) = J ij (C ij + m i m j ) log Z(J, h) m i = F MFA (J, h)/ h i F MFA (J, h)
11 Mean-field approximations (MFA) to the free-energy naive mean-field (nmf) F nmf = i 1+mi 1 mi H + H 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
12 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
13 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
14 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 = 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
15 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
16 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 F BA / c ij = 0 c ij (m i,m j,t ij ) = 1 2t ij 1+t 2 ij t ij = tanh(j ij ) (1 t 2 ij )2 4t ij (m i t ij m j )(m j t ij m i ) m i m j
17 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 i m (j) i t ij : magnetization of i in absence of j j
18 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 i m (j) i t ij : magnetization of i in absence of j 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)
19 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
20 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 J ij m j Jij(1 2 m 2 j)m i +...
21 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) C ij = m i h j, (C 1 ) ij = h i m j
22 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) C ij = m i h j, (C 1 ) ij = h i m j (C 1 nmf ) ij = (C 1 TAP ) ij = δ ij 1 m 2 J ij, i 1 1 m 2 + i k J 2 ik (1 m2 k ) δ ij J ij +2J 2 ijm i m j
23 Computing correlations by linear response in BA Analytic expression for the correlations (C 1 BA ) ij = 1 1 m 2 i k t ik f 2 (m k,m i,t ik ) 1 t 2 ik f(m k,m i,t ik ) 2 Coincide with the fixed point of Susceptibility Propagation No need to run any algorithm! δ ij t ijf 1 (m j,m i,t ij ) 1 t 2 ij f(m j,m i,t ij ) 2
24 Numerical results on estimating correlations C 1 N 2 (C ij Cij )2 1e+02 1 i,j 1e+00 1 C 1e-02 1e-04 1e-06 2D ferromagnet diluted (p=0.7) nmf TAP 3 rd 4 th BA 1e β
25 Improving correlations by a normalization trick 2D lattice In ferromagnetic models with loops, inferred correlations are too strong because of loops Usually C MFA ii > 1 Trick: enforce which is unphysical C MFA ii =1 by a normalization C ij C ij Cii C jj
26 2 More numerical results on estimating correlations 1e+02 1e MC data 10 6 MC data 1e-02 C 2D ferromagnet diluted (p=0.7) 1e-04 exact corr 1e-06 1e β TAP TAP norm BA BA norm
27 MFA for the inverse Ising problem
28 MFA for the inverse Ising problem (C 1 nmf ) ij = (C 1 TAP ) ij = δ ij 1 m 2 J ij, i 1 1 m 2 + i k J 2 ik (1 m2 k ) δ ij J ij +2J 2 ijm i m j
29 MFA for the inverse Ising problem (C 1 nmf ) ij = (C 1 TAP ) ij = δ ij 1 m 2 J ij, i 1 1 m 2 + i k J 2 ik (1 m2 k ) δ ij J ij +2J 2 ijm i m j
30 MFA for the inverse Ising problem (C 1 nmf ) ij = (C 1 TAP ) ij = δ ij 1 m 2 J ij, i 1 1 m 2 + i k = J nmf ij J 2 ik (1 m2 k ) = (C 1 ) ij δ ij J ij +2J 2 ijm i m j
31 MFA for the inverse Ising problem (C 1 nmf ) ij = (C 1 TAP ) ij = J TAP ij = δ ij = J nmf = (C 1 ) ij 1 m 2 J ij, ij i 1 1 m 2 + Jik 2 (1 m2 k ) δ ij J ij +2Jijm 2 i m j i k 1 8mi m j (C 1 ) ij 1 4m i m j
32 MFA for the inverse Ising problem J BA ij (C 1 nmf ) ij = (C 1 TAP ) ij = 2(C 1 ) ij J TAP ij = δ ij = J nmf = (C 1 ) ij 1 m 2 J ij, ij i 1 1 m 2 + Jik 2 (1 m2 k ) δ ij J ij +2Jijm 2 i m j i k 1 8mi m j (C 1 ) ij 1 4m i m j 1 = atanh 2(C (1 m 2 i ) )(1 m2 j )(C 1 ) 2 ij m im j ij (1 m 2 i )(1 m2 j )(C 1 ) 2 ij 2m im j (C 1 ) ij 4(C 1 ) 2 ij
33 Normalization trick for the inverse Ising problem TAP with m=0 J TAP ij = (C 1 ) ij
34 Normalization trick for the inverse Ising problem TAP with m=0 (C 1 TAP ) ij = 1+ k J TAP ij = (C 1 ) ij (Jik TAP ) 2 δ ij J TAP ij = C TAP = C
35 Normalization trick for the inverse Ising problem TAP with m=0 (C 1 TAP ) ij = 1+ k J TAP ij = (C 1 ) ij (Jik TAP ) 2 δ ij J TAP ij = C TAP = C ( C TAP ) ij = (C TAP ) ij (CTAP ) ii (C TAP ) jj = λ i λ j (C TAP ) ij λ i are N new parameters to fix s.t. CTAP = C
36 More approximations 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 J SM ij = (C 1 ) ij + J IP ij C ij (1 m 2 i )(1 m2 j ) (C ij) 2 [for m=0 can be derived in 2 lines, ask me if interested]
37 More approximations 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 J SM ij = (C 1 ) ij + J IP ij C ij (1 m 2 i )(1 m2 j ) (C ij) 2 [for m=0 can be derived in 2 lines, ask me if interested]
38 Numerical results for the inverse Ising problem J = i<j (J ij J ij) 2 i<j J 2 ij 10 1 ferromagnet N=20 on Bethe lattice (c=4) 0.1 J M = 10 4 M = 10 5 M = 10 6 exact BA SM β
39 Numerical results for the inverse Ising problem 10 2D ferromagnet N=7 2 diluted (p=0.7) 1 J β IP TAP SM BA BA norm
40 A major problem for frustrated models in a field J TAP ij = 1 8mi m j (C 1 ) ij 1 4m i m j The resulting coupling is real only if TAP 1 8m i m j (C 1 ) ij 0 The same happens also within the BA BA = 1 + 4(1 m 2 i )(1 m2 j )(C 1 ) 2 ij 2m im j (C 1 ) ij 2 4(C 1 ) 2 ij
41 A major problem for frustrated models in a field Three spins interacting with a coupling J in a constant field h P (s 1,s 2,s 3 ) exp[j(s 1 s 2 + s 2 s 3 + s 3 s 1 )+h(s 1 + s 2 + s 3 )] 5 TAP BA 4 < h 1 > J
42 Pseudo-likelihood method (PLM) Approximate P i (s i s \i ) P (s) i Define the pseudo-log-likelihood as PL(h, J s) =log P (s h, J) = i log P i (s h i, J i ) = i f i f i = h i m i + j J ij (C ij + m i m j ) log 2 cosh s i (h i + j J ij s j ) Maximize to estimate and For sparse models use an L1 regularization and maximize f i h i J ij f i λ j J ij
43 PLM vs. MFA 10 2D ferromagnet N=7 2 =49 diluted (p=0.7) M=5000 samples J β IP TAP SM BA PLM BA norm
44 Couplings inferred by PLM ferro 2D N=7 2 diluted (p=0.7) β=0.6 PLM λ=
45 Couplings inferred by PLM ferro 2D N=7 2 diluted (p=0.7) β=0.6 J=0 J=1 PLM λ=
46 Couplings inferred by PLM ferro 2D N=7 2 diluted (p=0.7) β=0.9 PLM λ=
47 Couplings inferred by PLM ferro 2D N=7 2 diluted (p=0.7) β=0.9 J=0 J=1 PLM λ=
48 Decimation procedure Run PLM Set to zero a constant fraction of couplings (those inferred to be the smallest) Re-run PLM only on couplings not set to zero (this is impossible within a MFA) Iterate until...
49 PLM + decimation β = J β = β = 0.6 β = num. iterations
50 PLM + decimation β = J β = β = 0.6 β = num. iterations
51 PLM + decimation PL β = 0.9 β = 0.8 β = 0.6 β = num. iterations
52 PLM + decimation PL log(2) β=0.9 β=0.8 β=0.6 β= #DOF
53 PLM + decimation β=0.9 β=0.8 β=0.6 β= tilted PL log(2) #DOF
54 Summary of recent results about the inverse Ising pb. Analytical expressions for the Bethe approx. Comparison of MFA in a wide temperature range Serious limitation of TAP and Bethe in a field Improvement in the inferred couplings by: normalization trick (in weakly frustrated models) PLM + decimation procedure JSTAT (2012) P08015
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