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Relationship between long timescales and the static free-energy in the Hopfield model

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LETTER TO THE EDITOR

Relationship between long timescales and the static free-energy in the Hopfield model

G Biroli and R Monasson†

CNRS-Laboratoire de Physique Th´eorique de l’ENS, 24 rue Lhomond, 75231 Paris cedex 05, France

Received 9 March 1998

Abstract. The Glauber dynamics of the Hopfield model at low storage level is considered.

We analytically derive the spectrum of relaxation times for large system sizes. The longest timescales are gathered in families, each family being in one-to-one correspondence with a stationary (not necessarily stable) point of the static mean-field free energy. Inside a family, the timescales are given by the reciprocals (of the absolute values) of the eigenvalues of the free-energy Hessian matrix.

Recently, there has been a renewal of interest for the long-time dynamics of spin-glass models, which may already exhibit nontrivial features, for example, violation of the fluctuation–dissipation relations, ageing phenomena, at the mean-field level [1]. In this context, it is natural to study the spectrum of relaxation times occuring in the Glauber dynamics of disordered Ising spin models [2]. Unfortunately, numerical investigations have so far been limited to very small sizes [4]. Moreover, to the best of our knowledge, no analytical studies of the relaxation times spectra have yet been performed owing to the technical difficulties arising in the diagonalization of the Glauber matrix.

In this letter, we focus on the Glauber dynamics of the Hopfield model at low storage [6]. This system is simple enough to be analytically solved along the lines of [5]. Our main result is that the longest timescales are gathered in families, each family being in one-to-one correspondence with a stationary (not necessarily stable) point of the static mean-field free energy. Inside a family, the timescales are given by the reciprocals of the absolute values of the eigenvalues of the free-energy Hessian matrix. As a consequence, our study strengthens the close relationship between long-time dynamics and static properties in disordered mean- field models [1].

We consider a Hopfield model includingN Ising spinsSi,i=1, . . . , N andpquenched patterns ξiµ, µ=1, . . . , p. In addition to its intrinsic interest as a paradigm for attractor neural networks [7], the Hopfield model may be seen as a spin-glass system smoothly interpolating between the Mattis model [8] (when p = 1) and the much more involved Sherrington–Kirpatrick model (for infinite p) [3]. Its Hamiltonian depends on the spin configurationS =(S1, . . . , SN)through the set ofp overlapsmµ(S)= N1 PN

i=1Siξiµ and reads

H (S)= −N 2

Xp

µ=1

mµ(S)2. (1)

E-mail addresses: biroli@physique.ens.fr; monasson@physique.ens.fr.

0305-4470/98/210391+06$19.50 c 1998 IOP Publishing Ltd L391

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The thermodynamics of the above Hamiltonian has been studied in [7]. At low storage level, that is whenpremains finite as the sizeN is sent to infinity, the free-energy density is particularly simple,

f (β)= −1 2

Xp

µ=1

(mµ)2+ 1 β

ln

2 cosh

β

Xp

µ=1

mµξµ

(2) where β is the inverse temperature and hhF ({ξµ})ii ≡ N1 PN

i=1F ({ξiµ}) denotes the site average. In the above expression (2), the overlaps are the solutions of the saddle-point equations

mµ=

ξµtanh

β Xp ν=1

mνξν

(3) with the smallest free energy. At high temperature, the system lies in the paramagnetic phase mµ =0, ∀µ. Below the critical temperature 1/βc = 1, many other solutions with nonzero overlaps appear. The low-temperature phase is characterized by a condensation along some particular patterns, while spurious mixtures give rise to metastable states.

These equilibrium states may be seen as the attractors of a retrieval dynamics [2] starting from a given initial configuration (which is often but not necessarily chosen as a corrugated version of the quenched patterns). One possible evolution scheme is provided by the usual Glauber dynamics: given a configurationS, the spin at siteiis flipped with probability rate w(S, i)= 12(1Sitanh[βhi(S)])and left unchanged with rate 1−w. The instantaneous magnetic fieldhi reads from (1),

hi(S)= X

j (6=i)

1 N

Xp

µ=1

ξiµξjµ

Sj = Xp

µ=1

ξiµmµ(S)p

NSi. (4)

The probability distributionP (S, t )satisfies the master equation d

dtP (S, t )= XN

i=1

(w(Si, i)P (Si, t )w(S, i)P (S, t )) (5) where the spin configuration Si is obtained from S by flipping spin i. Due to its linear structure, equation (5) may be recast in a more suitable operator formalism. To each configurationS is associated a vector|Si, which form the basis of a 2N-dimensional linear spaceV. The probability distribution then becomes a vector|P (t )i =P

SP (S, t )|Si. We now define the spin–flip and magnetization operators, respectively denoted by σix andσiz through σix|Si = |Sii and σiz|Si =Si|Si. It is easy to check that the vectorial operator

1

2σ= 12x, σyxσz, σz)satisfies the usual commutation relations for a quantum spin

1

2. The master equation (5) now acquires the following compact form d

dt|P (t )i =W|P (t )i W = 1 2

XN i=1

ix−1)(1−σiztanh(βhi)) (6) where the local field operator hi is obtained by replacing in (4) the spins Si with their corresponding magnetization operators σiz. Due to the detailed balance conditions, the operatorWs=exp(βH /2)Wexp(−βH /2)is Hermitian and reads

Ws = −N 2 +1

2 XN

i=1

ix(cosh(βhi))1+σiztanh(βhi)]. (7) We shall now analyse the spectrum ofWs, which is the same asW’s one. From a rigorous point of view, all eigenvalues are strictly negative except the zero mode corresponding to the

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equilibrium Gibbs measure. First, as far as long-time dynamics is concerned, all large (in absolute value) eigenvalues, that are associated with fast decaying modes, may be discarded.

Second, for very large system sizes (N 1), indeed the limit of physical interest, the zero eigenvalue of Ws is expected to become almost degenerate when β > βc. Dynamical processes taking place over exponentially large times, or more generally times diverging withN will not be considered. We therefore focus only on long but finite timescales.

To do so, we shall perform an expansion of Ws in powers of 1/√

N and only keep the nonvanishing terms in the thermodynamical limit. This expansion is made possible by the mean-field nature of the Hopfield Hamiltonian (1). For a given configuration S, the dominant contributionsHi ≡P

µξiµmµ(S)to the local fieldshi (4) take only 2p different values when scanning all sites i. Let us partition the N sites in 2p blocks Gτ where τ is a p binary component vector; site i belongs to the only group Gτ such that Hi equals Hτ =P

µτµmµ(S)[9]. We restrict ourselves to quenched patterns that differ from each other on a macroscopic number of sites, so each blockGτ contains a numberN cτ of sites of the order ofN; note that if the patterns are randomly drawn from an unbiased distribution cτ =2p in the limit of large sizes.

After some simple algebra, the evolution operator may be rewritten asWs =P

τWτs with

Wτs = −N

2cτ +Jτx(cosh(βHτ))1+Jτztanh(βHτ)βp

2 cτ(1−tanh(βHτ)2)+O Jτy

N

(8) where the angular momenta and the field operators respectively read

Jτ =1 2

X

iGτ

σi and Hτ = 2 N

X

η

Xp µ=1

τµηµJηz. (9) The last two terms on the r.h.s of (8) stem from the contributionspSi/N to the local fields (4). The angular momentaJτ andJτ0 commute for any two different groupsτ 6=τ0as does anyJτ withWs. Thus, the angular momenta may be used as quantum numbers labelling the different invariant subspaces.

It can be checked that the subspace Vs corresponding to the maximum value of each angular momentum Jτ = 12N cτ contains the totally symmetric vectors |{Mτ}i having magnetizations Mτ = −N cτ,N cτ +2, . . . , N cτ in blocks Gτ. Indeed, the vector

|{Mτ = N cτ}i = |+,+,· · ·,+i belongs to Vs since its angular momentum component along the z-axis equals N/2 and is maximal. Moreover, the lowering operators Jτ leave Vs invariant and |{N cτ −2dτ}i = ⊗(Jτ)dτ|{N cτ}i. In addition, the equilibrium Gibbs vector belongs toVs. In the following we shall see that not only the Gibbs mode but also all of the eigenvectors corresponding to the smallest eigenvalues lie inVs.

At infinite temperature, the system is purely diffusive and the equilibrium state |0i equals the normalized sum of all|Sis. The average values of the angular momentum read h0|Jy|0i = h0|Jz|0i = 0, h0|Jx|0i = N/2. For low-lying excitations |ωi, that is plane waves on theN-dimensional hypercube, the mean value of the angular momentumhω|J|ωi equals that in the equilibrium state up to nonextensive terms. The same picture holds at finite temperature, but not at phase transition boundaries. More precisely, equilibrium or quasistationary modes |0i are expected to correspond to a well-defined direction of the angular momenta Jτ in the three-dimensional space. This direction varies smoothly when exploring the excited modes slightly above |0i†. To capture the spectrum of low-

Activated processes between wells cannot be obtained this way, but they correspond to infinite timescales which we do not investigate here.

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lying states, we therefore proceed in two steps: first identify the preferred directions of the angular momentum corresponding to quasiequilibrium modes and then compute the fluctuations around these positions.

This approach, reminiscent of semiclassical approximations and spin-wave theory [11], may be made more precise in the following way. Consider the Holstein–Primakoff (HP) representation of the angular momentum vectorial operator [10] in terms of Bose operators

Lxτ =12N cτaτaτ

Lyτ = 1 2i

aτ

q

N cτaτaτ − q

N cτaτaτaτ

Lzτ =12

aτ q

N cτaτaτ + q

N cτaτaτaτ

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which is valid for fixed angular momentum 12N cτ and as long as the number of quanta does not exceedN cτ. The Bose operators fulfil the usual commutation relation [aτ, aτ0]=δτ0. The large N expansion will be consistent provided that we enforce the number of quanta to keep finite. To do so, we align the HP representations (10) along the quasiequilibrium directions of the angular momenta

Jτx =cosθτLxτ −sinθτLzτ Jτy =Lyτ Jτz=sinθτLxτ +cosθτLzτ (11) in every block. We may now expand the evolution operatorWs =N ·W0s+√

N ·W1s+ W2s+O(1/√

N )and find W0s= 1

2 X

τ

cτ

cosθτ

coshKτ

+sinθτtanhKτ −1

W1s=

cτ

2 coshKτ

(cosθτ sinhKτ −sinθτ)

×

aτ+aτ −X

η

Xp µ=1

τµηµcηcτ

cosθη coshKτ

(aη+aη)

(12) where

Kτ =β Xp

µ=1

τµX

η

cηsinθηηµ. (13)

Both W0s and W1s vanish when the angles are such as sinθτ = tanhKτ for all blocks τ. Expression (13) implies that the 2p angles may be computed through the knowledge of the p parametersmµ=P

ηcηsinθηηµ. It turns out that the resulting self-consistent equations for themµs are identical to the equilibrium saddle-point condition (3), as could be expected from the above discussion.

As a consequence, the smallest eigenvalues gather together in different families F, in one-to-one correspondence with the stationary points of the free energy (2). In each family F, we are left with the spin-wave part of the evolution operator

W2s= −14X

τ0

[(aτ +aτ)(O2−1)τ,τ0(aτ0+aτ0)]−X

τ

[aτaτ +12(1Oτ)] (14) where the 2p×2p matrixO reads

Oτ0=1τ,τ0β Xp

µ=1

cτ

coshKτ

τµ

cτ0

coshKτ0

τ0µ

. (15)

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At least 2pp eigenvalues of O equal one. A straigthforward calculation shows that the difference between the traces of thenth powers ofOand of the free-energy Hessian matrix, see (2) and (3),

Hµ,ν = 2f

∂mµ∂mν

F

(16) equals 2pp for all integers n. In the above expression, the Hessian depends on the particular familyF, i.e. saddle point under consideration. Therefore, the a priori nonunit eigenvalues of O, hereafter called λjs, j =1, . . . , p are precisely the eigenvalues of the second derivatives matrix of the static free energy.

To achieve the calculation of the spectrum of relaxation times, we now perform an orthogonal transformation over the Bose operators to diagonalizeO and obtain

W2s = − Xp

j=1

[14j2−1)(bj +bj)2+bjbj +12(1λj)]

2p

X

j=p+1

bjbj. (17) The first term on the r.h.s. of (17) includesp interacting bosons which can be decoupled by means of a Bogoliubov transformation

cj cj

=

coshφj sinhφj

sinhφj coshφj

bj bj

(18) whereφj = 12ln|λj|. We can finally rewriteW2s in a completely diagonal form,

W2s = − Xp

i=1

|λj|(cjcj +1j)

2p

X

j=p+1

bjbj (19)

where1j =1 ifλj <0 and 1j =0 if λj >0. In this form, the slow eigenmodes of the Glauber matrix are seen to correspond to the eigenvectors of a quantum harmonic oscillator and can be easily calculated [5]. Note that transformation (18) requires that no eigenvalue vanishes. Indeed, for a marginal saddle-point F, the large N expansion has to take into account that the number of quantaaa becomes of the order of√

N.

We can follow the same procedure to diagonalizeWsin the subspace labelled by angular momenta{Jτ = 12N cτjτ}where alljτs are finite; we obtain forWτs expression (19) plus the constant term−jτ. Therefore, the smallest eigenvalues do not belong to such subspaces.

If thejτs are not finite asN goes to infinity, the diverging evolution operatorsW0s andW1s can never be simultaneously cancelled through adequate rotations.

Starting from a given initial condition, the evolution in the parameter space {mµ}will be deterministic [12] and follow the macroscopic law

d

dtmµ(t )= − ∂f

∂mµ({mν(t )}) (20)

giving rise to a gradient descent of the free-energy landscape down to some local minimum.

When the numberpof patterns increases, this free-energy landscape becomes very complex, for example, the number of stationary points scales exponentially with p [7]. Each fixed point has a basin of attraction; a point for a local maximum, a line for points with only one stable direction,. . ., a manifold of dimensionNI for a stationary point of indexI. However, because of pure entropic effects, some stationary points have much wider basins of attractions than other ones, that is the number of microscopic configurations associated with them may be extremely large. For instance, a random and unbiased initial condition would fall into the basin of the paramagnetic saddle-pointmµ=0,∀µ.

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Generally speaking, when the starting point lies in a basin of attraction of a stationary point{mµ}F of the free energy, the system relaxes toward this stationary point. Later on, the order parameters remain stuck to their saddle-point values for all but finite times, even if F is unstable from a thermodynamical standpoint. This description is reminiscent of the scenario proposed in [13]. The timescale which we have calculated here govern the 1/√

N fluctuations around the stationary point. Fluctuations in the stable directions, that are related to positiveλjs, decay with relaxation times equal to 1/λj and eventually equilibrate (1j =0). As for fluctuations along the unstable directions associated with negative λjs, they exponentially grow(1j =1) with some characteristic ‘escape’ times 1/|λj|. In both cases, the number of affected spins is of the order of√

N and the intensive quantities remain unchanged.

References

[1] For a recent review, see Bouchaud J P, Cugliandolo L, M´ezard M and Kurchan J 1997 Out of Equilibrium Dynamics in Spin-glasses and Other Glassy Systems ed A P Young (Singapore: World Scientific) and references therein

[2] Glauber R J 1963 J. Math. Phys. 4 294–307

[3] Sherrington D and Kirkpatrick S 1975 Phys. Rev. Lett. 35 1792 [4] M´elin R 1996 J. Physique 6 469

Yoshino H, Hukushima K and Takayama H 1997 Relaxational modes and ageing in the Glauber dynamics of the Sherrington–Kirkpatrick model Prog. Theor. Phys. supplement 126

[5] Ruijgrok Th W, Tjon J A 1973 Physica 65 539 [6] Hopfield J J 1982 Proc. Natl. Acad. Sci. USA 79 2554

[7] Amit D J 1989 Modeling Brain Function (Cambridge: Cambridge University Press) [8] Mattis D C 1976 Phys. Lett. 56A 421

[9] van Hemmen J L, Grensing D, Huber A and K¨uhn R 1986 Z. Phys. B 65 53 [10] Holstein T and Primakoff H 1940 Phys. Rev. 58 1908

[11] Mattis D C 1981 The Theory of Magnetism (Springer Series in Solid State Sciences) vol 17 (Berlin: Springer) [12] Van Kampen N G 1961 Can. J. Phys. 39 551

[13] Kurchan J and Laloux L 1996 J. Phys. A: Math. Gen. 29 1929

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