The body of this paper (249 lines of TeX), from the line following "BODY" to the line preceding "ENDBODY", contains a total of 10001 characters. In the table below, this count is broken down by ASCII code; following one white space after the code is the corresponding character (intended for trouble shooting, in case things got garbled). 6310 lowercase letters 267 uppercase letters 193 digits 249 ASCII characters 10 1016 ASCII characters 32 3 ASCII characters 33 ! 262 ASCII characters 36 $ 5 ASCII characters 37 % 3 ASCII characters 38 & 4 ASCII characters 39 ' 64 ASCII characters 40 ( 64 ASCII characters 41 ) 1 ASCII characters 42 * 21 ASCII characters 43 + 116 ASCII characters 44 , 163 ASCII characters 45 - 77 ASCII characters 46 . 11 ASCII characters 47 / 4 ASCII characters 58 : 2 ASCII characters 59 ; 5 ASCII characters 60 < 35 ASCII characters 61 = 3 ASCII characters 62 > 14 ASCII characters 91 [ 570 ASCII characters 92 \ 14 ASCII characters 93 ] 72 ASCII characters 94 ^ 126 ASCII characters 95 _ 4 ASCII characters 96 ` 153 ASCII characters 123 { 8 ASCII characters 124 | 153 ASCII characters 125 } 9 ASCII characters 126 ~ BODY % PAPER.TEX \magnification=\magstep1 \nopagenumbers \hoffset=-9pt \voffset=-13pt \font\twelvebf=cmbx10 scaled\magstep1 \font\eightrm=cmr8 \font\eightbf=cmbx8 \font\sevenrm=cmr7 \let\cl=\centerline \let\s=\scriptstyle \def\mean{{\rm I\kern-.18em E\,}} \def\natural{{\rm I\kern-.18em N}} \def\integer{{\rm Z\kern-.32em Z}} \def\real{{\rm I\kern-.18em R}} \def\tr{{\rm tr}} \def\half{{1\over 2}} \def\SS{{\cal S}} \def\OO{{\cal O}} \def\oo{{\s\cal O}} \def\intdpm{\Bigl({N\beta\over 2\pi}\Bigr)^{p/2}\!\!\int\!d^pm\,} % ---------------------- \def\hamilt{1} \def\efree{2} \def\fdelta{3} \def\ulbound{4} \def\amunu{5} \def\abound{6} \def\pfun{7} \def\lbound{8} \def\pizero{9} \def\cnofp{10} \def\etran{11} \def\rAGS{1} \def\rSTa{2} \def\rSTb{3} % ---------------------- \cl{{\twelvebf A Free Energy Bound for the Hopfield Model}} \vskip.7in \cl{Hans Koch \footnote{$^*$} {{\sevenrm Supported in Part by the National Science Foundation under Grant No. DMS--9103590.}}} \cl{Department of Mathematics, University of Texas at Austin} \cl{Austin, TX 78712} \vskip.8in\baselineskip=9.5pt {\narrower\noindent{\eightbf Abstract.\ }\eightrm We give a simple upper and lower bound on the free energy density of the Hopfield model of size $\s N$ with $\s p$ stored patterns, in the limit where $\s N$ and $\s p$ tend to infinity with $\s p/N\to\alpha<1$. The two bounds coincide for $\s\alpha=0$.\par} %----------------------- \vskip.9in\noindent \baselineskip=11.5pt plus 1pt minus 1pt \advance\hsize by 18pt The Hamiltonian of the Hopfield model with $p$ stored $N$--bit patterns is a random function on the set $\SS=\{+1,-1\}^N$, with values given by the equation $$ H_{p,N,\xi}(S)=-{1\over 2N}\sum_{i,j=1}^N\sum_{\mu=1}^p \xi_i^\mu\xi_j^\mu S_iS_j\,,\qquad S\in\SS\,, \eqno(\hamilt) $$ where $\xi$ is a $p\times N$ matrix whose elements $\xi_i^\mu$ are independent random variables with values $\pm 1$ and mean zero. The quantity considered here is the expected value of the free energy density $$ f_{p,N,\beta}(\xi)=-{1\over\beta N}\ln\Bigl[\sum_{S\in\SS} e^{-\beta H_{p,N,\xi}(S)}\Bigr] \eqno(\efree) $$ at positive inverse temperatures $\beta$. In order to state our result, we define $$ \phi_\delta(\beta,h)=-{1\over\beta}\ln\bigl[2\cosh(\beta h)\bigr] +{1-\delta\over 2}h^2\,. \eqno(\fdelta) $$ \vskip.1in\noindent{\bf Theorem.\ } {\it Let $0\le\alpha<1$ and $\delta<1$ such that $\delta-4\sqrt{\alpha}(1-\delta)$ is positive. Then } $$ \min_{h\in\real}\phi_\delta(\beta,h) +{\alpha\over 2\beta}\ln[\delta-4\sqrt{\alpha}(1-\delta)] \le\lim_{\scriptstyle N,p\to\infty\atop\scriptstyle p/N\to\alpha} \mean f_{p,N,\beta}(\xi) \le \min_{h\in\real}\phi_0(\beta,h)\,. \eqno(\ulbound) $$ \advance\vsize by 26pt \medskip\noindent{\bf Remarks.\ } The upper bound in (\ulbound) is well known: It is the free energy density of the Curie--Weiss model, which is obtained by dropping all terms with $\mu>1$ from the Hamiltonian (\hamilt). For $\alpha=0$ the lower bound in (\ulbound) coincides with the upper bound as $\delta\downarrow 0$. In this case we recover a recent result of Shcherbina and Tirozzi [\rSTa]. For $\beta<1$, the lower bound with $\delta=1-\beta$ agrees with the correct free energy density up to $\OO(\alpha^{3/2})$; see e.g. the references [\rAGS] and [\rSTb]. \bigskip\noindent{\bf Proof.\ } Consider the overlap matrix $A$, $$ A_{\mu\nu}={1\over N}\sum_{i=1}^N\xi_i^\mu\xi_i^\nu-\delta_{\mu\nu}\,, \qquad \mu,\nu=1,\ldots,p. \eqno(\amunu) $$ An estimate in [\rSTa] implies that there exists a real number $\lambda$ and an even positive integer $n$, both depending on $N$, such that $$ \lambda^{-n}\mean\tr A^n\le\oo(N^{-1})\,,\qquad\lambda=4\sqrt{\alpha}+\oo(1)\,, \eqno(\abound) $$ as $N\to\infty$. A stronger version of this estimate (inequality (\etran)), which could be of independent interest, will be proved below. Assume now that $\alpha$ and $\delta$ satisfy the hypothesis of the theorem, and that $N$ is sufficiently large such that $\delta-\lambda(1-\delta)>0$. Denote by $\langle.,.\rangle$ the standard inner product on $\real^p$, and denote by $\xi_i$ the $i$--th column of the matrix $\xi$. If $\xi$ is such that the operator norm of $A$ is less than or equal to $\lambda$, we get an upper bound on the sum in (\efree) from the identity $$ \eqalign{ \sum_{S\in\SS}e^{-\beta H_{p,N,\xi}(S)} &=\intdpm e^{-\half N\beta\langle m,m\rangle} \sum_{S\in\SS}e^{\beta\sum_{i=1}^N\langle m,\xi_i\rangle S_i}\cr &=\intdpm e^{-\half N\beta\langle m,m\rangle} \prod_{i=1}^N2\cosh\bigl(\beta\langle m,\xi_i\rangle\bigr)\cr &=\intdpm e^{-\half N\beta\langle m,[\delta+(1-\delta)A]m\rangle -\beta\sum_{i=1}^N\phi_\delta(\beta,\langle m,\xi_i\rangle)},\cr} \eqno(\pfun) $$ if we replace the matrix $A$ by $-\lambda$ and the function $\phi_\delta(\beta,.)$ by its minimum value. The resulting lower bound on the free energy density can be extended to all patterns $\xi$, by adding to it the trivial bound $-[p/2+const]$, multiplied by the positive factor $\lambda^{-n}\tr A^n$ which is larger than $1$ whenever $||A||>\lambda\,$: $$ f_{p,N}(\beta,\xi)\ge\min_{h\in\real} \phi_\delta(\beta,h) +{p\over 2\beta N}\ln[\delta-\lambda(1-\delta)] -[p/2+const]\lambda^{-n}\tr A^n\,. \eqno(\lbound) $$ The assertion now follows from (\lbound) and (\abound). To complete the proof of our theorem we will now derive a bound that implies (\abound). Let $n\ge 4$. Then $N^n\mean\tr A^n$ is equal to the number of ordered products $$ \Pi_0\equiv\xi_{i_1}^{\mu_1}\xi_{i_1}^{\mu_2}\xi_{i_2}^{\mu_2}\xi_{i_2}^{\mu_3} \cdots \xi_{i_{k-1}}^{\mu_k}\bigl\{\xi_{i_k}^{\mu_k} \cdots \xi_{i_l}^{\mu_l}\bigr\}\xi_{i_l}^{\mu_{l+1}} \cdots \xi_{i_n}^{\mu_n}\xi_{i_n}^{\mu_1}\,, \eqno(\pizero) $$ (ignoring the braces) with $1\le\mu_k\le p$ and $1\le i_k\le N$ for all k, subject to the constraint that $\mu_1\ne\mu_2\ne\ldots\ne\mu_n\ne\mu_1$, and that each random variable $\xi_i^\mu$ appears an even number of times. Such products will be referred to as {\it contributing} products. Note that the constraint implies that the cardinalities $u=|\{\mu_1,\ldots,\mu_n\}|$ and $v=|\{i_1,\ldots,i_n\}|$ satisfy $u+v\le n+1$ and $2v\le n$. A contributing product will be called {\it simple} if its $2n$ factors can be grouped into $n$ pairs of identical random variables in such a way that if each pair is marked by braces as shown in equation (\pizero) for the pair $(\xi_{i_k}^{\mu_k},\xi_{i_l}^{\mu_l})$, then the ``sets'' corresponding to different pairs are either nested or disjoint. Note that such pairings can be specified by giving only the $n$ left braces. Thus, since each pairing determines whether or not any two indices can have different values, the number of simple products can be bounded by $$ c_n(p)\equiv{\textstyle{2n\choose n}} \max_{\scriptstyle u+v\le n+1\atop\scriptstyle v\le n/2} p^uN^v\,. \eqno(\cnofp) $$ Assume now that $\Pi_0$ is a non--simple contributing product. Below we will give a procedure for cutting $\Pi_0$ between some top--linked pairs (adjacent factors that have a common upper index, or the first and last factor) and regrouping the pieces into a simple product $\Pi_m$ that has the following property $P$: {\it The first factor of $\Pi_m$ is the same as that of $\Pi_0$; and if $\omega$ is any closed walk on the factors of $\Pi_m$ such that every odd--numbered (even--numbered) step is between two factors that are top--linked in $\Pi_m$ ($\Pi_0$), then, with one possible exception, all pairs connected by an odd--numbered step of $\omega$ are pairs of identical factors.} In particular, if we only consider walks whose first step goes to the right, and for which all but the last (or all) odd--numbered steps are between identical pairs, then we can find a set $W$ of walks of this type, such that every factor of $\Pi_m$ is visited by exactly one walk in $W$, and only once. Thus, it is possible to encode the original product $\Pi_0$ in a modified version $\Pi_m^\prime$ of $\Pi_m$, where, along each walk $(\omega_0,\omega_1,\ldots,\omega_\ell=\omega_0)\in W$ of length $\ell\ge 4$, the upper index in the identical factors $\omega_{2k-2}$ and $\omega_{2k-1}$ has been replaced by a pointer to the factor $\omega_{2k}$, for $1\le k<\ell/2$. Since the upper indices of $\Pi_m^\prime$ can take at most $p+2n$ different ``values'', it follows that the number of contributing products is bounded by $c_n(p+2n)$, and hence $$ \mean {\rm tr} A^n \le N^{-n}c_n(p+2n) \le 4^n(p+2n)\bigl({p+2n\over N}\bigr)^{n/2}\,, \qquad\qquad p+2n\le N. \eqno(\etran) $$ We get (\abound) by taking e.g. $\lambda=4(1+n^{-1/2})\bigl({p+2n\over N}\bigr)^{1/2}$ with $n=$ twice the integer part of $\sqrt{N}$. Finally, to construct $\Pi_m$ from $\Pi_0$ we iterate the following step. Assume that $\Pi_{m-1}$ is non--simple. Then, in this product, it is possible to find an odd number $\ge 3$ of consecutive factors $\xi_{i^\prime}^\mu\xi_i^\mu\cdots\xi_{j^\prime}^\mu$ such that $\xi_{j^\prime}^\mu$ is top--linked to a factor $\xi_j^\mu$ that is identical to $\xi_i^\mu$, with $i\ne i^\prime$ and $j\ne j^\prime$. After choosing such a sub--product, we now reverse the order of the factors $\xi_i^\mu\cdots\xi_{j^\prime}^\mu$ in $\Pi_{m-1}$ and define $\Pi_m$ to be the result of this operation. Note that this creates a new top--linked pair $(\xi_i^\mu\,,\xi_j^\mu)$ of identical factors. Thus, the iteration ends (with a simple product) after less than $n$ steps. It is easy to check that each step preserves the abovementioned property $P$, and that contributing products are mapped to contributing products. % ---------------------- \bigskip\noindent {\bf References}\hfil\break \smallskip \item{[\rAGS]} D.J.~Amit, H.~Gutfreund, and H.~Sompolinsky, {\it Spin--Glass Models of Neural Networks}, Phys.~Rev.~A {\bf 32}:1007 (1985). \item{[\rSTa]} M.~Shcherbina and B.~Tirozzi, {\it Exact Mean Field Theory for Some Hopfield Model}, \break Preprint (1991). \item{[\rSTb]} $X$.~Scacciatelli and B.~Tirozzi, {\it Fluctuation of the Free Energy in the Hopfield Model}, \break Preprint (1991). \bye