In [7] Krotov and Hopfield suggest a generalized version of the well-known
Hopfield model of associative memory. In their version they consider a
polynomial interaction function and claim that this increases the storage
capacity of the model. We prove this claim and take the "limit" as the degree
of the polynomial becomes infinite, i.e. an exponential interaction function.
With this interaction we prove that model has an exponential storage capacity
in the number of neurons, yet the basins of attraction are almost as large as
in the standard Hopfield model.

该论文使用多项式交互函数推广了著名的 Hopfield 模型，证明了该模型的存储能力随多项式次数增加呈指数级增长，并且具有与标准 Hopfield 模型几乎相等的吸引域大小。

具有大容量存储的联想记忆模型

On a model of associative memory with huge storage capacity

Motivated by recent progress in using restricted Boltzmann machines as
preprocessing algorithms for deep neural network, we revisit the mean-field
equations (belief-propagation and TAP equations) in the best understood such
machine, namely the Hopfield model of neural networks, and we explicit how they
can be used as iterative message-passing algorithms, providing a fast method to
compute the local polarizations of neurons. In the "retrieval phase" where
neurons polarize in the direction of one memorized pattern, we point out a
major difference between the belief propagation and TAP equations : the set of
belief propagation equations depends on the pattern which is retrieved, while
one can use a unique set of TAP equations. This makes the latter method much
better suited for applications in the learning process of restricted Boltzmann
machines. In the case where the patterns memorized in the Hopfield model are
not independent, but are correlated through a combinatorial structure, we show
that the TAP equations have to be modified. This modification can be seen
either as an alteration of the reaction term in TAP equations, or, more
interestingly, as the consequence of message passing on a graphical model with
several hidden layers, where the number of hidden layers depends on the depth
of the correlations in the memorized patterns. This layered structure is
actually necessary when one deals with more general restricted Boltzmann
machines.

通过重新审视神经网络 Hopfield 模型中的均场方程（置信度传播和 TAP 方程），我们提出如何将其作为迭代消息传递算法，以提供计算局部神经元极化的快速方法，并显示 TAP 方程在处理与组合结构相关的存储器模式时要进行修改。

Hopfield 模型及其推广中的均场消息传递方程

Mean-field message-passing equations in the Hopfield model and its  generalizations

The Sherrington--Kirkpatrick model of spin glasses, the Hopfield model of
neural networks and the Ising spin glass are all models of binary data
belonging to the one-parameter exponential family with quadratic sufficient
statistic. Under bare minimal conditions, we establish the
$\sqrt{N}$-consistency of the maximum pseudolikelihood estimate of the natural
parameter in this family, even at critical temperatures. Since very little is
known about the low and critical temperature regimes of these extremely
difficult models, the proof requires several new ideas. The author's version of
Stein's method is a particularly useful tool. We aim to introduce these
techniques into the realm of mathematical statistics through an example and
present some open questions.

通过作者版本的 Stein 方法，我们建立了一种二进制数据模型的最大伪似然估计方法的根号 N 一致性，并且在关键温度下仍然有效，从而增强了这些模型的数学统计分析。

自旋玻璃中的估计：第一步

Estimation in spin glasses: A first step

We study the Hopfield model on a random graph in scaling regimes where the
average number of connections per neuron is a finite number and where the spin
dynamics is governed by a synchronous execution of the microscopic update rule
(Little-Hopfield model).We solve this model within replica symmetry and by
using bifurcation analysis we prove that the spin-glass/paramagnetic and the
retrieval/paramagnetictransition lines of our phase diagram are identical to
those of sequential dynamics.The first-order retrieval/spin-glass transition
line follows by direct evaluation of our observables using population dynamics.
Within the accuracy of numerical precision and for sufficiently small values of
the connectivity parameter we find that this line coincides with the
corresponding sequential one. Comparison with simulation experiments shows
excellent agreement.

研究了 Hopfield 模型在随机图中的缩放规律下，当每个神经元的平均连接数为有限数时，并且自旋动力学由微观更新规则的同步执行（Little-Hopfield 模型）所支配。该模型在复制对称性的范围内得到了解决，通过分岔分析证明了我们相图的自旋玻璃 / 顺磁和检索 / 顺磁转换线与顺序动力学的转换线相同。