The Hopfield recurrent neural network is a classical auto-associative model
of memory, in which collections of symmetrically-coupled McCulloch-Pitts
neurons interact to perform emergent computation. Although previous researchers
have explored the potential of this network to solve combinatorial optimization
problems and store memories as attractors of its deterministic dynamics, a
basic open problem is to design a family of Hopfield networks with a number of
noise-tolerant memories that grows exponentially with neural population size.
Here, we discover such networks by minimizing probability flow, a recently
proposed objective for estimating parameters in discrete maximum entropy
models. By descending the gradient of the convex probability flow, our networks
adapt synaptic weights to achieve robust exponential storage, even when
presented with vanishingly small numbers of training patterns. In addition to
providing a new set of error-correcting codes that achieve Shannon's channel
capacity bound, these networks also efficiently solve a variant of the hidden
clique problem in computer science, opening new avenues for real-world
applications of computational models originating from biology.

通过最小化概率流量，设计出一种具有指数级噪声容忍内存的 Hopfield 循环神经网络，该网络不仅能够实现 Shannon 信道容量界限，还可以高效地解决计算机科学中的隐藏社团问题，为来自生物学的计算模型的实际应用敞开了新的大门。

Hopfield 网络的鲁棒指数记忆

Robust exponential memory in Hopfield networks

Consider an Erd\"os-Renyi random graph in which each edge is present
independently with probability 1/2, except for a subset $\sC_N$ of the vertices
that form a clique (a completely connected subgraph). We consider the problem
of identifying the clique, given a realization of such a random graph.
The best known algorithm provably finds the clique in linear time with high
probability, provided $|\sC_N|\ge 1.261\sqrt{N}$ \cite{dekel2011finding}.
Spectral methods can be shown to fail on cliques smaller than $\sqrt{N}$. In
this paper we describe a nearly linear time algorithm that succeeds with high
probability for $|\sC_N|\ge (1+\eps)\sqrt{N/e}$ for any $\eps>0$. This is the
first algorithm that provably improves over spectral methods.
We further generalize the hidden clique problem to other background graphs
(the standard case corresponding to the complete graph on $N$ vertices). For
large girth regular graphs of degree $(\Delta+1)$ we prove that `local'
algorithms succeed if $|\sC_N|\ge (1+\eps)N/\sqrt{e\Delta}$ and fail if
$|\sC_N|\le(1-\eps)N/\sqrt{e\Delta}$.

本文提出了一种新的算法，能在近线性时间内解决 Erdos-Renyi 随机图中的大小不小于 (1+eps) sqrt (N/e) 的 clique 识别问题，并在大环正则图的情况下通过 “local” 算法成功识别大小不小于 (1+eps) N/sqrt (eDelta) 的 clique。