In \cite{Hop82}, Hopfield introduced a \emph{Hebbian} learning rule based
neural network model and suggested how it can efficiently operate as an
associative memory. Studying random binary patterns, he also uncovered that, if
a small fraction of errors is tolerated in the stored patterns retrieval, the
capacity of the network (maximal number of memorized patterns, $m$) scales
linearly with each pattern's size, $n$. Moreover, he famously predicted
$\alpha_c=\lim_{n\rightarrow\infty}\frac{m}{n}\approx 0.14$. We study this very
same scenario with two famous pattern's basins of attraction:
\textbf{\emph{(i)}} The AGS one from \cite{AmiGutSom85}; and
\textbf{\emph{(ii)}} The NLT one from
\cite{Newman88,Louk94,Louk94a,Louk97,Tal98}. Relying on the \emph{fully lifted
random duality theory} (fl RDT) from \cite{Stojnicflrdt23}, we obtain the
following explicit capacity characterizations on the first level of lifting:
\begin{equation}
\alpha_c^{(AGS,1)} = \left ( \max_{\delta\in \left ( 0,\frac{1}{2}\right )
}\frac{1-2\delta}{\sqrt{2} \mbox{erfinv} \left ( 1-2\delta\right )} -
\frac{2}{\sqrt{2\pi}} e^{-\left ( \mbox{erfinv}\left ( 1-2\delta \right )\right
)^2}\right )^2 \approx \mathbf{0.137906} \end{equation}
\begin{equation}
\alpha_c^{(NLT,1)} = \frac{\mbox{erf}(x)^2}{2x^2}-1+\mbox{erf}(x)^2 \approx
\mathbf{0.129490}, \quad 1-\mbox{erf}(x)^2-
\frac{2\mbox{erf}(x)e^{-x^2}}{\sqrt{\pi}x}+\frac{2e^{-2x^2}}{\pi}=0.
\end{equation}
A substantial numerical work gives on the second level of lifting
$\alpha_c^{(AGS,2)} \approx \mathbf{0.138186}$ and $\alpha_c^{(NLT,2)} \approx
\mathbf{0.12979}$, effectively uncovering a remarkably fast lifting
convergence. Moreover, the obtained AGS characterizations exactly match the
replica symmetry based ones of \cite{AmiGutSom85} and the corresponding
symmetry breaking ones of \cite{SteKuh94}.

文中介绍了一种基于赫布学习规则的神经网络模型，并研究了模式检索和容量特性，使用完全升级随机对偶理论得到精确的容量特性，并展示了快速升级收敛的结果。

赫布规范 - 霍普菲尔德网络关联记忆的容量

Capacity of the Hebbian-Hopfield network associative memory

We consider the memorization capabilities of multilayered \emph{sign}
perceptrons neural networks (SPNNs). A recent rigorous upper-bounding capacity
characterization, obtained in \cite{Stojnictcmspnncaprdt23} utilizing the
Random Duality Theory (RDT), demonstrated that adding neurons in a network
configuration may indeed be very beneficial. Moreover, for particular
\emph{treelike committee machines} (TCM) architectures with $d\leq 5$ neurons
in the hidden layer, \cite{Stojnictcmspnncaprdt23} made a very first
mathematically rigorous progress in over 30 years by lowering the previously
best known capacity bounds of \cite{MitchDurb89}. Here, we first establish that
the RDT bounds from \cite{Stojnictcmspnncaprdt23} scale as $\sim \sqrt{d}$ and
can not on their own \emph{universally} (over the entire range of $d$) beat the
best known $\sim \log(d)$ scaling of the bounds from \cite{MitchDurb89}. After
recognizing that the progress from \cite{Stojnictcmspnncaprdt23} is therefore
promising, but yet without a complete concretization, we then proceed by
considering the recently developed fully lifted RDT (fl RDT) as an alternative.
While the fl RDT is indeed a powerful juggernaut, it typically relies on heavy
numerical evaluations. To avoid such heavy numerics, we here focus on a
simplified, \emph{partially lifted}, variant and show that it allows for very
neat, closed form, analytical capacity characterizations. Moreover, we obtain
the concrete capacity bounds that \emph{universally} improve for \emph{any} $d$
over the best known ones of \cite{MitchDurb89}.

对多层 sign 感知器神经网络（SPNNs）的记忆能力进行研究，并通过随机对偶理论（RDT）得出的容量表征结果证明，网络配置中增加神经元确实非常有益。此外，对于隐藏层具有 d≤5 个神经元的特定树状委员会机器（TCM）结构，该研究首次在 30 多年中通过降低之前最佳已知容量界限，取得了重大数学进展。