Capacity of the Hebbian-Hopfield network associative memory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914701052477440 |
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| author | Stojnic, Mihailo |
| author_facet | Stojnic, Mihailo |
| contents | 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 $α_c=\lim_{n\rightarrow\infty}\frac{m}{n}\approx 0.14$. 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}
α_c^{(AGS,1)} = \left ( \max_{δ\in \left ( 0,\frac{1}{2}\right ) }\frac{1-2δ}{\sqrt{2} \mbox{erfinv} \left ( 1-2δ\right )} - \frac{2}{\sqrt{2π}} e^{-\left ( \mbox{erfinv}\left ( 1-2δ\right )\right )^2}\right )^2 \approx \mathbf{0.137906} \end{equation}
\begin{equation}
α_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}}{\sqrtπx}+\frac{2e^{-2x^2}}π=0. \end{equation}
A substantial numerical work gives on the second level of lifting $α_c^{(AGS,2)} \approx \mathbf{0.138186}$ and $α_c^{(NLT,2)} \approx \mathbf{0.12979}$, 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01907 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Capacity of the Hebbian-Hopfield network associative memory Stojnic, Mihailo Machine Learning Disordered Systems and Neural Networks Information Theory Probability 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 $α_c=\lim_{n\rightarrow\infty}\frac{m}{n}\approx 0.14$. 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} α_c^{(AGS,1)} = \left ( \max_{δ\in \left ( 0,\frac{1}{2}\right ) }\frac{1-2δ}{\sqrt{2} \mbox{erfinv} \left ( 1-2δ\right )} - \frac{2}{\sqrt{2π}} e^{-\left ( \mbox{erfinv}\left ( 1-2δ\right )\right )^2}\right )^2 \approx \mathbf{0.137906} \end{equation} \begin{equation} α_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}}{\sqrtπx}+\frac{2e^{-2x^2}}π=0. \end{equation} A substantial numerical work gives on the second level of lifting $α_c^{(AGS,2)} \approx \mathbf{0.138186}$ and $α_c^{(NLT,2)} \approx \mathbf{0.12979}$, 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}. |
| title | Capacity of the Hebbian-Hopfield network associative memory |
| topic | Machine Learning Disordered Systems and Neural Networks Information Theory Probability |
| url | https://arxiv.org/abs/2403.01907 |