xLSTM: Extended Long Short-Term Memory — part 20/30

Cleaned-text code points [288023, 301912). Bibliography omitted here; source.md includes it.

egin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{ {{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-16.39468pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -22.69 0)} \pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ {}{}{}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}}}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}{\hbox to21.44pt{\vbox to4.11pt{\pgfpicture\makeatletter\hbox{\hskip 10.72217pt\lower-1.0pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} 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state (28) 𝒉 t \displaystyle\bm{h}_{t}\ = ⊙ 𝒉 ~ t , \displaystyle=\ \mathchoice{\hbox to33.5pt{\vbox to5.94pt{\pgfpicture\makeatletter\hbox{\hskip 16.75018pt\lower-1.5pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{ {{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-16.75018pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -23.18 0)} \pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ 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\pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ {}{}{}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}}}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}{\hbox to21.96pt{\vbox to4.11pt{\pgfpicture\makeatletter\hbox{\hskip 10.97885pt\lower-1.0pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{ {{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-10.97885pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -15.19 0)} \pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ {}{}{}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}}}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}{\hbox to17.37pt{\vbox to2.97pt{\pgfpicture\makeatletter\hbox{\hskip 8.68321pt\lower-0.75pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{ {{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-8.68321pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -12.01 0)} \pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ {}{}{}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}}}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}\odot\tilde{\bm{h}}_{t}\ , 𝒉 ~ t \displaystyle\tilde{\bm{h}}_{t}\ = ψ ⁡ ( ) \displaystyle=\ \psi\left(\mathchoice{\hbox to30.11pt{\vbox to5.94pt{\pgfpicture\makeatletter\hbox{\hskip 15.0539pt\lower-1.5pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{ {{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-15.0539pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -20.83 0)} \pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ {}{}{}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}}}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}{\hbox to30.11pt{\vbox to5.94pt{\pgfpicture\makeatletter\hbox{\hskip 15.0539pt\lower-1.5pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{ {{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-15.0539pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -20.83 0)} \pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ {}{}{}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}}}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}{\hbox to19.57pt{\vbox to4.11pt{\pgfpicture\makeatletter\hbox{\hskip 9.78363pt\lower-1.0pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{ {{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-9.78363pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -13.54 0)} \pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ {}{}{}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}}}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}{\hbox to14.55pt{\vbox to2.97pt{\pgfpicture\makeatletter\hbox{\hskip 7.27402pt\lower-0.75pt\hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} \lxSVG@begingroup@{stroke=#000000} \lxSVG@begingroup@{fill=#000000} \lxSVG@setlinewidth{\the\pgflinewidth}\lxSVG@begingroup@{stroke-width=0.4pt} \lx@inpgf@ignorespaces\nullfont\lxSVG@begingroup@{_scopebegin=1} \lxSVG@closescope \hbox to0.0pt{\lxSVG@begingroup@{_scopebegin=1} {}{ {{}}\lx@inpgf@ignorespaces\hbox{\hbox{{\lxSVG@begingroup@{_scopebegin=1} {{}{}{{ {}{}}}{ {}{}} {{}{{\lx@inpgf@ignorespaces}}}{{}{\lx@inpgf@ignorespaces}}{}{{}{\lx@inpgf@ignorespaces}} {\lx@inpgf@ignorespaces }{{{{\lx@inpgf@ignorespaces}}\lxSVG@begingroup@{_scopebegin=1} \lxSVG@transformcm{1.0}{0.0}{0.0}{1.0}{-7.27402pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -10.07 0)} \pgfsys@hbox{65}\lxSVG@closescope }}} \lxSVG@closescope }}} } \lxSVG@closescope \hbox to0.0pt{}{{ {}{}{}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}}}{\lx@inpgf@ignorespaces}{\lx@inpgf@ignorespaces}\hss}\lxSVG@discardpath\lxSVG@closescope \hss}}\lxSVG@closescope\endpgfpicture}}}\right) hidden state (29) = φ ⁡ ( 𝒛 ~ t ) , \displaystyle=\ \varphi\left(\tilde{\bm{z}}_{t}\right)\ , 𝒛 ~ t \displaystyle\tilde{\bm{z}}_{t}\ = 𝑾 𝒛 ​ 𝒙 t + 𝑹 𝒛 ​ 𝒉 t − 1 + 𝒃 𝒛 \displaystyle=\ \bm{W}_{\bm{z}}\ \bm{x}_{t}\ +\ \bm{R}_{\bm{z}}\ \bm{h}_{t-1}\ +\ \bm{b}_{\bm{z}}\ \ cell input (30) = σ ⁡ ( 𝐢 ~ t ) , \displaystyle=\ \sigma\left(\tilde{\mathbf{i}}_{t}\right)\ , 𝐢 ~ t \displaystyle\tilde{\mathbf{i}}_{t}\ = 𝑾 𝐢 ​ 𝒙 t + 𝑹 𝐢 ​ 𝒉 t − 1 + 𝒃 𝐢 \displaystyle=\ \bm{W}_{\mathbf{i}}\ \bm{x}_{t}\ +\ \bm{R}_{\mathbf{i}}\ \bm{h}_{t-1}\ +\ \bm{b}_{\mathbf{i}}\ \ input gate (31) = σ ⁡ ( 𝐟 ~ t ) , \displaystyle=\ \sigma\left(\tilde{\mathbf{f}}_{t}\right)\ , 𝐟 ~ t \displaystyle\tilde{\mathbf{f}}_{t}\ = 𝑾 𝐟 ​ 𝒙 t + 𝑹 𝐟 ​ 𝒉 t − 1 + 𝒃 𝐟 \displaystyle=\ \bm{W}_{\mathbf{f}}\ \bm{x}_{t}\ +\ \bm{R}_{\mathbf{f}}\ \bm{h}_{t-1}\ +\ \bm{b}_{\mathbf{f}}\ \ forget gate (32) = σ ⁡ ( 𝐨 ~ t ) , \displaystyle=\ \sigma\left(\tilde{\mathbf{o}}_{t}\right)\ , 𝐨 ~ t \displaystyle\tilde{\mathbf{o}}_{t}\ = 𝑾 𝐨 ​ 𝒙 t + 𝑹 𝐨 ​ 𝒉 t − 1 + 𝒃 𝐨 \displaystyle=\ \bm{W}_{\mathbf{o}}\ \bm{x}_{t}\ +\ \bm{R}_{\mathbf{o}}\ \bm{h}_{t-1}\ +\ \bm{b}_{\mathbf{o}}\ \ output gate (33)

The matrices 𝑾 𝒛 \bm{W}_{\bm{z}} , 𝑾 𝐢 \bm{W}_{\mathbf{i}} , 𝑾 𝐟 \bm{W}_{\mathbf{f}} , and 𝑾 𝐨 \bm{W}_{\mathbf{o}} correspond to the input weights between inputs 𝒙 t \bm{x}_{t} and cell input, input gate, forget gate, and output gate, respectively. The matrices 𝑹 𝒛 \bm{R}_{\bm{z}} , 𝑹 𝐢 \bm{R}_{\mathbf{i}} , 𝑹 𝐟 \bm{R}_{\mathbf{f}} , and 𝑹 𝐨 \bm{R}_{\mathbf{o}} correspond to the recurrent weights between hidden state 𝒉 t − 1 \bm{h}_{t-1} and cell input, input gate, forget gate, and output gate, respectively. 𝒃 𝒛 \bm{b}_{\bm{z}} , 𝒃 𝐢 \bm{b}_{\mathbf{i}} , 𝒃 𝐟 \bm{b}_{\mathbf{f}} , and 𝒃 𝐨 \bm{b}_{\mathbf{o}} are the corresponding bias vectors. φ \varphi and ψ \psi are the cell input and hidden state activation functions (typically tanh \tanh ). ψ \psi is used to normalize or squash the cell state, which would be unbounded otherwise.

### A.2 sLSTM