xLSTM: Extended Long Short-Term Memory — part 19/30

Cleaned-text code points [272023, 288023). Bibliography omitted here; source.md includes it.

We have partly answered our simple question: How far do we get in language modeling when scaling LSTM to billions of parameters? So far, we can answer: “At least as far as current technologies like Transformers or State Space Models”. We have enhanced LSTM to xLSTM by exponential gating with memory mixing and a new memory structure. xLSTM models perform favorably on language modeling when compared to state-of-the-art methods like Transformers and State Space Models. The scaling laws indicate that larger xLSTM models will be serious competitors to current Large Language Models that are built with the Transformer technology. xLSTM has the potential to considerably impact other fields like Reinforcement Learning, Time Series Prediction, or the modeling of physical systems.

## Acknowledgements

We thank Sebastian Lehner, Daniel Klotz, Thomas Adler, Matthias Dellago, Gerald Gutenbrunner, Fabian Paischer, Vihang Patil, Niklas Schmidinger, Benedikt Alkin, Kajetan Schweighofer, Anna Zimmel, Lukas Aichberger, Lukas Hauzenberger, Bernhard Schäfl and Johannes Lehner for helpful discussions and feedback.

## Appendix A Extended Long Short-Term Memory

### A.1 Vanilla Long Short-Term Memory Formulation: Vector Notation

The vanilla LSTM memory cell update rules ( Greff et al., 2015 ) at time step t t extend the scalar cell state formulation to a vector of cell states: = ⊙ + ⊙ \displaystyle=\ \mathchoice{\hbox to25.88pt{\vbox to8.44pt{\pgfpicture\makeatletter\hbox{\hskip 12.9391pt\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}{-12.9391pt}{0.0pt}\lxSVG@begingroup@{transform=matrix(1.0 0.0 0.0 1.0 -17.9 0)} 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