M$^2$RNN: Non-Linear RNNs with Matrix-Valued States for Scalable Language Modeling

arXiv:2603.14360v3 Announce Type: replace-cross Abstract: Transformers are highly parallel but are limited to computations in the TC$^0$ complexity class, excluding tasks such as entity tracking and code execution that provably require greater expressive power. Motivated by this limitation, we revisit non-linear Recurrent Neural Networks (RNNs) for language modeling and introduce Matrix-to-Matrix RNN (M$^2$RNN): an architecture with matrix-valued hidden states and expressive non-linear state transitions. We demonstrate that the language modeling performance of non-linear RNNs is limited by the

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