{"paper":{"title":"M$^2$RNN: Non-Linear RNNs with Matrix-Valued States for Scalable Language Modeling","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Non-linear RNNs with matrix-valued states achieve perfect unseen-length state tracking and outperform equivalent attention hybrids by 0.4-0.5 perplexity points while using three times smaller recurrent states.","cross_cats":["cs.AI"],"primary_cat":"cs.LG","authors_text":"Ion Stoica, Joseph Gonzalez, Mayank Mishra, Shawn Tan, Tri Dao","submitted_at":"2026-03-15T12:53:09Z","abstract_excerpt":"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 their state size, and show how the state size expansion mecha"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"M²RNN achieves perfect state tracking generalization at sequence lengths not seen during training. Hybrid M²RNN outperforms equivalent Gated DeltaNet hybrids by 0.4-0.5 perplexity points on a 7B MoE model while using 3× smaller state sizes for the recurrent layers.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the non-linear matrix-valued state transitions and state size expansion mechanism provide the claimed expressive power and efficiency gains without introducing training instability or hidden computational costs at scale.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"M²RNN achieves perfect state tracking at unseen lengths and outperforms Gated DeltaNet hybrids by 0.4-0.5 perplexity on 7B models with 3x smaller recurrent states.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Non-linear RNNs with matrix-valued states achieve perfect unseen-length state tracking and outperform equivalent attention hybrids by 0.4-0.5 perplexity points while using three times smaller recurrent states.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0ce4faf365470d0038d64ae0ba772f1335f485919b6313b1b9f7fcd21e69a076"},"source":{"id":"2603.14360","kind":"arxiv","version":2},"verdict":{"id":"e83a3215-3f10-4d25-8adb-1f59661d8aa8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T11:32:37.524446Z","strongest_claim":"M²RNN achieves perfect state tracking generalization at sequence lengths not seen during training. Hybrid M²RNN outperforms equivalent Gated DeltaNet hybrids by 0.4-0.5 perplexity points on a 7B MoE model while using 3× smaller state sizes for the recurrent layers.","one_line_summary":"M²RNN achieves perfect state tracking at unseen lengths and outperforms Gated DeltaNet hybrids by 0.4-0.5 perplexity on 7B models with 3x smaller recurrent states.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the non-linear matrix-valued state transitions and state size expansion mechanism provide the claimed expressive power and efficiency gains without introducing training instability or hidden computational costs at scale.","pith_extraction_headline":"Non-linear RNNs with matrix-valued states achieve perfect unseen-length state tracking and outperform equivalent attention hybrids by 0.4-0.5 perplexity points while using three times smaller recurrent states."},"references":{"count":46,"sample":[{"doi":"10.1145/3620665.3640366.https://doi.org/10.1145/3620665.3640366","year":null,"title":"PyTorch 2: Faster machine learning through dynamic Python bytecode transformation and graph compilation","work_id":"abd261ad-4aca-4ea3-8090-344914daba35","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1109/72.279181","year":2019,"title":"Simard, and Paolo Frasconi","work_id":"a6a7d71d-d4b9-4f4a-bd34-e010d6b09479","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1901,"title":"Language models are few-shot learners.Advances in neural information processing systems, 33:1877–1901","work_id":"84780adb-76cf-44ac-a8b7-e24d4fa5c592","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"On the properties of neural machine translation: Encoder-decoder approaches","work_id":"a6dc7801-0da7-423e-ba1d-00d246875f48","ref_index":4,"cited_arxiv_id":"1409.1259","is_internal_anchor":true},{"doi":"","year":null,"title":"Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling","work_id":"c7f2f5a9-ae4b-48db-aff0-24b9d0528995","ref_index":5,"cited_arxiv_id":"1412.3555","is_internal_anchor":true}],"resolved_work":46,"snapshot_sha256":"801929d3413b61bd7956e72c02f99a2a95ae4cbdc001ebc78201f1e7f5d95ef8","internal_anchors":26},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}