{ "id": "AIBD-0020", "title": "AlphaTensor discovers faster matrix multiplication algorithms over GF(2), immediately built upon by independent mathematicians", "summary": "AlphaTensor found a 47-multiplication 4x4 scheme over GF(2) and a 96 for 5x5; Kauers and Moosbauer improved the latter to 95.", "date_observed": "2022-10-05", "date_basis": "unassessed", "date_precision": "unassessed", "benefit_status": null, "source_collection": null, "source_entry": null, "account": "On 5 October 2022 Nature published AlphaTensor, a reinforcement-learning system from DeepMind (Fawzi, Balog, Huang, Hubert, Romera-Paredes and others). It treats the search for matrix multiplication algorithms as a one-player game, TensorGame. A deep neural network guides a planning procedure derived from AlphaZero. It found an algorithm multiplying two 4x4 matrices over the field with two elements in 47 multiplications. That beat Strassen's two-level algorithm for the first time in about 50 years. 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Nature reports the 4x4 improvement for matrices “in a finite field” and Kauers and Moosbauer give the 5x5 record of 96 multiplications over Z2." } ], "ai_role": { "status": "assessed", "statement": "The reinforcement-learning agent found the decompositions, which are explicit and checkable. The independent preprint treats the 96-multiplication 5x5 result as established and improves it, which corroborates the result's existence. The 47-multiplication 4x4 scheme rests on the vendor paper's published decomposition." }, "evidence": [ { "kind": "peer-reviewed study", "citation": "Fawzi, A. et al. (2022). Discovering faster matrix multiplication algorithms with reinforcement learning. Nature 610, 47–53.", "title": "Discovering faster matrix multiplication algorithms with reinforcement learning", "url": "https://www.nature.com/articles/s41586-022-05172-4", "date": "2022-10-05", "written_by": "makers", "quote": "Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor's algorithm improves on Strassen's two-level algorithm for the first time, to our knowledge, since its discovery 50 years ago." }, { "kind": "preprint", "citation": "Kauers, M. and Moosbauer, J. (2022). The FBHHRBNRSSSHK-Algorithm for Multiplication in Z2^(5x5) is still not the end of the story. arXiv:2210.04045.", "title": "The FBHHRBNRSSSHK-Algorithm for Multiplication in Z2^(5x5) is still not the end of the story", "url": "https://arxiv.org/abs/2210.04045", "date": "2022-10-08", "written_by": "independent", "quote": "In response to a recent Nature article which announced an algorithm for multiplying 5×5-matrices over ℤ₂ with only 96 multiplications, two fewer than the previous record" } ], "replication": { "produced_by": [ "Formulation of algorithm discovery as a tensor-decomposition game", "Neural-network-guided search with exact verification of each decomposition", "Publication of explicit decompositions that others can check and extend" ], "must_hold": [ "Decompositions must be verified exactly over the stated ring or field", "Independent reproduction or extension by researchers outside the vendor" ], "echoes": [ { "date": "2025-05-14", "setting": "AlphaEvolve (Google DeepMind) found a scheme multiplying 4x4 complex-valued matrices in 48 scalar multiplications; Dumas, Pernet and Sedoglavic (arXiv:2506.13242, submitted 16 June 2025) derived a 48-multiplication scheme with rational coefficients from AlphaEvolve's complex-valued decomposition, valid over any ring except characteristic 2", "model_version": "AlphaEvolve", "outcome": "recurred", "source": "https://deepmind.google/discover/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/ ; https://arxiv.org/abs/2506.13242" } ] }, "paired_incident": null, "headline": "AlphaTensor finds a 47-multiplication 4x4 scheme over GF(2)", "metric": "multiplying two 4x4 matrices over the field with two elements in 47 multiplications", "comparator": null, "later": { "status": "persisted", "date": "2022-10-08", "note": "Three days after publication, Kauers and Moosbauer took AlphaTensor's 96-multiplication 5x5 scheme over Z2 as a starting point and improved it to 95, confirming and extending the result.", "quote": "In response to a recent Nature article which announced an algorithm for multiplying 5×5-matrices over ℤ2 with only 96 multiplications, two fewer than the previous record, we present an algorithm that does the job with only 95 multiplications.", "url": "https://arxiv.org/abs/2210.04045" }, "value": null, "developer_country": "United Kingdom", "deployer_country": "United Kingdom", "unrated_because": null, "brief_notes": null, "brief": null, "revisions": [ { "date": "2026-10-20", "change": "Record drafted from the cited sources.", "source": null, "editor": "AIBD" }, { "date": "2026-10-20", "change": "Replaced the stitched Nature quote with the abstract sentence as printed, corrected the AlphaEvolve echo to say that the rational 48-multiplication scheme was projected from AlphaEvolve's complex-valued decomposition rather than derived independently, and confined the societal rating reason to what the cited sources state.", "source": "https://www.nature.com/articles/s41586-022-05172-4 ; https://arxiv.org/abs/2506.13242 ; https://arxiv.org/abs/2210.04045", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Added what happened since from the cited sources.", "source": "https://arxiv.org/abs/2210.04045", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Country of the developer and the deployer added from the organization's own site or a registry.", "source": "https://en.wikipedia.org/wiki/Google_DeepMind; https://en.wikipedia.org/wiki/Google_DeepMind", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Prose pass: a rating reason, account, the system's part rewritten in shorter sentences with abbreviations expanded; facts, figures and quotations unchanged.", "source": null, "editor": "AIBD" } ], "confidence": "reported", "shade": "medium", "claim_status": "Independently reported", "claim_basis": "Sources: one independent preprint and the makers' peer-reviewed study. 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