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The essential difference, when compared to standard gradient descent methods, is to replace the usual update matrix $M=U\\Sigma V^\\top$ with its polar factor $UV^\\top$. In this work, we consider a class of Muon-like updates, where we replace the update $M$ with $U\\Sigma^p V^\\top$ for some parameter $p$. We call this a \"spectral-shaping\" operation, and develop a theory of how to pick $p$ which depends on (a) local curvature of the loss function, (b) noise stemming from stochastic gradients and label noise, and (c) training stage. Our theory and experimentation reveal a previously overlooked behavior: positive $p$ helps early by emphasizing high-curvature directions and accelerating signal contraction, while mildly negative $p$ helps later by reallocating update strength toward low-curvature directions that still con","title":"DynMuon: A Dynamic Spectral Shaping View of Muon","url":"https://arxiv.org/abs/2605.17109","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.17109v1 Announce Type: cross \nAbstract: In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly. The essential difference, when compared to standard gradient descent methods, is to replace the usual update matrix $M=U\\Sigma V^\\top$ with its polar factor $UV^\\top$. In this work, we consider a class of Muon-like updates, where we replace the update $M$ with $U\\Sigma^p V^\\top$ for some parameter $p$. We call this a \"spectral-shaping\" operation, and develop a theory of how to pick $p$ which depends on (a) local curvature of the loss function, (b) noise stemming from stochastic gradients and label noise, and (c) training stage. Our theory and experimentation reveal a previously overlooked behavior: positive $p$ helps early by emphasizing high-curvature directions and accelerating signal contraction, while mildly negative $p$ helps later by reallocating update strength toward low-curvature directions that still con","title":"DynMuon: A Dynamic Spectral Shaping View of Muon","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-19T04:43:36Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.17109"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:8ff106705ecc7d990f976ad5fb6e0ee02d6ac9a58d7cec6aa041d9634c8c8f7cbde7851bc242867c63058040ac3ff7d9c4ff078647f4a400ba333cf45977370f","signer":"crovia.substrate","subject":{"observed_at":"2026-05-19T04:43:36Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.17109"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"2d76e540c3de04220f0cc0ea5f1af38f74fabaa1507e64b34e8b326a1b06b194","leaf_index":142689,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"670fe34aa8ee5eaaba37956fb0994d3ecbe831b52beb54f72329d0577b9615ba","side":"left"},{"sibling":"5afa533799b768eaa6199aeea9e4cd53816e99c2a122159141abeafae040ce38","side":"right"},{"sibling":"522acb2d994ce6371afd5772893a99c40bef627008f47bc30cc5525a4f6e7a6c","side":"right"},{"sibling":"8e4307c9c57f93efe18cd19cd24975822845ec148db2b9ad88163eea496e123b","side":"right"},{"sibling":"7f568bea1499e07faffdc8d29a1d5ebf42084cb38edddaa0813be733a8c7e0b0","side":"right"},{"sibling":"42677723a066cf123b6a402facb5b3ce9057d62dbe62cab130c32ab10890f9be","side":"left"},{"sibling":"a702afd5696d81cad4f2475034b940a5e42d4b9aea3e2927b84acf32e052e97d","side":"left"},{"sibling":"0681a3895a908b2cd98c077a0abe5cdd8542f0ecb3a6294a35366c915b2e2e3e","side":"right"},{"sibling":"f1c796d3bd453570426dcee9ab20072f19762202f84b7993afba2c7c0ee9ec3b","side":"left"},{"sibling":"2e0ce989d789c88e796991ef014ce7e4e1f96c0d4ede9da8d31afcf5ba6a8f46","side":"right"},{"sibling":"202f1bead178ef3785968d50d3d188264a95192a077654c331612e04a34cbfbe","side":"left"},{"sibling":"72249c8c8b068386e35d16f4bd0bbeb9ba820ca217ef0f0d28396c9fe493f5f0","side":"left"},{"sibling":"ea64599340f7ffdf17ad0cbc1d9401ef8870a347e3847bdc106d06b1673df09c","side":"right"},{"sibling":"8f4c0fbe56b6c010fbb8c782ebcd478079bb3f991d8704e2534209a075d9163c","side":"left"},{"sibling":"4db1f363729507e27a60851cf6ed334d7b9acdef194ed7d419aba4d2bd367a4a","side":"right"},{"sibling":"a86ee18c45e7fcc408b6007eaece05aa75b2d9ae30252e9e878462b4dffbef7b","side":"right"},{"sibling":"1d18e7663d43ccff0122ecc7ee12645bb16afb607b218e81b1ea2408f863cb78","side":"right"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":143302,"merkle_root":"999156d40a7c61d9ddd52b7338f3cbda3e68f53bace070c7b616ea194e23b123","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260519T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-19T05:37:30Z","sig_algorithm":"ed25519","signature":"b1a252cc66ff32bed1d10dd88a6b2a200e3856d3dbcfcc4ee55e02e00f3d548e854ed9c544704b222bd5d315492c4a935ba2d90d727c585a67899b0ad602fc05","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_2d7ce97bf9e28a87efcb475d7007acd117f1287fd06d377b4b938754484f1c3d"}}