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To theoretically explain this phenomenon, existing works have shown that activation sparsity does not result from the data properties or data fitting but from the implicit bias of the training process. However, these connections are obtained with strong assumptions, which cannot be applied to deep models standardly trained with a large number of steps. Different from these works, we find that the flatness of loss landscapes is also closely related to the MLP activation sparsity and can serve as a weaker and naturally emerging assumption standard deep networks. Specifically, we find that 1) the MLP activation sparsity equals a ratio between \"augmented flatness\" (a weighted sum of flatness measures) and the product of the input norm and activation ","title":"Towards the Connection between Activation Sparsity and Flat Minima","url":"https://arxiv.org/abs/2605.25612","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.25612v1 Announce Type: cross \nAbstract: The observation that activation sparsity emerges in MLP blocks of standardly trained Transformers offers an opportunity to drastically reduce computation costs without sacrificing performance. To theoretically explain this phenomenon, existing works have shown that activation sparsity does not result from the data properties or data fitting but from the implicit bias of the training process. However, these connections are obtained with strong assumptions, which cannot be applied to deep models standardly trained with a large number of steps. Different from these works, we find that the flatness of loss landscapes is also closely related to the MLP activation sparsity and can serve as a weaker and naturally emerging assumption standard deep networks. Specifically, we find that 1) the MLP activation sparsity equals a ratio between \"augmented flatness\" (a weighted sum of flatness measures) and the product of the input norm and activation ","title":"Towards the Connection between Activation Sparsity and Flat Minima","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-26T04:43:39Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.25612"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:d2d9fb0048177d81dfe644a2a378797e385d70496f391e353d9f5cc57b9956b927c8a11e2f07aaffc3538a1d4b62419991e570b9cfb761a58a581818dd18160f","signer":"crovia.substrate","subject":{"observed_at":"2026-05-26T04:43:39Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.25612"},"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":"bb3f7288f389e07af94915e6df4451ffca43b50fb02d27086211b4594ba1c561","leaf_index":151709,"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":"30f737b79d2e4fb685b555ceb89f90864697dca4a21332f53333e7e1c9d00b19","side":"left"},{"sibling":"98c8e81be006bb99371b28894b7bdffb34b500e77544b8e78aeba7d43fc982c8","side":"right"},{"sibling":"529afa61159dd2f18dfea8aa95954d9b94659cbadcb27f16770ee6fb1c60359b","side":"left"},{"sibling":"ddfca04deb04262ac5506b9bd3813317f794c228cd15e2e46838aca207027488","side":"left"},{"sibling":"0ab2ff7dd1be8f93ad56dbb1a795d5e58abfc300dfe3206007dafee8ed73a4aa","side":"left"},{"sibling":"3ed1b63a9ef71bc0b867033d2b26f2565035b22f65e2cc42d33645257e905332","side":"right"},{"sibling":"59c025ecd2dec1777c83a364f63a0af602c1b8c1069e8b09513a2e68a6f64e32","side":"right"},{"sibling":"f9db7ff3c5db7011b2738b446bad7b165ec8fd162c800ba3bad08298201edaa3","side":"left"},{"sibling":"aff657821100efc777fe98abc63d25a13c2d813d2a00f85a278e295ee3a166b3","side":"right"},{"sibling":"879666fab72e779ab55d0564eaabd64b00534fd6bba7f18412c7f31f61ffd09f","side":"right"},{"sibling":"f40ccedd90c323817e961adc0a2e2db82b8aabe192b6c9d5a373ff988987b207","side":"right"},{"sibling":"b85ea61ae405eed84392a7b6b1eee5536f5a38d6b04070638e23ec7b71e3443a","side":"right"},{"sibling":"d415e6939aee710631f5062799379b547d2c3e3d9a68f263bbb5a693285ab2ca","side":"left"},{"sibling":"e5893793e3591ed7f5e58ca94ffcfba46bb30f69fb1c25d5ba8ef49eb99f9126","side":"right"},{"sibling":"35ca36cee447f0ef7064a25d55f59357c66901e31427729c4c1d8b14aa8adb6c","side":"left"},{"sibling":"e3eecaf996dbe7229a7bb1d234c97aea97a252c5f7c89f8547b6d091db0f0e40","side":"right"},{"sibling":"55bcbd4da3e20d93931f7e58673f10232e81a5b1514d7396cb4b71e8f95788d0","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":152106,"merkle_root":"ac5182c6f3dd09931f2a689df5f4be36df7b55e55bcf106e195671f5ed55fd8f","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260526T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-26T05:37:34Z","sig_algorithm":"ed25519","signature":"a401243fbd2c077d29a623c6ef616c78fbd5c8ce1930afa7af7165b2386e5a8cf15d5094983a1c962e71b27b911a3f3ce09c8cfab9393be2ce5ce8ee6513da06","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_bbbef87c86117facc89ebc3913912cd1759c22012bc500d0775e5d0f7d5555c8"}}