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By decomposing this matrix into its symmetric and skew-symmetric parts, we interpret the symmetric component as governing the structure of the energy landscape, and the skew-symmetric component as driving circulation on that landscape. Leveraging the energy formulation induced by the symmetric component, we derive Hopfield-style stability measures that quantify the stability of retrieved features. We observe meaningful correlations between Hopfield-style stability measures and the fidelity-diversity trade-offs in generation. Finally, we propose a controllable knob to modulate this trade-off by modifying the circulation of the underlying dynamics. Code is available at our GitHub (https://github.com/hyeon-cho/Attention-Symmetric-Decomposition).","title":"Balancing Fidelity and Diversity in Diffusion Models via Symmetric Attention Decomposition: Hopfield Perspective","url":"https://arxiv.org/abs/2605.27476","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.27476v1 Announce Type: cross \nAbstract: We characterize the pre-softmax attention matrix $\\mathbf{QK^\\top}$ in transformers as an associative memory matrix encoding pairwise associations between input features. By decomposing this matrix into its symmetric and skew-symmetric parts, we interpret the symmetric component as governing the structure of the energy landscape, and the skew-symmetric component as driving circulation on that landscape. Leveraging the energy formulation induced by the symmetric component, we derive Hopfield-style stability measures that quantify the stability of retrieved features. We observe meaningful correlations between Hopfield-style stability measures and the fidelity-diversity trade-offs in generation. Finally, we propose a controllable knob to modulate this trade-off by modifying the circulation of the underlying dynamics. Code is available at our GitHub (https://github.com/hyeon-cho/Attention-Symmetric-Decomposition).","title":"Balancing Fidelity and Diversity in Diffusion Models via Symmetric Attention Decomposition: Hopfield Perspective","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-28T04:43:38Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.27476"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:0330ea3461f8c0d4bcb9b6042b83277e2aef5efebe119c279780f760ad627a69b1d166f61f6236a86995d9dfefdebdc1a1365411d082e016a470d9275c1fe502","signer":"crovia.substrate","subject":{"observed_at":"2026-05-28T04:43:38Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.27476"},"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":"7b3aee489d931cb16939062e44fb8d0a52e38f7d7a5f4b20e785015af3a94667","leaf_index":155779,"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":"e14d456742a197fdfc695f3dfdaf030cebb28c90126a7feebce4c86038e31b48","side":"left"},{"sibling":"7793e0bce83fda2bb5d0ef50bb8a426b41d05d3763fe7e36928431b58d36ad94","side":"left"},{"sibling":"78d7010eee00dab85903a3009d7197711590918390fe4737dfd4b0be6407076d","side":"right"},{"sibling":"bfd19709c85004c2378bdc230027e1d5371eeca6a198fbbba79c62fd620a105a","side":"right"},{"sibling":"6880d25fe07c485813e1548fa5621c010fda55c8e8e70a74255c9a4bf579649b","side":"right"},{"sibling":"1018a361170fe7e1f85a70e5458d9908e8d96d60f88b65cc77443b218552271b","side":"right"},{"sibling":"ee7da65b5a86af06f9a3335d17a9d3cfc91a80b40810cef933dac1f3ef2460ba","side":"right"},{"sibling":"753018fa645affe1bfe31d1c496292a6e72167116d741961e921b85a693e4153","side":"left"},{"sibling":"0adb35dd3ad9bfc4a781c02606acf0a3ce1e28e1a88066209cc1a35814e790ad","side":"right"},{"sibling":"f292d3278e493ec60902181b8c0bd5c89c0a1168ef928222fe6161287982f7f0","side":"right"},{"sibling":"2209295faf1a5bf51c97c6fd5a839a8181a4ea44f490420381530f35df7d9b2f","side":"right"},{"sibling":"5784576a15214ea9fc3569e6e1cff1ef443c0b1fc0d036028489088af089de27","side":"right"},{"sibling":"311772ec218efcb2da5a337f9e9f042fe1cc0028643adb0a354787e4ea7911b7","side":"right"},{"sibling":"66331bac84ca0f8983eb09fac7eaf95af234f1b82680b793eabff4ee25caac40","side":"left"},{"sibling":"35ca36cee447f0ef7064a25d55f59357c66901e31427729c4c1d8b14aa8adb6c","side":"left"},{"sibling":"7b6f0bea4291a5e63574dfca9aa0f9756f450c9478c3d07474d39a7ababb51f9","side":"right"},{"sibling":"1d39fe14b21e2ebbfb87e882423b24ee9469eae1e4c77af5b799ac4db9537467","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":156177,"merkle_root":"c5705a0243d16afd8b1ebfd731b7aa304079c442c2a7906493c5bbed374c69ec","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260528T053701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-28T05:37:36Z","sig_algorithm":"ed25519","signature":"f087e13febc8bb6a2e0812610de64cebc65be92915518d9c4b230799c3b161839b04c4eb1b02741f938f35545a76ab76555b04c782bdc2f9a44852d171d65909","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_2e715d2e836d0cf86aa36f8f667ba01f6fc3235ff66f382b9624c5b12ff6592d"}}