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While CBMs use supervision to align activations with human-labeled concepts, SAEs rely on sparse coding to uncover emergent ones. We show that both paradigms instantiate the same geometric structure: each learns a set of linear directions in activation space whose nonnegative combinations form a concept cone. Supervised and unsupervised methods thus differ not in kind but in how they select this cone. Building on this view, we propose an operational bridge between the two paradigms. CBMs provide human-defined reference geometries, while SAEs can be evaluated by how well their learned cones approximate or contain those of CBMs. This containment framework yields quantitative metrics linking inductive ","title":"A Geometric Unification of Concept Learning with Concept Cones","url":"https://arxiv.org/abs/2512.07355","vendor":"arxiv_cs_ai"},"summary":"arXiv:2512.07355v2 Announce Type: replace \nAbstract: Two traditions of interpretability have evolved side by side but seldom spoken to each other: Concept Bottleneck Models (CBMs), which prescribe what a concept should be, and Sparse Autoencoders (SAEs), which discover what concepts emerge. While CBMs use supervision to align activations with human-labeled concepts, SAEs rely on sparse coding to uncover emergent ones. We show that both paradigms instantiate the same geometric structure: each learns a set of linear directions in activation space whose nonnegative combinations form a concept cone. Supervised and unsupervised methods thus differ not in kind but in how they select this cone. Building on this view, we propose an operational bridge between the two paradigms. CBMs provide human-defined reference geometries, while SAEs can be evaluated by how well their learned cones approximate or contain those of CBMs. This containment framework yields quantitative metrics linking inductive ","title":"A Geometric Unification of Concept Learning with Concept Cones","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-09T04:43:45Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2512.07355"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:35309efb1ed82b3126c69cabaa59e532b99b9f3aee3eeb818b861e71935ec2521d5aa7185a173db971578d61344497f7a6a6e0052b34b2acb5977756b3e19603","signer":"crovia.substrate","subject":{"observed_at":"2026-06-09T04:43:45Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2512.07355"},"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":"e8a87f47aa10aaf17ef8c198c51dc41f4546d03d1776e837acd63a4fa5a5d9ef","leaf_index":224458,"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":"607f157dc9d575fe88caf93108af65380472c8e9b23be06815813177c42dbf35","side":"right"},{"sibling":"6c23eec07257fb610cffc2ae12ef5a07b019805ff32503d7445d59e9e0234575","side":"left"},{"sibling":"3cca94d278a14dd2d187ce50eb9c14105dcf94fae1a54a7a69da5b786996413d","side":"right"},{"sibling":"5aa8b68fb1f1a2763d3c6e7da3ee25d49f3abac37035607d40d340774039320a","side":"left"},{"sibling":"dcfa109865e99f057cf9154701a897872f8b893b161c05113b0e77cfa34cb785","side":"right"},{"sibling":"cda7f11a1cff7b91dee93d3661fadda29240d7915568285d6aace5cc6c68149c","side":"right"},{"sibling":"6544944fc5c0722251532a5665622a7bb9e1790fe437c54f30cf1b0f097f4768","side":"left"},{"sibling":"0dba1b4aa8c69a39c01dbd9d9884405788a8121b073c56815159112e9ccba4bd","side":"left"},{"sibling":"f6fb234a4e2f067b22329eec05b093a8b38f0411de9434d5a8eb55c2f70f1a2f","side":"right"},{"sibling":"b2df6a4bb3e928f0b447931cc688ae01d2415773a2b07cfed0b1cba689078aed","side":"right"},{"sibling":"b1c9ec856caa0fd46bb47b46f18c59ebcd295d774ca17adb3b46f05d394a6a5d","side":"left"},{"sibling":"24fdc29d461691aedb6fa920758206b5bb43851f477ef7a04c34aaed84b8971b","side":"left"},{"sibling":"036922da4e1e2c46d948f070454bfad299b7406fb00735ea9d8bd1e687f5f445","side":"right"},{"sibling":"533d82482604463aa4a281b9d8b85917b383b7c5f924b2b494039524c55e8797","side":"left"},{"sibling":"b2590791b920ca2a4ed39de126d2c0b1a10d9e7e62f572f12425f214e767b6e1","side":"left"},{"sibling":"87c6b850dfec08ac35a693d9db3a3315250a68adb1cfab9b1015f212b63b15bd","side":"right"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"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":224761,"merkle_root":"e9f7b49b652e869ab97ffba9c5a31356b2d0e3dc5d00bb28944adf737c46b1e7","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260609T103805Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-09T14:15:34Z","sig_algorithm":"ed25519","signature":"8ad8076fb12c8e486ae1d1559a9a7ba8e2ee996a9ad3d8ba7bcdbdbd88ab3a15bcb429707aca6d3e9d8b97e2ba755b3dcc77b1abb6601ccb829842719a6fb30d","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_de16db92e64a977495058c7d105caf7b7b87df4487bab068791c482201c92f9e"}}