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Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as \\textit{HilbNets}. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in p","title":"Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves","url":"https://arxiv.org/abs/2605.06395","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.06395v1 Announce Type: cross \nAbstract: Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as \\textit{HilbNets}. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in p","title":"Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-08T04:43:40Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2605.06395"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:7e1f807aef382c137d20761a3e0b41c50f0ef0552a6bebe78e7bc0fd23668a6d89daf5f786b75d1f4184ecd1d26607e55d5430b30e9deca90400c8388d6f1c01","signer":"crovia.substrate","subject":{"observed_at":"2026-05-08T04:43:40Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2605.06395"},"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":"eb1c3aaffe8a148778c0fcc896cb22daea463dfb3bc308c9c30cd58450cf911a","leaf_index":120321,"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":"e7ad662bae39b99cebad57e5e4f627bb55390914759a622a51a64ade585bd310","side":"left"},{"sibling":"b29f520f03d583912b55b019cfd6fa6760b401e736d041fce4e5fcd9079bfb9c","side":"right"},{"sibling":"220a1a8390baaad67bce6da66ad383739aabddc7464b3e4937716e76088f8e75","side":"right"},{"sibling":"7c5e24eabf07e4e180bf547e18126344ad8416fb03ae4a7368b982567fc835c6","side":"right"},{"sibling":"1dd410888f5c572059febdba5a7907c59838d5b9b159eb0c9399184179bda760","side":"right"},{"sibling":"ee48c805c85a4e43d4a9d14a4f80b704a7a8eb604db6cb321fb8c6fe08a82186","side":"right"},{"sibling":"ff6dfc3e3819926f225fca4697d75146861a953457867e78f1a01b86559feac7","side":"right"},{"sibling":"621ce3a316503da6017cc9f955ea0dca3f50bbc1e8fc10ed72f522e60e814e3f","side":"right"},{"sibling":"7cb9b06d4f6372fac19d788263daadfaf6def57cc0a73d46bed8b284ec0fe014","side":"right"},{"sibling":"42ade783b0aede9d0209c5be94ac7979e635da84dd65ecb4d383719029bbe8f4","side":"left"},{"sibling":"143f33d3924b3840fd6dd8ba12566fc35bf86189e663ef0ad4884d676f295e3a","side":"left"},{"sibling":"b1ed99341c327c7c9ab2489f40af2547ab3b3b4b6a74fb684210164fb891a413","side":"right"},{"sibling":"6ee3be9bdfc9bee55d32f7dbb0075f02fe87d20887d563d3e300caf36b1b88c7","side":"left"},{"sibling":"8ccd9937a2c0d5c04044d07d1557791b7d07bb31eac41a39a675608d44b38f23","side":"right"},{"sibling":"3a5e69cf0803f4c91f3895ed7c9a95748fef240bec4422e167c05300f79f06c0","side":"left"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"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":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","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_26c631f49d9715cbb524919bc6076537b149d69507517e7028e7c48eaa8a8a8d"}}