{"paper":{"title":"Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Sampling a manifold with a Hilbert bundle induces a cellular sheaf whose Laplacian converges in probability to the connection Laplacian, enabling consistent discrete networks for infinite-dimensional signals.","cross_cats":["cs.AI","eess.SP"],"primary_cat":"cs.LG","authors_text":"Alejandro Ribeiro, Claudio Battiloro, Francesca Dominici, Julian Gould, Kartik Tandon, Tanishq Bhatia","submitted_at":"2026-05-07T15:08:58Z","abstract_excerpt":"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 \\"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin & Niyogi convergence result for the graph Laplacian to the manifold Laplacian","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Sampling the manifold induces a Hilbert Cellular Sheaf with edge-wise coupling rules that preserve the necessary structure for the Laplacian convergence to hold in the infinite-dimensional Hilbert bundle case.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"HilbNets discretize Hilbert bundle convolutions through Hilbert Cellular Sheaves whose Laplacians converge to the continuous connection Laplacian, enabling consistent learning across samplings.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Sampling a manifold with a Hilbert bundle induces a cellular sheaf whose Laplacian converges in probability to the connection Laplacian, enabling consistent discrete networks for infinite-dimensional signals.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4e6ef27b4aad7372157746d0960500911638fc918797be271de245b945b6aa1d"},"source":{"id":"2605.06395","kind":"arxiv","version":2},"verdict":{"id":"fbfadf8f-521c-45af-8384-cba0bb58207b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T12:48:21.143336Z","strongest_claim":"we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin & Niyogi convergence result for the graph Laplacian to the manifold Laplacian","one_line_summary":"HilbNets discretize Hilbert bundle convolutions through Hilbert Cellular Sheaves whose Laplacians converge to the continuous connection Laplacian, enabling consistent learning across samplings.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Sampling the manifold induces a Hilbert Cellular Sheaf with edge-wise coupling rules that preserve the necessary structure for the Laplacian convergence to hold in the infinite-dimensional Hilbert bundle case.","pith_extraction_headline":"Sampling a manifold with a Hilbert bundle induces a cellular sheaf whose Laplacian converges in probability to the connection Laplacian, enabling consistent discrete networks for infinite-dimensional signals."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.06395/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T12:42:03.970960Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-20T07:43:22.876064Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T18:31:19.164260Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T12:43:47.236347Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c8a7cdea3631ebace37fb1f2669590e3f825de61882be8178076bfd918fab78e"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}