{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:SAH3XESM4CCNNG6KAWXAXF5D5H","short_pith_number":"pith:SAH3XESM","schema_version":"1.0","canonical_sha256":"900fbb924ce084d69bca05ae0b97a3e9df2ab81f3a0b8f856e9663f8b3fedd12","source":{"kind":"arxiv","id":"2602.09297","version":3},"attestation_state":"computed","paper":{"title":"Laplacian Heads Improve Transformers by Smoothing Token Representations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Replacing some attention heads with Laplacian operators improves Transformer performance by enabling beneficial smoothing of token representations.","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Vardan Papyan, Yuchong Zhang","submitted_at":"2026-02-10T00:27:45Z","abstract_excerpt":"Transformers update token representations through multi-head attention and residual connections as $X \\leftarrow X + \\sum_{i} P^{(i)}XW_{V_i}W_{o_i}$, where $P^{(i)}$ is the softmax attention matrix in head $i$. We propose replacing a subset of $P^{(i)}$'s with the Laplacian $I - P^{(i)}$, giving $X \\leftarrow X + \\sum_{i \\in \\mathcal{A}} P^{(i)}XW_{V_i}W_{o_i} + \\sum_{i \\in \\mathcal{L}} (I - P^{(i)})XW_{V_i}W_{o_i}$. Our proposal has two motivations. First, it allows attention heads to update the mean of token representations, while Laplacian heads can directly control within-sequence varianc"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":true},"canonical_record":{"source":{"id":"2602.09297","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-02-10T00:27:45Z","cross_cats_sorted":[],"title_canon_sha256":"81ee95e16ae3ba4d92318d7139e17d4b6c70c2ebeaa1fef635dd73709e30eed4","abstract_canon_sha256":"dc3fd782ca89fbf7dced4a2c33c9570a63fc9516e542b48be878274c8040a072"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:00:34.094861Z","signature_b64":"HznruQ6eJYguhcvSDloRWmJRVGzDvkeHSCz/7vB4nN4NPtdKEaiFiRs3EYs4kL/83s/S0aOQgPT9FRQBD5LjBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"900fbb924ce084d69bca05ae0b97a3e9df2ab81f3a0b8f856e9663f8b3fedd12","last_reissued_at":"2026-05-20T00:00:34.094185Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:00:34.094185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Laplacian Heads Improve Transformers by Smoothing Token Representations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Replacing some attention heads with Laplacian operators improves Transformer performance by enabling beneficial smoothing of token representations.","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Vardan Papyan, Yuchong Zhang","submitted_at":"2026-02-10T00:27:45Z","abstract_excerpt":"Transformers update token representations through multi-head attention and residual connections as $X \\leftarrow X + \\sum_{i} P^{(i)}XW_{V_i}W_{o_i}$, where $P^{(i)}$ is the softmax attention matrix in head $i$. We propose replacing a subset of $P^{(i)}$'s with the Laplacian $I - P^{(i)}$, giving $X \\leftarrow X + \\sum_{i \\in \\mathcal{A}} P^{(i)}XW_{V_i}W_{o_i} + \\sum_{i \\in \\mathcal{L}} (I - P^{(i)})XW_{V_i}W_{o_i}$. Our proposal has two motivations. First, it allows attention heads to update the mean of token representations, while Laplacian heads can directly control within-sequence varianc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that this simple modification improves performance across supervised learning, language modeling, and self-supervised learning tasks.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the performance gains are caused by the smoothing mechanism of the Laplacian heads rather than incidental changes in optimization dynamics or capacity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Laplacian heads replace some attention matrices with I minus P in Transformers, improving performance by smoothing tokens and challenging the view that oversmoothing is always detrimental.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Replacing some attention heads with Laplacian operators improves Transformer performance by enabling beneficial smoothing of token representations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e881345f8b9b9bd81d3144dc3112e5dedd84537f262765eebd224b425ac7811d"},"source":{"id":"2602.09297","kind":"arxiv","version":3},"verdict":{"id":"15d7ed43-ee0d-42e3-8478-5e2d7cf14c55","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T06:11:12.810571Z","strongest_claim":"We show that this simple modification improves performance across supervised learning, language modeling, and self-supervised learning tasks.","one_line_summary":"Laplacian heads replace some attention matrices with I minus P in Transformers, improving performance by smoothing tokens and challenging the view that oversmoothing is always detrimental.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the performance gains are caused by the smoothing mechanism of the Laplacian heads rather than incidental changes in optimization dynamics or capacity.","pith_extraction_headline":"Replacing some attention heads with Laplacian operators improves Transformer performance by enabling beneficial smoothing of token representations."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2602.09297/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"4ac687a462bde77cea9bc8bd6e98d73ae86a2377d60b3150c529080fcdff1f67"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2602.09297","created_at":"2026-05-20T00:00:34.094291+00:00"},{"alias_kind":"arxiv_version","alias_value":"2602.09297v3","created_at":"2026-05-20T00:00:34.094291+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.09297","created_at":"2026-05-20T00:00:34.094291+00:00"},{"alias_kind":"pith_short_12","alias_value":"SAH3XESM4CCN","created_at":"2026-05-20T00:00:34.094291+00:00"},{"alias_kind":"pith_short_16","alias_value":"SAH3XESM4CCNNG6K","created_at":"2026-05-20T00:00:34.094291+00:00"},{"alias_kind":"pith_short_8","alias_value":"SAH3XESM","created_at":"2026-05-20T00:00:34.094291+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H","json":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H.json","graph_json":"https://pith.science/api/pith-number/SAH3XESM4CCNNG6KAWXAXF5D5H/graph.json","events_json":"https://pith.science/api/pith-number/SAH3XESM4CCNNG6KAWXAXF5D5H/events.json","paper":"https://pith.science/paper/SAH3XESM"},"agent_actions":{"view_html":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H","download_json":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H.json","view_paper":"https://pith.science/paper/SAH3XESM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2602.09297&json=true","fetch_graph":"https://pith.science/api/pith-number/SAH3XESM4CCNNG6KAWXAXF5D5H/graph.json","fetch_events":"https://pith.science/api/pith-number/SAH3XESM4CCNNG6KAWXAXF5D5H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H/action/storage_attestation","attest_author":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H/action/author_attestation","sign_citation":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H/action/citation_signature","submit_replication":"https://pith.science/pith/SAH3XESM4CCNNG6KAWXAXF5D5H/action/replication_record"}},"created_at":"2026-05-20T00:00:34.094291+00:00","updated_at":"2026-05-20T00:00:34.094291+00:00"}