{"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"}