{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:53UDTA6NHFGGBUS3EYHDQDCXIM","short_pith_number":"pith:53UDTA6N","canonical_record":{"source":{"id":"2310.04064","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.DS","submitted_at":"2023-10-06T07:42:39Z","cross_cats_sorted":["cs.CC","cs.CL","cs.LG","stat.ML"],"title_canon_sha256":"3ed25a0e222618c79a70f82e75a81b0921b8ed889f1c1acb1c4fe9552b6e8c93","abstract_canon_sha256":"5ce5e7a3305b1fbbf1bff7bdc5c006779f145fd9d36815b0ca642f6a634d17db"},"schema_version":"1.0"},"canonical_sha256":"eee83983cd394c60d25b260e380c57431331db571164e030d07d2ca6da026c2a","source":{"kind":"arxiv","id":"2310.04064","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2310.04064","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"arxiv_version","alias_value":"2310.04064v1","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2310.04064","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"pith_short_12","alias_value":"53UDTA6NHFGG","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"pith_short_16","alias_value":"53UDTA6NHFGGBUS3","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"pith_short_8","alias_value":"53UDTA6N","created_at":"2026-07-05T06:57:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:53UDTA6NHFGGBUS3EYHDQDCXIM","target":"record","payload":{"canonical_record":{"source":{"id":"2310.04064","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.DS","submitted_at":"2023-10-06T07:42:39Z","cross_cats_sorted":["cs.CC","cs.CL","cs.LG","stat.ML"],"title_canon_sha256":"3ed25a0e222618c79a70f82e75a81b0921b8ed889f1c1acb1c4fe9552b6e8c93","abstract_canon_sha256":"5ce5e7a3305b1fbbf1bff7bdc5c006779f145fd9d36815b0ca642f6a634d17db"},"schema_version":"1.0"},"canonical_sha256":"eee83983cd394c60d25b260e380c57431331db571164e030d07d2ca6da026c2a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:57:57.827244Z","signature_b64":"kepdVP3ZFfEF359bmJ2EKTpHM7QwWJuf5I7J1ZjSc0TBCpsIIfSrWmBoAbKv5R2exaSdmfvi9b+HBFfLdHuZBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eee83983cd394c60d25b260e380c57431331db571164e030d07d2ca6da026c2a","last_reissued_at":"2026-07-05T06:57:57.826752Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:57:57.826752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2310.04064","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:57:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7DWhe8sFTAhyqKpCATIFVSo/wWjDAxQzFO6LsU6ZqLdVtIiEEc/qg3YyzafvnbMDcTgp4DaxqwRxQBDC2ocJCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T16:22:11.045420Z"},"content_sha256":"4577cba89b8366017bda7a94887f551fb56938ad04b97383573451c72e8eaf79","schema_version":"1.0","event_id":"sha256:4577cba89b8366017bda7a94887f551fb56938ad04b97383573451c72e8eaf79"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:53UDTA6NHFGGBUS3EYHDQDCXIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"How to Capture Higher-order Correlations? Generalizing Matrix Softmax Attention to Kronecker Computation","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.CC","cs.CL","cs.LG","stat.ML"],"primary_cat":"cs.DS","authors_text":"Josh Alman, Zhao Song","submitted_at":"2023-10-06T07:42:39Z","abstract_excerpt":"In the classical transformer attention scheme, we are given three $n \\times d$ size matrices $Q, K, V$ (the query, key, and value tokens), and the goal is to compute a new $n \\times d$ size matrix $D^{-1} \\exp(QK^\\top) V$ where $D = \\mathrm{diag}( \\exp(QK^\\top) {\\bf 1}_n )$. In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.04064","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2310.04064/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":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T06:57:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ePvlmuYjXplhz0hIBQ3MBCza2xwm9DLktzdFVXl9cdWtyKhdlAojrbpAKOKv1TcI3OYWaEYvPn/Z58aB2TB4Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T16:22:11.046083Z"},"content_sha256":"9de17f17164f756a8f5c84626e7f4bd01f91a3bbf2a3f689efd8d192027c97f2","schema_version":"1.0","event_id":"sha256:9de17f17164f756a8f5c84626e7f4bd01f91a3bbf2a3f689efd8d192027c97f2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/53UDTA6NHFGGBUS3EYHDQDCXIM/bundle.json","state_url":"https://pith.science/pith/53UDTA6NHFGGBUS3EYHDQDCXIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/53UDTA6NHFGGBUS3EYHDQDCXIM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-08T16:22:11Z","links":{"resolver":"https://pith.science/pith/53UDTA6NHFGGBUS3EYHDQDCXIM","bundle":"https://pith.science/pith/53UDTA6NHFGGBUS3EYHDQDCXIM/bundle.json","state":"https://pith.science/pith/53UDTA6NHFGGBUS3EYHDQDCXIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/53UDTA6NHFGGBUS3EYHDQDCXIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:53UDTA6NHFGGBUS3EYHDQDCXIM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5ce5e7a3305b1fbbf1bff7bdc5c006779f145fd9d36815b0ca642f6a634d17db","cross_cats_sorted":["cs.CC","cs.CL","cs.LG","stat.ML"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.DS","submitted_at":"2023-10-06T07:42:39Z","title_canon_sha256":"3ed25a0e222618c79a70f82e75a81b0921b8ed889f1c1acb1c4fe9552b6e8c93"},"schema_version":"1.0","source":{"id":"2310.04064","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2310.04064","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"arxiv_version","alias_value":"2310.04064v1","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2310.04064","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"pith_short_12","alias_value":"53UDTA6NHFGG","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"pith_short_16","alias_value":"53UDTA6NHFGGBUS3","created_at":"2026-07-05T06:57:57Z"},{"alias_kind":"pith_short_8","alias_value":"53UDTA6N","created_at":"2026-07-05T06:57:57Z"}],"graph_snapshots":[{"event_id":"sha256:9de17f17164f756a8f5c84626e7f4bd01f91a3bbf2a3f689efd8d192027c97f2","target":"graph","created_at":"2026-07-05T06:57:57Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2310.04064/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In the classical transformer attention scheme, we are given three $n \\times d$ size matrices $Q, K, V$ (the query, key, and value tokens), and the goal is to compute a new $n \\times d$ size matrix $D^{-1} \\exp(QK^\\top) V$ where $D = \\mathrm{diag}( \\exp(QK^\\top) {\\bf 1}_n )$. In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are e","authors_text":"Josh Alman, Zhao Song","cross_cats":["cs.CC","cs.CL","cs.LG","stat.ML"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.DS","submitted_at":"2023-10-06T07:42:39Z","title":"How to Capture Higher-order Correlations? Generalizing Matrix Softmax Attention to Kronecker Computation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.04064","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4577cba89b8366017bda7a94887f551fb56938ad04b97383573451c72e8eaf79","target":"record","created_at":"2026-07-05T06:57:57Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5ce5e7a3305b1fbbf1bff7bdc5c006779f145fd9d36815b0ca642f6a634d17db","cross_cats_sorted":["cs.CC","cs.CL","cs.LG","stat.ML"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.DS","submitted_at":"2023-10-06T07:42:39Z","title_canon_sha256":"3ed25a0e222618c79a70f82e75a81b0921b8ed889f1c1acb1c4fe9552b6e8c93"},"schema_version":"1.0","source":{"id":"2310.04064","kind":"arxiv","version":1}},"canonical_sha256":"eee83983cd394c60d25b260e380c57431331db571164e030d07d2ca6da026c2a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eee83983cd394c60d25b260e380c57431331db571164e030d07d2ca6da026c2a","first_computed_at":"2026-07-05T06:57:57.826752Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:57:57.826752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kepdVP3ZFfEF359bmJ2EKTpHM7QwWJuf5I7J1ZjSc0TBCpsIIfSrWmBoAbKv5R2exaSdmfvi9b+HBFfLdHuZBA==","signature_status":"signed_v1","signed_at":"2026-07-05T06:57:57.827244Z","signed_message":"canonical_sha256_bytes"},"source_id":"2310.04064","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4577cba89b8366017bda7a94887f551fb56938ad04b97383573451c72e8eaf79","sha256:9de17f17164f756a8f5c84626e7f4bd01f91a3bbf2a3f689efd8d192027c97f2"],"state_sha256":"e672d733f81a1ba9cc45e09048e4f4242c9a7ea7a960e35bfcd4111817d710b3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9NrS9pxEkD4IjSuRJdylydYoPFwn+42NuKXpgkzIezo706Spjyuv5yOECe8UgLwmWPlNVNIzaMLf/c5zBNasCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T16:22:11.052230Z","bundle_sha256":"11b9a8326951a9b0e4172f50309856466de6faae890b25d28312a55a9b5f77a1"}}