{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:F5FUJCWZCCPGCQTJWB4WEUOVDK","short_pith_number":"pith:F5FUJCWZ","schema_version":"1.0","canonical_sha256":"2f4b448ad9109e614269b0796251d51a88dc8d238e789b300413def0c28c82c8","source":{"kind":"arxiv","id":"2605.19458","version":1},"attestation_state":"computed","paper":{"title":"Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Guido Montufar, Tom Jacobs","submitted_at":"2026-05-19T07:10:18Z","abstract_excerpt":"We study the max-margin solutions reached by mirror flow in deep neural networks with homogeneous activation functions. Extending classical results on gradient flow, we derive a novel balance equation for mirror flow from convex duality, enabling a characterization of the horizon function governing the induced margin. We further establish max-margin characterizations together with convergence rates and norm growth estimates. Finally, we support our theory through experiments on synthetic datasets and standard vision tasks. Concretely, we show that: (1) distinct non-homogeneous mirror maps can "},"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":false},"canonical_record":{"source":{"id":"2605.19458","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-19T07:10:18Z","cross_cats_sorted":[],"title_canon_sha256":"28225b9ea58d212551b5ea95d20fc9100af9320a4da9753635556de20e7433c0","abstract_canon_sha256":"729219aae3f1361c58377fc15b9c9118380de812471fc9d52ad24247801a039c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:05:46.859368Z","signature_b64":"TBEDwGxjY2gTyL4cwKIgkYGHrs0XGebv3iXyK6QeYq8PCYQ7ErhXXoeQb5xskEE/QlwyQOglw1ttfY1smVHkAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f4b448ad9109e614269b0796251d51a88dc8d238e789b300413def0c28c82c8","last_reissued_at":"2026-05-20T01:05:46.858554Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:05:46.858554Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Guido Montufar, Tom Jacobs","submitted_at":"2026-05-19T07:10:18Z","abstract_excerpt":"We study the max-margin solutions reached by mirror flow in deep neural networks with homogeneous activation functions. Extending classical results on gradient flow, we derive a novel balance equation for mirror flow from convex duality, enabling a characterization of the horizon function governing the induced margin. We further establish max-margin characterizations together with convergence rates and norm growth estimates. Finally, we support our theory through experiments on synthetic datasets and standard vision tasks. Concretely, we show that: (1) distinct non-homogeneous mirror maps can "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.19458","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/2605.19458/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.19458","created_at":"2026-05-20T01:05:46.858687+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.19458v1","created_at":"2026-05-20T01:05:46.858687+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.19458","created_at":"2026-05-20T01:05:46.858687+00:00"},{"alias_kind":"pith_short_12","alias_value":"F5FUJCWZCCPG","created_at":"2026-05-20T01:05:46.858687+00:00"},{"alias_kind":"pith_short_16","alias_value":"F5FUJCWZCCPGCQTJ","created_at":"2026-05-20T01:05:46.858687+00:00"},{"alias_kind":"pith_short_8","alias_value":"F5FUJCWZ","created_at":"2026-05-20T01:05:46.858687+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK","json":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK.json","graph_json":"https://pith.science/api/pith-number/F5FUJCWZCCPGCQTJWB4WEUOVDK/graph.json","events_json":"https://pith.science/api/pith-number/F5FUJCWZCCPGCQTJWB4WEUOVDK/events.json","paper":"https://pith.science/paper/F5FUJCWZ"},"agent_actions":{"view_html":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK","download_json":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK.json","view_paper":"https://pith.science/paper/F5FUJCWZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.19458&json=true","fetch_graph":"https://pith.science/api/pith-number/F5FUJCWZCCPGCQTJWB4WEUOVDK/graph.json","fetch_events":"https://pith.science/api/pith-number/F5FUJCWZCCPGCQTJWB4WEUOVDK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK/action/storage_attestation","attest_author":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK/action/author_attestation","sign_citation":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK/action/citation_signature","submit_replication":"https://pith.science/pith/F5FUJCWZCCPGCQTJWB4WEUOVDK/action/replication_record"}},"created_at":"2026-05-20T01:05:46.858687+00:00","updated_at":"2026-05-20T01:05:46.858687+00:00"}