{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:FBXUXUXNNL4NUSJBM2XPASTTOQ","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":"1a7ef1d4d28a387e813d4cd121f5aeaa907a2a1e3327f608cf7b24fbf321cd44","cross_cats_sorted":["eess.IV"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2020-11-15T22:44:25Z","title_canon_sha256":"f349f991124de629086d7ecac6d338acb7e92900135a97606d2582741cb4f6fc"},"schema_version":"1.0","source":{"id":"2011.07643","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.07643","created_at":"2026-07-05T01:51:51Z"},{"alias_kind":"arxiv_version","alias_value":"2011.07643v1","created_at":"2026-07-05T01:51:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.07643","created_at":"2026-07-05T01:51:51Z"},{"alias_kind":"pith_short_12","alias_value":"FBXUXUXNNL4N","created_at":"2026-07-05T01:51:51Z"},{"alias_kind":"pith_short_16","alias_value":"FBXUXUXNNL4NUSJB","created_at":"2026-07-05T01:51:51Z"},{"alias_kind":"pith_short_8","alias_value":"FBXUXUXN","created_at":"2026-07-05T01:51:51Z"}],"graph_snapshots":[{"event_id":"sha256:080dd9c95ef68b5fb832d5f3c2acfb7b213178d88ea3178d5fbd13423bab5f4f","target":"graph","created_at":"2026-07-05T01:51:51Z","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/2011.07643/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we study an emerging class of neural networks based on the morphological operators of dilation and erosion. We explore these networks mathematically from a tropical geometry perspective as well as mathematical morphology. Our contributions are threefold. First, we examine the training of morphological networks via Difference-of-Convex programming methods and extend a binary morphological classifier to multiclass tasks. Second, we focus on the sparsity of dense morphological networks trained via gradient descent algorithms and compare their performance to their linear counterparts","authors_text":"Nikolaos Dimitriadis, Petros Maragos","cross_cats":["eess.IV"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2020-11-15T22:44:25Z","title":"Advances in the training, pruning and enforcement of shape constraints of Morphological Neural Networks using Tropical Algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.07643","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":"sha256:87395b95a70b99e7ed416ac2982a57c3712568530cb6a9a2c49e449e3c1d5736","target":"record","created_at":"2026-07-05T01:51:51Z","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":"1a7ef1d4d28a387e813d4cd121f5aeaa907a2a1e3327f608cf7b24fbf321cd44","cross_cats_sorted":["eess.IV"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2020-11-15T22:44:25Z","title_canon_sha256":"f349f991124de629086d7ecac6d338acb7e92900135a97606d2582741cb4f6fc"},"schema_version":"1.0","source":{"id":"2011.07643","kind":"arxiv","version":1}},"canonical_sha256":"286f4bd2ed6af8da492166aef04a737415c1672c6f1e04c61a321f817cae1cf1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"286f4bd2ed6af8da492166aef04a737415c1672c6f1e04c61a321f817cae1cf1","first_computed_at":"2026-07-05T01:51:51.934698Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:51:51.934698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gkCAVdwqRRcTfA2Rre2sOQUB8FCF6QOUCyjA/jJM6qverFxWvLmyJ79cQpqpTDmbgNgHq3BQOiQbEw62sI3fDA==","signature_status":"signed_v1","signed_at":"2026-07-05T01:51:51.935150Z","signed_message":"canonical_sha256_bytes"},"source_id":"2011.07643","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87395b95a70b99e7ed416ac2982a57c3712568530cb6a9a2c49e449e3c1d5736","sha256:080dd9c95ef68b5fb832d5f3c2acfb7b213178d88ea3178d5fbd13423bab5f4f"],"state_sha256":"4d47e891e56404c0a1e1c68d9ba4593253fea1bab6f38042984e9269d61babf1"}