{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:VD7LAQHKYQ2TX4VSAX5H23WKXK","short_pith_number":"pith:VD7LAQHK","schema_version":"1.0","canonical_sha256":"a8feb040eac4353bf2b205fa7d6ecaba8d322792a3301a0ab998ac82d44c0fb0","source":{"kind":"arxiv","id":"2106.02154","version":2},"attestation_state":"computed","paper":{"title":"Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CV","cs.LG"],"primary_cat":"stat.ML","authors_text":"Ali Ghodsi, Benyamin Ghojogh, Fakhri Karray, Mark Crowley","submitted_at":"2021-06-03T22:10:40Z","abstract_excerpt":"This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation of Laplacian. Then, we cover the cuts of graph and spectral clustering which applies clustering in a subspace of data. Different optimization variants of Laplacian eigenmap and its out-of-sample extension are explained. Thereafter, we introduce the locality preserving projection and its kernel variant as linear special cases of Laplacian eigenmap. Versions of"},"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":"2106.02154","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2021-06-03T22:10:40Z","cross_cats_sorted":["cs.CV","cs.LG"],"title_canon_sha256":"8383c8638399cf61f945e063a227c81b728637cefef144b346addcdb30b50809","abstract_canon_sha256":"81ce796c74611254fdd15388d7c1f4ffb5242d99eb5d3b9e090be0bab55b4e4b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:46:31.730401Z","signature_b64":"lPunjGJqtLZ0FG4/BlTJxpSCrNuSkd4fYX6bEavlbR1NmDKQ1XOKIZLP2NeZxO6JsbT1B5+iuKgbgBMZqfJ9AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8feb040eac4353bf2b205fa7d6ecaba8d322792a3301a0ab998ac82d44c0fb0","last_reissued_at":"2026-07-05T04:46:31.729916Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:46:31.729916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CV","cs.LG"],"primary_cat":"stat.ML","authors_text":"Ali Ghodsi, Benyamin Ghojogh, Fakhri Karray, Mark Crowley","submitted_at":"2021-06-03T22:10:40Z","abstract_excerpt":"This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation of Laplacian. Then, we cover the cuts of graph and spectral clustering which applies clustering in a subspace of data. Different optimization variants of Laplacian eigenmap and its out-of-sample extension are explained. Thereafter, we introduce the locality preserving projection and its kernel variant as linear special cases of Laplacian eigenmap. Versions of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.02154","kind":"arxiv","version":2},"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/2106.02154/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":"2106.02154","created_at":"2026-07-05T04:46:31.729972+00:00"},{"alias_kind":"arxiv_version","alias_value":"2106.02154v2","created_at":"2026-07-05T04:46:31.729972+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.02154","created_at":"2026-07-05T04:46:31.729972+00:00"},{"alias_kind":"pith_short_12","alias_value":"VD7LAQHKYQ2T","created_at":"2026-07-05T04:46:31.729972+00:00"},{"alias_kind":"pith_short_16","alias_value":"VD7LAQHKYQ2TX4VS","created_at":"2026-07-05T04:46:31.729972+00:00"},{"alias_kind":"pith_short_8","alias_value":"VD7LAQHK","created_at":"2026-07-05T04:46:31.729972+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.27917","citing_title":"Graph Dimensionality Reduction for Contextual Bandits: Structure-Specific Regret Bounds under Approximate Smoothness and Noisy Eigenspaces","ref_index":15,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK","json":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK.json","graph_json":"https://pith.science/api/pith-number/VD7LAQHKYQ2TX4VSAX5H23WKXK/graph.json","events_json":"https://pith.science/api/pith-number/VD7LAQHKYQ2TX4VSAX5H23WKXK/events.json","paper":"https://pith.science/paper/VD7LAQHK"},"agent_actions":{"view_html":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK","download_json":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK.json","view_paper":"https://pith.science/paper/VD7LAQHK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2106.02154&json=true","fetch_graph":"https://pith.science/api/pith-number/VD7LAQHKYQ2TX4VSAX5H23WKXK/graph.json","fetch_events":"https://pith.science/api/pith-number/VD7LAQHKYQ2TX4VSAX5H23WKXK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK/action/storage_attestation","attest_author":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK/action/author_attestation","sign_citation":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK/action/citation_signature","submit_replication":"https://pith.science/pith/VD7LAQHKYQ2TX4VSAX5H23WKXK/action/replication_record"}},"created_at":"2026-07-05T04:46:31.729972+00:00","updated_at":"2026-07-05T04:46:31.729972+00:00"}