{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:RP7FL2TLY472L7MWP2XN4FFF3A","short_pith_number":"pith:RP7FL2TL","canonical_record":{"source":{"id":"0909.5000","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2009-09-28T04:25:03Z","cross_cats_sorted":["cs.NA","cs.NE","math.NA"],"title_canon_sha256":"5b87db7b9f2c2b9f39a61d5a4553e13d9a890d6da9a4f0443e0c5bc3934ce611","abstract_canon_sha256":"a51b280d2ced4d98b8f9f3bb6b0bae3645d7a66300a40bf8b278865ed21e5f1d"},"schema_version":"1.0"},"canonical_sha256":"8bfe55ea6bc73fa5fd967eaede14a5d810c025d2808d279eddceb95c85672c4a","source":{"kind":"arxiv","id":"0909.5000","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.5000","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"arxiv_version","alias_value":"0909.5000v1","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.5000","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"pith_short_12","alias_value":"RP7FL2TLY472","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"pith_short_16","alias_value":"RP7FL2TLY472L7MW","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"pith_short_8","alias_value":"RP7FL2TL","created_at":"2026-06-03T23:06:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:RP7FL2TLY472L7MWP2XN4FFF3A","target":"record","payload":{"canonical_record":{"source":{"id":"0909.5000","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2009-09-28T04:25:03Z","cross_cats_sorted":["cs.NA","cs.NE","math.NA"],"title_canon_sha256":"5b87db7b9f2c2b9f39a61d5a4553e13d9a890d6da9a4f0443e0c5bc3934ce611","abstract_canon_sha256":"a51b280d2ced4d98b8f9f3bb6b0bae3645d7a66300a40bf8b278865ed21e5f1d"},"schema_version":"1.0"},"canonical_sha256":"8bfe55ea6bc73fa5fd967eaede14a5d810c025d2808d279eddceb95c85672c4a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T23:06:29.215785Z","signature_b64":"DqRd/6/mljawfoYUjffflZDz9g98jroBJ9sKluCPp4OZJwTY0sC3T7QUhOlzVchAMue07pdUi6WmXqVc3dz9Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8bfe55ea6bc73fa5fd967eaede14a5d810c025d2808d279eddceb95c85672c4a","last_reissued_at":"2026-06-03T23:06:29.215263Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T23:06:29.215263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0909.5000","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":"2026-06-03T23:06:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tSCpzb0cUq/R6Pt8wBGSBL1yrC4gVhAAfRU+TELU6QN6StPlGDKz1/FYebR36uCLPeOSwJv17fIVX1glfsQxDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T10:00:12.851206Z"},"content_sha256":"c1e8fb63c481d8e934724d85fdc804d3421ffdc043624d626785fa295920fea1","schema_version":"1.0","event_id":"sha256:c1e8fb63c481d8e934724d85fdc804d3421ffdc043624d626785fa295920fea1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:RP7FL2TLY472L7MWP2XN4FFF3A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Eignets for function approximation on manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","cs.NE","math.NA"],"primary_cat":"cs.LG","authors_text":"H. N. Mhaskar","submitted_at":"2009-09-28T04:25:03Z","abstract_excerpt":"Let $\\XX$ be a compact, smooth, connected, Riemannian manifold without boundary, $G:\\XX\\times\\XX\\to \\RR$ be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form $\\sum_{j=1}^M a_jG(\\circ,y_j)$, where $a_j\\in\\RR$, $y_j\\in\\XX$, $1\\le j\\le M$. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in $L^p(\\mu;\\XX)$ for a general class of measures $\\mu$ and kernels $G$. Our algorithm yields linear operators. Using the minimal separation amongst the centers $y_j$ as the cost of approximation, we give modulus o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.5000","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/0909.5000/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":"2026-06-03T23:06:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cQH+CHEQwn7On1D/HD7JwvYSc3Cy5biDI+6eUPKHqIvY/lzdsV4eUVP816v8b1AVg+rH/meSLyQ7gONE5jmLCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T10:00:12.851608Z"},"content_sha256":"2de8cb5ee04c758507d2cd56785e0ee642da7f0ba9f3ea45abbf15e62533905d","schema_version":"1.0","event_id":"sha256:2de8cb5ee04c758507d2cd56785e0ee642da7f0ba9f3ea45abbf15e62533905d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RP7FL2TLY472L7MWP2XN4FFF3A/bundle.json","state_url":"https://pith.science/pith/RP7FL2TLY472L7MWP2XN4FFF3A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RP7FL2TLY472L7MWP2XN4FFF3A/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":"2026-06-22T10:00:12Z","links":{"resolver":"https://pith.science/pith/RP7FL2TLY472L7MWP2XN4FFF3A","bundle":"https://pith.science/pith/RP7FL2TLY472L7MWP2XN4FFF3A/bundle.json","state":"https://pith.science/pith/RP7FL2TLY472L7MWP2XN4FFF3A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RP7FL2TLY472L7MWP2XN4FFF3A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:RP7FL2TLY472L7MWP2XN4FFF3A","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":"a51b280d2ced4d98b8f9f3bb6b0bae3645d7a66300a40bf8b278865ed21e5f1d","cross_cats_sorted":["cs.NA","cs.NE","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2009-09-28T04:25:03Z","title_canon_sha256":"5b87db7b9f2c2b9f39a61d5a4553e13d9a890d6da9a4f0443e0c5bc3934ce611"},"schema_version":"1.0","source":{"id":"0909.5000","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.5000","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"arxiv_version","alias_value":"0909.5000v1","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.5000","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"pith_short_12","alias_value":"RP7FL2TLY472","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"pith_short_16","alias_value":"RP7FL2TLY472L7MW","created_at":"2026-06-03T23:06:29Z"},{"alias_kind":"pith_short_8","alias_value":"RP7FL2TL","created_at":"2026-06-03T23:06:29Z"}],"graph_snapshots":[{"event_id":"sha256:2de8cb5ee04c758507d2cd56785e0ee642da7f0ba9f3ea45abbf15e62533905d","target":"graph","created_at":"2026-06-03T23:06:29Z","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/0909.5000/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\XX$ be a compact, smooth, connected, Riemannian manifold without boundary, $G:\\XX\\times\\XX\\to \\RR$ be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form $\\sum_{j=1}^M a_jG(\\circ,y_j)$, where $a_j\\in\\RR$, $y_j\\in\\XX$, $1\\le j\\le M$. We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in $L^p(\\mu;\\XX)$ for a general class of measures $\\mu$ and kernels $G$. Our algorithm yields linear operators. Using the minimal separation amongst the centers $y_j$ as the cost of approximation, we give modulus o","authors_text":"H. N. Mhaskar","cross_cats":["cs.NA","cs.NE","math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2009-09-28T04:25:03Z","title":"Eignets for function approximation on manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.5000","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:c1e8fb63c481d8e934724d85fdc804d3421ffdc043624d626785fa295920fea1","target":"record","created_at":"2026-06-03T23:06:29Z","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":"a51b280d2ced4d98b8f9f3bb6b0bae3645d7a66300a40bf8b278865ed21e5f1d","cross_cats_sorted":["cs.NA","cs.NE","math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2009-09-28T04:25:03Z","title_canon_sha256":"5b87db7b9f2c2b9f39a61d5a4553e13d9a890d6da9a4f0443e0c5bc3934ce611"},"schema_version":"1.0","source":{"id":"0909.5000","kind":"arxiv","version":1}},"canonical_sha256":"8bfe55ea6bc73fa5fd967eaede14a5d810c025d2808d279eddceb95c85672c4a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8bfe55ea6bc73fa5fd967eaede14a5d810c025d2808d279eddceb95c85672c4a","first_computed_at":"2026-06-03T23:06:29.215263Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T23:06:29.215263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DqRd/6/mljawfoYUjffflZDz9g98jroBJ9sKluCPp4OZJwTY0sC3T7QUhOlzVchAMue07pdUi6WmXqVc3dz9Bw==","signature_status":"signed_v1","signed_at":"2026-06-03T23:06:29.215785Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.5000","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c1e8fb63c481d8e934724d85fdc804d3421ffdc043624d626785fa295920fea1","sha256:2de8cb5ee04c758507d2cd56785e0ee642da7f0ba9f3ea45abbf15e62533905d"],"state_sha256":"d99af5ea5b082f3a2eb5e166caba0c01eb9867cec70be22b183d4f8f786b5656"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3yAaus3r5bJlJM2af6dY+XIvVeXE4ia8khbi2SzHzVTwmGEclVdJtdwqDqZGkt88zdP5gTMyN4Bfkcyc2WpaDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T10:00:12.853624Z","bundle_sha256":"da7fb00d4a5da349ef68345f000e9f2ed2d450313122a36292ea8f2edd39e144"}}