{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:CUI6XZWVNZMTL2WJERJYQTPBAD","short_pith_number":"pith:CUI6XZWV","canonical_record":{"source":{"id":"1206.2061","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.NA","submitted_at":"2012-06-10T22:13:45Z","cross_cats_sorted":["cs.CV"],"title_canon_sha256":"266a82ef6810c08d4e2106e542b6fafce923f904d426136e1e9a55bfb92a0615","abstract_canon_sha256":"7f59bd7ed8aa713781badcd10acda88bfbf387bff91bb32813e4397d55d20ace"},"schema_version":"1.0"},"canonical_sha256":"1511ebe6d56e5935eac92453884de100d9eb221216dd3de61bfbc7465dd07175","source":{"kind":"arxiv","id":"1206.2061","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.2061","created_at":"2026-05-18T03:53:53Z"},{"alias_kind":"arxiv_version","alias_value":"1206.2061v1","created_at":"2026-05-18T03:53:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.2061","created_at":"2026-05-18T03:53:53Z"},{"alias_kind":"pith_short_12","alias_value":"CUI6XZWVNZMT","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CUI6XZWVNZMTL2WJ","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CUI6XZWV","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:CUI6XZWVNZMTL2WJERJYQTPBAD","target":"record","payload":{"canonical_record":{"source":{"id":"1206.2061","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.NA","submitted_at":"2012-06-10T22:13:45Z","cross_cats_sorted":["cs.CV"],"title_canon_sha256":"266a82ef6810c08d4e2106e542b6fafce923f904d426136e1e9a55bfb92a0615","abstract_canon_sha256":"7f59bd7ed8aa713781badcd10acda88bfbf387bff91bb32813e4397d55d20ace"},"schema_version":"1.0"},"canonical_sha256":"1511ebe6d56e5935eac92453884de100d9eb221216dd3de61bfbc7465dd07175","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:53.550943Z","signature_b64":"SNpaPBJ3l4xvIFHCDy91YKCMhDgBhilazylNpqCFlej80VZflgAbzEqixJdTUyu549Dbe4+DfqWe6rw79Yk8Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1511ebe6d56e5935eac92453884de100d9eb221216dd3de61bfbc7465dd07175","last_reissued_at":"2026-05-18T03:53:53.550491Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:53.550491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.2061","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":"2026-05-18T03:53:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9e4+CEllr0WJR3d3LoF1FM/BhhBA4zlkgOZT89cKLAdWpdYtWD4NMAx+Zp/XLSt2XXJZ7jD+eSGE7xckMaTaAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:06:09.199043Z"},"content_sha256":"21ab2b382d4071c83a7f79f2c4e93b8c817963d7777bc391292cb04cc631b6f7","schema_version":"1.0","event_id":"sha256:21ab2b382d4071c83a7f79f2c4e93b8c817963d7777bc391292cb04cc631b6f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:CUI6XZWVNZMTL2WJERJYQTPBAD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Comments on \"On Approximating Euclidean Metrics by Weighted t-Cost Distances in Arbitrary Dimension\"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CV"],"primary_cat":"cs.NA","authors_text":"Fatih Celiker, Hassan A. Kingravi, M. Emre Celebi","submitted_at":"2012-06-10T22:13:45Z","abstract_excerpt":"Mukherjee (Pattern Recognition Letters, vol. 32, pp. 824-831, 2011) recently introduced a class of distance functions called weighted t-cost distances that generalize m-neighbor, octagonal, and t-cost distances. He proved that weighted t-cost distances form a family of metrics and derived an approximation for the Euclidean norm in $\\mathbb{Z}^n$. In this note we compare this approximation to two previously proposed Euclidean norm approximations and demonstrate that the empirical average errors given by Mukherjee are significantly optimistic in $\\mathbb{R}^n$. We also propose a simple normaliza"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2061","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":""},"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":"2026-05-18T03:53:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m3NK4ydgI5r8NATbuvQmA6+ciZPFKbPFYJowc+PE+X7HnFbL8ZL7Qd5kqsvc/3avOvm84J8tYbnGOu2K2nhjAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:06:09.199396Z"},"content_sha256":"d457f972b38d09ddc374e1ec9dfa6b7f6fd1211443cde51a55ee7c11b1ebfca3","schema_version":"1.0","event_id":"sha256:d457f972b38d09ddc374e1ec9dfa6b7f6fd1211443cde51a55ee7c11b1ebfca3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CUI6XZWVNZMTL2WJERJYQTPBAD/bundle.json","state_url":"https://pith.science/pith/CUI6XZWVNZMTL2WJERJYQTPBAD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CUI6XZWVNZMTL2WJERJYQTPBAD/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":"2026-06-03T07:06:09Z","links":{"resolver":"https://pith.science/pith/CUI6XZWVNZMTL2WJERJYQTPBAD","bundle":"https://pith.science/pith/CUI6XZWVNZMTL2WJERJYQTPBAD/bundle.json","state":"https://pith.science/pith/CUI6XZWVNZMTL2WJERJYQTPBAD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CUI6XZWVNZMTL2WJERJYQTPBAD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CUI6XZWVNZMTL2WJERJYQTPBAD","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":"7f59bd7ed8aa713781badcd10acda88bfbf387bff91bb32813e4397d55d20ace","cross_cats_sorted":["cs.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.NA","submitted_at":"2012-06-10T22:13:45Z","title_canon_sha256":"266a82ef6810c08d4e2106e542b6fafce923f904d426136e1e9a55bfb92a0615"},"schema_version":"1.0","source":{"id":"1206.2061","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.2061","created_at":"2026-05-18T03:53:53Z"},{"alias_kind":"arxiv_version","alias_value":"1206.2061v1","created_at":"2026-05-18T03:53:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.2061","created_at":"2026-05-18T03:53:53Z"},{"alias_kind":"pith_short_12","alias_value":"CUI6XZWVNZMT","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CUI6XZWVNZMTL2WJ","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CUI6XZWV","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:d457f972b38d09ddc374e1ec9dfa6b7f6fd1211443cde51a55ee7c11b1ebfca3","target":"graph","created_at":"2026-05-18T03:53:53Z","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"},"paper":{"abstract_excerpt":"Mukherjee (Pattern Recognition Letters, vol. 32, pp. 824-831, 2011) recently introduced a class of distance functions called weighted t-cost distances that generalize m-neighbor, octagonal, and t-cost distances. He proved that weighted t-cost distances form a family of metrics and derived an approximation for the Euclidean norm in $\\mathbb{Z}^n$. In this note we compare this approximation to two previously proposed Euclidean norm approximations and demonstrate that the empirical average errors given by Mukherjee are significantly optimistic in $\\mathbb{R}^n$. We also propose a simple normaliza","authors_text":"Fatih Celiker, Hassan A. Kingravi, M. Emre Celebi","cross_cats":["cs.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.NA","submitted_at":"2012-06-10T22:13:45Z","title":"Comments on \"On Approximating Euclidean Metrics by Weighted t-Cost Distances in Arbitrary Dimension\""},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2061","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":"sha256:21ab2b382d4071c83a7f79f2c4e93b8c817963d7777bc391292cb04cc631b6f7","target":"record","created_at":"2026-05-18T03:53:53Z","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":"7f59bd7ed8aa713781badcd10acda88bfbf387bff91bb32813e4397d55d20ace","cross_cats_sorted":["cs.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.NA","submitted_at":"2012-06-10T22:13:45Z","title_canon_sha256":"266a82ef6810c08d4e2106e542b6fafce923f904d426136e1e9a55bfb92a0615"},"schema_version":"1.0","source":{"id":"1206.2061","kind":"arxiv","version":1}},"canonical_sha256":"1511ebe6d56e5935eac92453884de100d9eb221216dd3de61bfbc7465dd07175","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1511ebe6d56e5935eac92453884de100d9eb221216dd3de61bfbc7465dd07175","first_computed_at":"2026-05-18T03:53:53.550491Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:53:53.550491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SNpaPBJ3l4xvIFHCDy91YKCMhDgBhilazylNpqCFlej80VZflgAbzEqixJdTUyu549Dbe4+DfqWe6rw79Yk8Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:53:53.550943Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.2061","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21ab2b382d4071c83a7f79f2c4e93b8c817963d7777bc391292cb04cc631b6f7","sha256:d457f972b38d09ddc374e1ec9dfa6b7f6fd1211443cde51a55ee7c11b1ebfca3"],"state_sha256":"b99de2af783b44501815037435adcef4ad4ce932641421c8cea739f8eaafc828"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gddm8386fyurGRgWQGH3Q7ve9Kn6wTBkzqRdi2ATSG1hU0LKaynfTIOXKx+gvyev57TlxJpiRVvGjUYhvuYVAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T07:06:09.201256Z","bundle_sha256":"88fb9498e91d48bb3967f24606479ae09cca2202a36490fb3845f40b224eb6d1"}}