rank-faithful concept

Definition	Of a map, preserving the ordering of distances without any bound on the stretch. Chapter 0 section 0.10. Also geodesic-rank, angle-only.
Example	A compression under which every one of the 19900 pair rankings is kept is rank-faithful.
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `rank-faithful`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 0.21.
Assumptions and scope	A map is rank-faithful when it preserves the ordering of distances with no bound on the stretch. Every strictly increasing transform of the distance is rank-faithful, so a reader that ranks, the nearest-neighbour reader among them, returns the same answer under it. Rank-faithful and bi-Lipschitz are separate words. The square preserves every ordering and stretches large distances beyond any fixed factor, and the substrate table measures rank agreement, not stretch.
Prior art	none recorded
Evidence	`geometric-observation/chapters/ch09_legibility.md:31-41`, `geometric-observation/chapters/ch03_historical_precursors.md:98-110`, `geometric-observation/chapters/ch09_legibility.md:10-41`, `lean/DataMiningAsObservation/RankFaithful.lean`
Reviewed	not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page.

Compressed distances that keep the exact ranking.

Equation

Book equation 0.21.

\[\frac1K\,d(x,y)\ \le\ d'\big(f(x),f(y)\big)\ \le\ K\,d(x,y)\qquad\text{for all }x,y.\]

Conditions

A map is rank-faithful when it preserves the ordering of distances with no bound on the stretch. Every strictly increasing transform of the distance is rank-faithful, so a reader that ranks, the nearest-neighbour reader among them, returns the same answer under it.
Rank-faithful and bi-Lipschitz are separate words. The square preserves every ordering and stretches large distances beyond any fixed factor, and the substrate table measures rank agreement, not stretch.

Conditions are curated in entries.toml rather than read from a record.

Ledger

none

First stated

Volume 14, chapter 9, geometric-observation/chapters/ch09_legibility.md:10-41, DOI 10.5281/zenodo.21776291, and chapter 0 section 0.10 and chapter 3 section 3.3 of Data Mining as Observation.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 3 section 3.3	row normalization is the projection onto the read subspace of the geodesic-rank consumer	`geometric-observation/chapters/ch09_legibility.md:31-41`; `geometric-observation/chapters/ch03_historical_precursors.md:98-110`
chapter 9 section 9.1	the geodesic-rank reader discards the radius, row normalization is the angular projection	`geometric-observation/chapters/ch09_legibility.md:10-41`; `the-angular-observer/theorem.md:110-146`

Failures and corrections

none

Invariance envelope

none declared

Machine checked

lean/DataMiningAsObservation/RankFaithful.lean, theorems isRankFaithful_of_strictMono, closer_set_eq, nearest_eq, sq_rankFaithful_on_nonneg, sq_not_bi_lipschitz, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation chapters 0, 3.

bi-Lipschitz; rank certificate; recognizer; quotient.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.

← rank certificateread direction →