Euclidean distance concept
| Definition | The straight-line distance between two vectors, which reads every coordinate at the scale it arrives in. Chapter 3. Also Euclidean. |
|---|---|
| Example | The distance between (0, 0) and (3, 4) is 5. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id euclidean-distance, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | none |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/EuclideanDistance.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
none
Conditions
- The straight-line distance between two vectors, whose square is the identity reader’s quadratic form on the difference, so it reads every coordinate at the scale it arrives in. It is symmetric and zero exactly between a row and itself.
- Rescaling one coordinate changes which of two rows is nearer. A choice of scale is a choice of reader, and reconstruction error, the Euclidean distance between a row and its code, is the identity reader’s number and not the consumer’s.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 3 section 3.2 of Data Mining as Observation, where a distance is a choice of what to ignore.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/EuclideanDistance.lean,
theorems distSq_eq_quad_one, distSq_comm,
distSq_eq_zero_iff, ranking_flips, at
observation-data-mining f3914f0; what the check covers is stated in the
book’s appendix
C.
Used in
Data Mining as Observation primer L, chapters 0, 1, 2, 3, 6, 10, 11, 12.
Related
identity reader; dot product; Mahalanobis distance; distance concentration.
See also
Book equations stated beside the entry’s terms, not defining it: 0.1, 4.1.
Ledger rows that cite the entry’s records without naming it: NEG-2.
Sources-table rows that share a record with the entry without naming it: chapter 1 section 1.4, chapter 2 section 2.5, chapter 3 section 3.2, chapter 8 section 8.2, chapter 11 section 11.1, chapter 11 section 11.2.
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.