isotropic concept
| Definition | Of a covariance, having the same variance in every direction. Chapter 0. Also anisotropic. |
|---|---|
| Example | Covariance diag(1, 1) is isotropic, and a reader at any angle sees variance 1. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id isotropic, 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/Isotropy.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
none
Conditions
- An isotropic covariance has the same variance in every direction, so every unit reader reads it the same and no flip is possible. An anisotropic one has different variances in different directions, and two unit readers can read it differently.
- The flip needs anisotropy. Two codes with the variances exchanged are read the same by every reader exactly when the two variances agree, which is the boundary the coupling null names from the other side.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.3 and chapter 3 section 3.2 of Data Mining as Observation, with the isotropic-Gaussian control of chapter 10.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Isotropy.lean,
theorems isotropic_reads_same,
isotropic_no_flip, anisotropic_readers_differ,
flip_iff_anisotropic, 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, 2, 3, 4, 7, 9, 10, 11, 12.
Related
the flip; coupling null; covariance matrix; alignment.
See also
Book equations stated beside the entry’s terms, not defining it: 3.1, 4.6.
Ledger rows that cite the entry’s records without naming it: GO-2 (neg. half: not reconstruction).
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.