intrinsic dimension concept
| Definition | The number of directions a dataset varies along locally, whatever the number of its coordinates and whatever the number of its covariance directions. Chapter 0 section 0.10. Also Weyl. |
|---|---|
| Example | A circle drawn in three coordinates has intrinsic dimension 1; the recognizer read 1.81, 2.80, and 1.74 on its three templates. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id intrinsic-dimension, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.31. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | geometric-observation/chapters/ch11_the_recognizer.md:1-95, lean/DataMiningAsObservation/IntrinsicDimension.lean |
| Reviewed | semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.31.
\[N(\lambda)=\#\{k:\lambda_k\le\lambda\}\ \sim\ C_d\,\lambda^{d/2}.\]
Book equation 9.2.
\[N(\lambda)=\#\{k:\lambda_k\le\lambda\}\ \sim\ C_d\,\lambda^{d/2}\qquad\Rightarrow\qquad d=2\,\frac{d\log N}{d\log\lambda}.\]
Conditions
- The number of directions a dataset varies along locally, whatever its number of coordinates. Read from the slope of the Laplacian eigenvalue count on log axes by Weyl’s law, it is an asymptotic estimate that depends on the graph’s construction and the range of eigenvalues fitted. It is not the number of covariance directions, since a circle has intrinsic dimension one and two covariance directions.
- The recognizer reads dimension before shape, and the battery’s dimensions of 1.81, 2.80, and 1.74 were read against sealed bars on templates built for the purpose.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Weyl’s law, 1911, as chapter 0 section 0.15 states it, applied in the
recognizer of Volume 14 chapter 11,
geometric-observation/chapters/ch11_the_recognizer.md:1-95,
and chapter 9 section 9.2 of Data Mining as Observation.
Measurements
| Where the book states it | Numbers, as the book’s sources table records them | Source |
|---|---|---|
| chapter 9 section 9.2 | the recognizer’s mechanism, low multiplets and angular distances, dimension before shape by Weyl’s law, refusal, the growth gotcha | geometric-observation/chapters/ch11_the_recognizer.md:1-95 |
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/IntrinsicDimension.lean,
theorems log_weyl, dimension_from_slope,
weyl_double, 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, 8, 9, 11.
Related
recognizer; effective rank; distance concentration; vacuity threshold.
See also
Book equations stated beside the entry’s terms, not defining it: 0.7.
Ledger rows that cite the entry’s records without naming it: GO-3.
Sources-table rows that share a record with the entry without naming it: chapter 3 section 3.3, chapter 9 section 9.1, chapter 9 section 9.2, chapter 9 section 9.4, chapter 11 section 11.1, chapter 14 section 14.3, chapter 14 section 14.5, chapter 14 section 14.7.
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.