manifold concept
| Definition | A curved surface of some dimension that looks flat when viewed closely. A dataset lies on one when it is locally low-dimensional, which is not the same as varying along fewer covariance directions, since a circle is one-dimensional and uses two coordinates. Chapter 0 section 0.10. |
|---|---|
| Example | A circle in the plane is a one-dimensional manifold that uses both coordinates. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id manifold, 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/IntrinsicDimension.lean |
| Reviewed | semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
none
Conditions
- A space that is locally like flat space of some dimension. A dataset lies on a manifold when it is locally low-dimensional, which does not mean it varies globally along fewer covariance directions. A circle in the plane is one-dimensional and uses both coordinates, so no linear projection recovers it.
- Convergence of the graph Laplacian to the manifold’s holds under stated conditions on sampling density, graph construction, kernel bandwidth, normalization, and scaling, after Belkin and Niyogi, and a finite graph spectrum names a shape only against a finite list of candidates, which is the recognizer’s limitation and applies here.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.10 and chapter 9 section 9.2 of Data Mining as Observation, with the recognizer of Volume 14 chapter 11 and its battery in the-angular-observer.
Measurements
| Where the book states it | Numbers, as the book’s sources table records them | Source |
|---|---|---|
| chapter 3 section 3.3 | hyperbolic rejected, curvature negative 0.98 to negative 0.14, trend 1.09 minus 0.157 d across 20 manifolds and 5 families | the-angular-observer\README.md:111-147,183-185 |
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, 9.
Related
intrinsic dimension; recognizer; graph; Laplacian; template match.
See also
Book equations stated beside the entry’s terms, not defining it: 0.31, 9.2, 0.32.
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 11 section 11.1.
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.