principal component analysis instrument
| Definition | Projection of the centred data onto the eigenvectors of its covariance with the largest eigenvalues, the linear orthogonal projection of rank k that minimizes reconstruction error for the identity reader. Chapter 4. Also principal component. |
|---|---|
| Example | With eigenvalues 4, 3, 2, and 1, keeping the top two components drops an error of 3. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id principal-component-analysis, kind instrument. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | none |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | turboquant-pro/CLAIMS.md:28-49, lean/DataMiningAsObservation/PCA.lean |
| Reviewed | semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
none
Conditions
- Projection of the centred data onto the eigenvectors of its covariance with the largest eigenvalues. Among linear orthogonal projections of rank k it minimizes the reconstruction error. In the eigenbasis the variance along a unit direction lies between the smallest and the largest eigenvalue, a basis direction attains its own eigenvalue, and the error of keeping k components is the sum of the dropped eigenvalues.
- It is the reduction that minimizes reconstruction error for the identity reader, which weighs all directions equally, which is to say for no particular reader at all. At matched reconstruction the consumer-aware code beat it in twelve of twelve domains, and truncation loses on compact sets.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Pearson, on lines and planes of closest fit, 1901, and Hotelling,
1933, as chapter 4 section 4.1 of Data Mining as Observation
reads them, with the program’s truncation record in
turboquant-pro/benchmarks/RESULTS_glove.md:1-40.
Measurements
| Where the book states it | Numbers, as the book’s sources table records them | Source |
|---|---|---|
| chapter 11 section 11.5 | 9.6x at recall 0.999 CI-gated on GloVe 1.18M; 32x at 0.9993 on private 199k LaBSE, ties OPQ, beats RaBitQ, 20x build; 27.7x and 114x reported; PCA truncation loses on compact sets; 20x at 199k and 4x at 1M over OPQ; RaBitQ builds in under a second | turboquant-pro/CLAIMS.md:28-49 |
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/PCA.lean,
theorems varAlong_basis, varAlong_le,
varAlong_ge, dropped_eq,
dropped_nonneg, 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, 4, 6, 12.
Related
covariance matrix; explained variance; effective rank; projection; identity reader.
See also
Book equations stated beside the entry’s terms, not defining it: 0.5, 0.7, 4.2.
Ledger rows that cite the entry’s records without naming it: GO-1, GO-6.
Sources-table rows that share a record with the entry without naming it: chapter 4 section 4.2, chapter 4 section 4.5.
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.