effective rank concept
| Definition | The number of directions a matrix really uses, the square of the sum of its eigenvalues over the sum of their squares, defined for a nonnegative spectrum that is not all zero, with no cutoff to choose. Equation 0.7. Also participation ratio. |
|---|---|
| Example | Eigenvalues (1, 1, 1, 1) have effective rank 4, and (1, 0.12, 0.06, 0.02) have effective rank about 1.4. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id effective-rank, kind concept. |
| Status | measures [demonstrated]. Corrections: none recorded. |
| Defining equation | Book equation 0.7. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | geometric-observation/claims/LEDGER.md:30, readscope/readscope/spectrum.py:35-70, turboquant-pro/turboquant_pro/read_allocation.py:244-307, gtc-prototype/docs/SPECTRUM_FINDINGS.md:10-18, lean/DataMiningAsObservation/EffectiveRank.lean |
| Reviewed | semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.7.
\[r_{\mathrm{eff}}=\frac{\big(\sum_i\lambda_i\big)^{2}}{\sum_i\lambda_i^{2}}.\]
Conditions
- The effective rank is the square of the sum of the eigenvalues over the sum of their squares, between one and the dimension for a nonnegative spectrum, with no cutoff to choose.
- It is not invariant under a change of basis of the input, unlike the trace pairing and the rank, and the ledger row that says so is the one that licenses comparing read operators across coordinate systems.
- An allocation over a spectrum with effective rank below two concentrates on one direction, and the allocation report says so before giving a gain.
- It is defined for a nonnegative spectrum that is not all zero, and it lies between one and the dimension.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
- measures. OT-7
[demonstrated]. The damage form, trace pairing, rank, and loading covariance are GL(d)-invariant underP' = A⁻ᵀPA⁻¹, while spectrum, effective rank, principal angles, and water-filling are O(d)-only — consumer-weighted damage is a geometric scalar, and …geometric-observation/claims/LEDGER.md:30.
First stated
The participation ratio of a spectrum, applied to read operators in
readscope, readscope/readscope/spectrum.py:35-70, and
chapter 0 section 0.4 of Data Mining as Observation.
Measurements
| Where the book states it | Numbers, as the book’s sources table records them | Source |
|---|---|---|
| chapter 4 section 4.2 | effective rank as participation ratio, energy rank | readscope/readscope/spectrum.py:35-70 |
| chapter 4 section 4.2 | allocation report, gain over uniform, concentration caution below effective rank 2 | turboquant-pro/turboquant_pro/read_allocation.py:244-307 |
| chapter 9 section 9.4 | explained ratios, effective rank 5.19 of 8, convergence with a second method, property of the representation not the space | gtc-prototype/docs/SPECTRUM_FINDINGS.md:10-18 |
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/EffectiveRank.lean,
theorems sq_sum_le, effRank_le,
sum_sq_le_sq_sum, one_le_effRank,
effRank_const, effRank_single, 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, 4, 9, 11, 14.
Related
read operator; read subspace; water-filling; recognizer.
See also
Book equations stated beside the entry’s terms, not defining it: 4.2.
Sources-table rows that share a record with the entry without naming it: 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.