concentrated concept
| Definition | Of a spectrum, having a few eigenvalues that carry most of the total. Chapter 0 section 0.4. Also concentration caution. |
|---|---|
| Example | A spectrum of 1.0, 0.12, 0.06, and 0.02 puts 83 percent of its total in one direction. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id concentrated, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | none |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | turboquant-pro/turboquant_pro/read_allocation.py:244-307, lean/DataMiningAsObservation/Concentrated.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
none
Conditions
- Of a spectrum, having a few eigenvalues that carry most of the total. If one eigenvalue carries a fraction f of the total, the effective rank is at most one over f squared, so half the mass in one direction gives effective rank at most four.
- The allocation report warns below effective rank two, which is the same statement read from the rank side, and an allocation over a concentrated spectrum spends almost everything on one direction.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.4 of Data Mining as Observation, with
the allocation report’s concentration caution in
turboquant-pro/turboquant_pro/read_allocation.py:244-307.
Measurements
| Where the book states it | Numbers, as the book’s sources table records them | Source |
|---|---|---|
| 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 |
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Concentrated.lean,
theorems sq_le_sum_sq, effRank_le_of_fraction,
effRank_le_four, 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, 4, 6, 10, 11, 13.
Related
spectrum; effective rank; water-filling; isotropic.
See also
Book equations stated beside the entry’s terms, not defining it: 0.7, 0.5.
Ledger rows that cite the entry’s records without naming it: OT-7.
Sources-table rows that share a record with the entry without naming it: chapter 4 section 4.2.
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.