kernel concept
| Definition | The directions a matrix sends to zero. The kernel of the read operator is the nuisance. Chapter 0 section 0.5. |
|---|---|
| Example | The read operator of the consumer x1 + x2 sends (1, −1) to zero, so (1, −1) is in its kernel. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id kernel, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.9. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/ReadOperator.lean, lean/DataMiningAsObservation/Rank.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.9.
\[P_C=\mathbb E\!\left[g\,g^{\top}\right],\qquad g=\nabla C(x).\]
Book equation 0.12b.
\[x\sim_{\bar P_{C,\mu}} x'\quad\Longleftrightarrow\quad x-x'\in\ker\bar P_{C,\mu}.\]
Conditions
- The directions a matrix sends to zero. The kernel of the read operator is the nuisance, the directions the consumer never reads, and a direction is in it exactly when every row’s sensitivity is orthogonal to it. The kernel’s dimension is the dimension less the rank.
- The reconstruction coder spends bits on the kernel and the output coder does not, and the gap between them is governed by the share of the error’s entropy that lies in the kernel.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.5 and section 0.6 of Data Mining as
Observation, with the kernel entropy share in
geometric-observation/claims/LEDGER.md row GO-6.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/ReadOperator.lean,
theorems rank_one_reads_one_direction,
readOp_mulVec, quad_readOp,
quad_readOp_nonneg, readOp_mulVec_eq_zero_iff,
readOp_diag, readOp_offdiag,
readOp_symm, readOp_neg,
affine_const_along_nuisance, readOp_affine,
readOp_sqLength_basis, at observation-data-mining f3914f0;
what the check covers is stated in the book’s appendix
C.
lean/DataMiningAsObservation/Rank.lean,
theorems rank_le_width, rank_le_height,
rank_outer_le_one, rank_mul_le,
rank_zero, rank_transpose, 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, 2, 4, 6, 11.
Related
nuisance; read operator; read subspace; quotient; rank.
See also
Book equations stated beside the entry’s terms, not defining it: 0.12a.
Ledger rows that cite the entry’s records without naming it: OT-7, GO-6.
Sources-table rows that share a record with the entry without naming it: chapter 2 section 2.2, 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.