Jacobian concept
| Definition | The matrix of partial derivatives of a vector-valued consumer, whose transpose times itself averages to the read operator. A discrete stage has none. Chapter 0 section 0.5 and chapter 12. Also not differentiable. |
|---|---|
| Example | A consumer mapping (x1, x2) to (x1 + x2, x1 − x2) has Jacobian with rows (1, 1) and (1, −1). |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id jacobian, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 2.1. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Pipeline.lean, lean/DataMiningAsObservation/ReadOperator.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 2.1.
\[P_{C_2\circ C_1}(x)=J_1(x)^{\top}\,P_{C_2}\big(C_1(x)\big)\,J_1(x),\qquad \operatorname{rank}P_{C_2\circ C_1}(x)\le\operatorname{rank}P_{C_2}\big(C_1(x)\big)\quad\text{at each row } x.\]
Book equation 0.11.
\[P_C(x)=J(x)^{\top}G\big(C(x)\big)\,J(x),\qquad J(x)=\frac{\partial C}{\partial x}(x),\qquad \bar P_{C,\mu}=\mathbb E_{\mu}\!\left[P_C(x)\right].\]
Conditions
- The matrix of partial derivatives of a vector-valued consumer, whose transpose times itself averages to the read operator. Along a chain of stages the rank of the composed operator is at most the rank of either stage, and a stage that identifies two inputs identifies them for every stage after it.
- Chunking and indexing have no Jacobian, since a document goes to a set of pieces and a query to a candidate list, and what survives without a derivative is the statement about distinctions.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.5 and chapter 2 section 2.2 of Data Mining as
Observation, with the pullback composition and the rank bound in
geometric-observation/chapters/ch06_mathematical_preliminaries.md:10-27.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Pipeline.lean,
theorems quotient_inherited,
quotient_inherited_chain, rank_comp_le_first,
rank_comp_le_second, at observation-data-mining f3914f0;
what the check covers is stated in the book’s appendix
C.
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.
Used in
Data Mining as Observation primer L, chapters 0, 2, 12.
Related
read operator; sensitivity; Hessian; retrieval-augmented pipeline.
See also
Book equations stated beside the entry’s terms, not defining it: 0.9.
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 2 section 2.2, chapter 12 section 12.2.
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.