read subspace concept

Definition	The directions of the data a consumer can distinguish, spanned by the eigenvectors of its read operator with nonzero eigenvalue. Chapter 0 section 0.5.
Example	The consumer 3x1 + 4x2 has a one-dimensional read subspace spanned by (3, 4).
Book	Data Mining as Observation, draft 0.2, commit `f3914f0`; entry id `read-subspace`, kind concept.
Status	no ledger row names this entry. Corrections: none recorded.
Defining equation	Book equation 0.9.
Assumptions and scope	The read subspace is spanned by the eigenvectors of the read operator with nonzero eigenvalue, and it is small. A linear classifier reads one direction, an attention head a handful, and the input has hundreds or thousands. Pointwise, a scalar-output consumer reads one direction at each row, and the averaged operator can still have full rank, as the consumer that reads the squared length of a row shows. Recovering it from a black box costs two consumer calls per input dimension per row and is a cliff at the full dimension.
Prior art	The active subspace of Constantine and Gleich when the output is scalar and Euclidean.
Evidence	`geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:7-48`, `geometric-observation/chapters/ch06_mathematical_preliminaries.md:10-27`, `lean/DataMiningAsObservation/ReadOperator.lean`
Reviewed	semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page.

The directions the consumer reads.

Equation

Book equation 0.9.

\[P_C=\mathbb E\!\left[g\,g^{\top}\right],\qquad g=\nabla C(x).\]

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 6.1.

\[s(x)=w\cdot x+b,\qquad P_C=\mathbb E\big[\sigma'(s)^{2}\big]\,w\,w^{\top},\qquad \operatorname{rank}P_C=1.\]

Conditions

The read subspace is spanned by the eigenvectors of the read operator with nonzero eigenvalue, and it is small. A linear classifier reads one direction, an attention head a handful, and the input has hundreds or thousands.
Pointwise, a scalar-output consumer reads one direction at each row, and the averaged operator can still have full rank, as the consumer that reads the squared length of a row shows.
Recovering it from a black box costs two consumer calls per input dimension per row and is a cliff at the full dimension.

Conditions are curated in entries.toml rather than read from a record.

Ledger

none

First stated

Volume 14, chapter 5, geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:7-48, DOI 10.5281/zenodo.21776291, where the read subspace is the range of the read operator.

Measurements

Where the book states it	Numbers, as the book’s sources table records them	Source
chapter 2 section 2.2	read subspace small, operator local, pullback composition, rank cannot increase	`geometric-observation/chapters/ch05_the_read_metric_and_the_quotient.md:7-48`; `geometric-observation/chapters/ch06_mathematical_preliminaries.md:10-27`

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.

Used in

Data Mining as Observation chapters 0, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13.

nuisance; read operator; blind probe; budget cliff.

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.

← read operatorrecall at k →