outer product concept
| Definition | The matrix whose entry in row i and column j is the product of a vector's i-th and j-th coordinates, of rank at most one and positive semidefinite for a vector with itself. Chapter 0 section 0.5. |
|---|---|
| Example | (1, 2) with itself gives the matrix with rows (1, 2) and (2, 4), which has rank one. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id outer-product, 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/PositiveSemidefinite.lean, 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 6.1.
\[s(x)=w\cdot x+b,\qquad P_C=\mathbb E\big[\sigma'(s)^{2}\big]\,w\,w^{\top},\qquad \operatorname{rank}P_C=1.\]
Book equation 0.3.
\[\Sigma_{ij}=\mathbb E\big[(x_i-\mu_i)(x_j-\mu_j)\big],\qquad \operatorname{tr}\Sigma=\sum_{i}\Sigma_{ii}.\]
Conditions
- The matrix whose entry in row i and column j is the product of the i-th and j-th coordinates of a vector. Its action on any vector is a multiple of the vector it was built from, it has rank at most one, and the outer product of a vector with itself is positive semidefinite.
- The read operator is the weighted mean of the outer product of the sensitivity with itself, and a linear classifier’s read operator is one outer product scaled by how steep the score is.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.5 of Data Mining as Observation.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/PositiveSemidefinite.lean,
theorems psd_add, psd_smul,
psd_outer, psd_readOp,
psd_diag_nonneg, 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.
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, 2, 6.
Related
read operator; rank; positive semidefinite; covariance matrix; sensitivity.
See also
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 6 section 6.1.
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.