DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Rank-q principal component analysis by Frobenius reconstruction error

Definition

For a finite data matrix $X \in R^{n \times p}$ with $n, p \geq 1$ , set $H_{n} = I_{n} - n^{- 1} 1_{n} 1_{n}^{T}$ and $X_{c} = H_{n} X$ . For an integer $q$ with $1 \leq q \leq p$ , a rank- $q$ PCA frame is a matrix $Q \in R^{p \times q}$ with $Q^{T} Q = I_{q}$ . Its reconstruction is $X_{c} Q Q^{T}$ , and its squared Frobenius reconstruction error is

$∥ X_{c} - X_{c} Q Q^{T} ∥_{F}^{2} = \sum_{i < n} \sum_{j < p} (X_{c} - X_{c} Q Q^{T})_{i j}^{2} .$

A rank- $q$ PCA solution is any frame minimizing this error over all such $Q$ ; its score matrix is $X_{c} Q$ . A minimizing frame need not be unique when the relevant eigenvalues are tied.

Definition references

Finite centering matrix and centered data

Frobenius matrix norm

Hastie, Tibshirani, and Friedman, ESL II, §14.5.1, printed pp. 534–541

Cameron, Linear and Nonlinear Dimensionality Reduction, §§2.1–2.2, PDF pp. 2–4

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)
Maria Cameron, Linear and Nonlinear Dimensionality Reduction, lecture notes (standard reference, not scraped)