DefinitionDefinition: AI-adaptedProof: 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.

Fixed-design least squares as orthogonal projection

Definition

Let $n, d \in N$ with $n, d \geq 1$ , and let $X \in R^{n \times d}$ be a fixed design matrix, and write $V_{X} : = {X θ : θ \in R^{d}} \subseteq R^{n}$ for its column space. For an observed vector $y \in R^{n}$ , the fixed-design least-squares fitted vector is

${\hat{μ}}_{X} (y) : = P_{V_{X}} y,$

where $P_{V_{X}}$ is the orthogonal projection onto $V_{X}$ , as defined in The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$ . It is uniquely defined even when $X$ is rank deficient. The nearest-point characterization in The orthogonal projection is the unique nearest point in the subspace says that ${\hat{μ}}_{X} (y)$ minimizes $∥ y - v ∥_{2}$ over $v \in V_{X}$ . This item defines the fitted vector; it does not select a coefficient representation $\hat{θ}$ .

Depends on

Used by

Constrained support-search least squares Algorithm
Penalized support-search least squares Algorithm
Collinear columns identify a fitted mean, not its coefficients Counterexample
Series least squares with a spline basis Definition
Zero training residual and positive test error Example
Exact squared-error risk of an isotropic projection fit Theorem
Expected empirical training risk for spline series least squares Theorem
Sub-Gaussian projection risk bounds Theorem

Dependency tree · two levels

15 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

Philippe Rigollet and Jan-Christian Hütter, High-Dimensional Statistics: Lecture Notes, Chapter 2, §§7–8 (standard reference, not scraped)
Sylvain Arlot, Model/estimator selection, Lecture on model selection, §§1–2 (standard reference, not scraped)