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.

Minimum-norm affine reconstruction weights for a finite LLE neighborhood

Definition

Let $x_{1}, \dots, x_{n} \in R^{p}$ , $n \geq 2$ , and $1 \leq k < n$ . For each $i$ , let $N_{i}$ be the first $k$ indices $j \neq i$ under the increasing lexicographic order of $(∥ x_{i} - x_{j} ∥_{2}, j)$ , so equal distances are broken by sample index; the norm on $R^{p}$ is $∥ z ∥_{2} = (\sum_{a = 1}^{p} z_{a}^{2})^{1 / 2}$ . The local affine reconstruction weights are the vector $w_{i} = (w_{i j})_{j \in N_{i}} \in R^{k}$ minimizing

${∥ x_{i} - \sum_{j \in N_{i}} w_{i j} x_{j} ∥}_{2}^{2} subject to \sum_{j \in N_{i}} w_{i j} = 1,$

Among all minimizers, choose the one of smallest Euclidean norm $∥ w_{i} ∥_{2} = (\sum_{j \in N_{i}} w_{i j}^{2})^{1 / 2}$ . This minimum-norm rule fixes the weights even when the local covariance matrix is singular.

Definition references

Deterministic finite neighbor lists

Hastie, Tibshirani, and Friedman, ESL II, §14.9, printed pp. 573–575

Cameron, Linear and Nonlinear Dimensionality Reduction, §7, PDF pp. 14–16

Depends on

Deterministic finite nearest-neighbor lists and graph symmetrizations

Used by

Dependency tree · one level

1 result within one dependency step 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)