AlgorithmAlgorithm: 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.

Finite-sample k-nearest-neighbor classifier with fixed tie-breaking

Definition

Let $d, n, C$ be positive integers, let $(x_{i}, y_{i})_{i = 1}^{n}$ be a finite labeled sample with $x_{i} \in R^{d}$ and $y_{i} \in {1, \dots, C}$ , let $M \in R^{d \times d}$ be symmetric with $v^{T} M v \geq 0$ for every $v \in R^{d}$ , let $1 \leq k \leq n$ , and let $x \in R^{d}$ be a query. For each sample index set $δ_{i} = \sqrt{(x - x_{i})^{T} M (x - x_{i})}$ . Order indices $i$ by increasing pairs $(δ_{i}, i)$ in lexicographic order, and let $K_{k} (x)$ be the first $k$ indices in this order. For each class $c \in {1, \dots, C}$ compute $v_{c} (x) = \sum_{i \in K_{k} (x)} 1_{{y_{i} = c}}$ and return the smallest class label attaining $\max_{1 \leq c \leq C} v_{c} (x)$ . Thus equal distances are resolved by the fixed sample-index order, and equal class votes by the smallest class label.

Definition references

Mahalanobis squared distance and its nonnegative square root

Finite-sum convention

Weinberger and Saul, §2.1, printed pp. 209–210, and §3.1, pp. 214–216

Bellet, Habrard, and Sebban, §1 and §3.2, printed pp. 1–2 and 12–13

Depends on

Used by

Nothing in the library uses this result yet.

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

Kilian Q. Weinberger and Lawrence K. Saul, Distance Metric Learning for Large Margin Nearest Neighbor Classification, JMLR 10 (2009), 207–244 (standard reference, not scraped)
Aurélien Bellet, Amaury Habrard, and Marc Sebban, A Survey on Metric Learning for Feature Vectors and Structured Data, arXiv:1306.6709v4 (2014) (standard reference, not scraped)