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.

K-means objective

Definition

Let $n, p$ be positive integers, let $1 \leq K \leq n$ , let $x_{1}, \dots, x_{n} \in R^{p}$ be observed vectors, and let $μ_{1}, \dots, μ_{K} \in R^{p}$ be centers. A labeling is a vector $z = (z_{1}, \dots, z_{n}) \in {1, \dots, K}^{n}$ , and its $k$ th cluster is $C_{k} (z) : = {i : z_{i} = k}$ . For $v = (v_{1}, \dots, v_{p}) \in R^{p}$ , set $∥ v ∥_{2}^{2} : = \sum_{j = 1}^{p} v_{j}^{2}$ . The K-means objective is

$J (z, μ_{1}, \dots, μ_{K}) : = \sum_{i = 1}^{n} ∥ x_{i} - μ_{z_{i}} ∥_{2}^{2} = \sum_{k = 1}^{K} \sum_{i \in C_{k} (z)} ∥ x_{i} - μ_{k} ∥_{2}^{2} .$

An empty cluster contributes zero to the sum.

Used by

Lloyd's k-means algorithm with explicit tie handling Algorithm
Tie retention gives distinct Lloyd fixed points Example
Lloyd iteration is objective-decreasing and terminates Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, 2nd ed., corrected 12th printing (standard reference, not scraped)
Christopher M. Bishop, Pattern Recognition and Machine Learning (standard reference, not scraped)