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.

Geometric median of block means

Definition

Let $H$ be a separable real Hilbert space, let $n \geq 1$ , and let $1 \leq k \leq n$ be an integer. Given $(x_{1}, \dots, x_{n}) \in H^{n}$ , put $q = ⌊ n / k ⌋$ and $r = n - k q$ . Form consecutive blocks $B_{1}, \dots, B_{k}$ in index order, with $∣ B_{j} ∣ = q + 1$ for $j \leq r$ and $∣ B_{j} ∣ = q$ for $j > r$ . Compute the block means $m_{j} = ∣ B_{j} ∣^{- 1} \sum_{i \in B_{j}} x_{i}$ . Return the unique minimum-norm point of

\underset{z \in H}{argmin} \sum_{j = 1}^{k} ∥ z - m_{j} ∥ .

The objective minimizers are called geometric medians of the block means. The minimum-norm rule makes the output single-valued. Existence, uniqueness of that selected point, and Borel measurability as a function of the input tuple are proved in Measurable minimum-norm geometric-median selection ↗.

Depends on

Used by

Dependency tree · two levels

13 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

Stanislav Minsker, Geometric Median and Robust Estimation in Banach Spaces, Lemma 2.1 and §4.1 (standard reference, not scraped)
Gábor Lugosi, Mean Estimation in High Dimension, §3 (standard reference, not scraped)