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.

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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Proper Frobenius covering numbers

Definition

Let $n, m \geq 1$ , let $U$ be a nonempty subset of the real matrix space $R^{n \times m}$ , and let $ϵ > 0$ . A proper $ϵ$ -cover of $U$ in Frobenius norm is a finite subset $V \subseteq U$ such that every $X \in U$ has some $Y \in V$ with $∥ X - Y ∥_{F} \leq ϵ$ . The proper Frobenius covering number $N_{F} (U, ϵ)$ is the least cardinality of such a cover when one exists, and is $+ \infty$ otherwise. The cover is proper because its centers belong to $U$ .

Definition references

The Frobenius norm $∥ A ∥_{F} = (\sum_{i, j} ∣ a_{i j} ∣^{2})^{1 / 2}$ on real or complex matrices: defines $∥ X - Y ∥_{F}$ as the Euclidean norm of the full matrix-entry list.

Bartlett, Foster, and Telgarsky, §3.2: defines the covering number using finite centers in the covered set.

Depends on

The Frobenius norm $\lVert A\rVert_F=(\sum_{i,j}|a_{ij}|^2)^{1/2}$ on real or complex matrices

Used by

Dependency tree · two levels

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Sources

Peter L. Bartlett, Dylan J. Foster, and Matus Telgarsky, Spectrally-normalized margin bounds for neural networks, §3.2 (standard reference, not scraped)