LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

The Huber covariance contamination modulus has a quadratic lower bound

Statement

For fixed $M > 0$ , $0 \leq ϵ \leq 1$ , and $p \geq 1$ , the squared operator loss $L (Σ_{0}, Σ_{1}) = ∥ Σ_{0} - Σ_{1} ∥_{o p}^{2}$ has contamination modulus at least $c_{M} ϵ^{2}$ on $F_{p} (M)$ and on $F_{p, s} (M)$ for every $1 \leq s \leq p / 2$ .

Facts & Assumptions

Given: The bounded covariance classes, the contamination modulus, the Gaussian Huber covariance model, Pinsker's inequality, and the local Gaussian KL formula.

[F1]

Huber contamination modulus: the modulus uses pairs of clean laws whose total-variation distance is at most $ϵ / (1 - ϵ)$ ; at $ϵ = 1$ the threshold is infinite.

[F2]

Pinsker’s inequality with event-supremum total variation: $T V (P, Q) \leq \sqrt{D (P ∥ Q) / 2}$ under $T V = \sup_{A} ∣ P (A) - Q (A) ∣$ .

[F3]

Gaussian Huber contamination model for covariance estimation: the clean Gaussian law is $N_{p} (0, Σ)$ with covariance parameter $Σ$ .

[F4]

Kullback–Leibler divergence between nondegenerate centered Gaussian laws: the KL divergence of the positive-definite Gaussian pair is given by its trace-determinant formula.

[F5]

Bounded-spectrum covariance class and Single-block sparse covariance class and restricted directions: a diagonal positive semidefinite matrix with largest eigenvalue at most $M$ lies in the dense class and in every defined single-block class.

Proof

1.1F3F5algebra

For $0 < ϵ \leq 1$ , set $Σ_{0} = (M / 2) I$ and $Σ_{1} = (M / 2) I + (M ϵ / 4) e_{1} e_{1}^{T}$ . Both are positive definite, their largest eigenvalues are at most $3 M / 4$ , and they are diagonal. Thus they belong to the dense class and to every block-sparse class in the Statement by [F5]. Their squared operator separation is $M^{2} ϵ^{2} / 16$ .

2.1F2F4step 1.1algebra

Put $r = ϵ / 2$ . By [F4], $D (N_{p} (0, Σ_{0}) ∥ N_{p} (0, Σ_{1})) = \frac{1}{2} [\log (1 + r) + (1 + r)^{- 1} - 1] \leq \frac{r^{2}}{4} = \frac{ϵ^{2}}{16} .$ The derivative of the bracket is $r / (1 + r)^{2} \leq r$ ; integration from $0$ proves the inequality. Pinsker [F2] gives $T V \leq ϵ / \sqrt{32} \leq ϵ \leq ϵ / (1 - ϵ)$ for $0 < ϵ < 1$ . At $ϵ = 1$ , the modulus threshold is infinite.

3.1F1step 1.1step 2.1∎

The pair from step 1.1 is admissible in the supremum defining the contamination modulus [F1]. Its loss is $M^{2} ϵ^{2} / 16$ , so the modulus on both classes is at least that amount. At $ϵ = 0$ the claimed lower bound is zero.

Depends on

Used by

Dependency tree · two levels

45 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

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, proof of Theorems 3.2 and 3.7 (standard reference, not scraped)