ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generated

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.

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A covariance matrix with one correlated block

Example

Let $p = 4$ , $s = 2$ , $M = 2$ , $ϵ = 1 / 10$ , and $Σ = I_{4} + \frac{1}{4} (e_{1} e_{2}^{T} + e_{2} e_{1}^{T}), Q = δ_{0} .$ The observations have law $P_{ϵ, Σ, Q}$ .

Facts & Assumptions

Given: The matrix $Σ$ , block size $s = 2$ , bound $M = 2$ , and contamination law in the Example.

[F1]

Bounded-spectrum covariance class defines $F_{p} (M)$ by symmetry, positive semidefiniteness and operator norm at most $M$ . Single-block sparse covariance class and restricted directions additionally requires all off-diagonal entries to lie in one common coordinate block of size at most $s$ and defines $F_{p, s} (M)$ .

[F2]

Gaussian Huber contamination model for covariance estimation defines the Gaussian Huber mixture.

[F3]

The Gaussian Huber covariance mixture is a probability law proves the mixture's probability-measure property.

[F4]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis: the eigenvalues of a real symmetric matrix determine its positive semidefiniteness and operator norm.

[F5]

The Dirac set function at a point and A Dirac set function is a probability measure: a point mass is a probability measure on the Borel space.

Verification

1.1F1F4algebra

The only nonzero off-diagonal entries of $Σ$ are $Σ_{12} = Σ_{21} = 1 / 4$ , both supported in $S = {1, 2}$ . Its eigenvalues are $5 / 4, 3 / 4, 1, 1$ , so $Σ ⪰ 0$ and $∥ Σ ∥_{o p} = 5 / 4 \leq M$ by [F4]. Since $Σ$ is symmetric, these checks give $Σ \in F_{4} (2)$ and, with the support check, $Σ \in F_{4, 2} (2)$ by [F1].

2.1F2F3F5algebra∎

The contaminant $Q = δ_{0}$ is a probability measure by [F5], and $ϵ \in [0, 1]$ . The mixture is a well-defined Huber model by [F2] and [F3]. The structure is one correlated block; it does not assert the column-wise weak- $ℓ_{q}$ sparsity condition from the separate sparse covariance model.

Depends on

Used by

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Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, §3.4 (standard reference, not scraped)