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Dense Gaussian Huber covariance model

Example

Let $p = 2$ , $M = 2$ , $ϵ = 1 / 10$ , $Σ = (\begin{array}{cc} 1 & 1 / 4 \\ 1 / 4 & 1 \end{array}), Q = δ_{(2, 0)} .$ Take independent observations with the Gaussian Huber law $P_{ϵ, Σ, Q}$ from Gaussian Huber contamination model for covariance estimation.

Facts & Assumptions

Given: The matrix $Σ$ , $M = 2$ , $ϵ = 1 / 10$ , and the point-mass contaminant $Q$ in the Example.

[F1]

Gaussian Huber contamination model for covariance estimation defines the mixture $(1 - ϵ) N_{p} (0, Σ) + ϵ Q$ for $0 \leq ϵ \leq 1$ .

[F2]

Bounded-spectrum covariance class defines $F_{p} (M)$ by positive semidefiniteness and the operator-norm bound.

[F3]

The Gaussian Huber covariance mixture is a probability law proves the displayed mixture is a probability law.

[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

1.1F2F4algebra

The eigenvalues of $Σ$ are $5 / 4$ and $3 / 4$ , so $Σ$ is positive definite and $∥ Σ ∥_{o p} = 5 / 4 \leq 2$ by [F4]. Thus $Σ \in F_{2} (2)$ .

2.1F1F3F5algebra∎

The Dirac law $Q = δ_{(2, 0)}$ is a probability measure by [F5], and $0 < ϵ < 1$ . By [F1] and [F3], $P_{ϵ, Σ, Q}$ is a well-defined Huber sampling law. The target remains the clean Gaussian covariance $Σ$ , not the covariance of the mixture.

Depends on

Used by

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Dependency tree · two levels

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Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, §2 (standard reference, not scraped)