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Dense Gaussian Huber covariance: high-probability upper and capped lower bounds

Statement

Assume AC. Fix $M > 0$ . Let $X_{1}, \dots, X_{n}$ be independent with common law $P_{ϵ, Σ, Q}$ from Gaussian Huber contamination model for covariance estimation, where $Σ \in F_{p} (M)$ and $Q$ is arbitrary. If $0 \leq ϵ < 1 / 5$ , $0 < δ < 1 / 2$ , and $C_{1} (p + \log (1 / δ)) / n < 1$ for a sufficiently large universal $C_{1}$ , then every Borel measurable dense matrix-depth estimator, from either the global positive semidefinite correspondence or the bounded-class correspondence, $\hat{Σ} = \hat{Γ} / β$ from Matrix depth and covariance estimators satisfies $∥ \hat{Σ} - Σ ∥_{o p}^{2} \leq C_{M} [(\frac{p}{n} \lor ϵ^{2}) + \frac{\log (1 / δ)}{n}]$ with probability at least $1 - 2 δ$ , uniformly over $Q$ and $Σ \in F_{p} (M)$ .

Such Borel maximizing selectors exist by Measurable selection of a matrix-depth maximizer.

There are constants $c_{M}, c_{M}^{'} > 0$ such that for every $p, n \geq 1$ and every $0 \leq ϵ \leq 1$ , $\inf_{\tilde{Σ}} \sup_{Σ \in F_{p} (M)} \sup_{Q} P_{ϵ, Σ, Q}^{\otimes n} ⁣ {∥ \tilde{Σ} - Σ ∥_{o p}^{2} \geq c_{M} \min {1, \frac{p}{n} \lor ϵ^{2}}} \geq c_{M}^{'} .$ The infimum is over all Borel maps $(R^{p})^{n} \to R^{p \times p}$ . The constants may depend on $M$ .

Facts & Assumptions

Given: AC, the Gaussian Huber model, $Σ \in F_{p} (M)$ , arbitrary contaminating law $Q$ , and the stated confidence and sample-size conditions.

[F1]

Operator error bound for the Gaussian Huber matrix-depth estimator gives the directional operator-error bound for the dense sphere family, for a Borel statistic satisfying the empirical-depth comparison.

[F2]

Clean Gaussian covariance has a capped squared operator lower rate gives, under AC, the clean tail-risk lower bound of order $\min {1, p / n}$ for all $p, n \geq 1$ , retaining the original small-ratio result.

[F3]

The Huber covariance contamination modulus has a quadratic lower bound gives $ω (ϵ, F_{p} (M)) \geq d_{M} ϵ^{2}$ .

[F4]

Huber covariance modulus lower bound for unrestricted matrix-valued actions: under AC, a clean tail-risk lower bound for all matrix-valued estimators and a finite contamination modulus give the Huber lower bound at threshold $\frac{1}{16} (M_{0} \lor ω)$ , for the same unrestricted action space.

[F5]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis: for real symmetric $A$ , $∥ A ∥_{o p} = \sup_{∥ u ∥_{2} = 1} ∣ u^{T} A u ∣$ .

[F6]

Bounded-spectrum covariance class: every $Σ \in F_{p} (M)$ is positive semidefinite and has operator norm at most $M$ .

[F7]

Measurable selection of a matrix-depth maximizer supplies Borel maximizers for the global and bounded-class correspondences. A global maximizer dominates $β Σ$ in depth; in the bounded case $β Σ$ belongs to the search class by the target spectrum bound.

Proof

1.1F1F5F7algebra

By [F7], Borel maximizers exist in both versions and every such maximizer satisfies the comparison premise in [F1]. Take $U = S^{p - 1}$ , one support $S_{1} = [p]$ , and $r = p$ in [F1]. For a real symmetric matrix $A$ , [F5] gives $∥ A ∥_{o p} = \sup_{∥ u ∥_{2} = 1} ∣ u^{T} A u ∣$ . Thus $∥ \hat{Σ} - Σ ∥_{o p} \leq C_{M} (ϵ + \sqrt{(p + \log (1 / δ)) / n})$ . Squaring and using $(a + b)^{2} \leq 2 a^{2} + 2 b^{2}$ gives the stated upper rate.

1.2F2F3F4F6algebra

The all-regime clean lower bound [F2] supplies constants $a_{M}, b_{0} > 0$ such that the clean tail-risk premise in [F4] holds for every $p, n \geq 1$ with $M_{0} = a_{M} \min {1, p / n}$ and $c_{0} = b_{0}$ . By [F3], the contamination modulus is at least $d_{M} ϵ^{2}$ . It is finite because $∥ Σ_{0} - Σ_{1} ∥_{o p}^{2} \leq 4 M^{2}$ for every pair in the bounded class.

2.1F2F3F4step 1.2algebra∎

Apply [F4] with the same unrestricted matrix-valued estimator class as in the Statement and the inputs from step 1.2. Its threshold $\frac{1}{16} (M_{0} \lor ω)$ is at least $\frac{1}{16} \min {a_{M}, d_{M}} (\min {1, p / n} \lor ϵ^{2})$ . Since $0 \leq ϵ \leq 1$ , $\min {1, p / n} \lor ϵ^{2} = \min {1, (p / n) \lor ϵ^{2}}$ . Lowering to this capped threshold only enlarges the event, and the probability lower bound is $\min (b_{0}, 1 / 2)$ . Renaming the two positive constants proves the all-dimension lower assertion. Its original uncapped rate remains valid whenever $p / n \leq 1$ .

Authoring status

The lower proof uses the current batch-17 and batch-31 supplier bodies. The local measurable-selection proof discharges the selector-existence obligation for both global and bounded-class matrix-depth versions.

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Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, Theorems 3.1, 3.2 and 5.1 (standard reference, not scraped)
Philippe Rigollet, 18.S997 High-Dimensional Statistics Lecture Notes, MIT OpenCourseWare, Chapter 5 (standard reference, not scraped)