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Huber covariance modulus lower bound for unrestricted matrix-valued actions

Statement

Assume AC. Fix $p, n \geq 1$ , $M > 0$ , $0 \leq ϵ \leq 1$ , and let $Θ$ be either $F_{p} (M)$ from Bounded-spectrum covariance class or $F_{p, s} (M)$ from Single-block sparse covariance class and restricted directions with $1 \leq s \leq p / 2$ , equipped with its induced Borel sigma-algebra. Let $E_{n}$ be the set of Borel maps $(R^{p})^{n} \to R^{p \times p}$ . The clean family is $P_{Σ} = N_{p} (0, Σ)$ . For $Σ \in Θ$ and matrix action $A$ , put $ℓ (Σ, A) = ∥ Σ - A ∥_{o p}^{2} .$ Its restriction to $Θ \times Θ$ defines the contamination modulus $ω (ϵ, Θ)$ through Huber contamination modulus. Suppose $ω (ϵ, Θ) < \infty$ , $0 \leq M_{0} < \infty$ , $0 < c_{0} \leq 1$ , and $\inf_{\hat{Σ} \in E_{n}} \sup_{Σ \in Θ} P_{Σ}^{\otimes n} {ℓ (Σ, \hat{Σ}) \geq M_{0}} \geq c_{0} .$ Then $\inf_{\hat{Σ} \in E_{n}} \sup_{Σ \in Θ} \sup_{Q} P_{ϵ, Σ, Q}^{\otimes n} {ℓ (Σ, \hat{Σ}) \geq \frac{1}{16} (M_{0} \lor ω (ϵ, Θ))} \geq \min (c_{0}, 1 / 2) .$

Facts & Assumptions

Given: AC, the bounded covariance class, the Gaussian Huber model, its contamination modulus, and the clean tail-risk premise in the Statement.

[F1]

Gaussian Huber contamination model for covariance estimation: one observation has law $(1 - ϵ) N_{p} (0, Σ) + ϵ Q$ and the sample consists of iid observations from this law.

[F2]

Huber contamination modulus: for $ϵ < 1$ , pairs in the modulus satisfy $T V (P_{Σ_{0}}, P_{Σ_{1}}) \leq ϵ / (1 - ϵ)$ ; at $ϵ = 1$ all pairs are allowed.

[F3]

TV-close laws have identical Huber mixtures at the contamination threshold: under AC, every pair satisfying [F2] admits contaminants making the two one-observation Huber laws equal.

[F4]

For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique: finite product laws are determined by their factor laws and rectangle values.

[F5]

Measurable spaces and measurable sets, A measurable function between measurable spaces, and Composition with a Borel measurable outer map preserves measurability: a Borel estimator composed with the continuous loss section gives measurable tail events.

[F6]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv|| and The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product define the matrix and vector norms. The inner-product norm is definite, homogeneous, and satisfies the triangle inequality gives the vector-norm triangle inequality: for every unit $v$ , $∥ (B + C) v ∥_{2} \leq ∥ B v ∥_{2} + ∥ C v ∥_{2} \leq ∥ B ∥_{o p} + ∥ C ∥_{o p}$ . Taking the supremum gives $∥ B + C ∥_{o p} \leq ∥ B ∥_{o p} + ∥ C ∥_{o p}$ , without symmetry assumptions on $B, C$ .

Proof

1.1F1givenalgebra

Fix any $\hat{Σ} \in E_{n}$ . For each $Σ \in Θ$ , choosing $Q = P_{Σ}$ makes the one-observation Huber law equal to the clean law by [F1]. If $ω \leq M_{0}$ , then $\frac{1}{16} (M_{0} \lor ω) \leq M_{0}$ , so the robust supremum probability at the smaller threshold is at least the clean supremum probability at $M_{0}$ . The clean premise therefore gives the claimed bound $c_{0}$ for this estimator. Taking the infimum preserves it.

1.2F1F2F3F4choose

Suppose $ω > M_{0}$ . By the definition of the finite supremum in [F2], there are $Σ_{0}, Σ_{1} \in Θ$ with $ℓ (Σ_{0}, Σ_{1}) > ω / 2$ and satisfying the total-variation condition there; when $ϵ = 1$ every pair is admissible. By [F3], choose contaminants $Q_{0}, Q_{1}$ whose one-observation Huber laws coincide at a law $R_{1}$ . Their iid sample laws both equal $R_{1}^{\otimes n}$ by [F1] and [F4].

2.1F5F6step 1.2algebra

For every matrix action $A$ , the triangle inequality gives $\max_{i = 0, 1} ℓ (Σ_{i}, A) \geq \frac{1}{4} ℓ (Σ_{0}, Σ_{1}) > \frac{1}{8} ω (ϵ, Θ) > \frac{1}{16} (M_{0} \lor ω (ϵ, Θ)) .$ Indeed, one of the two distances from $A$ is at least half the distance between $Σ_{0}$ and $Σ_{1}$ . The two measurable loss events at threshold $\frac{1}{4} ℓ (Σ_{0}, Σ_{1})$ therefore cover the sample space under the common law; their probabilities sum to at least one. One has probability at least $1 / 2$ , and its event is contained in the event at the smaller displayed threshold. The robust supremum probability for this estimator is at least $1 / 2$ .

3.1step 1.1step 1.2step 2.1algebra∎

Step 1.1 applies when $ω \leq M_{0}$ and steps 1.2–2.1 apply when $ω > M_{0}$ . The bound holds for every Borel matrix-valued estimator, so taking their infimum proves the statement.

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Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, §5 and supplement (standard reference, not scraped)
Philippe Rigollet, 18.S997 High-Dimensional Statistics Lecture Notes, MIT OpenCourseWare, Chapter 5, §5.3 (standard reference, not scraped)