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Single-block sparse Gaussian Huber covariance: high-probability upper and low-ratio lower bounds

Statement

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

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

There are constants $c_{M}, c_{M}^{'}, c_{1} > 0$ such that whenever $s \log (e p / s) / n \leq c_{1}$ , for every $0 \leq ϵ \leq 1$ , $\inf_{\tilde{Σ}} \sup_{Σ \in F_{p, s} (M)} \sup_{Q} P_{ϵ, Σ, Q}^{\otimes n} ⁣ {∥ \tilde{Σ} - Σ ∥_{o p}^{2} \geq c_{M} (\frac{s \log (e p / s)}{n} \lor ϵ^{2})} \geq c_{M}^{'} .$ The constants may depend on $M$ . The class is one common correlated block, not a row-sparsity class. The infimum is over all Borel maps $(R^{p})^{n} \to R^{p \times p}$ .

Facts & Assumptions

Given: AC, $1 \leq s \leq p / 2$ , $Σ \in F_{p, s} (M)$ , the Gaussian Huber model, and the stated confidence and sample-size restrictions.

[F1]

Operator error bound for the Gaussian Huber matrix-depth estimator gives the directional error bound over a union of coordinate spheres, for a Borel statistic satisfying the empirical-depth comparison.

[F2]

Clean block-sparse Gaussian covariance has squared operator lower rate s log(ep/s) over n gives, under AC, the clean tail-risk lower bound of order $s \log (e p / s) / n$ when its sample-size condition holds.

[F3]

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

[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]

Single-block sparse covariance class and restricted directions: off-diagonal support lies in one common coordinate block of size at most $s$ , and the operator norm is bounded by $M$ .

[F8]

The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ , $(\binom{n}{k}) k! (n - k)! = n!$ for $k \leq n$ ; hence $(\binom{n}{k}) k! = n^{\underline{k}}$ , the quotient $n! / (k! (n - k)!)$ is a natural number, and $(\binom{n}{k}) = (\binom{n}{n - k})$ , The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, and Laws of finite sums and products in $N$ , and $ι (\sum_{k < n} a_{k}) = \sum_{k < n} ι (a_{k})$ : identifying naturals with their real embeddings, for integers $1 \leq k \leq p$ , $(\binom{p}{k}) = p^{\underline{k}} / k!$ and $p^{\underline{k}} = p (p - 1) \dots (p - k + 1)$ .

[F9]

The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , The real exponential function and the number $e$ by a power series, The exponential function is strictly increasing, Integer powers $a^{m}$ , and Laws of integer exponents: $\log$ is increasing, $\log e = 1$ , and the integral of $\log x$ on $[1, k]$ is $k \log k - k + 1$ by the antiderivative $x \log x - x$ .

[F10]

Every nonempty finite set of reals has a maximum and a minimum: a nonempty finite set of real numbers has a maximum; applying this to the absolute diagonal entries gives a coordinate attaining their largest absolute value.

[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

1.1F1F5F6F7F8F9F10algebra

By [F7], Borel maximizers exist in both versions and satisfy [F1]'s empirical-depth comparison. Put $k = 2 s$ and $m = (\binom{p}{k})$ . By [F8], $m \leq p^{k} / k!$ . Since $\log x \leq \log j$ for $x \in [j - 1, j]$ and $2 \leq j \leq k$ , summing over these intervals and using [F9] gives $\log k! = \sum_{j = 2}^{k} \log j \geq \int_{1}^{k} \log x d x = k \log k - k + 1 \geq k \log k - k$ . Exponentiating with [F9] yields $k! \geq (k / e)^{k}$ , hence $m \leq (e p / k)^{k}$ and $\log m \leq 2 s \log (e p / (2 s)) \leq 2 s \log (e p / s)$ . Enumerate the coordinate sets of size $2 s$ and take $U = U_{s}$ , $r = 2 s$ in [F1]. If $A = \hat{Σ} - Σ$ , choose blocks of size at most $s$ supporting the off-diagonal entries of its two summands, and let $T$ be their union. Then $∣ T ∣ \leq 2 s$ , entries between $T$ and $T^{c}$ vanish, and the block on $T^{c}$ is diagonal. Thus $A$ is block diagonal on $R^{T} \oplus R^{T^{c}}$ , with symmetric block $A_{T}$ and diagonal block $D$ on $T^{c}$ . Its operator norm is the larger block norm: for $x = x_{T} + x_{T^{c}}$ , $∥ A x ∥_{2}^{2}$ is the sum of the squared norms of the two block outputs, and a norm-maximizing vector in either nonempty finite-dimensional block attains its block norm. If $T \neq \emptyset$ and either $T^{c} = \emptyset$ or $∥ A_{T} ∥_{o p} \geq ∥ D ∥_{o p}$ , [F5] gives a unit eigenvector attaining $∥ A_{T} ∥_{o p}$ , supported in $T$ . If $T = \emptyset$ and $∥ D ∥_{o p} = 0$ , then $A = 0$ , and any coordinate unit vector belongs to $U_{s}$ because $p \geq 2 s$ ; it attains $∥ A ∥_{o p} = 0$ . In every remaining case $T^{c} \neq \emptyset$ and $∥ D ∥_{o p} > ∥ A_{T} ∥_{o p}$ (with the empty-block norm set to zero). Write $D = diag (d_{i} : i \in T^{c})$ . By [F10] choose $j \in T^{c}$ with $∣ d_{j} ∣ = \max_{i \in T^{c}} ∣ d_{i} ∣$ . For every unit $x \in R^{T^{c}}$ , $∣ x^{T} D x ∣ = ∣ \sum_{i} d_{i} x_{i}^{2} ∣ \leq \max_{i} ∣ d_{i} ∣ \sum_{i} x_{i}^{2} = ∣ d_{j} ∣$ , and the coordinate vector $e_{j}$ attains equality. By [F5], $e_{j}$ therefore attains $∥ D ∥_{o p}$ . In either case some unit vector attaining $∥ A ∥_{o p}$ has support of size at most $2 s$ . Since $p \geq 2 s$ , its support extends to a set of size exactly $2 s$ , so the vector belongs to $U_{s}$ . Conversely every unit vector in $U_{s}$ is among all unit vectors, hence $∥ A ∥_{o p} = \sup_{u \in U_{s}} ∣ u^{T} A u ∣$ by [F5]. Finally, $2 s + \log (m / δ) \leq 4 s \log (e p / s) + \log (1 / δ) \leq 4 (s \log (e p / s) + \log (1 / δ)) .$ Squaring [F1]'s bound proves the displayed upper rate after adjusting $C_{M}$ .

1.2F2F3F6algebra

By [F2], the clean lower premise holds at $M_{0} = b_{M} s \log (e p / s) / n$ for sufficiently small $s \log (e p / s) / n$ , with a fixed positive probability $b_{0}$ . 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 all pairs in the bounded class.

2.1F2F3F4step 1.2algebra∎

Apply [F4] with these clean and modulus bounds and the unrestricted matrix-valued estimator class in the Statement. Its threshold $\frac{1}{16} (M_{0} \lor ω)$ is at least a constant multiple of $s \log (e p / s) / n \lor ϵ^{2}$ , and the probability lower bound is $\min (b_{0}, 1 / 2)$ . Lowering the threshold only enlarges the event, which proves the lower assertion for the event probabilities in the Statement.

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.

Depends on

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Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, Theorems 3.6, 3.7 and 5.1 (standard reference, not scraped)
Philippe Rigollet, 18.S997 High-Dimensional Statistics Lecture Notes, MIT OpenCourseWare, Chapter 5 (standard reference, not scraped)