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Capped thresholded minimax rate for weak-lq sparse covariance

Statement

Assume AC. Fix $β > 1$ , $M, τ > 0$ , and $0 \leq q < 1$ . For all integers $n \geq 1$ , $p \geq 2$ with $p \geq n^{β}$ , and every radius $0 \leq c = c_{n, p} \leq M n^{(1 - q) / 2} (\log p)^{- (3 - q) / 2},$ put $a = \log (p) / n$ and $r_{n, p} = c^{2} a^{1 - q} + a$ . Over the law class $P_{q} (τ, c)$ of Weak-lq covariance class with directional sub-Gaussian tails, the minimax squared operator risk of Minimax squared operator risk satisfies $C_{1} \min {1, r_{n, p}} \leq R_{n} (P_{q} (τ, c)) \leq C_{2} \min {1, r_{n, p}},$ where $0 < C_{1} \leq C_{2} < \infty$ depend only on $β, M, τ, q$ . For $a \leq 1$ , the unmodified estimator of Entrywise-thresholded sample covariance, with $λ = 8192 τ \sqrt{a}$ , has expected squared operator risk at most $C (q, τ) r_{n, p}$ . A rule attaining the capped upper bound in every dimension is explicit: use that estimator when $r_{n, p} < 1$ , and output the zero matrix when $r_{n, p} \geq 1$ . All original dimension regimes, including super-exponential $p$ , are retained; the risk saturates instead of diverging. The raw threshold estimator alone is not claimed to attain the cap outside its stated regime.

Facts & Assumptions

Given: The fixed parameters, nonnegative radius, dimension growth, law class and risk conventions in the statement.

[F1]

In the regime $a \leq 1$ , under the n-fold product sample law $P^{\otimes n}$ , the raw threshold estimator with divisor $n$ and sample centering has expected risk at most $C (q, τ) (c^{2} a^{1 - q} + a)$ . Minimax squared operator risk Expected squared operator error of thresholded sparse covariance.

[F2]

The diagonal Gaussian subfamily gives risk at least $b τ^{2} \min {1, a}$ , for every radius including zero. Diagonal covariance lower bound with bounded-risk saturation.

[F3]

For fixed parameters, sufficiently large $p$ and $c / [2 (s v \sqrt{a})^{q}] \geq 2$ , the explicit sparse Gaussian mixture gives risk at least $b_{1} c^{2} a^{1 - q}$ , with $s = τ / 4$ and fixed $v > 0$ . Gaussian mixture lower bound for weakly sparse covariance.

[F4]

The class imposes the unit-direction tail bound $P (∣ v^{T} (X - E X) ∣ > t) \leq e^{- t^{2} / (2 τ)}$ . Weak-lq covariance class with directional sub-Gaussian tails.

[F5]

The covariance matrix is $Σ = E [(X - E X) (X - E X)^{T}]$ , so it is symmetric and positive semidefinite. Sample covariance estimators and squared operator loss.

[F6]

For nonnegative $T$ , $E [T] = \int_{0}^{\infty} P (T > u) d u$ . Layer-cake formulas for random variables.

[F7]

For a real symmetric matrix, the operator norm is the maximum absolute Rayleigh quotient. A symmetric matrix operator norm is its largest absolute Rayleigh quotient.

Proof

1.1givenF1F4F5F6F7algebra

For any unit vector $v$ , put $Y = v^{T} (X - E X)$ . By [F4] and layer cake [F6], $v^{T} Σ v = E [Y^{2}] = \int_{0}^{\infty} P (Y^{2} > u) d u \leq \int_{0}^{\infty} e^{- u / (2 τ)} d u = 2 τ$ . By [F5], $Σ$ is symmetric positive semidefinite, so [F7] gives $∥ Σ ∥_{o p} \leq 2 τ$ . The constant zero estimator therefore has squared loss at most $4 τ^{2}$ . If $r_{n, p} \geq 1$ , this proves the capped upper bound for the specified switch. If $r_{n, p} < 1$ , then $a < 1$ , and [F1] proves it for the threshold branch. The switch is measurable, since its branch depends only on the known dimensions and class radius; the threshold operations are measurable entrywise.

1.2givenF2F3algebra

Let $P_{0}$ and $v$ be the fixed constants in [F3]. For $p < P_{0}$ , $n \leq p^{1 / β} < P_{0}^{1 / β}$ , so $a \geq \log 2 / P_{0}^{1 / β}$ . Thus [F2] is bounded below by a positive fixed-parameter constant, which dominates a constant multiple of $\min {1, r_{n, p}} \leq 1$ . For $a \geq 1$ , [F2] likewise gives this constant lower bound, regardless of $p$ . These arguments cover all finite-size and saturated-dimension cases without a hidden asymptotic restriction.

2.1givenF2F3step 1.1step 1.2algebra∎

It remains to consider $p \geq P_{0}$ and $a < 1$ . If $c / [2 (s v \sqrt{a})^{q}] < 2$ , then $c^{2} a^{1 - q} < 16 (s v)^{2 q} a$ , and [F2] alone bounds the risk below by a positive fixed-parameter multiple of $r_{n, p}$ ; this includes all effective sparse counts rounding to zero and $q = 0$ . Otherwise [F3] supplies the sparse term, while [F2] supplies the $a$ term. The maximum of two nonnegative lower bounds is at least half their sum, giving a fixed-parameter multiple of $r_{n, p}$ and hence of its cap. Combining with step 1.2 and the upper bound proves the result.

Remarks

The uncapped claim copied from the source fails when $\log p / n$ diverges; The stated sparse-covariance minimax rate is unbounded for super-exponential dimension retains the precise obstruction. The cap corrects that claim without removing any dimension regime. The local sparse-mixture lemma proves its technical remainder estimate directly; no source supplement lemma is used without proof.

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Sources

T. Tony Cai and Harrison H. Zhou, Optimal Rates of Convergence for Sparse Covariance Matrix Estimation, Theorem 1 (standard reference, not scraped)