Extreme value theory for singular subspace estimation in the matrix denoising model

TL;DR

This work develops a novel extreme-value inference framework for singular subspace estimation in the Gaussian matrix denoising model, centering on the maximum row-wise error $\|\widehat{\mathbf{U}}\mathbf{R}_{\mathbf{U}} - \mathbf{U}\|_{2,\infty}$ and its Gumbel limit after appropriate centering and scaling. By combining row-wise perturbation analysis, random matrix theory, and saddle-point approximations, the authors show tail equivalence to a generalized gamma distribution and derive explicit normalizers, enabling a principled, data-driven plug-in test based on de-biased singular values. They develop asymptotic Type I error control and power results for hypothesis tests on singular subspaces, with empirical evidence showing enhanced power for row-localized alternatives compared to Frobenius-norm based tests. The framework demonstrates robustness to certain non-Gaussian noise regimes and provides a practical approach for testing low-rank structure in high-dimensional denoising tasks. Overall, the paper advances extreme-value theory in matrix denoising and offers a powerful, fine-grained testing methodology for subspace structure with potential extensions to heteroskedastic and non-Gaussian settings.

Abstract

This paper studies fine-grained singular subspace estimation in the matrix denoising model where a deterministic low-rank signal matrix is additively perturbed by a stochastic matrix of Gaussian noise. We establish that the maximum Euclidean row norm (i.e., the two-to-infinity norm) of the aligned difference between the leading sample and population singular vectors approaches the Gumbel distribution in the large-matrix limit, under suitable signal-to-noise conditions and after appropriate centering and scaling. We apply our novel asymptotic distributional theory to test hypotheses of low-rank signal structure encoded in the leading singular vectors and their corresponding principal subspace. We provide de-biased estimators for the corresponding nuisance signal singular values and show that our proposed plug-in test statistic has desirable properties. Notably, compared to using the Frobenius norm subspace distance, our test statistic based on the two-to-infinity norm empirically has higher power to detect structured alternatives that differ from the null in only a few matrix entries or rows. Our main results are obtained by a novel synthesis of and technical analysis involving row-wise matrix perturbation analysis, extreme value theory, saddle point approximation methods, and random matrix theory. Our contributions complement the existing literature for matrix denoising focused on minimaxity, mean squared error analysis, unitarily invariant distances between subspaces, component-wise asymptotic distributional theory, and row-wise uniform error bounds. Numerical simulations illustrate our main results and demonstrate the robustness properties of our testing procedure to non-Gaussian noise distributions.

Figures (8)

• Figure 1: Empirical test statistic distributions for simulated data under $\operatorname{H}_{0}$ in \ref{['body-eq:T1']}. Left column: test statistic ${T_{{\mathbf{s}},n}}$ using the signal singular values. Middle column: test statistic ${\widetilde{T}_{\tilde{\mathbf{s}},n}}$ using the de-biased sample singular values. Right column: test statistic ${\widetilde{T}_{\hat{\mathbf{s}},n}^{\operatorname{uc}}}$ using the uncorrected sample singular values. See \ref{['body-section:simulation-setup-convergence']}.
• Figure 2: Contour plots depicting empirical rejection rates for the tests with rejection rules ${\widetilde{T}_{\tilde{\mathbf{s}},n}} \geq F^{-1}_{G}(0.95)$ and ${\widetilde{T}_{\operatorname{Frob}}} \geq F_{Z}^{-1}(0.95)$, respectively, for various combinations of the signal strength $s_{r}$ and the row discrepancy $\xi$. See \ref{['body-section:power-against-local-alternatives']}.
• Figure 3: Empirical rejection rates using ${\widetilde{T}_{\tilde{\mathbf{s}},n}}$ for various pairs of $(n,d_{n})$. Above the dotted line is the exterior of the parameter space $\Theta$, where the assumptions for \ref{['body-theorem:power-analysis']} do not hold. The dashed line represents the threshold for consistent testing. See \ref{['body-section:power-phase-transition']}.
• Figure 4: Values of ${\widetilde{T}_{\tilde{\mathbf{s}},n}}$ under noise from different standardized $t$ distributions. See \ref{['body-section:departures-from-Gaussianity']}.
• Figure 5: An illustration of the function $K'$. The saddle point $\widehat{t}_{x}$ is the value of $t$ at which $K'(t) = x$. The value of $\widehat{t}_{x}$ is uniquely defined on $t<1$, and approaches $1$ from below as $x$ grows to infinity.

Theorems & Definitions (53)

• Definition 1: Signal matrix quantities
• Lemma 1: First-order approximation of the left principal subspace
• Lemma 2: Moment generating function of $X_{i}^{2}$
• Definition 2: Tail equivalence
• Proposition 3: Tail equivalence of $X$ and $H$
• Theorem 4: Extreme value asymptotics for singular subspace estimation
• Remark 1: Scaling property of the generalized gamma distribution
• Theorem 5: Extreme value CDF convergence rate for singular subspace estimation
• Proposition 6: De-biased singular value estimators
• Theorem 7: Data-driven plug-in version of \ref{['body-theorem:Gumbel-convergence']}