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Precise sample covariance spectral norm error -- an RDT view

arXiv 2607.14460 ↗First posted 16 Jul 2026math.ST cs.IT math.ITcited by 0

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Abstract (arXiv)

We study the sample covariance error of centered Gaussians. A remarkable breakthrough [66] established the correct error scaling order and explicitly revealed the critical role of both the effective rank and the true covariance spectrum. In this work, we move beyond scaling characterizations and determine the precise limiting value of the error's spectral norm. To do so, we develop a generic framework based on Random Duality Theory (RDT). Within this framework, we first determine closed-form, explicit RDT-based upper bounds. We then establish complementary lower bounds by introducing a novel bilinear-quadratic RDT lower-bounding mechanism. By combining this mechanism with a two-replica systems bounding strategy, we show that our lower and upper bounds match in large-dimensional contexts. Our theoretical results are supplemented with numerical evaluations and simulations, demonstrating an excellent agreement already for problem sizes on the order of thousands.

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References · 23

  1. 2019Mean estimation and regression under heavy-tailed distributions--a surveycited by 136
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  11. 2022Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tailscited by 14
  12. 2021Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticitycited by 8
11 more
  1. 2021Robust low-rank covariance matrix estimation with a general pattern of missing valuescited by 8
  2. 2023Optimal Algorithms for the Inhomogeneous Spiked Wigner Modelcited by 8
  3. 2018Optimal Sparse Singular Value Decomposition for High-dimensional High-order Datacited by 6
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  5. 2025A CLuP algorithm to practically achieve ~ 0.76 SK--model ground state free energycited by 5
  6. 2018Estimation of large block structured covariance matrices: Application to "multi-omic" approaches to study seed qualitycited by 3
  7. 2019Structured Channel Covariance Estimation from Limited Samples in Massive MIMOcited by 3
  8. 2023Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussianscited by 3
  9. 2025Binary perceptron computational gap -- a parametric fl RDT viewcited by 3
  10. 2014Estimation of functionals of sparse covariance matricescited by 2
  11. 2016Minimax Rate-optimal Estimation of High-dimensional Covariance Matrices with Incomplete Datacited by 2

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