# Precise sample covariance spectral norm error -- an RDT view

arXiv: https://arxiv.org/abs/2607.14460 · HTML: https://mlsanity.com/paper/2607.14460/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 134 references that are in the map.
- First posted on arXiv 16 Jul 2026.

## Builds on (most cited first, 23 of 23)

- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
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- [Outlier Robust Mean Estimation with Subgaussian Rates via Stability](https://mlsanity.com/paper/2007.15618/) (2020)
- [Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model](https://mlsanity.com/paper/1506.00691/) (2015)
- [An Optimal Statistical and Computational Framework for Generalized Tensor Estimation](https://mlsanity.com/paper/2002.11255/) (2020)
- [Statistical limits of spiked tensor models](https://mlsanity.com/paper/1612.07728/) (2016)
- [Regularly random duality](https://mlsanity.com/paper/1303.7295/) (2013)
- [Covariance Estimation in High Dimensions via Kronecker Product Expansions](https://mlsanity.com/paper/1302.2686/) (2013)
- [User-Friendly Covariance Estimation for Heavy-Tailed Distributions](https://mlsanity.com/paper/1811.01520/) (2018)
- [Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails](https://mlsanity.com/paper/2205.08494/) (2022)
- [Fundamental limits for rank-one matrix estimation with groupwise heteroskedasticity](https://mlsanity.com/paper/2106.11950/) (2021)
- [Robust low-rank covariance matrix estimation with a general pattern of missing values](https://mlsanity.com/paper/2107.10505/) (2021)
- [Optimal Algorithms for the Inhomogeneous Spiked Wigner Model](https://mlsanity.com/paper/2302.06665/) (2023)
- [Optimal Sparse Singular Value Decomposition for High-dimensional High-order Data](https://mlsanity.com/paper/1809.01796/) (2018)
- [Implicit High-Order Moment Tensor Estimation and Learning Latent Variable Models](https://mlsanity.com/paper/2411.15669/) (2024)
- [A CLuP algorithm to practically achieve ~ 0.76 SK--model ground state free energy](https://mlsanity.com/paper/2507.09247/) (2025)
- [Estimation of large block structured covariance matrices: Application to "multi-omic" approaches to study seed quality](https://mlsanity.com/paper/1806.10093/) (2018)
- [Structured Channel Covariance Estimation from Limited Samples in Massive MIMO](https://mlsanity.com/paper/1910.14467/) (2019)
- [Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians](https://mlsanity.com/paper/2301.09024/) (2023)
- [Binary perceptron computational gap -- a parametric fl RDT view](https://mlsanity.com/paper/2511.01037/) (2025)
- [Estimation of functionals of sparse covariance matrices](https://mlsanity.com/paper/1408.5087/) (2014)
- [Minimax Rate-optimal Estimation of High-dimensional Covariance Matrices with Incomplete Data](https://mlsanity.com/paper/1605.04358/) (2016)

## 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.