# A Novel Gaussian Min-Max Theorem and its Applications

arXiv: https://arxiv.org/abs/2402.07356 · HTML: https://mlsanity.com/paper/2402.07356/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 12 Feb 2024.

## Builds on (most cited first, 18 of 18)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [A Rewriting System for Convex Optimization Problems](https://mlsanity.com/paper/1709.04494/) (2017)
- [Universality of empirical risk minimization](https://mlsanity.com/paper/2202.08832/) (2022)
- [The role of regularization in classification of high-dimensional noisy Gaussian mixture](https://mlsanity.com/paper/2002.11544/) (2020)
- [Learning Gaussian Mixtures with Generalised Linear Models: Precise Asymptotics in High-dimensions](https://mlsanity.com/paper/2106.03791/) (2021)
- [Precise Statistical Analysis of Classification Accuracies for Adversarial Training](https://mlsanity.com/paper/2010.11213/) (2020)
- [A framework to characterize performance of LASSO algorithms](https://mlsanity.com/paper/1303.7291/) (2013)
- [The Impact of Regularization on High-dimensional Logistic Regression](https://mlsanity.com/paper/1906.03761/) (2019)
- [Theoretical Insights Into Multiclass Classification: A High-dimensional Asymptotic View](https://mlsanity.com/paper/2011.07729/) (2020)
- [The generalization error of max-margin linear classifiers: Benign overfitting and high dimensional asymptotics in the overparametrized regime](https://mlsanity.com/paper/1911.01544/) (2019)
- [Universality laws for Gaussian mixtures in generalized linear models](https://mlsanity.com/paper/2302.08933/) (2023)
- [Scaling laws for learning with real and surrogate data](https://mlsanity.com/paper/2402.04376/) (2024)
- [On the Inherent Regularization Effects of Noise Injection During Training](https://mlsanity.com/paper/2102.07379/) (2021)
- [Sharp Asymptotics and Optimal Performance for Inference in Binary Models](https://mlsanity.com/paper/2002.07284/) (2020)
- [The Performance Analysis of Generalized Margin Maximizer (GMM) on Separable Data](https://mlsanity.com/paper/2010.15379/) (2020)
- [Regularized Linear Regression for Binary Classification](https://mlsanity.com/paper/2311.02270/) (2023)
- [Wasserstein Distributionally Robust Estimation in High Dimensions: Performance Analysis and Optimal Hyperparameter Tuning](https://mlsanity.com/paper/2206.13269/) (2022)
- [Uniform Convergence with Square-Root Lipschitz Loss](https://mlsanity.com/paper/2306.13188/) (2023)

## Built on by (most cited first, 4 of 4)

- [Characterization of Gaussian Universality Breakdown in High-Dimensional Empirical Risk Minimization](https://mlsanity.com/paper/2604.03146/) (2026)
- [One-Bit Quantization for Random Features Models](https://mlsanity.com/paper/2510.16250/) (2025)
- [Asymptotic Behavior of Multi--Task Learning: Implicit Regularization and Double Descent Effects](https://mlsanity.com/paper/2603.05060/) (2026)
- [Precise Performance of Linear Denoisers in the Proportional Regime](https://mlsanity.com/paper/2603.18483/) (2026)

## Abstract (arXiv)

A celebrated result by Gordon allows one to compare the min-max behavior of two Gaussian processes if certain inequality conditions are met. The consequences of this result include the Gaussian min-max (GMT) and convex Gaussian min-max (CGMT) theorems which have had far-reaching implications in high-dimensional statistics, machine learning, non-smooth optimization, and signal processing. Both theorems rely on a pair of Gaussian processes, first identified by Slepian, that satisfy Gordon's comparison inequalities. In this paper, we identify such a new pair. The resulting theorems extend the classical GMT and CGMT Theorems from the case where the underlying Gaussian matrix in the primary process has iid rows to where it has independent but non-identically-distributed ones. The new CGMT is applied to the problems of multi-source Gaussian regression, as well as to binary classification of general Gaussian mixture models.