# Classification of Data Generated by Gaussian Mixture Models Using Deep ReLU Networks

arXiv: https://arxiv.org/abs/2308.08030 · HTML: https://mlsanity.com/paper/2308.08030/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 41 references that are in the map.
- First posted on arXiv 15 Aug 2023.

## Builds on (most cited first, 10 of 10)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [Fast convergence rates of deep neural networks for classification](https://mlsanity.com/paper/1812.03599/) (2018)
- [Convergence rates of deep ReLU networks for multiclass classification](https://mlsanity.com/paper/2108.00969/) (2021)
- [Multi-class SVMs: From Tighter Data-Dependent Generalization Bounds to Novel Algorithms](https://mlsanity.com/paper/1506.04359/) (2015)
- [Learning Ability of Interpolating Deep Convolutional Neural Networks](https://mlsanity.com/paper/2210.14184/) (2022)
- [Training shallow ReLU networks on noisy data using hinge loss: when do we overfit and is it benign?](https://mlsanity.com/paper/2306.09955/) (2023)
- [Bias of Homotopic Gradient Descent for the Hinge Loss](https://mlsanity.com/paper/1907.11746/) (2019)

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

- [A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative Models](https://mlsanity.com/paper/2401.07187/) (2024)
- [Generalization analysis with deep ReLU networks for metric and similarity learning](https://mlsanity.com/paper/2405.06415/) (2024)
- [Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets](https://mlsanity.com/paper/2309.09258/) (2023)
- [Model-agnostic information transfer and fusion for classification with label noise](https://mlsanity.com/paper/2604.25845/) (2026)

## Abstract (arXiv)

This paper studies the binary classification of unbounded data from mathbb Rd generated under Gaussian Mixture Models (GMMs) using deep ReLU neural networks. We obtain unicodex2013 for the first time unicodex2013 non-asymptotic upper bounds and convergence rates of the excess risk (excess misclassification error) for the classification without restrictions on model parameters. The convergence rates we derive do not depend on dimension d, demonstrating that deep ReLU networks can overcome the curse of dimensionality in classification. While the majority of existing generalization analysis of classification algorithms relies on a bounded domain, we consider an unbounded domain by leveraging the analyticity and fast decay of Gaussian distributions. To facilitate our analysis, we give a novel approximation error bound for general analytic functions using ReLU networks, which may be of independent interest. Gaussian distributions can be adopted nicely to model data arising in applications, e.g., speeches, images, and texts; our results provide a theoretical verification of the observed efficiency of deep neural networks in practical classification problems.