# Asymptotic Bayes risk for Gaussian mixture in a semi-supervised setting

arXiv: https://arxiv.org/abs/1907.03792 · HTML: https://mlsanity.com/paper/1907.03792/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2019.

## Built on by (most cited first, 15 of 15)

- [Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeed](https://mlsanity.com/paper/2102.11742/) (2021)
- [Gaussian Universality of Perceptrons with Random Labels](https://mlsanity.com/paper/2205.13303/) (2022)
- [Classification of Heavy-tailed Features in High Dimensions: a Superstatistical Approach](https://mlsanity.com/paper/2304.02912/) (2023)
- [Bias in Motion: Theoretical Insights into the Dynamics of Bias in SGD Training](https://mlsanity.com/paper/2405.18296/) (2024)
- [Random Matrix Analysis to Balance between Supervised and Unsupervised Learning under the Low Density Separation Assumption](https://mlsanity.com/paper/2310.13434/) (2023)
- [When and How Unlabeled Data Provably Improve In-Context Learning](https://mlsanity.com/paper/2506.15329/) (2025)
- [PCA-based Multi Task Learning: a Random Matrix Approach](https://mlsanity.com/paper/2111.00924/) (2021)
- [The Quantum Path Kernel: a Generalized Quantum Neural Tangent Kernel for Deep Quantum Machine Learning](https://mlsanity.com/paper/2212.11826/) (2022)
- [Asymptotic Bayes risk of semi-supervised multitask learning on Gaussian mixture](https://mlsanity.com/paper/2303.02048/) (2023)
- [Universal Lower Bounds and Optimal Rates: Achieving Minimax Clustering Error in Sub-Exponential Mixture Models](https://mlsanity.com/paper/2402.15432/) (2024)
- [The Effect of Optimal Self-Distillation in Noisy Gaussian Mixture Model](https://mlsanity.com/paper/2501.16226/) (2025)
- [Bayesian Inference with Nonlinear Generative Models: Comments on Secure Learning](https://mlsanity.com/paper/2201.09986/) (2022)
- [Asymptotic Bayes risk of semi-supervised learning with uncertain labeling](https://mlsanity.com/paper/2403.17767/) (2024)
- [Analysis of High-dimensional Gaussian Labeled-unlabeled Mixture Model via Message-passing Algorithm](https://mlsanity.com/paper/2411.19553/) (2024)
- [Optimal Exact Recovery in Semi-Supervised Learning: A Study of Spectral Methods and Graph Convolutional Networks](https://mlsanity.com/paper/2412.13754/) (2024)

## Abstract (arXiv)

Semi-supervised learning (SSL) uses unlabeled data for training and has been shown to greatly improve performance when compared to a supervised approach on the labeled data available. This claim depends both on the amount of labeled data available and on the algorithm used. In this paper, we compute analytically the gap between the best fully-supervised approach using only labeled data and the best semi-supervised approach using both labeled and unlabeled data. We quantify the best possible increase in performance obtained thanks to the unlabeled data, i.e. we compute the accuracy increase due to the information contained in the unlabeled data. Our work deals with a simple high-dimensional Gaussian mixture model for the data in a Bayesian setting. Our rigorous analysis builds on recent theoretical breakthroughs in high-dimensional inference and a large body of mathematical tools from statistical physics initially developed for spin glasses.