A Chinese associate professor born in the 1990s has broken through a 30-year-old mathematical conjecture, and the conclusion is directly related to generative AI.

The Talagrand's convolution conjecture, which has puzzled the mathematical community for over 30 years, has been solved by a Chinese mathematician born in the 1990s!

Yuansi Chen from the Swiss Federal Institute of Technology in Zurich has just published his latest research results on arXiv:

The paper proves Talagrand's convolution conjecture on the Boolean hypercube, with the result accurate to a log log η factor.

This result has attracted a great deal of attention. Simply put, it provides a mathematical argument for understanding the smoothing in high - dimensional discrete spaces.

In addition, this research is also closely related to machine learning:

It theoretically supports the concept of regularization in machine learning;

It provides direct mathematical tools and physical intuition for developing generative AI models for processing discrete data.

Cracking a 30 - year - old mathematical problem

Talagrand's convolution conjecture was proposed in 1989 by Michel Talagrand, the winner of the Abel Prize, known as the "Nobel Prize of mathematics".

Let's first understand two concepts. One is "heating smoothing":

Imagine a very high - dimensional space, such as a huge multi - dimensional chessboard, where the state of each square is a binary choice. There is a function, which may be very "sharp", with extremely large values in some places and extremely small values in others.

The mathematical operations of "convolution" or "heat semigroup" are like "heating" this function, allowing the heat to spread and the high values to flow to the surrounding low - value areas. As a result, the function becomes smooth, and the peaks are flattened.

The second is Markov's inequality:

Markov's inequality tells us that the probability that a non - negative random variable takes an extremely large value is very small. For example, if the average value is 1, the probability that the value exceeds 100 (η) is at most 1% (i.e., 1/η).

Talagrand's conjecture is that after performing the "heating smoothing" (convolution) operation on a function in a probability space such as the Gaussian space or the Boolean hypercube, the probability that this function takes an extremely large value should be much lower than that predicted by Markov's inequality.

He believes that this probability is not only controlled by 1/η but should also be additionally divided by a factor related to

That is, Talagrand's convolution conjecture holds that the possibility of extreme outliers in the smoothed data is lower than that predicted by general theory by a specific magnitude.

△

Previously, the Gaussian form (continuous space) of this conjecture has been solved by mathematicians. However, extending it to discrete spaces such as the Boolean hypercube remains a huge challenge.

The solution of the Gaussian form is based on the smoothness and the completeness of tools provided by calculus and stochastic differential equations in the continuous space. These characteristics cannot be directly transferred to discrete spaces.

In response, Yuansi Chen's solution is to draw on the framework of stochastic analysis in the Gaussian space and use the characteristics of the reverse heat process to design perturbations to adapt to the discrete characteristics of the Boolean hypercube.

Specifically, the new coupling construction uses perturbations along the stochastic process. The perturbation term δ is not a constant but depends on the state and coordinates.

The paper finally proves:

It shows that the core idea of Talagrand's convolution conjecture is correct.

This result solves the original conjecture with an accuracy that differs only by a log log η factor. Since the growth of log log η is extremely slow, it can be considered that the Talagrand's convolution conjecture is nearly completely solved.

It is worth noting that this paper is a pure mathematical study on probability theory, but its results are directly related to machine learning and even generative AI technology.

First, the "reverse heat process" used in the paper is the counterpart of the diffusion model on the Boolean hypercube, and the two are highly similar.

This means that this research may help to understand or develop diffusion generative models for discrete data.

Second, the core of Talagrand's convolution conjecture is to quantify the regularization effect brought by the convolution operation. In machine learning, regularization is a key means to prevent model overfitting and improve generalization ability.

This result provides theoretical support for "why smoothing or adding noise can make the model more stable in complex high - dimensional spaces".

In addition, in machine learning, many data are essentially discrete and high - dimensional. This research helps to understand the geometric properties of high - dimensional discrete spaces and is valuable for developing learning theories for binary data or logical functions.

A Chinese mathematician born in the 1990s

The author of the paper, Yuansi Chen, was born in July 1990 and is from Ningbo, Zhejiang.

His main research areas include statistical machine learning, Markov chain Monte Carlo methods, applied probability, high - dimensional geometry, etc.

In 2019, he graduated with a doctorate from the University of California, Berkeley, under the supervision of the Chinese statistician Bin Yu.

After two years of postdoctoral research at the Swiss Federal Institute of Technology in Zurich, he joined Duke University from 2021 to 2024 as an assistant professor in the Department of Statistical Science. In early 2024, he transferred to the Swiss Federal Institute of Technology in Zurich as an associate professor.

Google Scholar shows that his papers have been cited 1623 times, and his h - index is 13.

He is also the recipient of the Sloan Research Fellowship in 2023.

Previously, his work on the KLS conjecture also attracted a lot of attention: A Chinese statistics doctoral student solved the "apple - cutting" problem that had puzzled mathematicians for 25 years.

Paper link: https://arxiv.org/abs/2511.19374

This article is from the WeChat public account "QbitAI". The author focuses on cutting - edge technology. 36Kr is authorized to publish it.