A long-standing mathematical challenge, Talagrand’s Convolution Conjecture, which has confounded mathematicians for over three decades, has finally been cracked by a Chinese associate professor born in the 1990s.

In 1989, French mathematician Michel Talagrand put forth a conjecture concerning the regularization effects of applying convolution to the L function on Boolean hypercubes.

The paper provides a proof for Talagrand’s Convolution Conjecture on Boolean hypercubes, with the results being accurate up to a factor of log log η.

This conjecture, proposed by Abel Prize laureate Michel Talagrand in 1989, has long been regarded as a formidable problem in the realm of high-dimensional probability and analysis.

To comprehend this conjecture, it's essential to first grasp two fundamental concepts.

The first is heat smoothing. Picture an incredibly high-dimensional space, akin to a vast multidimensional chessboard, where each square has only two possible states. A function is defined on this space, and it could be quite erratic, with some points being extremely high and others unusually low.

In mathematical terms, convolution or heat semigroup operations are similar to heating up this function:

Heat spreads to the surrounding areas, with high values flowing to low values, thus smoothing out the entire function and flattening any sharp peaks.

The second concept is Markov's inequality, which states that the probability of a non-negative random variable taking on an extremely large value is small.

For instance, if the mean value is 1, the probability of the variable exceeding 100 is at most 1% (expressed as 1/η).

Talagrand hypothesized that in probability spaces such as Gaussian spaces and Boolean hypercubes, after applying heat smoothing to a function, the probability of the function taking on extremely large values should not only be controlled by 1/η but also be further reduced by a factor related to the function's range.

In simpler terms, after smoothing, the data should exhibit even fewer extreme outliers than what is generally theorized.

While the conjecture for continuous spaces (Gaussian case) has already been resolved, extending it to discrete spaces (such as Boolean hypercubes) was deemed a daunting task.

The first reason is that continuous spaces rely on tools like calculus and stochastic differential equations. The second reason is that discrete spaces lack these 'smooth structures,' making it impossible to directly adapt the methods used in continuous spaces.

Consequently, the problem remained unsolved for an extended period.

Yuansi Chen's core strategy was to draw inspiration from stochastic analysis in Gaussian spaces and devise a reverse heat process that could be adapted to discrete structures.

Key innovations in his approach include:

Firstly, a new coupling construction introduces perturbations along the stochastic process. However, the perturbation δ is not constant; instead, it varies depending on the state and coordinates.

Secondly, this non-uniform perturbation enables the concepts of heating and cooling to be reestablished on Boolean hypercubes, thereby reconstructing high-dimensional analytical tools.

Ultimately, the paper demonstrates that the core idea of Talagrand's conjecture is correct, with an error margin of only a nearly negligible log log η factor.

Although the paper is rooted in pure mathematics, its theories have a natural connection to modern machine learning, particularly generative AI.

Diffusion models serve as the foundational mechanism of generative AI. The 'reverse heat process' described in the paper is its discrete counterpart. This breakthrough could drive advancements in:

The theoretical design of discrete data generation models.

More powerful generative methods for binary and logical functions.

Talagrand's conjecture essentially provides a quantitative description of the 'regularization effects brought about by convolution.'

In machine learning, regularization plays a pivotal role in enhancing generalization and preventing overfitting.

Why do operations such as adding noise, smoothing, and diffusion contribute to the stability of high-dimensional models? This result offers a deeper theoretical explanation:

Much of the data (including text, binary features, and logical structures) is inherently high-dimensional and discrete. This research will aid in understanding their geometric structures and developing new learning theories.

The paper was authored by Chinese mathematician Yuansi Chen, born in 1990.

Hailing from Ningbo, Zhejiang, Yuansi Chen focuses his research on statistical machine learning, Markov chain Monte Carlo methods, applied probability, and high-dimensional geometry.

In 2019, he earned his Ph.D. from the University of California, Berkeley, under the guidance of renowned statistician Bin Yu.

After conducting two years of postdoctoral research at ETH Zurich, he served as an assistant professor in the Department of Statistical Science at Duke University from 2021 to 2024. In 2024, he returned to ETH Zurich as an associate professor.

References:

https://arxiv.org/abs/2511.19374