AI helped prove the KLS conjecture

AI helped prove the KLS conjecture — a 1995 problem

Zhao Sun and Xinchi Zhang (Microsoft Research) proved the 1995 KLS conjecture: the Cheeger constant ψn of any isotropic log-concave distribution in R^n is bounded below by a universal constant independent of dimension — the key idea was found with the help of AI agent search and human selection.

Compiled by Sergey KostenchukPublished 2026-10-22Updated 2026-10-22

2026-10-22 Research Google

🤖 A 1995 problem proved with the help of AI

Zhao Sun and Xinchi Zhang from Microsoft Research proved the KLS conjecture: the Cheeger constant ψn of any isotropic log-concave distribution in R^n is bounded by a universal constant independent of dimension. AI agents explored over 100 approaches, with the authors filtering out weak ones; model trials without human involvement yielded no results.

🌍 The KLS conjecture is a central open problem in multidimensional convex geometry. The O(1) bound establishes the foundation for the convergence speed of random walks and MCMC sampling: the dimensionality barrier has been removed.

👤 For MCMC practitioners, convergence guarantees for log-concave distributions no longer depend on dimension via KLS. Other teams have independently confirmed the result, and verification is underway in Lean 4.

Source 1: https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/ Source 2: https://arxiv.org/abs/2610.01447

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