A Microsoft researcher and Zhao Song prove the KLS conjecture

Microsoft researcher Xinzhi Zhang and collaborator Zhao Song posted a proof on October 4 that resolves the Kannan-Lovász-Simonovits (KLS) conjecture, a 1995 problem about how efficiently random walks explore high-dimensional probability distributions, Microsoft Research announced on October 21. Their paper establishes a bound that does not grow with the number of dimensions, the defining claim of the conjecture, after a human-directed AI search helped identify the key mathematical criterion while human judgment shaped and completed the proof.

A Microsoft researcher and Zhao Song prove the KLS conjecture The October proof removes dimension dependence from a 1995 bound, after a human-directed AI search helped identify the key mathematical criterion. By Ryan Merket https://runtimewire.com/author/ryan-merket · Published · Updated Primary source: Microsoft Research https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/ Why it matters The proof turns a widely used assumption about high-dimensional sampling into a theorem. Its AI contribution is also specific: guided exploration helped researchers find a useful criterion, while human mathematical judgment shaped and completed the proof. A proof posted on October 4th resolves the Kannan-Lovász-Simonovits conjecture, a 1995 problem about how efficiently random walks can explore high-dimensional probability distributions. Microsoft Research announced the result on October 21st, crediting Microsoft researcher Xinzhi Zhang https://www.microsoft.com/en-us/research/people/xinzhizhang/ and collaborator Zhao Song https://simons.berkeley.edu/people/zhao-song . Their paper https://arxiv.org/abs/2610.01447 establishes a bound that does not grow with the number of dimensions, the defining claim of the conjecture.… When you purchase through links in our articles, we may earn a small commission https://runtimewire.com/affiliate-monetization-standards . This doesn’t affect our editorial independence.