{"slug": "a-microsoft-researcher-and-zhao-song-prove-the-kls-conjecture", "title": "A Microsoft researcher and Zhao Song prove the KLS conjecture", "summary": "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.", "body_md": "# A Microsoft researcher and Zhao Song prove the KLS conjecture\n\n**The October proof removes dimension dependence from a 1995 bound, after a human-directed AI search helped identify the key mathematical criterion.**\n\n        By [Ryan Merket](https://runtimewire.com/author/ryan-merket)\n        · Published \n        · Updated \n\nPrimary source: [Microsoft Research](https://www.microsoft.com/en-us/research/blog/solving-a-31-year-old-grand-challenge-in-mathematics-the-kls-conjecture/)\n\n## Why it matters\n\nThe 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.\n\nA 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.…\n\nWhen 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.", "url": "https://wpnews.pro/news/a-microsoft-researcher-and-zhao-song-prove-the-kls-conjecture", "canonical_source": "https://runtimewire.com/article/microsoft-research-kls-conjecture-proof", "published_at": "2026-10-21 17:45:31+00:00", "updated_at": "2026-10-21 21:13:05.384343+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research", "machine-learning"], "entities": ["Microsoft Research", "Xinzhi Zhang", "Zhao Song", "Kannan-Lovász-Simonovits conjecture"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/a-microsoft-researcher-and-zhao-song-prove-the-kls-conjecture", "markdown": "https://wpnews.pro/news/a-microsoft-researcher-and-zhao-song-prove-the-kls-conjecture.md", "text": "https://wpnews.pro/news/a-microsoft-researcher-and-zhao-song-prove-the-kls-conjecture.txt", "jsonld": "https://wpnews.pro/news/a-microsoft-researcher-and-zhao-song-prove-the-kls-conjecture.jsonld"}}