A team of researchers, including Xinzhi Zhang from Microsoft Research, has achieved a major breakthrough by resolving the Kannan-Lovász-Simonovits (KLS) conjecture, a 31-year-old problem in high-dimensional geometry and probability. The KLS conjecture, proposed in 1995, questions whether flat or hyperplane cuts are nearly as effective as cuts of any shape for a broad class of high-dimensional probability distributions, regardless of the number of dimensions. This resolution establishes a universal geometric principle that simplifies the understanding of high-dimensional spaces, showing that their obstacles do not grow without bound as dimensions increase.
The breakthrough was achieved through a collaborative process involving human-directed AI systems. The researchers explored over 100 candidate approaches, many involving complex calculations with higher-order tensors. The final proof involved a recursive refinement method, inspired by earlier work on fast algorithms for linear programming, which eliminated the dependence on the number of dimensions. AI tools assisted in exploring candidate approaches, performing detailed derivations, and checking proofs, but the mathematical direction and final argument were determined by the researchers.
Independent confirmation of the research has strengthened its validity. On the same day the authors posted their complete proof, separate groups of experts developed parallel proofs, and a Lean 4 formalization was subsequently released. These independent proofs demonstrate that the result is not dependent on a single chain of reasoning, providing strong validation of the underlying principle. The researchers plan to release a shortened and verified version of their proof after final review.
The resolution of the KLS conjecture has significant implications for high-dimensional geometry, probability, and theoretical computer science. It removes a major theoretical obstacle to understanding the efficiency of randomized algorithms used across science and computing. The research also highlights the power of human-AI collaboration, where researchers direct AI systems to explore mathematical possibilities while evaluating and pruning the search.