Microsoft Solves 31-Year-Old KLS Geometry Conjecture
Microsoft researchers solved the 31-year-old KLS conjecture using a human-directed AI workflow, establishing a fundamental geometric principle for high-dimensional algorithms.
Microsoft Research's Xinzhi Zhang and co-author Zhao Song have proven the Kannan-Lovász-Simonovits (KLS) conjecture, resolving a 31-year-old open problem in high-dimensional geometry and probability. First proposed in 1995, the conjecture posits that the narrowest bottlenecks of log-concave probability distributions can be captured by simple hyperplane cuts within a universal constant bound, regardless of dimension. By proving this dimension-free property, the researchers eliminated a long-standing theoretical obstacle that constrained the analysis of high-dimensional sampling and randomized algorithms.
The discovery relied on a human-directed agentic research process where AI systems evaluated more than 100 candidate mathematical approaches involving higher-order tensor calculations. While unguided runs of frontier AI models failed under identical token budgets, human researchers directed agents iteratively—pruning unpromising paths and feeding refined results back into recursive arguments. An October 1, 2026 draft lowered dimension dependence to an iterated logarithm, before the researchers applied a repeated recursive technique borrowed from fast linear programming algorithms. The final October 4, 2026 version proved that accumulated errors stay bounded by a fixed constant, completing the proof.
The breakthrough received rapid validation from the mathematics community. On the day the manuscript appeared, two separate expert groups posted parallel proofs using the team's key criterion, and the argument was subsequently machine-verified using the Lean 4 proof assistant. The researchers are now preparing a streamlined version of the proof for publication.
For computer scientists and statisticians, this proof removes theoretical dependencies across high-dimensional geometry and optimization. Theorems that previously assumed the validity of KLS can now be stated unconditionally, improving theoretical foundations for randomized exploration algorithms and high-dimensional sampling techniques used across machine learning and scientific computing.
This is our own summary of reporting by Microsoft Research Blog
