Researchers at Microsoft Research have proved the 31-year-old Kannan-Lovász-Simonovits, or KLS, conjecture, a long-standing open problem regarded as central to high-dimensional geometry. The work was led by Zhang Xinzhi and colleagues in the organization’s Machine Learning and Optimization group, according to Techub News. The result establishes a general geometric principle for log-concave distributions, showing that even in very high dimensions, simple hyperplane cuts can capture the narrowest bottleneck within a constant factor. The conjecture has implications across statistics, optimization, algorithms, and computer science.
Techub said the breakthrough came out of a human-AI research process. Researchers guided an AI system to explore mathematical possibilities, then evaluated, pruned, and redirected those lines of inquiry. The result has since been checked by multiple independent researchers and also completed machine-checked formal verification. First proposed in 1995, the KLS conjecture had stood as a major theoretical barrier to understanding the efficiency of random algorithms widely used in science and computing. Earlier best-known results allowed the bound to grow slowly with dimension, while the new proof shows the relationship is bounded by a constant.
Microsoft Research researcher Zhang Xinzhi and colleagues in the company’s Machine Learning and Optimization group have proved the Kannan-Lovász-Simonovits, or KLS, conjecture, according to Techub News.
The conjecture had remained open for 31 years and is widely seen as a central problem in high-dimensional geometry, with links to statistics, optimization, algorithms, and computer science. The new result establishes a general geometric principle for log-concave distributions. It shows that no matter how large the dimension becomes, a simple hyperplane cut can still capture the narrowest bottleneck within a constant factor.
Human-AI collaboration shaped the research process
Techub said the breakthrough emerged from a human-AI collaboration. Researchers guided an AI system to explore possible mathematical paths, then evaluated the outputs, pruned unpromising branches, and adjusted the direction of the work.
The result has already been verified by multiple independent researchers. It has also undergone machine-checked formal verification.
A long-standing barrier for random algorithm theory
The KLS conjecture was proposed in 1995. Its proof removes a major theoretical obstacle in understanding the efficiency of random algorithms that are widely used in science and computing.
The previous best result allowed the bound to grow slowly as dimension increased. The new work proves that the relationship is bounded by a constant, indicating that high-dimensional space behaves better than previously known.
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