But when it comes to computer science applications, Bubeck said, “you don’t need the full conjecture to get the full impact.”Ĭhen is not a convex geometer by training - instead, he is a statistician who became interested in the KLS conjecture because he wanted to get a handle on random sampling. Chen’s work doesn’t quite prove the full KLS conjecture. And for a wide range of different computer science problems, he said, “the most important subroutine in the algorithm is, you want to sample a random point.”Ĭhen’s new result gives instant improvements to the known running times of algorithms for tasks such as computing the volume of a convex shape or sampling from an assortment of machine learning models. Random walks are pretty much the only effective methods available for sampling random points, Eldan said. What’s more, the KLS conjecture lies at the heart of many questions in statistics and computer science, such as how long it will take for heat to diffuse through a convex shape, or how many steps a random walker must take from a starting point before reaching a truly random location. This 25-year-old conjecture, which asks about the best way to slice a shape into two equal portions, implies Bourgain’s conjecture. The new paper, by Yuansi Chen - a postdoctoral researcher at the Swiss Federal Institute of Technology Zurich who is about to join the statistical science faculty at Duke University - gets at the Bourgain slicing problem via an even more far-reaching question about convex geometry called the KLS conjecture. Now, Bourgain’s guess has been vindicated: A paper posted online in November has proved, not quite Bourgain’s full conjecture, but a version so close that it puts a strict limit on high-dimensional weirdness, for all practical purposes.īourgain, said Klartag, “would have dreamt” of achieving a result this strong. “The beauty of high-dimensional geometry is exactly that it doesn’t look anything like dimension two,” said Sébastien Bubeck of Microsoft Research in Redmond, Washington.īourgain’s slicing conjecture is a vote for high-dimensional tameness - a guess that high-dimensional shapes conform to our intuition in at least some ways. For example, in dimensions 10 and up, it is possible to build a cube and a ball such that the cube has larger volume than the ball, but every slice through the center of the cube has smaller area than the corresponding slice through the center of the ball. The difficulty is that high-dimensional shapes often behave in ways that defy our human, low-dimensional intuition. “And then, the more you think about it, the more you understand how delicate it really is.” “Come on - how hard can it be?” Ronen Eldan, a high-dimensional geometer at the Weizmann Institute remembers thinking when he first heard of the problem. After all, if the shape were extremely skinny in every direction, how could it have enough substance to form one unit of volume? In particular, he conjectured that there is some universal constant, independent of the dimension, such that every shape contains at least one slice with area greater than this constant.Īt first glance, Bourgain’s conjecture might seem obviously true. Moreover, Inter-GPS incorporates theorem knowledge as conditional rules and performs symbolic reasoning step by step.Bourgain guessed that some of these lower-dimensional slices must have substantial area. Inter-GPS parses the problem text and diagram into formal language automatically via rule-based text parsing and neural object detecting, respectively. Inter-GPS is the first geometry problem solver that achieves automatic program parsing and interpretable symbolic reasoning. We further propose a novel geometry solving approach with formal language and symbolic reasoning, called Interpretable Geometry Problem Solver ( Inter-GPS). Four data examples in the Geometr圓K dataset are shown below: We define 91 predicates and their corresponding literal templates to describe each problem. We construct a new large-scale benchmark, Geometr圓K, which consists of 3,002 geometry problems with dense annotation in formal language. Code and data for ACL 2021 Paper " Inter-GPS: Interpretable Geometry Problem Solving with Formal Language and Symbolic Reasoning".
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |