Year: 2024
Type: conference-paper
Abstract: We study the approximability of general convex sets in$\mathbb{R}^{n}$by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution and the complexity of an approximation is the number of halfspaces used. While a large body of research has considered the approximation of convex sets by intersections of halfspaces under distance metrics such as the Lebesgue measure and Hausdorff distance, prior to our work there has not been a systematic study of convex approximation under the Gaussian distribution. We establish a range of upper and lower bounds, both for general convex sets and for specific natural convex sets that are of particular interest. Our results demonstrate that the landscape of approximation is intriguingly different under the Gaussian distribution versus previously studied distance measures. Our results are proved using techniques from many different areas. These include classical results on convex polyhedral approximation, Cramér-type bounds on large deviations from probability theory, and-perhaps surprisingly-a range of topics from computational complexity, including computational learning theory, unconditional pseudorandomness, and the study of influences and noise sensitivity in the analysis of Boolean functions.
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: Collaborative Research: AF: Medium: Continuous Concrete Complexity, AF: Medium: The Trace Reconstruction Problem, AF: Medium: Research in Algorithms and Complexity: Total Functions, Games, and the Brain, AF: Small: Threshold Functions--Derandomization, Testing and Applications, BIGDATA: F: Big Data Analysis via Non-Standard Property Testing, CAREER: Learning and property testing -- a complexity theoretic perspective, AF: Medium: Smoothed Analysis for Optimization and Games
OpenAlex ID: W4404848362