题目：Stability of Least Square Approximation under Random Sampling
主讲人：许志强
时间：2024-11-06 09:00
地点：#腾讯会议：367-561-52
举办部门：数学与计算机学院
讲座要点：
In this talk, we investigate the stability of the least squares approximation within the univariate polynomial space of a certain degree. This approximation involves identifying a polynomial that closely approximates a function over a specific domain, based on samples taken from that function at randomly selected points,according to a designated measure. Our primary aim is to deter mine the sampling rate needed to ensure the stability of this approximation. Under the assumption that the sampling points are independent and identically distributed with respect to a Jacobi weight function, we present the necessary sampling rates for maintaining stability. In particular, we show that for uniform random sampling, a sampling rate proportional to the square of the degree is required to ensure stability. By integrating these findings with previous work by Cohen-Davenport-Leviatan, we conclude that, for uniform random sampling, the optimal sampling rate for guaranteeing the stability of the approximation is similarly proportional to the square of the degree, with an additional logarithmic factor. Motivated by this result, weextend an existing impossibility theorem, which was initially applicable to equally spaced samples, to the case of random samples.This extension highlights the trade-off between accuracy and stability when it comes to recovering analytic functions.
主讲人简介：
许志强, 中国科学院数学与系统科学研究院研究员, 冯康首席研究员 （2021-2023). 研究领域包括逼近论、计算调和分析、离散数学等, 尤其对采样理论, 压缩感知, 球面布点以及相位恢复等领域感兴趣. 一方面, 他将纯粹数学中的研究方法引入到计算调和分析, 系统发展了相位恢复的代数簇方法, 从而对一些困难问题得到实质性进展;另一方面, 将逼近论中样祥条函数和代数理论相结, 从而解决了多个猜想和公开问题. 2020年获得国家杰出青年基金资助. 现担任IEEE Trans. Information Theory, 数学学报 （英文版）， J. Comp Math., Numerical Mathematics: Theory, Methods and Applications等国际期刊编。