【95周年校庆系列讲座】On A Phase Transition in General Order Spline Regression

时间:2020-07-16         阅读:

光华讲坛——社会名流与企业家论坛第 5757 期

(线上讲座)

主题On A Phase Transition in General Order Spline Regression

主讲人华盛顿大学 韩放助理教授

主持人统计学院 常晋源教授

时间2020年7月17日(周五)10:00-11:20

直播平台及会议ID腾讯会议,会议ID:131 504 931

主办单位:统计研究中心 数据科学与商业智能联合实验室 统计学院 科研处

主讲人简介:

Dr. Fang Han is an assistant professor of statistics at the University of Washington. He obtained his Ph.D. from the Department of Biostatistics, Johns Hopkins University in 2015. Previously, he received his B.S. (Mathematics) from Peking University and M.S. (Biostatistics) from University of Minnesota. His main methodology/theory interests lie in rank-based methods, nonparametric regression methods, and hierarchical models.

韩放,华盛顿大学统计系助理教授。2015年获得约翰霍普金斯大学统计学博士学位。在此之前,他在北京大学获得数学学士学位,在明尼苏达大学获得生物统计学硕士学位。他的主要研究兴趣是基于秩次的方法、非参数回归方法和分层模型。

内容提要:

In the Gaussian sequence model, we study the fundamental limit of approximating the signal by a class of (generalized) splines with free knots. Our results give the minimax rate of estimation and reveal the corresponding phase transition. The transition boundary demonstrates the critical role of the order of differentiability at each inner knot in the separation between a faster loglog(16n) and a slower log(en) rate. We further show that, once encouraging an additional ‘d-monotonicity’ shape constraint (including monotonicity for d = 0 and convexity for d=1), the above phase transition is eliminated and the faster kloglog(16n/k) rate can be achieved for all k. These results provide theoretical support for developing L_0-penalized (shape-constrained) spline regression procedures as useful alternatives to L_1- and L_2-penalized ones.

在高斯序列模型中,本文研究了用一种具有自由结(free knots)的广义样条函数来近似估算信号的基本极限。本文的结果给出了估计的极大极小率,并揭示了相应的相变。过渡边界演示了在更快的loglog(16n)和log(en)率的分离中,每个内结的可微性阶数的关键作用。本文进一步表明,一旦鼓励额外的‘d-monotonicity’形状约束(包括d=0时的单调性和d=1时的凸性),上述的相变就可被消除,并对所有k都可以得到更快的kloglog(16n/k)率。这些结果为开发L_0-penalized的(形状约束)样条回归程序提供了理论支撑,它和L_1-和L_2-penalized的样条回归一样有用。

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