High-Dimensional Statistics and Graph Neural Networks

Abstract: Semi-supervised learning is devoted to using unlabeled data to improve the performance of machine learning algorithms. In this paper, we study the semi-supervised generalized linear model (GLM) in the distributed setup. In the cases of single or multiple machines containing unlabeled data, we propose two distributed semi-supervised algorithms based on distributed approximate Newton method. When the labeled local sample size is small, our algorithms still give a consistent estimation, while the fully supervised methods fail to converge. Moreover, we theoretically prove that the convergence rate is greatly improved when sufficient unlabeled data exists. Therefore the proposed method requires much fewer rounds of communications to achieve the optimal rate than its fully-supervised counterpart. In the case of the linear model, we prove the rate lower bound after one round of communication, which shows that rate improvement is essential. Finally, several simulation analyses and real data studies are provided to demonstrate the effectiveness of our method.

Abstract: We will provide a brief summary of some results appeared in kernel regression/deep neural network and propose some potential directions.

Abstract: 近年来，由于图结构数据的强大表达能力，用机器学习方法分析和挖掘图数据的研究越来越受到重视。图神经网络（Graph Neural Networks）是一类基于深度学习的处理图数据的方法，在众多领域展现出了卓越的性能，其已经成为一种广泛应用的图分析方法。谱域图神经网络是图神经网络研究中一类重要的方法，它们在拉普拉斯谱域中设计和学习不同的图卷积，具有良好的理论保证和可解释性。本报告拟先介绍图神经网络的任务和一些前沿应用，然后从图傅里叶变换、图卷积的设计和图谱滤波器的多项式近似等方面探讨谱域图神经网络的理论基础，最后将讨论我们在谱域图神经网络所做的一些工作和对未来工作的展望。

Abstract: This paper proposes a flexible framework for inferring large-scale time-varying and time-lagged correlation networks from multivariate or high-dimensional non-stationary time series with piecewise smooth trends. Built on a novel and unified multiple-testing procedure of time-lagged cross-correlation functions with a fixed or diverging number of lags, our method can accurately disclose flexible time-varying network structures associated with complex functional structures at all time points. We broaden the applicability of our method to the structure breaks by developing difference-based nonparametric estimators of cross-correlations, achieve accurate family-wise error control via a bootstrap-assisted procedure adaptive to the complex temporal dynamics, and enhance the probability of recovering the time-varying network structures using a new uniform variance reduction technique. We prove the asymptotic validity of the proposed method and demonstrate its effectiveness in finite samples through simulation studies and empirical applications.