Nearly Optimal Stochastic Approximation for Online Principal Subspace Estimation

Processing streaming data as they arrive is often necessary for high dimensional data analysis. In this paper, we analyse the convergence of a subspace online PCA iteration, as a followup of the recent work of Li, Wang, Liu, and Zhang [Math. Program., Ser. B, DOI 10.1007/s10107-017-1182-z] who considered the case for the most significant principal component only, i.e., a single vector. Under the sub-Gaussian assumption, we obtain a finite-sample error bound that closely matches the minimax information lower bound of Vu and Lei [Ann. Statist. 41:6 (2013), 2905-2947].Comment: 37 page

Emergence of hysteresis loop in social contagions on complex networks

Understanding the spreading mechanisms of social contagions in complex network systems has attracted much attention in the physics community. Here we propose a generalized threshold model to describe social contagions. Using extensive numerical simulations and theoretical analyses, we find that a hysteresis loop emerges in the system. Specifically, the steady state of the system is sensitive to the initial conditions of the dynamics of the system. In the steady state, the adoption size increases discontinuously with the transmission probability of information about social contagions, and trial size exhibits a non-monotonic pattern, i.e., it first increases discontinuously then decreases continuously. Finally we study social contagions on heterogeneous networks and find that network topology does not qualitatively affect our results.This work was funded in part by the National Key Research and Development Program (Grant No. 2016YFB0800602), the National Natural Science the Foundation of China (Grant Nos 61472045,61573067), and the China Scholarship Council. (2016YFB0800602 - National Key Research and Development Program; 61472045 - National Natural Science the Foundation of China; 61573067 - National Natural Science the Foundation of China; China Scholarship Council)Published versio