Tuning-Free Heterogeneity Pursuit in Massive Networks

TL;DR

This paper introduces a tuning-free framework for detecting heterogeneity across multiple large-scale Gaussian networks, providing new tests and algorithms with optimal theoretical properties and practical scalability.

Contribution

It proposes a novel tuning-free heterogeneity pursuit method with new tests and an efficient algorithm for high-dimensional network data analysis.

Findings

The proposed tests are asymptotically optimal.

The tuning-free algorithm converges globally and efficiently.

Method performs well in simulations and real data applications.

Abstract

Heterogeneity is often natural in many contemporary applications involving massive data. While posing new challenges to effective learning, it can play a crucial role in powering meaningful scientific discoveries through the understanding of important differences among subpopulations of interest. In this paper, we exploit multiple networks with Gaussian graphs to encode the connectivity patterns of a large number of features on the subpopulations. To uncover the heterogeneity of these structures across subpopulations, we suggest a new framework of tuning-free heterogeneity pursuit (THP) via large-scale inference, where the number of networks is allowed to diverge. In particular, two new tests, the chi-based test and the linear functional-based test, are introduced and their asymptotic null distributions are established. Under mild regularity conditions, we establish that both tests are optimal in achieving the testable region boundary and the sample size requirement for the latter test is minimal. Both theoretical guarantees and the tuning-free feature stem from efficient multiple-network estimation by our newly suggested approach of heterogeneous group square-root Lasso (HGSL) for high-dimensional multi-response regression with heterogeneous noises. To solve this convex program, we further introduce a tuning-free algorithm that is scalable and enjoys provable convergence to the global optimum. Both computational and theoretical advantages of our procedure are elucidated through simulation and real data examples.