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Abstract:
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Heterogeneity is often natural in many contemporary applications involving massive data. It can play a crucial role in powering scientific discoveries through the understanding of differences among sub-populations of interest. In this talk, we exploit multiple networks with Gaussian graphs to encode the connectivity patterns of a large number of features on the subpopulations. To uncover the sub-population heterogeneity, we suggest a new framework of tuning-free heterogeneity pursuit (THP) via large-scale inference. 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 conditions, we show 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 multiple-network estimation by newly suggested approach of heterogeneous group square-root Lasso (HGSL) for multi-response regression with heterogeneous noises. To solve HGSL, we further introduce an algorithm that is scalable and enjoys provable convergence to the global optimum.
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