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We consider adaptive estimation and statistical inference forhigh-dimensional graph-based linear models. In our model, the coordinates ofregression coefficients correspond to an underlying undirected graph.Furthermore, the given graph governs the piecewise polynomial structure of theregression vector. In the adaptive estimation part, we apply graph-basedregularization techniques and propose a family of locally adaptive estimatorscalled the Graph-Piecewise-Polynomial-Lasso. We further study a one-step updateof the Graph-Piecewise-Polynomial-Lasso for the problem of statisticalinference. We develop the corresponding theory, which includes the fixed designand the sub-Gaussian random design. Finally, we illustrate the superiorperformance of our approaches by extensive simulation studies and conclude withan application to an Arabidopsis thaliana microarray dataset.

Adaptive Estimation and Statistical Inference for High-Dimensional Graph-Based Linear Models