In the field of high-dimensional statistics, it is commonly assumed that only a small subset of the variables are relevant and sparse estimators are pursued to exploit this assumption. Sparse estimation methodologies are often straightforward to construct, and indeed there is a full spectrum of sparse algorithms covering almost all statistical learning problems. In contrast, theoretical developments are more limited and often focus on asymptotic theories. In applications, non-asymptotic results may be more relevant.

The goal of this work is to show how non-asymptotic statistical theory can be developed for sparse estimation problems that assume group sparsity. We discuss three different problems: principal component analysis (PCA), sliced inverse regression (SIR) and multivariate regression. For PCA, we study a two-stage thresholding algorithm and provide theories that go beyond the common spiked-covariance model. SIR is then related to PCA in some special settings, and it is shown that the theory of sparse PCA can be modified to work for SIR. Regression represents another important research direction in high-dimensional analysis. We study a linear regression model in which both the response and predictors are grouped, as an extension of group Lasso.

Despite the distinctions in these problems, the proofs of consistency and support recovery share some common elements: concentration inequalities and union probability bounds, which are also the foundation of most existing sparse estimation theories. The proofs are presented in modules in order to clearly reveal how most sparse estimators can be theoretically justified. Moreover, we identify those modules that are possibly not optimized to show the limitation of the existing proof techniques and how they could be extended.