Statistical and machine learning theory has developed several conditions ensuring that popular estimators such as the Lasso or the Dantzig selector perform well in high-dimensional sparse regression, including the restricted eigenvalue, compatibility, and q sensitivity properties. However, some of the central aspects of these conditions are not well understood. For instance, it is unknown if these conditions can be checked efficiently on any given dataset. This is problematic, because they are at the core of the theory of sparse regression. Here we provide a rigorous proof that these conditions are NP-hard to check. This shows that the conditions are computationally infeasible to verify, and raises some questions about their practical applications. However, by taking an average-case perspective instead of the worst-case view of NP-hardness, we show that a particular condition, q sensitivity, has certain

Heterogeneity is an unwanted variation when analyzing aggregated datasets from multiple sources. Though different methods have been proposed for heterogeneity adjustment, no systematic theory exists to justify these methods. In this work, we propose a generic framework named ALPHA (short for Adaptive Low-rank Principal Heterogeneity Adjustment) to model, estimate, and adjust heterogeneity from the original data. Once the heterogeneity is adjusted, we are able to remove the batch effects and to enhance the inferential power by aggregating the homogeneous residuals from multiple sources. Under a pervasive assumption that the latent heterogeneity factors simultaneously affect a fraction of observed variables, we provide a rigorous theory to justify the proposed framework. Our framework also allows the incorporation of informative covariates and appeals to the Bless of Dimensionality. As an illustrative

Many statistical models seek relationship between variables via subspaces of reduced dimensions. For instance, in factor models, variables are roughly distributed around a low dimensional subspace determined by the loading matrix; in mixed linear regression models, the coefficient vectors for different mixtures form a subspace that captures all regression functions; in multiple index models, the effect of covariates is summarized by the effective dimension reduction space.

We prove a sharp Bernstein inequality for general-state-space and not necessarily reversible Markov chains. It is sharp in the sense that the variance proxy term is optimal. Our result covers the classical Bernstein's inequality for independent random variables as a special case.

Suppose we lhave n observationis ftom Y= X+ c, where e is nieasuremiienit error with kiowin distributioni, and the denesity f of X is uniknowii withi thte nloln-paraniietric conistrainit