Testability of high-dimensional linear models with non-sparse structures

TL;DR

This paper investigates the limits of statistical testing in high-dimensional linear models with non-sparse structures, revealing that testability depends on feature correlation rather than sparsity of coefficients.

Contribution

It introduces new concepts of uniform and essentially uniform non-testability, establishing minimax testability results independent of coefficient sparsity.

Findings

Testability depends on the sparsity of the precision matrix rows, not the regression coefficients.

New minimax testability results show tests can achieve √n rate regardless of sparsity.

Tradeoffs between testability and feature correlation are identified.

Abstract

Understanding statistical inference under possibly non-sparse high-dimensional models has gained much interest recently. For a given component of the regression coefficient, we show that the difficulty of the problem depends on the sparsity of the corresponding row of the precision matrix of the covariates, not the sparsity of the regression coefficients. We develop new concepts of uniform and essentially uniform non-testability that allow the study of limitations of tests across a broad set of alternatives. Uniform non-testability identifies a collection of alternatives such that the power of any test, against any alternative in the group, is asymptotically at most equal to the nominal size. Implications of the new constructions include new minimax testability results that, in sharp contrast to the current results, do not depend on the sparsity of the regression parameters. We identify new tradeoffs between testability and feature correlation. In particular, we show that, in models with weak feature correlations, minimax lower bound can be attained by a test whose power has the $\sqrt{n}$ rate, regardless of the size of the model sparsity.