Testability of high-dimensional linear models with non-sparse structures
This work addresses limitations in high-dimensional inference for statisticians and data scientists, offering a novel theoretical framework that shifts focus from coefficient sparsity to precision matrix structure, though it is incremental in refining existing testability concepts.
The paper tackles the problem of statistical inference in high-dimensional linear models with non-sparse structures, showing that testability depends on the sparsity of the precision matrix rows rather than regression coefficients, and identifies tradeoffs where minimax lower bounds can be achieved with a test having a √n rate in weakly correlated models.
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.