A Residual Bootstrap for High-Dimensional Regression with Near Low-Rank Designs
This work addresses the challenge of statistical inference in high-dimensional settings for researchers in statistics and machine learning, offering an incremental extension of bootstrap methods to near low-rank designs.
The paper tackles the problem of extending the residual bootstrap method to high-dimensional linear regression where the number of predictors is comparable to the sample size, by proposing a version that uses ridge regression residuals and assuming a near low-rank design structure. It shows that this method consistently approximates the distribution of linear contrasts and confidence intervals for mean responses without requiring sparsity assumptions on coefficients.
We study the residual bootstrap (RB) method in the context of high-dimensional linear regression. Specifically, we analyze the distributional approximation of linear contrasts $c^{\top} (\hatβ_ρ-β)$, where $\hatβ_ρ$ is a ridge-regression estimator. When regression coefficients are estimated via least squares, classical results show that RB consistently approximates the laws of contrasts, provided that $p\ll n$, where the design matrix is of size $n\times p$. Up to now, relatively little work has considered how additional structure in the linear model may extend the validity of RB to the setting where $p/n\asymp 1$. In this setting, we propose a version of RB that resamples residuals obtained from ridge regression. Our main structural assumption on the design matrix is that it is nearly low rank --- in the sense that its singular values decay according to a power-law profile. Under a few extra technical assumptions, we derive a simple criterion for ensuring that RB consistently approximates the law of a given contrast. We then specialize this result to study confidence intervals for mean response values $X_i^{\top} β$, where $X_i^{\top}$ is the $i$th row of the design. More precisely, we show that conditionally on a Gaussian design with near low-rank structure, RB simultaneously approximates all of the laws $X_i^{\top}(\hatβ_ρ-β)$, $i=1,\dots,n$. This result is also notable as it imposes no sparsity assumptions on $β$. Furthermore, since our consistency results are formulated in terms of the Mallows (Kantorovich) metric, the existence of a limiting distribution is not required.