Sparse Regression Distilled from a Single Robust Fit
Abstract
Robust linear fits can resist response contamination yet remain too dense or unstable for useful global explanations. We propose penalized distillation, which fits a smoothly clipped absolute deviation (SCAD) estimator to a robust initial estimator's empirical fitted surface along a safeguarded coordinate-descent path and evaluates candidate states separately for fidelity, parsimony, perturbation stability, and held-out prediction. The new results attach to the states the algorithm actually computes. Conditional on a fixed uncontaminated design, deterministic bounds transfer response-replacement boundedness from the initial fit to every retained path state. Turning to fixed dimension, we characterize the oracle-support branch by its empirical-Gram projection and influence function, give conditions for covariance-weighted least-squares approximation equivalence, and establish a path-conditional generalized information criterion. By contrast, at large dimension-to-sample ratios the full-coordinate robust fit collapses without warning, and screening restores the construction. Under a sure-screening framework, the robustness bound and the support and selection guarantees transfer to the screened fit. Simulations separate robustness transfer from support recovery, efficiency, and computation across the dimension-to-sample ratio, with p up to 240, and the signal density, which isolates what the sparse stage adds once the screen over-selects. In a duplicate-grouped superconductivity study, the distilled estimator remains predictively stable under prespecified training-response shifts but retains 66.8--68.8 of 81 slopes. Stronger sparsification reduces the model to 12.6--14.0 slopes only at visible fidelity and prediction cost. Distillation therefore preserves predictive stability on these data without substantiating a compact coordinate-level explanation.
Related
- Byzantine-Robust Sparse Regression with Missing Covariates
- Spatial-Sign based Robust Sparse Precision Matrix Estimation and its Applications
- Sparse-group Convoluted Smoothed Quantile Regression in High Dimension
- Decentralized Sparse Quantile Regression
- Component-weighted sparse group LASSO for finite mixture regression models
BibTeX
@misc{shin2026sparse,
title = {{Sparse Regression Distilled from a Single Robust Fit}},
author = {Wooyoung Shin and Seunghwan Park},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23937},
eprint = {2609.23937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23937},
}
RIS
TY - UNPB
TI - Sparse Regression Distilled from a Single Robust Fit
AU - Wooyoung Shin
AU - Seunghwan Park
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23937
UR - https://arxiv.org/abs/2609.23937
AB - Robust linear fits can resist response contamination yet remain too dense or unstable for useful global explanations. We propose penalized distillation, which fits a smoothly clipped absolute deviation (SCAD) estimator to a robust initial estimator's empirical fitted surface along a safeguarded coordinate-descent path and evaluates candidate states separately for fidelity, parsimony, perturbation stability, and held-out prediction. The new results attach to the states the algorithm actually computes. Conditional on a fixed uncontaminated design, deterministic bounds transfer response-replacement boundedness from the initial fit to every retained path state. Turning to fixed dimension, we characterize the oracle-support branch by its empirical-Gram projection and influence function, give conditions for covariance-weighted least-squares approximation equivalence, and establish a path-conditional generalized information criterion. By contrast, at large dimension-to-sample ratios the full-coordinate robust fit collapses without warning, and screening restores the construction. Under a sure-screening framework, the robustness bound and the support and selection guarantees transfer to the screened fit. Simulations separate robustness transfer from support recovery, efficiency, and computation across the dimension-to-sample ratio, with p up to 240, and the signal density, which isolates what the sparse stage adds once the screen over-selects. In a duplicate-grouped superconductivity study, the distilled estimator remains predictively stable under prespecified training-response shifts but retains 66.8--68.8 of 81 slopes. Stronger sparsification reduces the model to 12.6--14.0 slopes only at visible fidelity and prediction cost. Distillation therefore preserves predictive stability on these data without substantiating a compact coordinate-level explanation.
ER -