# On model selection consistency of regularized M-estimators

arXiv: https://arxiv.org/abs/1305.7477 · HTML: https://mlsanity.com/paper/1305.7477/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2013.

## Built on by (most cited first, 3 of 3)

- [Thresholded Graphical Lasso Adjusts for Latent Variables: Application to Functional Neural Connectivity](https://mlsanity.com/paper/2104.06389/) (2021)
- [Joint Estimation and Inference for Multi-Experiment Networks of High-Dimensional Point Processes](https://mlsanity.com/paper/2109.11634/) (2021)
- [Asymptotic Behavior of Adversarial Training Estimator under ℓ_∞-Perturbation](https://mlsanity.com/paper/2401.15262/) (2024)

## Abstract (arXiv)

Regularized M-estimators are used in diverse areas of science and engineering to fit high-dimensional models with some low-dimensional structure. Usually the low-dimensional structure is encoded by the presence of the (unknown) parameters in some low-dimensional model subspace. In such settings, it is desirable for estimates of the model parameters to be model selection consistent: the estimates also fall in the model subspace. We develop a general framework for establishing consistency and model selection consistency of regularized M-estimators and show how it applies to some special cases of interest in statistical learning. Our analysis identifies two key properties of regularized M-estimators, referred to as geometric decomposability and irrepresentability, that ensure the estimators are consistent and model selection consistent.