# Model Consistency of Partly Smooth Regularizers

arXiv: https://arxiv.org/abs/1405.1004 · HTML: https://mlsanity.com/paper/1405.1004/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2014.

## Built on by (most cited first, 6 of 6)

- [Implicit differentiation for fast hyperparameter selection in non-smooth convex learning](https://mlsanity.com/paper/2105.01637/) (2021)
- [A Normal Map-Based Proximal Stochastic Gradient Method: Convergence and Identification Properties](https://mlsanity.com/paper/2305.05828/) (2023)
- [A Morse-Bott Framework for Blind Inverse Problems: Local Recovery Guarantees and the Failure of the MAP](https://mlsanity.com/paper/2508.02923/) (2025)
- [A Unified Framework for Pattern Recovery in Penalized and Thresholded Estimation and its Geometry](https://mlsanity.com/paper/2307.10158/) (2023)
- [Anderson Acceleration in Nonsmooth Problems: Local Convergence via Active Manifold Identification](https://mlsanity.com/paper/2410.09420/) (2024)
- [Alternating minimization for square root principal component pursuit](https://mlsanity.com/paper/2501.00471/) (2024)

## Abstract (arXiv)

This paper studies least-square regression penalized with partly smooth convex regularizers. This class of functions is very large and versatile allowing to promote solutions conforming to some notion of low-complexity. Indeed, they force solutions of variational problems to belong to a low-dimensional manifold (the so-called model) which is stable under small perturbations of the function. This property is crucial to make the underlying low-complexity model robust to small noise. We show that a generalized "irrepresentable condition" implies stable model selection under small noise perturbations in the observations and the design matrix, when the regularization parameter is tuned proportionally to the noise level. This condition is shown to be almost a necessary condition. We then show that this condition implies model consistency of the regularized estimator. That is, with a probability tending to one as the number of measurements increases, the regularized estimator belongs to the correct low-dimensional model manifold. This work unifies and generalizes several previous ones, where model consistency is known to hold for sparse, group sparse, total variation and low-rank regularizations.