# Model Selection with Low Complexity Priors

arXiv: https://arxiv.org/abs/1307.2342 · HTML: https://mlsanity.com/paper/1307.2342/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jul 2013.

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

- [Screening for Sparse Online Learning](https://mlsanity.com/paper/2101.06982/) (2021)
- [A Unified Framework for Pattern Recovery in Penalized and Thresholded Estimation and its Geometry](https://mlsanity.com/paper/2307.10158/) (2023)
- [Low Complexity Regularized Phase Retrieval](https://mlsanity.com/paper/2407.16413/) (2024)

## Abstract (arXiv)

Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied regularization models with various types of low-dimensional structures. In such settings, the general approach is to solve a regularized optimization problem, which combines a data fidelity term and some regularization penalty that promotes the assumed low-dimensional/simple structure. This paper provides a general framework to capture this low-dimensional structure through what we coin partly smooth functions relative to a linear manifold. These are convex, non-negative, closed and finite-valued functions that will promote objects living on low-dimensional subspaces. This class of regularizers encompasses many popular examples such as the L1 norm, L1-L2 norm (group sparsity), as well as several others including the Linfty norm. We also show that the set of partly smooth functions relative to a linear manifold is closed under addition and pre-composition by a linear operator, which allows to cover mixed regularization, and the so-called analysis-type priors (e.g. total variation, fused Lasso, finite-valued polyhedral gauges). Our main result presents a unified sharp analysis of exact and robust recovery of the low-dimensional subspace model associated to the object to recover from partial measurements. This analysis is illustrated on a number of special and previously studied cases, and on an analysis of the performance of Linfty regularization in a compressed sensing scenario.