# Sparse Recovery via Differential Inclusions

arXiv: https://arxiv.org/abs/1406.7728 · HTML: https://mlsanity.com/paper/1406.7728/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jun 2014.

## Built on by (most cited first, 11 of 11)

- [Proximal Subgradient Norm Minimization of ISTA and FISTA](https://mlsanity.com/paper/2211.01610/) (2022)
- [Linear Convergence of ISTA and FISTA](https://mlsanity.com/paper/2212.06319/) (2022)
- [Adaptive Pruning of Pretrained Transformer via Differential Inclusions](https://mlsanity.com/paper/2501.03289/) (2025)
- [Incremental Learning in Mirror Flows](https://mlsanity.com/paper/2606.23198/) (2026)
- [Iterative regularization for low complexity regularizers](https://mlsanity.com/paper/2202.00420/) (2022)
- [Nonconvex Regularization for Feature Selection in Reinforcement Learning](https://mlsanity.com/paper/2509.15652/) (2025)
- [Iterative Regularization with k-support Norm: An Important Complement to Sparse Recovery](https://mlsanity.com/paper/2401.05394/) (2023)
- [A General Continuous-Time Formulation of Stochastic ADMM and Its Variants](https://mlsanity.com/paper/2404.14358/) (2024)
- [Robust Network Learning via Inverse Scale Variational Sparsification](https://mlsanity.com/paper/2409.18419/) (2024)
- [Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusions](https://mlsanity.com/paper/2607.04990/) (2026)
- [Statistical Benefits of Fine-Tuning from Pretrained Initialization in Diagonal Linear Networks](https://mlsanity.com/paper/2609.34756/) (2026)

## Abstract (arXiv)

In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which is based on the notion of inverse scale space (ISS) developed in applied mathematics. Our goal here is to bring this idea to address a challenging problem in statistics, i.e. finding the oracle estimator which is unbiased and sign-consistent using dynamics. We call our dynamics Bregman ISS and Linearized Bregman ISS. A well-known shortcoming of LASSO and any convex regularization approaches lies in the bias of estimators. However, we show that under proper conditions, there exists a bias-free and sign-consistent point on the solution paths of such dynamics, which corresponds to a signal that is the unbiased estimate of the true signal and whose entries have the same signs as those of the true signs, i.e. the oracle estimator. Therefore, their solution paths are regularization paths better than the LASSO regularization path, since the points on the latter path are biased when sign-consistency is reached. We also show how to efficiently compute their solution paths in both continuous and discretized settings: the full solution paths can be exactly computed piece by piece, and a discretization leads to Linearized Bregman iteration, which is a simple iterative thresholding rule and easy to parallelize. Theoretical guarantees such as sign-consistency and minimax optimal l2-error bounds are established in both continuous and discrete settings for specific points on the paths. Early-stopping rules for identifying these points are given. The key treatment relies on the development of differential inequalities for differential inclusions and their discretizations, which extends the previous results and leads to exponentially fast recovering of sparse signals before selecting wrong ones.