Sparse Recovery via Differential Inclusions

Stanley Osher, Feng Ruan, Jiechao Xiong, Yuan Yao, and Wotao Yin

Published in Applied and Computational Harmonic Analysis


In this paper, we recover sparse signals from their noisy linear measurements by solving nonlinear differential inclusions, which we call Bregman ISS and Linearized Bregman ISS. We show that under proper conditions, there exists a bias-free and sign-consistent point on their solution paths, 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. Therefore, their solution paths are regularization paths better than the LASSO regularization path, since the points on the latter path are biased. 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 faster 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.



S. Osher, F. Ruan, J. Xiong, Y. Yao, and W. Yin, Sparse Recovery via Differential Inclusions, Applied and Computational Harmonic Analysis, 41(2), 436-469, 2016.

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