Rice University, Sparse Optimization Project

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The Bregman Iterative Procedure

Linearized Bregman

Papers

  • NEW: W. Yin and S. Osher. "Error Forgetting of Bregman Iteration." Submitted to Journal of Scientific Computing, Special Issue. Rice CAAM Technical Report TR12-03 [pdf] [code]
    Explain why solving Bregman subproblems at low accuracies (1e-6) gives a Bregman solution at near the machine precision (1e-15).
  • W. Yin, S. Osher, J. Darbon, and D. Goldfarb. "Bregman Iterative Algorithms for Compressed Sensing and Related Problems." SIAM Journal on Imaging Sciences, 1(1):143-168, 2008. [pdf]
  • S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin. "An iterative regularization method for total variation-based image restoration." SIAM Journal on Multiscale Modeling and Simulation, 4(2):460-489, 2005. [pdf]
  • Do not confuse this with the linearized Bregman method!

Matlab Demo

  • Matlab demo of error forgetting [zip]

Slides

  • Review slides, IPAM, 02/12/2010: [pdf]
  • Older slides [pdf]

Two different usages of Bregman iteration:

  • To improve the regularization quality of nonsmooth regularizers such as L1, total variations, and their variants; see [slides 6-10] for a demo.
  • To give fast, accurate iterations for constrained L1-like minimization. It enjoys error forgetting and cancellation properties; see [slides 25-27] and the Matlab demo below.