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Enhancing Sparsity by Reweighted L-1 Minimization. Authors: Emmanuel Candes , Michael Wakin , and Stephen Boyd A review by Jeremy Watt. The Basis pursuit problem. The Basis pursuit problem. The related Lasso problem. The Basis pursuit problem. Measures: cardinality and magnitude

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enhancing sparsity by reweighted l 1 minimization

Enhancing Sparsity by Reweighted L-1 Minimization

Authors: Emmanuel Candes, Michael Wakin, and Stephen Boyd

A review by Jeremy Watt

slide3

The Basis pursuit problem

The related Lasso problem

slide4

The Basis pursuit problem

Measures: cardinality and

magnitude

(sensitive to outliers)

Measures: cardinality

ideal weightings compensate for magnitudes
Ideal weightings: compensate for magnitudes
  • Say we know and
  • Want to recover from
  • So ideal weightings are
ideal weightings
Ideal weightings
  • Only the support of can enter the model
  • Nullifies true magnitudes
  • Must stay feasible w.r.t. true support
algorithm for general problem1
Algorithm for general problem
  • Early iterations may find inaccurate signal estimates, but largest signal coefficients are likely to be identified as nonzero.
  • Once these locations are identified, their influence is downweighted in order to allow more sensitivity for identifying the remaining small but nonzero signal coefficients.
simulated example
Simulated example
  • Signal length = 512
  • # spikes = 130
  • indep normal entries
  • = 0.1
  • 2 iterations of the algorithm performed for perfect recovery
tv minimization image reconstruction
TV Minimization Image Reconstruction

Data: = sampled Fourier coefficients of image

= sampled Fourier matrix

Goal: Reconstruct original image

Leverage: Image gradient sparsity

tv minimization image reconstruction1
TV Minimization Image Reconstruction

Data: = sampled Fourier coefficients of image

= sampled Fourier matrix

Goal: Reconstruct original image

Leverage: Image gradient sparsity

concluding thoughts on reweighted l 1 minimization
Concluding thoughts on reweighted L-1 minimization
  • An attempt to nullify the ‘magnitude problem’ with L-1 norm (outliers in some sense)
    • Same sort of motivation leads to Iteratively Reweighted Least Squares
  • Many superior results over standard L-1
  • Generalizations to other sparsity problems
  • Deeper justification for efficacy as a Majorization-Minimization algorithm for solving alternative sparse recovery problem
epilogue majorization minimization mm justification
Epilogue: Majorization-Minimization (MM) justification

Standard

epigraph trick

  • Epigraph trick
  • smooths objective
  • Adds linear inequality constraints to model
epilogue majorization minimization mm justification1
Epilogue: Majorization-Minimization (MM) justification

In MM approach:

  • Majorize objective function (use first order approximation)
  • Form sub-problem with this objective and original constraints
  • Solve series of such sub-problems to solve original problem

In our case tth sub-problem takes reweighted L-1 form