In pursuit of a root

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Abstract

The basis pursuit technique is used to find a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise fits the least-squares problem only approximately, and a single parameter determines a curve that traces the trade-off between the least-squares fit and the one-norm of the solution. We show that the function that describes this curve is convex and continuously differentiable over all points of interest. The dual solution of a least-squares problem with an explicit one-norm constraint gives function and derivative information needed for a root-finding method. As a result, we can compute arbitrary points on this curve. Numerical experiments demonstrate that our method, which relies on only matrix-vector operations, scales well to large problems.

BiBTeX

@techreport{BergFrie:2007,
  Address = {University of British Columbia, Vancouver},
  Author = {E. {van den} Berg and M. P. Friedlander},
  Institution = {Department of Computer Science},
  Month = {June},
  Number = {TR-2007-19},
  Title = {In pursuit of a root},
  Type = {Tech. Rep.},
  Url = {http://www.optimization-online.org/DB_HTML/2007/06/1708.html},
  Year = 2007
}