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Sparse nonnegative solution of underdetermined linear equations by linear programming.

David L Donoho1, Jared Tanner

  • 1Department of Statistics, Stanford University, Stanford, CA 94305-4065, USA. donoho@stat.stanford.edu

Proceedings of the National Academy of Sciences of the United States of America
|June 25, 2005
PubMed
Summary
This summary is machine-generated.

Finding the sparsest nonnegative solution to linear systems is NP-hard. However, for many matrices, if the solution is sparse enough, linear programming can find it, explained by polytope theory.

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Area of Science:

  • Linear Algebra
  • Optimization
  • Computational Complexity

Background:

  • Underdetermined systems of linear equations (y = Ax) with known y and matrix A are common.
  • Seeking the nonnegative solution vector x with the fewest nonzeros is generally NP-hard.

Purpose of the Study:

  • To investigate conditions under which the sparsest nonnegative solution to y = Ax can be found using linear programming.
  • To establish a theoretical framework connecting polytope geometry to the recoverability of sparse solutions.

Main Methods:

  • Convex polytope theory, specifically the concept of 'outward k-neighborliness' of the polytope formed by the columns of matrix A.
  • Relating geometric properties of the polytope to the sparsity and uniqueness of solutions to linear systems.

Main Results:

  • Outward k-neighborliness of the polytope P (convex hull of A's columns) is equivalent to guaranteeing that any nonnegative solution with at most k nonzeros is the unique sparsest solution.
  • Weak neighborliness implies that most k-sparse nonnegative solutions are uniquely recoverable via linear programming.

Conclusions:

  • The geometric property of 'neighborliness' in the polytope of matrix columns provides a sufficient condition for exact recovery of sparse solutions using linear programming.
  • This framework explains the threshold phenomenon and offers guarantees for sparse signal recovery in underdetermined systems.