Computing the spark: mixed-integer programming for the (vector) matroid girth problem

We investigate the NP-hard problem of computing the spark of a matrix (i.e., the smallest number of linearly dependent columns), a key parameter in compressed sensing and sparse signal recovery. To that end, we identify polynomially solvable special cases, gather upper and lower bounding procedures,...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 74; no. 2; pp. 387 - 441
Main Author: Tillmann, Andreas M.
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2019
Springer Nature B.V
Subjects:
Algorithms
Computation
Convex and Discrete Geometry
Integer programming
Linear programming
Management Science
Mathematical programming
Mathematics
Mathematics and Statistics
Matrices (mathematics)
Operations Research
Operations Research/Decision Theory
Optimization
Signal reconstruction
Solvers
Statistics
Matroid girth
90C10
94A15
05B35
90C57
Mixed-integer programming
68Q17
90C11
Sparse recovery
Spark
Branch-and-cut
Compressed sensing
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:We investigate the NP-hard problem of computing the spark of a matrix (i.e., the smallest number of linearly dependent columns), a key parameter in compressed sensing and sparse signal recovery. To that end, we identify polynomially solvable special cases, gather upper and lower bounding procedures, and propose several exact (mixed-)integer programming models and linear programming heuristics. In particular, we develop a branch and cut scheme to determine the girth of a matroid, focussing on the vector matroid case, for which the girth is precisely the spark of the representation matrix. Extensive numerical experiments demonstrate the effectiveness of our specialized algorithms compared to general-purpose black-box solvers applied to several mixed-integer programming models.
Bibliography:ObjectType-Article-1
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-019-00114-9