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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| Veröffentlicht in: | Computational optimization and applications Jg. 74; H. 2; S. 387 - 441 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.11.2019
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| ISSN: | 0926-6003, 1573-2894 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Tillmann, Andreas M. |
| Author_xml | – sequence: 1 givenname: Andreas M. orcidid: 0000-0002-7744-836X surname: Tillmann fullname: Tillmann, Andreas M. email: andreas.tillmann@cs.rwth-aachen.de organization: Visual Computing Institute and Chair of Operations Research, RWTH Aachen University |
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| CitedBy_id | crossref_primary_10_1016_j_acha_2025_101786 crossref_primary_10_1007_s10489_022_04144_5 crossref_primary_10_1007_s10589_020_00235_6 crossref_primary_10_1007_s43670_023_00053_x crossref_primary_10_1109_TIT_2022_3159100 crossref_primary_10_1137_21M142770X |
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| Keywords | Matroid girth 90C10 94A15 05B35 90C57 Mixed-integer programming 68Q17 90C11 Sparse recovery Spark Branch-and-cut Compressed sensing |
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| Snippet | 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... |
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| SubjectTerms | 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 |
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| Title | Computing the spark: mixed-integer programming for the (vector) matroid girth problem |
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| Volume | 74 |
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