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TWO NEW LOWER BOUNDS FOR THE SPARK OF A MATRIX

Part of: Combinatorics

Published online by Cambridge University Press:  08 June 2017

HAIFENG LIU Open the ORCID record for HAIFENG LIU [Opens in a new window] ,
JIHUA ZHU  and
JIGEN PENG
HAIFENG LIU*
Affiliation:
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China email lhflkcc@126.com
JIHUA ZHU
Affiliation:
School of Software Engineering, Xi’an Jiaotong University, Xi’an 710049, China email zhujh@mail.xjtu.edu.cn
JIGEN PENG
Affiliation:
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China email jgpeng@mail.xjtu.edu.cn
*
✉lhflkcc@126.com
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Abstract

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The $l_{0}$-minimisation problem has attracted much attention in recent years with the development of compressive sensing. The spark of a matrix is an important measure that can determine whether a given sparse vector is the solution of an $l_{0}$-minimisation problem. However, its calculation involves a combinatorial search over all possible subsets of columns of the matrix, which is an NP-hard problem. We use Weyl’s theorem to give two new lower bounds for the spark of a matrix. One is based on the mutual coherence and the other on the coherence function. Numerical examples are given to show that the new bounds can be significantly better than existing ones.

Keywords

lower boundsparkmutual coherencecoherence functioncompressive sensing

MSC classification

Primary: 05A20: Combinatorial inequalities
Secondary: 90C27: Combinatorial optimization
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 3 , December 2017 , pp. 353 - 360
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

H. F. Liu, J. H. Zhu and J. G. Peng are supported by the Natural Science Foundation of China (Grant Nos. 11401463, 61573273 and 11131006) and the Fundamental Research Funds for the Central Universities.

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