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DCACO: an algorithm for designing incoherent redundant matrices

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Abstract

The mutual coherence of a matrix, defined as the maximum absolute value of the normalized inner-products between different columns, is an important property that characterizes the similarity between different matrix columns. Redundant matrices with very low mutual coherence are referred to as incoherent redundant matrices which play an important role in mathematical signal processing tasks. The problem of minimizing the mutual coherence in a give matrix space where each matrix has normalized columns is called the coherence optimization problem. In this paper, we transform equivalently the coherence optimization problem as a rank constrained semidefinite \(\ell _{\infty }\)-minimization problem. It is critical to analyze the projection operator onto the nonconvex set in the new matrix optimization constraints, i.e., the nonconvex set of symmetric positive semidefinite matrices whose rank is not greater than a give positive integer. By exploiting the projection operator, we express the nonconvex set mentioned above as the set of zero roots of a Difference of two Convex (DC) functions. For the convex function related to the projection operator in the DC function, we characterize its properties and obtain the concise form of its subdifferential. With the help of the DC function, a new algorithm based on DCA (DC Algorithms) is proposed to solve the Coherence Optimization problem, and thus the proposed algorithm is called DCACO for short. We also study the convergence analysis of DCACO. An advantage of DCACO is that subproblems in each iteration have closed-form solutions. Experimental results demonstrate that DCACO leads to state-of-art performance on generating highly incoherent redundant matrices, and DCACO can also compete with several other algorithms on designing optimized projection matrices for improving the performance of compressed sensing.

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Acknowledgements

The authors would like to thank the editor-in chief and the anonymous reviewers for their valuable comments and suggestions, which help us very much in improving the original version of this paper. The authors would like to thank Professor Bogdan Dumitrescu for providing the source code of ISPM in [31]. The authors also would like to thank Chongyang Wang for improving numerical experiments.

Funding

This research was supported by the National Natural Science Foundation of China under the Grant nos. 61901404, 12031003 and 11771347, and supported by the Nanhu Scholars Program for Young Scholars of XYNU.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

    Yongchao Yu

  2. School of Mathematics and Information Sciences, Guangzhou University, 230 GuangZhou University City Outer Ring Road, GuangZhou, 510006, China

    Jigen Peng

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  1. Yongchao Yu
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Correspondence to Yongchao Yu.

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All data and code generated or used during the study are available at https://github.com/sparseelad/DCACO.

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Yu, Y., Peng, J. DCACO: an algorithm for designing incoherent redundant matrices. Numer Algor 93, 785–813 (2023). https://doi.org/10.1007/s11075-022-01441-5

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