Sparse signal recovery using a new class of random matrices
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Enrico Au-Yeung
Abstract
We propose a new class of random matrices that enables the recovery of signals with sparse representation in a known basis with overwhelmingly high probability. To construct a matrix in this class, we begin with a fixed non-random matrix that satisfies two very general conditions. Then we decompose the matrix into pieces of sparse matrices. A random sum (involving Bernoulli random variables) of these pieces of sparse matrices is used to construct the final matrix. We say that the random matrix is the randomized Bernoulli transform of the original matrix. The random matrix is not created by filling all its entries with random variables, as in the case of Gaussian or Bernoulli matrices. Therefore, as a benefit, far fewer number of random variables are needed to generate this new type of random matrices. We prove that the number of samples needed to recover a random signal is proportional to the sparsity of the signal, up to a logarithmic factor, and hence this number is nearly optimal.
Acknowledgements
The author is grateful to his mentor Professor Özgür Yılmaz. His insight has substantially improved the quality of this paper. The author likes to thank the anonymous referee who has provided many helpful comments to improve the manuscript.
References
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Articles in the same Issue
- Frontmatter
- Sparse signal recovery using a new class of random matrices
- The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman--Opdam theory on ℝd
- Generalized variational-like inclusion involving relaxed monotone operators
- A study on the product set-labeling of graphs
- Optimal control problem and viscosity solutions for the Vlasov equation in Yang–Mills charged Bianchi models
- Jaco-type graphs and black energy dissipation
Articles in the same Issue
- Frontmatter
- Sparse signal recovery using a new class of random matrices
- The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman--Opdam theory on ℝd
- Generalized variational-like inclusion involving relaxed monotone operators
- A study on the product set-labeling of graphs
- Optimal control problem and viscosity solutions for the Vlasov equation in Yang–Mills charged Bianchi models
- Jaco-type graphs and black energy dissipation
