Functional Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909836439977984 |
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| author | Krishna, K. Mahesh |
| author_facet | Krishna, K. Mahesh |
| contents | Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} and Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution to P-Type $\ell_1$-minimization problem. In this paper, we extend their result to abstract Banach spaces using 1-approximate Schauder frames. We notice that the `normalized' condition for Hilbert spaces can be generalized to a larger extent when we consider Banach spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_09609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Functional Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem Krishna, K. Mahesh Functional Analysis Information Theory Optimization and Control 42C15, 94A12, 46B45 Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} and Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution to P-Type $\ell_1$-minimization problem. In this paper, we extend their result to abstract Banach spaces using 1-approximate Schauder frames. We notice that the `normalized' condition for Hilbert spaces can be generalized to a larger extent when we consider Banach spaces. |
| title | Functional Donoho-Elad-Gribonval-Nielsen-Fuchs Sparsity Theorem |
| topic | Functional Analysis Information Theory Optimization and Control 42C15, 94A12, 46B45 |
| url | https://arxiv.org/abs/2510.09609 |