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• W3030280132 abstract "Abstract The field of compressed sensing has become a major tool in high-dimensional analysis, with the realization that vectors can be recovered from relatively very few linear measurements as long as the vectors lie in a low-dimensional structure, typically the vectors that are zero in most coordinates with respect to a basis. However, there are many applications where we instead want to recover vectors that are sparse with respect to a dictionary rather than a basis. That is, we assume the vectors are linear combinations of at most s columns of a d × n matrix D, where s is very small relative to n and the columns of D form a (typically overcomplete) spanning set. In this direction, we show that as a matrix D stays bounded away from zero in norm on a set S and a provided map Φ comprised of i.i.d. subgaussian rows has number of measurements at least proportional to the square of w ( D S ) , the Gaussian width of the related set D S , then with high probability the composition Φ D also stays bounded away from zero. As a specific application, we obtain that the null space property of order s is preserved under such subgaussian maps with high probability. Consequently, we obtain stable recovery guarantees for dictionary-sparse signals via the l 1 -synthesis method with only O ( s log ⁡ ( n / s ) ) random measurements and a minimal condition on D, which complements the compressed sensing literature." @default.
• W3030280132 created "2020-06-05" @default.
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• W3030280132 date "2020-09-01" @default.
• W3030280132 modified "2023-10-12" @default.
• W3030280132 title "Preserving injectivity under subgaussian mappings and its application to compressed sensing" @default.
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• W3030280132 doi "https://doi.org/10.1016/j.acha.2020.05.006" @default.
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