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• W4313147713 abstract "We study the problem of reconstructing a high-dimensional signal <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$mathrm {x} in mathbb {R}^{n}$ </tex-math></inline-formula> from a low-dimensional noisy linear measurement <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$mathrm {y}=mathrm {M}mathrm {x}+mathrm {e} in mathbb {R}^{ell }$ </tex-math></inline-formula> , assuming x admits a certain structure. We model the measurement matrix as M = BA, with arbitrary <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$mathrm {B} in mathbb {R}^{ell times m}$ </tex-math></inline-formula> and sub-gaussian <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$mathrm {A} in mathbb {R}^{m times n}$ </tex-math></inline-formula> ; therefore allowing for a family of random measurement matrices which may have heavy tails, dependent rows and columns, and a large dynamic range for the singular values. The structure is either given as a non-convex cone <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$T subset mathbb {R}^{n}$ </tex-math></inline-formula> , or is induced via minimizing a given convex function <inline-formula xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink> <tex-math notation=LaTeX>$f(cdot)$ </tex-math></inline-formula> , hence our study is sparsity-free. We prove, in both cases, that an approximate empirical risk minimizer robustly recovers the signal if the effective number of measurements is sufficient, even in the presence of a model mismatch, i.e., the signal not exactly admitting the model’s structure. While in classical compressed sensing the number of independent (sub)-gaussian measurements regulates the possibility of a robust reconstruction, in our setting the effective number of measurements depends on the properties of B. We show that, in this model, the stable rank of B indicates the effective number of measurements, and an accurate recovery is guaranteed whenever it exceeds, to within a constant factor, the effective dimension of the structure set. We apply our results to the special case of generative priors, i.e., when x is close to the range of a Generative Neural Network (GNN) with ReLU activation functions. Also, if the GNN has random weights in the last layer, our theory allows a partial Fourier measurement matrix, thus taking the first step towards a theoretical analysis of compressed sensing MRI with GNN. Our work relies on a recent result in random matrix theory by Jeong et al. (2020)." @default.
• W4313147713 created "2023-01-06" @default.
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• W4313147713 date "2022-09-01" @default.
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• W4313147713 title "Sparsity-Free Compressed Sensing With Applications to Generative Priors" @default.
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• W4313147713 doi "https://doi.org/10.1109/jsait.2022.3219807" @default.
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