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Instance optimality in phase retrieval
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AI
DOI：10.1016/j.acha.2025.101818.png)
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Abstract
En 中文
Compressed sensing has demonstrated that a general signal x∈Fn ( F∈{R,C}) can be estimated from few linear measurements with an error proportional to the best k-term approximation error, a property known as instance optimality. In this paper, we investigate instance optimality in the context of phaseless measurements using the ℓp-minimization decoder, where p∈(0,1], for both real and complex cases. More specifically, we prove that (2,1) and (1,1)-instance optimality of order k can be achieved with m=O(klog(n/k)) phaseless measurements, paralleling results from linear measurements. These results imply that one can stably recover approximately k-sparse signals from m=O(klog(n/k)) phaseless measurements. Our approach leverages the phaseless bi-Lipschitz condition. Additionally, we present a non-uniform version of (2,2)-instance optimality result in probability applicable to any fixed vector x∈Fn. These findings reveal striking parallels between compressive phase retrieval and classical compressed sensing, enhancing our understanding of both phase retrieval and instance optimality.
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3.2
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