Instance optimality in phase retrieval
Yu Xia and Zhiqiang Xu
压缩感知已经证明，一般信号x∈𝔽^n（𝔽∈{ℝ,ℂ}）可以从少量线性测量中估计出来，其误差与最佳k项逼近误差成正比，这一性质被称为实例最优性。在本文中，我们研究了在无相位测量背景下使用ℓ_p最小化解码器（其中p ∈ (0, 1]）的实例最优性，涵盖了实数和复数情况。更具体地说，我们证明了通过m = O(k log(n/k))次无相位测量可以实现阶数为k的(2,1)和(1,1)-实例最优性，这与线性测量的结果相平行。这些结果表明，可以从m = O(k log(n/k))次无相位测量中稳定地恢复近似k稀疏信号。我们的方法利用了无相位双Lipschitz条件。此外，我们提出了一个适用于任意固定向量x∈𝔽^n的概率意义上的(2,2)-实例最优性结果的非均匀版本。这些发现揭示了压缩相位恢复与经典压缩感知之间的显著相似性，加深了我们对相位恢复和实例最优性的理解。
Compressed sensing has demonstrated that a general signal x∈𝔽^n (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 ℓ_p-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 achie...
