Spherical designs as a tool for derandomization: The case of PhaseLift
Spherical designs as a tool for derandomization: The case of PhaseLift
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球形设计作为去随机化的工具：PhaseLift 的案例
DOI：
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发表时间：
2015
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通讯作者：
F. Krahmer
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文献类型：
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作者：
R. Kueng;D. Gross;F. Krahmer
仅从幅度测量中检索相位信息的问题在上个世纪的许多科学学科中都出现了。 PhaseLift 是最近推出的一种相位恢复算法，该算法易于计算且数值稳定。然而，最初严格的性能保证特别依赖于高斯随机测量向量。迄今为止，尚不清楚测量的哪些属性可以使问题得到适定。考虑到这个问题，我们采用球形 t 设计的概念来实现 PhaseLift 的部分去随机化。球面设计是向量的集合，可再现复单位球面上均匀分布的前 2t 矩。因此，当 t 接近无穷大时，它们提供了“均匀分布”向量集的概念，范围从紧框架 (t = 1) 到整个球体。除了 PhaseLift 的具体案例之外，这一结果还凸显了球形设计在数据恢复方案去随机化方面的实用性。
The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally tractable and numerically stable. However, initial rigorous performance guarantees relied specifically on Gaussian random measurement vectors. To date, it remains unclear which properties of the measurements render the problem well-posed. With this question in mind, we employ the concept of spherical t-designs to achieve a partial derandomziation of PhaseLift. Spherical designs are ensembles of vectors which reproduce the first 2t moments of the uniform distribution on the complex unit sphere. As such, they provide notions of “evenly distributed” sets of vectors, ranging from tight frames (t = 1) to the full sphere, as t approaches infinity. Beyond the specific case of PhaseLift, this result highlights the utility of spherical designs for the derandomization of data recovery schemes.