Bispectrum Inversion with Application to Multireference Alignment.
Bispectrum Inversion with Application to Multireference Alignment.
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DOI：
10.1109/tsp.2017.2775591
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发表时间：
2018-02-15
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影响因子：
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通讯作者：
Singer A
中科院分区：
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作者：
Bendory T;Boumal N;Ma C;Zhao Z;Singer A
我们考虑了从自身的噪声循环平移版本中估计信号的问题，称为多参考对齐(MRA)。MRA的一种自然方法可能是首先估计观测数据的偏移，然后通过对数据进行对齐和平均来推断信号。相反，我们考虑一种基于直接估计信号的方法，使用信号在平移下不变的特征。具体地说，我们估计了观测信号的功率谱和双谱。在温和的假设下，这些不变特征包含了足够的信息来推断信号。特别地，双谱可以用来估计傅立叶相位。为此，我们提出并分析了几种算法。我们的主要方法包括光滑相流形上的非凸优化。根据经验，在没有噪声的情况下，这些非凸算法似乎在随机初始化的情况下收敛到目标信号。这些算法对噪声也有很强的稳健性。然后，我们建议另外三种方法。这些方法基于频率推进法、半定松弛法和整数规划法。前两种方法可以在没有噪声的情况下准确地恢复相位。在高噪声环境下，如果量测的数量与噪声方差的立方(即信息论比率)相似，则不变特征方法可以得到稳定的估计。此外，它只需要对数据进行一次遍历，这在低信噪比时是重要的，当观测数量必须很大时。
We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal–to–noise ratio when the number of observations must be large.