稀疏FFT中基于平均隔离的样本有效估计和恢复
Sample Efficient Estimation and Recovery in Sparse FFT via Isolation on Average
摘要：
快速计算其频谱由少量$k$频率支配的信号的傅立叶变换并且在时域中使用信号的少量样本的问题（稀疏FFT问题）最近受到了极大的关注。已知如何在$\approx k\log^2 n$时间中近似计算$k$ -稀疏傅立叶变换[Hassanieh等人的“STOC”12]，或者在时域中使用样本的最佳数量$O(k\log n)$ [Indyk等人的“FOCS”14]，或者同时进入这两个界限的$(\log\log n)^{O(1)}$因子内，但是没有已知在次线性时间中实现最佳$O(k\log n)$界限的算法。在本文中，我们提出了一种新的技术来分析噪声散列方案中出现的稀疏FFT，我们称之为隔离平均。我们将这种技术应用于稀疏FFT中的两个问题：使用少量样本估计频率列表的值和计算稀疏FFT本身，在$k\log^{O(1)} n$时间内实现样本最优结果。我们认为，我们的方法可能会在设计傅立叶采样方案更一般的设置（例如，基于模型的稀疏FFT）的兴趣。
英文摘要：
The problem of computing the Fourier Transform of a signal whose spectrum is dominated by a small number $k$ of frequencies quickly and using a small number of samples of the signal in time domain (the Sparse FFT problem) has received significant attention recently. It is known how to approximately compute the $k$-sparse Fourier transform in $\approx k\log^2 n$ time [Hassanieh et al'STOC'12], or using the optimal number $O(k\log n)$ of samples [Indyk et al'FOCS'14] in time domain, or come within $(\log\log n)^{O(1)}$ factors of both these bounds simultaneously, but no algorithm achieving the optimal $O(k\log n)$ bound in sublinear time is known. In this paper we propose a new technique for analysing noisy hashing schemes that arise in Sparse FFT, which we refer to as isolation on average. We apply this technique to two problems in Sparse FFT: estimating the values of a list of frequencies using few samples and computing Sparse FFT itself, achieving sample-optimal results in $k\log^{O(1)} n$ time for both. We feel that our approach will likely be of interest in designing Fourier sampling schemes for more general settings (e.g. model based Sparse FFT).
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