一种自适应次线性时间块稀疏傅里叶变换
An Adaptive Sublinear-Time Block Sparse Fourier Transform
摘要：
快速近似计算矢量$X$的$k$主傅立叶系数并且在时域中使用少量样本的问题被称为稀疏傅立叶变换（稀疏FFT）问题。关于稀疏FFT的一长串工作已经产生了具有$O(k\log n\log (n/k))$运行时间的算法[Hassanieh等人，STOC '12]和$O(k\log n)$样本复杂度[Indyk等人，FOCS '14]。使用非自适应算法证明了这些结果，并且后一个$O(k\log n)$样本复杂度结果基本上是单独在稀疏性假设下的最佳可能结果。本文重新审视稀疏FFT问题，增加了扭曲，稀疏系数近似服从$(k_0,k_1)$块稀疏模型。在该模型中，信号频率被聚集在傅立叶空间中具有宽度$k_1$的$k_0$间隔中，其中$k= k_0k_1$是总稀疏度。在应用中出现的信号通常通过该模型与$k_0\ll k$很好地近似。我们的主要结果是第一个稀疏FFT算法$(k_0, k_1)$块稀疏信号的样本复杂度为$O^*(k_0k_1 + k_0\log(1+ k_0)\log n)$在恒定的信号-噪声比，和次线性运行时间。类似的样本复杂度先前在使用随机高斯测量的基于模型的压缩感测的工作中实现，但是使用了$Ω(n)$运行时。据我们所知，我们的结果是基于模型的压缩感知的第一个次线性时间算法，以及第一个稀疏FFT结果，低于$O(k\log n)$样本复杂度界限。我们的算法至关重要地使用{\em adaptivity}来实现改进的样本复杂度界，并且我们证明了如果使用傅立叶测量，则自适应性实际上是必要的：任何非自适应算法必须使用$Ω(k_0k_1\log \frac{n}{k_0k_1})$样本用于$(k_0,k_1$）块稀疏模型，排除了对香草稀疏性假设的改进。
英文摘要：
The problem of approximately computing the $k$ dominant Fourier coefficients of a vector $X$ quickly, and using few samples in time domain, is known as the Sparse Fourier Transform (sparse FFT) problem. A long line of work on the sparse FFT has resulted in algorithms with $O(k\log n\log (n/k))$ runtime [Hassanieh et al., STOC'12] and $O(k\log n)$ sample complexity [Indyk et al., FOCS'14]. These results are proved using non-adaptive algorithms, and the latter $O(k\log n)$ sample complexity result is essentially the best possible under the sparsity assumption alone. This paper revisits the sparse FFT problem with the added twist that the sparse coefficients approximately obey a $(k_0,k_1)$-block sparse model. In this model, signal frequencies are clustered in $k_0$ intervals with width $k_1$ in Fourier space, where $k= k_0k_1$ is the total sparsity. Signals arising in applications are often well approximated by this model with $k_0\ll k$. Our main result is the first sparse FFT algorithm for $(k_0, k_1)$-block sparse signals with the sample complexity of $O^*(k_0k_1 + k_0\log(1+ k_0)\log n)$ at constant signal-to-noise ratios, and sublinear runtime. A similar sample complexity was previously achieved in the works on model-based compressive sensing using random Gaussian measurements, but used $Ω(n)$ runtime. To the best of our knowledge, our result is the first sublinear-time algorithm for model based compressed sensing, and the first sparse FFT result that goes below the $O(k\log n)$ sample complexity bound. Our algorithm crucially uses {\em adaptivity} to achieve the improved sample complexity bound, and we prove that adaptivity is in fact necessary if Fourier measurements are used: Any non-adaptive algorithm must use $Ω(k_0k_1\log \frac{n}{k_0k_1})$ samples for the $(k_0,k_1$)-block sparse model, ruling out improvements over the vanilla sparsity assumption.
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