近似最优稀疏傅里叶变换
Nearly Optimal Sparse Fourier Transform
摘要：
我们考虑的问题，计算k-稀疏近似的离散傅立叶变换的n维信号。我们展示：* 对于输入信号具有至多k个非零傅立叶系数的情况，是O（k log n）时间随机算法，并且 * 对于一般输入信号，是O（k log n log（n/k））时间随机算法。这两种算法都实现了o（n log n）的时间，因此对于任何k = o（n），都优于快速傅立叶变换。它们是第一个已知的算法，满足这个属性。此外，如果假设快速傅立叶变换是最优的，那么对于任何k = n^{Ω（1）}，用于精确k稀疏情况的算法都是最优的。我们补充我们的算法结果表明，任何算法计算稀疏傅立叶变换的一般信号必须使用至少Ω（k log（n/k）/ log log n）的信号样本，即使它被允许执行自适应采样。
英文摘要：
We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: * An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and * An O(k log n log(n/k))-time randomized algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = n^{Ω(1)}. We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log(n/k)/ log log n) signal samples, even if it is allowed to perform adaptive sampling.
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