稀疏傅里叶变换分析的随机化算法的理论与实验分析
Theoretical and Experimental Analysis of a Randomized Algorithm for Sparse Fourier Transform Analysis
AI总结：
提出并实现了一种改进的实用化随机化稀疏傅里叶分析算法（RAlSFA），在多项式时间内以高概率构造近最优B项稀疏表示，并在一维和二维中优于FFTW。
摘要：
我们分析了一种亚线性的RAlSFA（稀疏傅里叶分析的随机化算法），该算法遵循\cite{GGIMS}中的方法，在时间和空间复杂度poly(B,log(N))内，为给定长度为N的离散信号S找到近最优的B项稀疏表示R。其时间代价poly(log(N))应与快速傅里叶变换（FFT）的超线性O(N log N)时间需求进行比较。理论论文\cite{GGIMS}中给出的RAlSFA的直接实现在实践中非常缓慢。我们的主要结果是一个大幅改进且实用的RAlSFA。我们引入了若干新思想和技术来加速算法。给出了参数选择的严格和启发式论证。我们的RAlSFA以至少1-delta的概率，在时间poly(B)log(N)log(1/delta)/ epsilon^{2} log(M)内构造近最优的B项表示R，使得||S-R||^{2}<=(1+epsilon)||S-R_{opt}||^{2}。此外，该RAlSFA实现对于并非不合理的大N已经击败了FFTW。我们从理论和数值上将该算法扩展到高维情况。交叉点在一维为N=70000，在二维N*N网格上对于存在噪声的小B信号为N=900。
英文摘要：
We analyze a sublinear RAlSFA (Randomized Algorithm for Sparse Fourier Analysis) that finds a near-optimal B-term Sparse Representation R for a given discrete signal S of length N, in time and space poly(B,log(N)), following the approach given in \cite{GGIMS}. Its time cost poly(log(N)) should be compared with the superlinear O(N log N) time requirement of the Fast Fourier Transform (FFT). A straightforward implementation of the RAlSFA, as presented in the theoretical paper \cite{GGIMS}, turns out to be very slow in practice. Our main result is a greatly improved and practical RAlSFA. We introduce several new ideas and techniques that speed up the algorithm. Both rigorous and heuristic arguments for parameter choices are presented. Our RAlSFA constructs, with probability at least 1-delta, a near-optimal B-term representation R in time poly(B)log(N)log(1/delta)/ epsilon^{2} log(M) such that ||S-R||^{2}<=(1+epsilon)||S-R_{opt}||^{2}. Furthermore, this RAlSFA implementation already beats the FFTW for not unreasonably large N. We extend the algorithm to higher dimensional cases both theoretically and numerically. The crossover point lies at N=70000 in one dimension, and at N=900 for data on a N*N grid in two dimensions for small B signals where there is noise.
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