preview

Essay On How To Solve The Sparse Approximate Dft Problem

Good Essays

The first work that tried to solve the sparse approximate DFT problem can be found in \cite{NYN93}, in which they designed an algorithm based on Hadamard Transform, i.e. the Fourier transform over the Boolean cube. A polynomial time algorithm to interpolates a sparse polynomial was developed in \cite{Y95}. The method in this paper inspired the authors of \cite{GMS05}, in which they described an algorithm that can be used to approximate DFT when $N$ is a power of 2. In the early 2000s, people paid a lot of attention to the sparse approximation problem in Fourier space. The first algorithm with sub-linear runtime and the sub-sampling property was given in \cite{GGIMM02}. In which they give a randomized algorithm with runtime poly($s, \log N, …show more content…

However, it needs to point out that the runtime of the algorithm in \cite{AGS03} has a high dependence on sparsity compare with \cite{GGIMM02} and \cite{GMS05}.

All the SFT algorithms above are randomized algorithms. This means they have small probability to fail to give the correct or optimal recovery on each input signal. Thus, they are not appropriate for long-lived failure intolerant applications. The first deterministic sub-linear time SFT algorithm was developed in \cite{I08} based on the deterministic Compressed Sensing results of Cormode and Muthukrishnan (CM)\cite{RSR69}\cite{CM05}\cite{CM06}. A simpler optimized version of this algorithm was given in \cite{I10}, which has similar runtime/sampling bounds ($\mathcal{O}(s^2 \log ^4 N)$) to the one in \cite{GMS05}. Later, in \cite{I11}, a further modified SFT algorithm was provided. It showed simple methods for extending the improved sparse Fourier transforms to higher dimensional settings. More specifically, the algorithm can find the near optimal $s$-term approximation for any given input function, $f: [0,2\pi]^{D} \rightarrow \C$ in $\mathcal{O}(s^2 D^4)$ time (neglecting logarithmic factors). The algorithms in \cite{I08}\cite{I10}\cite{I11} are all aliasing-based search algorithm \cite{indyk_overview}, which means they rely on the combinatorial properties of aliasing among frequencies in sub-samples DFTs. The algorithms

Get Access