Friday, September 21, 2012

Another Super Fast FFT

We propose Fourier transform algorithms using QTT format for data-sparse approximate representation of one- and multi-dimensional vectors (m-tensors). Although the Fourier matrix itself does not have a low-rank QTT representation, it can be efficiently applied to a vector in the QTT format exploiting the multilevel structure of the Cooley-Tukey algorithm. The m-dimensional Fourier transform of an n×⋯×n vector with n=2 d has O(m d2 R3)complexity, where R is the maximum QTT-rank of input, output and all intermediate vectors in the procedure. For the vectors with moderate R and large n and m the proposed algorithm outperforms the O(nm logn) fast Fourier transform (FFT) algorithm and has asymptotically the same log-squared complexity as the superfast quantum Fourier transform (QFT) algorithm. By numerical experiments we demonstrate the examples of problems for which the use of QTT format relaxes the grid size constrains and allows the high-resolution computations of Fourier images and convolutions in higher dimensions without the ‘curse of dimensionality’. We compare the proposed method with Sparse Fourier transform algorithms and show that our approach is competitive for signals with small number of randomly distributed frequencies and signals with limited bandwidth.
No toolbox available. An intriguing aspect of this paper aside from providing a competitor to sFFT is the QTT format mentioned. More on that later.

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David Friedman said...

The paper seems to be behind a paywall.

Igor said...

Yes it is. I am sorry.

But i think the QTT decomposition apsect is important.


Laurent Duval said...

One should find an non-paywall version here: Superfast Fourier transform using QTT approximation

Igor said...

THanks Laurent. Let me update the entry as a result.