Nearly Optimal Sparse Fourier Transform (1201.2501v2)

Abstract: 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^{\Omega(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 \Omega(k log(n/k)/ log log n) signal samples, even if it is allowed to perform adaptive sampling.

• The paper presents two deterministic algorithms that compute a k-sparse DFT in O(k log n) and O(k log n log(n/k)) time, outperforming classic FFT for sparse signals.
• It leverages randomized sampling and frequency binning techniques to efficiently isolate and estimate nonzero Fourier coefficients from an n-dimensional signal.
• The study also establishes theoretical lower bounds for sampling efficiency, offering practical insights for improved implementations in signal processing and data compression.

Nearly Optimal Sparse Fourier Transform: A Summary

The paper "Nearly Optimal Sparse Fourier Transform" by Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price addresses the problem of efficiently computing a k-sparse approximation to the discrete Fourier transform (DFT) of an n-dimensional signal. This topic is fundamental in areas such as signal processing, communications, and data compression, where the Fourier Transform is widely applied. The authors propose algorithms that outperform the time complexity of the Fast Fourier Transform (FFT) for sparse signals, achieving sublinear runtime for specific cases, which marks a significant improvement over classical approaches.

Algorithmic Contributions

The authors present two deterministic algorithms:

1. Exact k-Sparse Fourier Transform:
   • This algorithm operates in O(k log n) time for the case where the input signal contains exactly k non-zero Fourier coefficients.
   • It achieves runtime efficiency better than the FFT's O(n log n) for k = o(n), thereby extending existing results for a wider range of k.
2. General Sparse Fourier Transform:
   • For general input signals, this algorithm runs in O(k log n log(n/k)) time.
   • It ensures a nearly optimal sublinear runtime while addressing a broader spectrum of input sparsities.

The algorithms make use of techniques such as randomized sampling and sublinear time frequency binning, where the Fourier coefficients are partitioned into a small number of bins. These methods take advantage of properties of the signals' frequency domain to rapidly isolate and estimate pertinent coefficients.

Theoretical Insights and Lower Bounds

In addition to the algorithmic innovations, the paper establishes theoretical lower bounds for the problem. It demonstrates that any k-sparse Fourier transform computation must use at least Ω(k log(n/k) / log log n) signal samples, even with adaptive sampling. This bound indicates that while the proposed algorithms are highly efficient, there remains a theoretical ceiling on the sampling efficiency that can be attained.

Practical Implications

For practical purposes, the paper's algorithms present opportunities for more efficient implementations of the Fourier transform in real-world applications, particularly those involving sparse data such as video signals or compressed sensing. The implementation details suggest that real improvements over existing FFT libraries like FFTW can be achieved for specific signal sizes (e.g., n = 2^17 and sparsity up to 2000).

Future Directions and Open Questions

While the results are compelling, several open questions remain, inviting further exploration:

• Can the algorithm for the general case be improved to achieve O(k log n) time?
• How can the sample complexity be reduced further while maintaining or improving runtime efficiency?
• Is it possible to extend these results to more generalized settings, such as different transformations or non-power-of-two signal dimensions?
• How might the algorithm's failure probability be decreased below the current constant level without significant overhead?

Conclusion

This paper advances the field of sparse computations of the Fourier transform by providing novel methods that improve on existing time complexities under certain sparsity conditions. The algorithms proposed contribute both theoretical advancements and practical implementations that hold promise for a wide array of applications outside traditional signal processing. As computational demand for efficient transform calculations grows, insights from this paper provide a foundation for developing more scalable and effective Fourier transform techniques.