The Complexity of Computing a Fourier Perturbation. Article Swipe

MAKE IMPACT

Nir Ailon , Gal Yehuda · YOU? · · 2016 · Open Access ·

The complexity of computing the Fourier transform is a longstanding open problem. Very recently, Ailon (2013, 2014, 2015) showed in a collection of papers that, roughly speaking, a speedup of the Fourier transform computation implies numerical ill-condition. The papers also quantify this tradeoff. The main method for proving these results is via a potential function called quasi-entropy, reminiscent of Shannon entropy. The quasi-entropy method opens new doors to understanding the computational complexity of the important Fourier transformation. However, it suffers from various drawbacks. This paper, motivated by one such drawback, eliminates it by extending the method. The argument goes as follows: If instead of computing the Fourier transform Fx of input x ∈ Rn we were to compute a Fourier e-perturbation, defined as (Id+eF )x, then the quasientropy method in the well-conditioned regime would, without any adjustments, lead to a linear algebraic lower bound of Ω(en logn) many operations (counting additions and multiplications on scalar variables). Had this bound been matched by an algorithm, then we would have been able to extract Fx in time O(en logn) by first computing (Id+eF )x, then subtracting x from the output and dividing the result by e. By taking e = Θ(1/ √ logn) we could artificially drive the computation time toward the trivial linear time lower bound. Such a scheme would suffer on a real computer from numerical errors, but this could be avoided by extending the computer word size by only Θ(log e) = Θ(log log n) bits. The end result is a Fourier algorithm in running time O(n log logn) (counting logical bit operations, and using fast integer multiplication). We generalize the quasi-entropy method so as to show that driving e down does not allow such a free ride in case of the Walsh-Hadamard Fourier transform, and that the linear algebraic lower bound is, in fact, Ω((n logn)/ log e). This exactly ‘cancels out’ the numerical accuracy overhead. It also strengthens our belief that, roughly speaking, Fourier computation requires Ω(n logn) time in a computational model that takes into account numerical accuracy and logical bit operations. ∗ [email protected] [email protected]

Type

preprint

Language

en

Landing Page

https://arxiv.org/pdf/1604.02557.pdf

OA Status

green

References

30

Related Works

20

OpenAlex ID

https://openalex.org/W2341760615

Raw OpenAlex JSON

OpenAlex ID

https://openalex.org/W2341760615

Canonical identifier for this work in OpenAlex

Title

The Complexity of Computing a Fourier Perturbation.

Work title

Type

preprint

OpenAlex work type

Language

en

Primary language

Publication year

2016

Year of publication

Publication date

2016-04-09

Full publication date if available

Authors

Nir Ailon, Gal Yehuda

List of authors in order

Landing page

https://arxiv.org/pdf/1604.02557.pdf

Publisher landing page

Open access

Yes

Whether a free full text is available

OA status

green

Open access status per OpenAlex

OA URL

https://arxiv.org/pdf/1604.02557.pdf

Direct OA link when available

Concepts

Fourier transform, Fast Fourier transform, Computation, Speedup, Computational complexity theory, Time complexity, Entropy (arrow of time), Mathematics, Upper and lower bounds, Discrete-time Fourier transform, Algorithm, Computer science, Discrete mathematics, Fourier analysis, Fractional Fourier transform, Mathematical analysis, Parallel computing, Physics, Quantum mechanics

Top concepts (fields/topics) attached by OpenAlex

Cited by

0

Total citation count in OpenAlex

References (count)

30

Number of works referenced by this work

Related works (count)

20

Other works algorithmically related by OpenAlex

Full payload