The Complexity of Computing (Almost) Unitary Matrices With $\eps$-Copies of the Fourier Transform

TL;DR

This paper investigates the computational complexity of approximating the Fourier transform using nearly unitary matrices with limited copies, exploring tradeoffs between speed and numerical stability.

Contribution

It advances understanding of Fourier transform complexity by addressing limitations of the quasi-entropy method and connecting computational complexity with group theory.

Findings

Identifies limitations of the quasi-entropy approach

Proposes new insights into the complexity of approximate Fourier transforms

Links complexity questions to group-theoretic problems

Abstract

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 obvious limitations. This paper, motivated by one such limitation, partly overcomes it, while at the same time sheds llight on new interesting, and problems on the intersection of computational complexity and group theory. The paper also explains why this research direction, if fruitful, has a chance of solving much bigger questions about the complexity of the Fourier transform.