Tighter Fourier Transform Complexity Tradeoffs

TL;DR

This paper establishes a fundamental tradeoff between the speed of Fourier transform algorithms and the information loss they incur, showing that significant speedups necessarily involve multiple ill-conditioned steps causing data overflow or underflow.

Contribution

It proves that any attempt to significantly speed up the FFT must involve multiple ill-conditioned bottlenecks, extending previous lower bounds and providing a quantitative tradeoff between speed and information loss.

Findings

Speedup factors imply multiple ill-conditioned steps.

Each bottleneck causes information overflow or underflow.

Provides a quantitative tradeoff between computation speed and information loss.

Abstract

The Fourier Transform is one of the most important linear transformations used in science and engineering. Cooley and Tukey's Fast Fourier Transform (FFT) from 1964 is a method for computing this transformation in time $O(n\log n)$. Achieving a matching lower bound in a reasonable computational model is one of the most important open problems in theoretical computer science. In 2014, improving on his previous work, Ailon showed that if an algorithm speeds up the FFT by a factor of $b=b(n)\geq 1$, then it must rely on computing, as an intermediate "bottleneck" step, a linear mapping of the input with condition number $\Omega(b(n))$. Our main result shows that a factor $b$ speedup implies existence of not just one but $\Omega(n)$ $b$-ill conditioned bottlenecks occurring at $\Omega(n)$ different steps, each causing information from independent (orthogonal) components of the input to either overflow or underflow. This provides further evidence that beating FFT is hard. Our result also gives the first quantitative tradeoff between computation speed and information loss in Fourier computation on fixed word size architectures. The main technical result is an entropy analysis of the Fourier transform under transformations of low trace, which is interesting in its own right.