Tighter Fourier Transform Complexity Tradeoffs

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 logn). From a lower bound perspective, Morgenstern’s result from 1974 provides an Ω(n logn) lower bound for the unnormalized Fourier Transform (of determinant n), assuming the linear computational model using numbers of at most constant modulus. Ailon shows in 2013 an Ω(n logn) for computing the normalized Fourier Transform (of determinant 1) assuming only unitary operations on two coordinates are allowed at each step, and no extra memory is allowed. In 2014, Ailon then improved the result to show that, essentially, if an algorithm speeds up the FFT by a factor of b(n) ≥ 1, then it must rely on computing, as an intermediate “bottleneck” step, a linear mapping M of the input with condition number Ω(b(n)). We improve [Ailon 2014] in two ways. Our secondary result shows that the bottleneck is more severe than that presented in [Ailon 2014]. Our primary result shows that a factor b speedup implies existence of a b-ill conditioned bottleneck occuring at Ω(n) different steps, each causing information from independent (orthogonal) components of the input to either overflow or underflow.