The Complexity of Computing a Fourier Perturbation

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 ε-perturbation, defined as (Id+εF )x, then the quasientropy method in the well-conditioned regime would, without any adjustments, lead to a linear algebraic lower bound of Ω(εn 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(εn logn) by first computing (Id+εF )x, then subtracting x from the output and dividing the result by ε. By taking ε = 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 ε) = Θ(log log n) bits. The end result is a Fourier algorithm in running time Õ(n log logn) (counting logical bit operations, and using fast integer multiplication). We generalize the quasi-entropy method so as to show that driving ε 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 ε). 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]