Theory Seminar: An $n\log n$ Lower Bound for Fourier Transform Computation in the Well Conditioned Model

Nir Ailon (CS, Technion)

Wednesday, 02.04.2014, 13:00

Taub 201

Obtaining a non-trivial (super-linear) lower bound for computation of the Fourier transform in the linear circuit model has been a long standing open problem for over 40 years. All lower bounds so far have made strong restrictions on the computational model.

An early result by Morgenstern from 1973, provides an $\Omega(n \log n)$ lower bound for the unnormalized Fourier transform when the constants used in the computation are bounded. The proof uses a potential function related to a determinant. The determinant of the unnormalized Fourier transform is $n^{n/2}$, and thus by showing that it can grow by at most a constant factor after each step yields the result. This classic result does not explain why the normalized Fourier transform (of unit determinant) should require $\Omega(n\log n)$ steps.

More recently, Ailon (2013) showed that if only unitary 2-by-2 gates are used, one obtains an $\Omega(n\log n)$ lower bound. (The normalized FFT works in this model.) In this work we take another step and show an $\Omega(n\log n)$ lower bound for computing the Fourier transform (with any scaling) in a considerably less restricted computational model. In this model, we allow the gates to perform any nonsingular (not necessarily unitary) transformation involving 2 variables. Our restriction is that the composition of all gates up to any given step is uniformly well conditioned. This is a practical requirement because an ill conditioned transformation must result in numerical instability for certain inputs.

The main technical technique is extending the notion of matrix entropy of Ailon (2013) to the nonsingular group. For the unitary subgroup this was easily defined as Shannon entropy of the distribution defined using the matrix elements squared. For general nonsingular matrices, it turns out that the correct way to define this potential is by using both the matrix and its inverse, and also to allow ``negative probabilities''.

An early result by Morgenstern from 1973, provides an $\Omega(n \log n)$ lower bound for the unnormalized Fourier transform when the constants used in the computation are bounded. The proof uses a potential function related to a determinant. The determinant of the unnormalized Fourier transform is $n^{n/2}$, and thus by showing that it can grow by at most a constant factor after each step yields the result. This classic result does not explain why the normalized Fourier transform (of unit determinant) should require $\Omega(n\log n)$ steps.

More recently, Ailon (2013) showed that if only unitary 2-by-2 gates are used, one obtains an $\Omega(n\log n)$ lower bound. (The normalized FFT works in this model.) In this work we take another step and show an $\Omega(n\log n)$ lower bound for computing the Fourier transform (with any scaling) in a considerably less restricted computational model. In this model, we allow the gates to perform any nonsingular (not necessarily unitary) transformation involving 2 variables. Our restriction is that the composition of all gates up to any given step is uniformly well conditioned. This is a practical requirement because an ill conditioned transformation must result in numerical instability for certain inputs.

The main technical technique is extending the notion of matrix entropy of Ailon (2013) to the nonsingular group. For the unitary subgroup this was easily defined as Shannon entropy of the distribution defined using the matrix elements squared. For general nonsingular matrices, it turns out that the correct way to define this potential is by using both the matrix and its inverse, and also to allow ``negative probabilities''.