  Given a time series vector, how can we efficiently compute a specified part
of Fourier coefficients? Fast Fourier transform (FFT) is a widely used
algorithm that computes the discrete Fourier transform in many machine learning
applications. Despite its pervasive use, all known FFT algorithms do not
provide a fine-tuning option for the user to specify one's demand, that is, the
output size (the number of Fourier coefficients to be computed) is
algorithmically determined by the input size. This matters because not every
application using FFT requires the whole spectrum of the frequency domain,
resulting in an inefficiency due to extra computation. In this paper, we
propose a fast Partial Fourier Transform (PFT), a careful modification of the
Cooley-Tukey algorithm that enables one to specify an arbitrary consecutive
range where the coefficients should be computed. We derive the asymptotic time
complexity of PFT with respect to input and output sizes, as well as its
numerical accuracy. Experimental results show that our algorithm outperforms
the state-of-the-art FFT algorithms, with an order of magnitude of speedup for
sufficiently small output sizes without sacrificing accuracy.
