Sample Efficient Estimation and Recovery in Sparse FFT via Isolation on Average

The problem of computing the Fourier Transform of a signal whose spectrum is dominated by a small number $k$ of frequencies quickly and using a small number of samples of the signal in time domain (the Sparse FFT problem) has received significant attention recently. It is known how to approximately compute the $k$-sparse Fourier transform in $\approx k\log^2 n$ time [Hassanieh et al'STOC'12], or using the optimal number $O(k\log n)$ of samples [Indyk et al'FOCS'14] in time domain, or come within $(\log\log n)^{O(1)}$ factors of both these bounds simultaneously, but no algorithm achieving the optimal $O(k\log n)$ bound in sublinear time is known. In this paper we propose a new technique for analysing noisy hashing schemes that arise in Sparse FFT, which we refer to as isolation on average. We apply this technique to two problems in Sparse FFT: estimating the values of a list of frequencies using few samples and computing Sparse FFT itself, achieving sample-optimal results in $k\log^{O(1)} n$ time for both. We feel that our approach will likely be of interest in designing Fourier sampling schemes for more general settings (e.g. model based Sparse FFT).