An adaptive sublinear-time block sparse fourier transform

The problem of approximately computing the k dominant Fourier coefficients of a vector X quickly, and using few samples in time domain, is known as the Sparse Fourier Transform (sparse FFT) problem. A long line of work on the sparse FFT has resulted in algorithms with O(klognlog(n/k)) runtime [Hassanieh et al., STOC'12] and O(klogn) sample complexity [Indyk et al., FOCS'14]. This paper revisits the sparse FFT problem with the added twist that the sparse coefficients approximately obey a (k0,k1)-block sparse model. In this model, signal frequencies are clustered in k0 intervals with width k1 in Fourier space, and k= k0k1 is the total sparsity. Our main result is the first sparse FFT algorithm for (k0, k1)-block sparse signals with a sample complexity of O*(k0k1 + k0log(1+ k0)logn) at constant signal-to-noise ratios, and sublinear runtime. Our algorithm crucially uses adaptivity to achieve the improved sample complexity bound, and we provide a lower bound showing that this is essential in the Fourier setting: Any non-adaptive algorithm must use Ω(k0k1logn/k0k1) samples for the (k0,k1)-block sparse model, ruling out improvements over the vanilla sparsity assumption. Our main technical innovation for adaptivity is a new randomized energy-based importance sampling technique that may be of independent interest.