Sparse fourier transform in any constant dimension with nearly-optimal sample complexity in sublinear time

We consider the problem of computing a k-sparse approximation to the Fourier transform of a length N signal. Our main result is a randomized algorithm for computing such an approximation (i.e. achieving ℓ2/ℓ2 sparse recovery guarantees using Fourier measurements) using Od(klogNloglogN) samples of the signal in time domain and Od(klogd+3 N) runtime, where d≥ 1 is the dimensionality of the Fourier transform. The sample complexity matches the Ω(klog(N/k)) lower bound for non-adaptive algorithms due to [DIPW] for any k≤ N1−δ for a constant δ>0 up to an O(loglogN) factor. Prior to our work a result with comparable sample complexity klogN logO(1)logN and sublinear runtime was known for the Fourier transform on the line [IKP], but for any dimension d≥ 2 previously known techniques either suffered from a (logN) factor loss in sample complexity or required Ω(N) runtime.