Sample-Optimal Fourier Sampling in Any Constant Dimension

We give an algorithm for ℓ₂/ℓ₂ sparse recovery from Fourier measurements using O(k log N) samples, matching the lower bound of Do Ba-Indyk-Price-Woodruff'10 for non-adaptive algorithms up to constant factors for any k ≤ N^1-δ. The algorithm runs in Õ(N) time. Our algorithm extends to higher dimensions, leading to sample complexity of Õ_d(k log N), which is optimal up to constant factors for any d = O(1). These are the first sample optimal algorithms for these problems. A preliminary experimental evaluation indicates that our algorithm has empirical sampling complexity comparable to that of other recovery methods known in the literature, while providing strong provable guarantees on the recovery quality.