Sparse recovery and Fourier sampling

In the last decade a broad literature has arisen studying sparse recovery, the estimation of sparse vectors from low dimensional linear projections. Sparse recovery has a wide variety of applications such as streaming algorithms, image acquisition, and disease testing. A particularly important subclass of sparse recovery is the sparse Fourier transform, which considers the computation of a discrete Fourier transform when the output is sparse. The sparse Fourier transform has applications to medical imaging, spectrum sensing, and purely computation tasks involving convolution. This thesis describes a coherent set of techniques that achieve optimal or near-optimal upper and lower bounds for a variety of sparse recovery problems. We give a number of state-of-the-art algorithms for recovery of an approximately k-sparse vector in n dimensions: • Two sparse Fourier transform algorithms that takeO(k log n log(n/k)) time andO(k log n log log n) samples, respectively. The latter is log log n-competitive with the optimal sample complexity when k < n1− . • An algorithm for adaptive sparse recovery using O(k log log(n/k)) measurements, showing that adaptivity can give substantial improvements when k is small. • An algorithm for C-approximate sparse recovery with O(k logC(n/k) log∗ k) measurements, which matches our lower bound up to the log∗ k factor and gives the first improvement for 1 C n . In the second part of this thesis, we give lower bounds for the above problems and more. Thesis Supervisor: Piotr Indyk Title: Professor