In modern data and signal acquisition, one main challenge arises from the growing scale of data. The data acquisition devices, however, are often limited by physical and hardware constraints, precluding sampling with the desired rate and precision. It is thus of great interest to reduce the sensing complexity while retaining recovery resolution. And that is why we are interested in reconstructing a signal from a small number of randomly observed time domain samples. The main contributions of this thesis are as follows.
First, we consider reconstructing a one-dimensional (1-D) spectrally sparse signal from a small number of randomly observed time-domain samples. The signal of interest is a linear combination of complex sinusoids at R distinct frequencies. The frequencies can assume any continuous values in the normalized frequency domain [0, 1). After converting the spectrally sparse signal into a low-rank Hankel structured matrix completion problem, we propose an efficient feasible point approach, named projected Wirtinger gradient descent (PWGD) algorithm, to efficiently solve this structured matrix completion problem. We give the convergence analysis of our proposed algorithms. We then apply this algorithm to a different formulation of structured matrix recovery: Hankel and Toeplitz mosaic structured matrix. The algorithms provide better recovery performance; and faster signal recovery than existing algorithms including atomic norm minimization (ANM) and Enhanced Matrix Completion (EMaC). We further accelerate our proposed algorithm by a scheme inspired by FISTA. Extensive numerical experiments are provided to illustrate the efficiency of our proposed algorithms. Different from earlier approaches, our algorithm can solve problems of very large dimensions very efficiently. Moreover, we extend our algorithms to signal recovery from noisy samples. Finally, we aim to reconstruct a two-dimension (2-D) spectrally sparse signal from a small size of randomly observed time-domain samples. We extend our algorithms to high-dimensional signal recovery from noisy samples and multivariate frequencies.