Special Seminar in Applied Mathematics
Most of the recent activity on sparse signal recovery has focused on signals with sparse representations in finite discrete dictionaries. However, signals encountered in applications such as imaging, radar, sonar, sensor array, communication, seismology, and remote sensing are usually specified by sparse parameters in continuous domains. In this talk, I will show that atomic minimization provides a general convex approach to directly enforce sparsity in the continuous domain, thus circumventing some of the computational and theoretical issues arising from discretization based approaches. I then specialize the framework to estimating the continuous frequencies and phases of a mixture of complex exponentials from incomplete, noisy, and corrupted time samples. I will demonstrate that the corresponding atomic minimization problems have exact semidefinite reformulations. For the natural random subsampling scheme, the number of samples necessary for accurate frequency estimation is proportional to the number of frequencies present in the signal, up to log factors. This is a direct generalization of compressive sensing with a discretized Fourier dictionary that is completely free of gridding. I will also discuss implications of the results in spectral estimation, signal sampling, and super-resolution imaging.