Spurious minimizers in non uniform Fourier sampling optimization

A recent trend in the signal/image processing literature is the optimization of Fourier sampling schemes for specific datasets of signals. In this paper, we explain why choosing optimal non Cartesian Fourier sampling patterns is a difficult nonconvex problem by bringing to light two optimization issues. The first one is the existence of a combinatorial number of spurious minimizers for a generic class of signals. The second one is a vanishing gradient effect for the high frequencies. We conclude the paper by showing how using large datasets can mitigate first effect and illustrate experimentally the benefits of using stochastic gradient algorithms with a variable metric.