  Finite Rate of Innovation (FRI) theory considers sampling and reconstruction
of classes of non-bandlimited continuous signals that have a small number of
free parameters, such as a stream of Diracs. The task of reconstructing FRI
signals from discrete samples is often transformed into a spectral estimation
problem and solved using Prony's method and matrix pencil method which involve
estimating signal subspaces. They achieve an optimal performance given by the
Cram\'{e}r-Rao bound yet break down at a certain peak signal-to-noise ratio
(PSNR). This is probably due to the so-called subspace swap event. In this
paper, we aim to alleviate the subspace swap problem and investigate
alternative approaches including directly estimating FRI parameters using deep
neural networks and utilising deep neural networks as denoisers to reduce the
noise in the samples. Simulations show significant improvements on the
breakdown PSNR over existing FRI methods, which still outperform learning-based
approaches in medium to high PSNR regimes.
