  Neural recordings, returns from radars and sonars, images in astronomy and
single-molecule microscopy can be modeled as a linear superposition of a small
number of scaled and delayed copies of a band-limited or diffraction-limited
point spread function, which is either determined by the nature or designed by
the users; in other words, we observe the convolution between a point spread
function and a sparse spike signal with unknown amplitudes and delays. While it
is of great interest to accurately resolve the spike signal from as few samples
as possible, however, when the point spread function is not known a priori,
this problem is terribly ill-posed. This paper proposes a convex optimization
framework to simultaneously estimate the point spread function as well as the
spike signal, by mildly constraining the point spread function to lie in a
known low-dimensional subspace. By applying the lifting trick, we obtain an
underdetermined linear system of an ensemble of signals with joint spectral
sparsity, to which atomic norm minimization is applied. Under mild randomness
assumptions of the low-dimensional subspace as well as a separation condition
of the spike signal, we prove the proposed algorithm, dubbed as AtomicLift, is
guaranteed to recover the spike signal up to a scaling factor as soon as the
number of samples is large enough. The extension of AtomicLift to handle noisy
measurements is also discussed. Numerical examples are provided to validate the
effectiveness of the proposed approaches.
