  Probabilities of vibronic transitions in molecules are referred to as
Franck-Condon factors (FCFs). Although several approaches for calculating FCFs
have been developed, such calculations are still challenging. Recently it was
shown that there exists a correspondence between the problem of calculating
FCFs and boson sampling. However, if the output photon number distribution of
boson sampling is sparse then it can be classically simulated. Exploiting these
results, we develop a method to approximately reconstruct the distribution of
FCFs of certain molecules. We demonstrate this method by applying it to formic
acid and thymine at $0$ K. In our method, we first obtain the marginal photon
number distributions for pairs of modes of a Gaussian state associated with the
molecular transition. We then apply a compressive sensing method called
polynomial-time matching pursuit to recover FCFs.
