  We apply two sparse reconstruction techniques, the least absolute shrinkage
and selection operator (LASSO) and the sparse exponential mode analysis (SEMA),
to two-dimensional (2D) spectroscopy. The algorithms are first tested on model
data, showing that both are able to reconstruct the spectra using only a
fraction of the data required by the traditional Fourier-based estimator.
Through the analysis of a sparsely sampled experimental fluorescence detected
2D spectra of LH2 complexes, we conclude that both SEMA and LASSO can be used
to significantly reduce the required data, still allowing to reconstruct the
multidimensional spectra. Of the two techniques, it is shown that SEMA offers
preferable performance, providing more accurate estimation of the spectral line
widths and their positions. Furthermore, SEMA allows for off-grid components,
enabling the use of a much smaller dictionary than the LASSO, thereby improving
both the performance and lowering the computational complexity for
reconstructing coherent multidimensional spectra.
