We explore whether medium-resolution stellar spectra can be reconstructed
from photometric observations, taking advantage of the highly compressible nature of the
spectra. We formulate the spectral reconstruction as
a least-squares problem with a sparsity constraint. In our test case using data from the
Sloan Digital Sky Survey, only three broad-band filters are used as input.
We demonstrate that reconstruction using three
principal components is feasible with these filters, leading to differences with respect
to the original spectrum smaller than 5%. We analyze the effect of uncertainties in the observed magnitudes and find
that the available high photometric precision induces very small errors in the reconstruction.
This process may facilitate the extraction of purely spectroscopic
quantities, such as the overall metallicity, for hundreds of millions of stars for which
only photometric information is available, using standard techniques applied
to the reconstructed spectra.