  We have shown that data-driven models are effective for inferring physical
attributes of stars (labels; Teff, logg, [M/H]) from spectra, even when the
signal-to-noise ratio is low. Here we explore whether this is possible when the
dimensionality of the label space is large (Teff, logg, and 15 abundances: C,
N, O, Na, Mg, Al, Si, S, K, Ca, Ti, V, Mn, Fe, Ni) and the model is non-linear
in its response to abundance and parameter changes. We adopt ideas from
compressed sensing to limit overall model complexity while retaining model
freedom. The model is trained with a set of 12,681 red-giant stars with high
signal-to-noise spectroscopic observations and stellar parameters and
abundances taken from the APOGEE Survey. We find that we can successfully train
and use a model with 17 stellar labels. Validation shows that the model does a
good job of inferring all 17 labels (typical abundance precision is 0.04 dex),
even when we degrade the signal-to-noise by discarding ~50% of the observing
time. The model dependencies make sense: the spectral derivatives with respect
to abundances correlate with known atomic lines, and we identify elements
belonging to atomic lines that were previously unknown. We recover
(anti-)correlations in abundance labels for globular cluster stars, consistent
with the literature. However we find the intrinsic spread in globular cluster
abundances is 3--4 times smaller than previously reported. We deliver 17 labels
with associated errors for 87,563 red giant stars, as well as open-source code
to extend this work to other spectroscopic surveys.
