  Chemical tagging of stars based on their similar compositions can offer new
insights about the star formation and dynamical history of the Milky Way. We
investigate the feasibility of identifying groups of stars in chemical space by
forgoing the use of model derived abundances in favour of direct analysis of
spectra. This facilitates the propagation of measurement uncertainties and does
not presuppose knowledge of which elements are important for distinguishing
stars in chemical space. We use ~16,000 red-giant and red-clump H-band spectra
from the Apache Point Observatory Galactic Evolution Experiment and perform
polynomial fits to remove trends not due to abundance-ratio variations. Using
expectation maximized principal component analysis, we find principal
components with high signal in the wavelength regions most important for
distinguishing between stars. Different subsamples of red-giant and red-clump
stars are all consistent with needing about 10 principal components to
accurately model the spectra above the level of the measurement uncertainties.
The dimensionality of stellar chemical space that can be investigated in the
H-band is therefore $\lesssim 10$. For APOGEE observations with typical
signal-to-noise ratios of 100, the number of chemical space cells within which
stars cannot be distinguished is approximately $10^{10\pm2} \times (5\pm
2)^{n-10}$ with $n$ the number of principal components. This high
dimensionality and the fine-grained sampling of chemical space are a promising
first step towards chemical tagging based on spectra alone.
