Quality of the (quadratic) PSM approximation: a single PSM was constructed using 250 or 1000 a.i. model spectra (cf. the absolute minimum number of 231), calculated at label points (“objects”) drawn randomly from those in the APOGEE survey (Alam et al. 2015; Holtzman et al. 2015). The panels illustrate different PSM—a.i. model comparisons, for 10,000 other objects drawn from the labels of the APOGEE survey. The top left panel shows for a limited wavelength section the average of the exact a.i. model spectra and of the PSM, which appear indistinguishable. The bottom left panel shows the ensemble average (absolute) difference between the a.i. model and the PSM flux (the approximation error), as a function of wavelength. For each one of the 10,000 objects there is a pixel-by-pixel distribution of these approximation errors, which is shown in the top right panel for the pixel-by-pixel average approximation error. The bottom right panel finally shows the distribution across all objects of their (pixel-by-pixel) median approximation error. Note that there are rare cases (objects of very high [Fe/H], where the approximation is only good to a median of 10⁻³). Taken together, however, this shows that a single PSM approximates the exact a.i. model spectra typically to within 10⁻³ for objects with a label distribution resembling that of the entire APOGEE survey (which merely serves as an illustration here), over the 10,000 labels of the median. Constructing the PSM from 1000 instead of 250 random label points leads to a better PSM approximation.