Functional Data Analysis for Extracting the Intrinsic Dimensionality of Spectra -- Application: Chemical Homogeneity in Open Cluster M67

High-resolution spectroscopic surveys of the Milky Way have entered the Big Data regime and have opened avenues for solving outstanding questions in Galactic Archaeology. However, exploiting their full potential is limited by complex systematics, whose characterization has not received much attention in modern spectroscopic analyses. In this work, we present a novel method to disentangle the component of spectral data space intrinsic to the stars from that due to systematics. Using functional principal component analysis on a sample of $18,933$ giant spectra from APOGEE, we find that the intrinsic structure above the level of observational uncertainties requires ${\approx}$10 Functional Principal Components (FPCs). Our FPCs can reduce the dimensionality of spectra, remove systematics, and impute masked wavelengths, thereby enabling accurate studies of stellar populations. To demonstrate the applicability of our FPCs, we use them to infer stellar parameters and abundances of 28 giants in the open cluster M67. We employ Sequential Neural Likelihood, a simulation-based Bayesian inference method that learns likelihood functions using neural density estimators, to incorporate non-Gaussian effects in spectral likelihoods. By hierarchically combining the inferred abundances, we limit the spread of the following elements in M67: $\mathrm{Fe} \lesssim 0.02$ dex; $\mathrm{C} \lesssim 0.03$ dex; $\mathrm{O}, \mathrm{Mg}, \mathrm{Si}, \mathrm{Ni} \lesssim 0.04$ dex; $\mathrm{Ca} \lesssim 0.05$ dex; $\mathrm{N}, \mathrm{Al} \lesssim 0.07$ dex (at 68% confidence). Our constraints suggest a lack of self-pollution by core-collapse supernovae in M67, which has promising implications for the future of chemical tagging to understand the star formation history and dynamical evolution of the Milky Way.