Blind source separation of the stellar halo

The stellar halo of the Milky Way comprises an abundance of chemical signatures from accretion events and \textit{in-situ} evolution, that form an interweaving tapestry in kinematic space. To untangle this, we consider the mixtures of chemical information, in a given region of integral of motion space, as a variant of the blind source separation problem using non-negative matrix factorisation (NMF). Specifically, we examine the variation in [Fe/H], [Mg/Fe], and [Al/Fe] distributions of APOGEE DR17 stars across the $(E,L_z)$ plane of the halo. When 2 components are prescribed, the NMF algorithm splits stellar halo into low- and high-energy components in the $(E,L_z)$ plane which approximately correspond to the accreted and \textit{in-situ} halo respectively. We use these components to define a boundary between the \textit{in-situ} and the accreted stellar halo, and calculate their fractional contribution to the stellar halo as a function of energy, galactocentric spherical radius ($r$), height ($z$), and galactocentric cylindrical radius ($R$). Using a stellar halo defined by kinematic cuts, we derive a boundary in $(E,L_z)$ space where the halo transitions from \textit{in-situ} dominated to accretion dominated. Spatially, we find that this transition happens at $(r,z,R) \approx (8.7, 3.0, 8.1)$ kpc. We find that between 34\% to 53\% of the stellar halo's content is of accreted origin. Upon prescribing more components to the NMF model, we find evidence for overlapping chemical evolution sequences. We examine features within these components that resemble known substructures in the halo, such as \textit{Eos} and \textit{Aurora}.