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 and utilise 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 two components to define a new boundary between the \textit{in-situ} and the accreted stellar halo. Moreover, we calculate the components fractional contribution to the stellar halo as a function of energy, galactocentric spherical radius, height, and galactocentric cylindrical radius. Using a stellar halo defined by kinematic cuts, we find that the halo transitions from \textit{in-situ} dominated to accretion dominated at $E \approx -1.67 \times 10^5$ (km/s)$^2$ (using the potential in McMillan 2017), and at $(r,z,R) \approx (8.7, 3.0, 8.1)$ kpc. The low-energy component is found to span a range of [Al/Fe] that falls beyond the typically accepted \textit{in-situ} floor of [Al/Fe] $=0$. Upon prescribing more components to the NMF model, we find hints of the existence of overlapping chemical evolution sequences that other techniques struggle to find. We also examine features within these components that resemble known substructures in the halo, such as \textit{Eos} and \textit{Aurora}. This work provides insight into their origin and the part they play in the Milky Way's formation.