  Theoretical work has shed light on the phase behavior of idealized mixtures
of many components with random interactions. But typical mixtures interact
through particular physical features, leading to a structured, non-random
interaction matrix of lower rank. Here we develop a theoretical framework for
such mixtures and derive mean-field conditions for thermodynamic stability and
critical behavior. Irrespective of the number of components and features, this
framework allows for a generally lower-dimensional representation in the space
of features and proposes a principled way to coarse-grain multicomponent
mixtures as binary mixtures. Moreover, it suggests a way to systematically
characterize different series of critical points and their codimensions in
mean-field. Since every pairwise interaction matrix can be expressed in terms
of features, our work is applicable to a broad class of mean-field models.
