Quotients Beyond Group Actions: Learning Variable Equivalence with Local Geometry

Learning equivalence classes removes redundancy by identifying different realizations that represent the same input. Existing group-based equivalence methods accommodate redundant variation through a shared transformation structure, using the group action to define both the equivalence classes and their quotient geometry. This restricts the resulting equivalence to classes generated by that action, limiting task-, context-, and realization-dependent or non-geometric equivalence. We propose a novel approach to representation learning that learns a quotient map directly from observed outcomes, instead of deriving it from predefined transformations, by assigning shared coordinates to equivalent realizations. In continuous domains, the learned quotient map realizes equivalence classes as smooth level sets, with its differential and the Riemannian metric defining the tangent spaces and orthogonal decomposition, yielding a local quotient geometry. We use this geometry with flow matching to generate preferred realizations within fixed equivalence classes and to define consistent motion across changing classes. Experiments show that the learned quotient captures equivalence across both image and language domains, while articulated system experiments demonstrate generation and dynamics on the learned quotient geometry.