G-INVARIANT REPRESENTATIONS USING COORBITS

Abstract

Consider a finite-dimensional real vector space and a finite group acting unitarily on it. We investigate the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding relies on subsets of sorted coorbits with respect to chosen window vectors.Our main injectivity results examine the conditions under which such embeddings are injective. We establish these results using semialgebraic techniques. Furthermore, our main stability result states and demonstrates that any embedding based on sorted coorbits is automatically bi-Lipschitz when injective. We establish this result using geometric function techniques. Our work has applications in data science, where certain systems exhibit intrinsic invariance to group actions. For instance, in graph deep learning, graph-level regression and classification models must be invariant to node labeling.