An equivariant homeomorphism between two representations of a finite group is called a topological similarity, and if the representations are not isomorphic, a non-linear similarity. Topological similarity implies isomorphism for groups of odd order by work of Hsiang-Pardon and Madsen-Rothenberg. We give another proof of this theorem (Corollary B) using techniques from bounded algebraic K- and L-theory. Our methods also apply to representations of even order groups, with certain restrictions on the isotropy (Theorem A). We obtain a new necessary condition for the existence of non-linear similarities (Corollary C).