One can use invariants to distinguish geometric objects up to symmetries. In Classical Invariant Theory one studies polynomial invariants. For noisy data with symmetries we would also like to consider a distance (a measure of similarity) between objects. Suppose the objects all lie in a Euclidean vector space V on which a compact group G acts by symmetries. The shortest distance between objects gives a metric on the orbit space V/G. A set of generating polynomial invariants gives an embedding from the orbit space V/G into a Euclidean space. Unfortunately, this embedding is usually not bi-Lipschitz and is not good for estimating the distance between objects. For working with noisy data, we would like to construct a bi-Lipschitz embedding of the orbit space V/G into a Euclidean space with small distortion. In this talk I will give an overview of bi-Lipschitz invariant and discuss recent work by Ben Blum-Smith, Dustin Mixon, Yousef Qaddura, Brantley Vose and the author.