Given a variety (or more generally a scheme) $X$ equipped with the action of a reductive algebraic group $G$, one can use geometric invariant theory to stratify $X$ by ``instability type''. When the points of $X$ correspond to algebro-geometric objects, the instability type of an object can often be interpreted as a certain canonical filtration. These filtrations have been intensely studied when the objects parameterized by $X$ live in an Abelian category. However, far less is known about these canonical filtrations for objects in non-Abelian categories. In this talk, we will investigate these filtrations for finite-dimensional Lie algebras and finite-dimensional associative algebras. We will see that automorphisms play a key role in this investigation. In fact, we will show how to compute the filtration using convex optimization when the automorphism group is sufficiently rich. While determining the canonical filtration is a difficult problem in general, we will characterize algebras for which the filtration is trivial and give some results for algebras whose canonical filtration is induced by a grading.