We develop tools for computing invariants of singular varieties and apply them to the classical theory of nodal curves and the complexity analysis of non-convex optimization problems.

The first result provides a method for computing the Segre class of a closed embedding X → Y in terms of the Segre classes of X and Y in an ambient space Z. This method is used to extend the classical Riemann-Kempf formula to the case of nodal curves.

Next we focus on techniques for computing the ED degree of a complex projective variety associated to an optimization problem. As a first application we consider the problem of scene reconstruction and find a degree 3 polynomial that computes the ED degree of the multiview variety as a function of the number of cameras. Our second application concerns the problem of weighted low rank approximation. We provide a characterization of the weight matrices for which the weighted 1-rank approximation problem has maximal ED degree.