The Far-Field Curvature Index

Critical points of an optimization problem are equilibria of the corresponding gradient flow, and having many such points can indicate a more complicated optimization landscape. In metric algebraic geometry, the Euclidean distance degree gives a count of the critical points of the squared distance function on an algebraic variety, over the complex numbers and for generic data.

We introduce the Far-Field Curvature Index as a different way to measure the number of real critical points, in an average sense. We consider globally subanalytic sets and a large family of loss functions. When the data points are sampled farther and farther away from the constrained set, we show that the average number of real critical points converges to an invariant determined by the geometry of the set. More precisely, this invariant is related to curvature of the set together with a contribution coming from the geometry at infinity, through the link at infinity.

This is joint work with Andrea Rosana.