On the Geometry of Deep Learning
Randall Balestriero, Ahmed Imtiaz Humayun, Richard G. Baraniuk
Notices of the American Mathematical Society
April 2025
In this paper, we overview one promising avenue of progress at the mathematical foundation of deep learning: the connection between deep networks and function approximation by affine splines (continuous piecewise linear functions in multiple dimensions). In particular, we overview work over the past decade on understanding certain geometrical properties of a deep network’s affine spline mapping, in particular how it tessellates its input space. The affine spline connection and geometrical viewpoint provide a powerful portal through which to view, analyze, and improve the inner workings of deep networks.