Codes and Expansions (CodEx) Seminar

Emily King (Colorado State University)
This Magic Moment: Frame Theory Meets Spherical Curves and Neurons

For certain finite subsets of the sphere, averaging any low-degree polynomial over the set results in the exact average of that polynomial over the entire sphere. Finite subsets for which this identity holds for every polynomial of degree at most (t) are called spherical (t)-designs. In the special case where (t=2), they are also known as centered tight frames. In this talk, I will describe two seemingly unrelated settings in which this geometry appears. First, spherical (t)-design curves are closed curves on the sphere whose arclength measure exactly integrates every spherical polynomial of degree at most (t). For (t>1), their first two moment conditions are continuous versions of the centering and tight-frame conditions, and we ask how short such exact quadrature curves can be. Second, I will discuss superposition in ReLU neural networks, where more features than neurons can be represented by an overcomplete dictionary. In a simple model, centering and tightness control the mean and variance of feature crosstalk, while low coherence helps control the tails needed for reliable support recovery and approximation. In both stories, frame geometry enables a form of compression: a one-dimensional curve can replace integration over a sphere for polynomials of bounded degree, while superposition allows more concepts than neurons. The two projects also provide contrasting case studies in “AI for math” and “math for AI”: generative AI played an integral role in the spherical-design-curve project, while the superposition work uses mathematics to explain an emergent neural-network phenomenon.