papers and preprints
With Robert Ghrist. Journal of Applied and Computational Topology. [Journal link]
This paper contains the theoretical foundations of what one might call “spectral sheaf theory,” an extension of spectral graph theory to sheaves on graphs and complexes. Introducing an inner product structure on the stalks of a cellular sheaf allows us to construct Hodge Laplacians, which behave analogously to the discrete Hodge Laplacians first introduced by Eckmann. A number of constructions in spectral graph theory and network engineering turn out to be examples of cellular sheaves, and we suggest further directions for applications of the theory.
With Robert Ghrist. Presented at the 2019 Allerton Conference on Communication, Control, and Computing.
Consider a network of independent agents , each with a locally computable real-valued function . Can the agents collaborate via connections in the network to find the minimum of ? The answer is yes; one approach involves converting the global optimization problem to a collection of local problems subject to an equality constraint represented by the graph Laplacian. Standard optimization techniques then produce an algorithm that only requires communication between neighboring agents in the graph. We generalize this method to optimization over the space of 0-cochains of a cellular sheaf subject to the constraint . Examples with connections to engineering problems are included.
With Robert Ghrist. Presented at ICASSP 2019.
How do we learn a network structure from data? That is, suppose we have a number of signals supported on the vertices of an unknown network and we wish to discover the network. Under the assumption that these signals are smooth, i.e., they have small variation over edges, this is a convex optimization problem. However, the smoothness assumption privileges constant signals on the graph, while the structure of the signals might come from more complicated interactions over edges. This paper shows that we can learn a sheaf Laplacian instead of a graph Laplacian, allowing us to extract networks with more complex behaviors.
With Jared Culbertson, Dan Guralnik, and Peter Stiller. Presented at ICData 2019. [Proceedings]
Carlsson and Mémoli showed that there exists a unique hierarchical clustering functor satisfying certain natural constraints. In this paper, we consider a generalization of clustering where we allow clusters to overlap, and show that this allows for a significantly greater variety of clustering functors.
Presented in the Applied Topology session of the 2019 Union College Mathematics Conference.
Presented at the 2019 SIAM Workshop on Network Science.
Presented at the AMS Eastern Sectional Meeting, September 2018.
An expository introduction to sheaves on graphs, oriented toward applications in engineering and network science. This is a work in progress and will be updated periodically. (Last update: 15 October 2019.)
A Julia package for learning sheaves from data. Based largely on the optimization problems considered in Learning Sheaf Laplacians from Smooth Signals.