This thesis presents an in-depth examination of the properties and efficient computation of Graph Diffusion Distance (GDD). This distance metric, defined for undirected graphs of unequal size, uses the eigenvalues of the graph Laplacian to quantify the difference in behavior between diffusion of heat running on the nodes of each graph. In Chapter 2, we defined a class of distance measures which use the eigenvalues of a graph Laplacian, and its matrix exponential, as a distinctive measurement of the properties of diffusion on the nodes of the graph. This family of related distance measures have several nice theoretical properties, which we examine in detail in Chapter 3. One specific nice property is that GDD and several of its variants are bounded above and below by expressions which depend on the eigenvalues of the two graphs (the spectral lower and upper bounds), making them possible to compute with real-valued (rather than combinatorial) optimization. However, these measures typically still require expensive matrix-valued optimizations to calculate. In Chapters 4 and 5 we present, and examine the numeric behavior of, a novel optimization algorithm which greatly reduces the number and size of matrix optimizations which need to be performed, resulting in a speedup of up to 1000x.

With this computational speedup, GDD can differentiate digits of MNIST, distinguish which of several 3D meshes a given graph is a discretization of, and be used to classify biological data (morphological graphs); we show some of these applications in Chapter 8. Furthermore, the $P$ matrix which is produced during the GDD calculation is useful as a prolongation/restriction operator to coarsen and refine computational graphs. An example of this latter application is the use of coarsening and refinement operators as part of a neural network architecture; these multiscale machine learning models are more accurate and more efficient than their single-level counterparts. In Chapters 6 and 7 we demonstrate the advantages of these machine learning approaches through a variety of numerical experiments. Specifically, $P$ matrices can be used to automatically coarsen the model architecture of a machine learning model. The resulting coarsened model learns more efficiently, and in the case of the experiments in Chapter 7 learns to emulate a dataset with lower error than a model operating only at one scale.

The graph Laplacian (along with its matrix exponential), is a fundamental object which captures structural information about a graph. This thesis presents a variety of methods for comparing such operators and accelerating machine learning models which are constructed around the graph Laplacian.

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