Collaborative Research: Integrating Algebraic Topology, Graph Theory, and Multiscale Analysis for Learning Complex and Diverse Datasets

Graph theory, a branch of discrete mathematics, concerns the relationship between objects. These objects can be either simple vertices, i.e., nodes and/or points (zero simplexes), or high-dimensional simplexes. Here, the relationship refers to connectivity with possible orientations. Graph theory has many branches, such as geometric graph theory, algebraic graph theory, and topological graph theory. The study of graph theory draws on many other areas of mathematics, including algebraic topology, knot theory, algebra, geometry, group theory, combinatorics, etc. For example, algebraic graph theory can be investigated by using either linear algebra, group theory, or graph invariants. Among them, the The interaction between spectral theory and differential geometry became one of the critical developments. For example, the spectral theory of the Laplacian on a compact Riemannian manifold is a central object of de Rham-Hodge theory. Note that the Hodge Laplacian spectrum contains the topological information of the underlying manifold. Specifically, the harmonic part of the Hodge Laplacian spectrum corresponds to topological cycles. Connections between topology and spectral graph theory also play a central role in understanding the connectivity properties of graphs. Similarly, as the topological invariants revealing the connectivity of a topological space, the multiplicity of 0 eigenvalues of a 0-combinatorial Laplacian matrix is the number of connected components of a graph. Indeed, the number of q-dimensional holes can also be unveiled from the number of 0 eigenvalues of the q-combinatorial Laplacian. Nonetheless, spectral graph theory offers additional non-harmonic spectral information beyond topological invariants. The objective of this proposal is to introduce persistent spectral graph as a new paradigm for the multiscale analysis of the topological invariants and geometric shapes of high-dimensional datasets. Motivated by the success of persistent homology and multiscale graphs in dealing with complex biomolecular data, we construct a family of spectral graphs induced by a filtration parameter. In a combination with a simple machine learning algorithm, this additional spectral information is found to provide a powerful new tool for the quantitative analysis of molecular data, including the prediction of a set of protein B-factors for which existing standard predictors fail to work.