Foundations of data science (Springfield, Mo.)最新文献

We establish a new theory which unifies various aspects of topological approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.

We present an application of topological data analysis (TDA) to discrete optimization problems, which we show can improve the performance of the 2-opt local search method for the traveling salesman problem by simply applying standard Vietoris-Rips construction to a data set of trials. We then construct a simplicial complex which is specialized for this sort of simulated data set, determined by a stochastic matrix with a steady state vector begin{document}$ (P,pi) $end{document}. When begin{document}$ P $end{document} is induced from a random walk on a finite metric space, this complex exhibits similarities with standard constructions such as Vietoris-Rips on the data set, but is not sensitive to outliers, as sparsity is a natural feature of the construction. We interpret the persistent homology groups in several examples coming from random walks and discrete optimization, and illustrate how higher dimensional Betti numbers can be used to classify connected components, i.e. zero dimensional features in higher dimensions.

The p-persistent q-combinatorial Laplacian defined for a pair of simplicial complexes is a generalization of the q-combinatorial Laplacian. Given a filtration, the spectra of persistent combinatorial Laplacians not only recover the persistent Betti numbers of persistent homology but also provide extra multiscale geometrical information of the data. Paired with machine learning algorithms, the persistent Laplacian has many potential applications in data science. Seeking different ways to find the spectrum of an operator is an active research topic, becoming interesting when ideas are originated from multiple fields. In this work, we explore an alternative approach for the spectrum of persistent Laplacians. As the eigenvalues of a persistent Laplacian matrix are the roots of its characteristic polynomial, one may attempt to find the roots of the characteristic polynomial by homotopy continuation, and thus resolving the spectrum of the corresponding persistent Laplacian. We consider a set of simple polytopes and small molecules to prove the principle that algebraic topology, combinatorial graph, and algebraic geometry can be integrated to understand the shape of data.

Control-type particle filters have been receiving increasing attention over the last decade as a means of obtaining sample based approximations to the sequential Bayesian filtering problem in the nonlinear setting. Here we analyse one such type, namely the feedback particle filter and a recently proposed approximation of the associated gain function based on diffusion maps. The key purpose is to provide analytic insights on the form of the approximate gain, which are of interest in their own right. These are then used to establish a roadmap to obtaining well-posedness and convergence of the finite begin{document}$ N $end{document} system to its mean field limit. A number of possible future research directions are also discussed.