Topology Distance: A Topology-Based Approach For Evaluating Generative Adversarial Networks

Automatic evaluation of the goodness of Generative Adversarial Networks (GANs) has been a challenge for the field of machine learning. In this work, we propose a distance complementary to existing measures: Topology Distance (TD), the main idea behind which is to compare the geometric and topological features of the latent manifold of real data with those of generated data. More specifically, we build Vietoris-Rips complex on image features, and define TD based on the differences in persistent-homology groups of the two manifolds. We compare TD with the most commonly used and relevant measures in the field, including Inception Score (IS), Frechet Inception Distance (FID), Kernel Inception Distance (KID) and Geometry Score (GS), in a range of experiments on various datasets. We demonstrate the unique advantage and superiority of our proposed approach over the aforementioned metrics. A combination of our empirical results and the theoretical argument we propose in favour of TD, strongly supports the claim that TD is a powerful candidate metric that researchers can employ when aiming to automatically evaluate the goodness of GANs' learning.Comment: Submitted to ICML 2020; 12 pages, 7 figure

Persistent homology of complex networks

Long-lived topological features are distinguished from short-lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and presented as a parameterized version of a Betti number. Complex networks with distinct degree distributions exhibit distinct persistent topological features. Persistent topological attributes, shown to be related to the robust quality of networks, also reflect the deficiency in certain connectivity properties of networks. Random networks, networks with exponential connectivity distribution and scale-free networks were considered for homological persistency analysis

Spectra of combinatorial Laplace operators on simplicial complexes

We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the weighted Laplacian, and the normalized graph Laplacian. This framework then allows us to define the normalized Laplace operator Delta(up)(i) on simplicial complexes which we then systematically investigate. We study the effects of a wedge sum, a join and a duplication of a motif on the spectrum of the normalized Laplace operator and identify some of the combinatorial features of a simplicial complex that are encoded in its spectrum

Cooperation, Conflict and Higher-Order Structures of Social Networks

Simplicial complexes represent powerful models of complex networks and complex systems in general. We explore the properties of spectra of combinatorial Laplacian operator of simplicial complexes in the context of connectivity of cliques in the simplicial clique complex associated with social networks. The necessity of higher order spectral analysis is discussed and compared with results for ordinary graph spectra. Methods and results are applied using social network of the Zachary karate club and the network of characters from Victor Hugos novel Les Miserables

Graphs, Simplicial Complexes and Hypergraphs: Spectral Theory and Topology

In this chapter we discuss the spectral theory of discrete structures such as graphs, simplicial complexes and hypergraphs. We focus, in particular, on the corresponding Laplace operators. We present the theoretical foundations, but we also discuss the motivation to model and study real data with these tools