Efficient Sparsification of Simplicial Complexes via Local Densities of States

Simplicial complexes (SCs), a generalization of graph models for relational data that account for higher-order relations between data items, have become a popular abstraction for analyzing complex data using tools from topological data analysis or topological signal processing. However, the analysis of many real-world datasets leads to dense SCs with a large number of higher-order interactions. Unfortunately, analyzing such large SCs often has a prohibitive cost in terms of computation time and memory consumption. The sparsification of such complexes, i.e., the approximation of an original SC with a sparser simplicial complex with only a log-linear number of high-order simplices while maintaining a spectrum close to the original SC, is of broad interest. In this work, we develop a novel method for a probabilistic sparsifaction of SCs. At its core lies the efficient computation of sparsifying sampling probability through local densities of states as functional descriptors of the spectral information. To avoid pathological structures in the spectrum of the corresponding Hodge Laplacian operators, we suggest a "kernel-ignoring" decomposition for approximating the sampling probability; additionally, we exploit error estimates to show asymptotically prevailing algorithmic complexity of the developed method. The performance of the framework is demonstrated on the family of Vietoris--Rips filtered simplicial complexes.

Convergence of gradient based training for linear Graph Neural Networks

Graph Neural Networks (GNNs) are powerful tools for addressing learning problems on graph structures, with a wide range of applications in molecular biology and social networks. However, the theoretical foundations underlying their empirical performance are not well understood. In this article, we examine the convergence of gradient dynamics in the training of linear GNNs. Specifically, we prove that the gradient flow training of a linear GNN with mean squared loss converges to the global minimum at an exponential rate. The convergence rate depends explicitly on the initial weights and the graph shift operator, which we validate on synthetic datasets from well-known graph models and real-world datasets. Furthermore, we discuss the gradient flow that minimizes the total weights at the global minimum. In addition to the gradient flow, we study the convergence of linear GNNs under gradient descent training, an iterative scheme viewed as a discretization of gradient flow.