Related papers: Gradient flows on graphons: existence, convergence…
This thesis analyze the Wasserstein gradient flow of a functional defined as a double convolution of a non-smooth repulsive interaction potential. To be more precise, the potential under investigation has a -|x| behavior close to the…
This paper provides results on Wasserstein gradient flows between measures on the real line. Utilizing the isometric embedding of the Wasserstein space $\mathcal P_2(\mathbb R)$ into the Hilbert space $L_2((0,1))$, Wasserstein gradient…
Graph convolutional networks (GCNs) are a widely used method for graph representation learning. We investigate the power of GCNs, as a function of their number of layers, to distinguish between different random graph models on the basis of…
The problem of deriving a gradient flow structure for the porous medium equation which is {\em thermodynamic}, in that it arises from the large deviations of some microscopic particle system, is studied. To this end, a rescaled zero-range…
For a harmonic function u on Euclidean space, this note shows that its gradient is essentially determined by the geometry of its level hypersurfaces. Specifically, the factor by which |grad(u)| changes along a gradient flow is completely…
Exchangeable random graphs, which include some of the most widely studied network models, have emerged as the mainstay of statistical network analysis in recent years. Graphons, which are the central objects in graph limit theory, provide a…
We construct a Wasserstein gradient flow of the maximum mean discrepancy (MMD) and study its convergence properties. The MMD is an integral probability metric defined for a reproducing kernel Hilbert space (RKHS), and serves as a metric on…
Many signals evolve in time as a stochastic process, randomly switching between states over discretely sampled time points. Here we make an explicit link between the underlying stochastic process of a signal that can take on a bounded…
We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\mathcal F_\nu := \text{MMD}_K^2(\cdot, \nu)$ towards given target measures $\nu$ on the real line, where we focus on the…
This article overviews how gradient flows, and discretizations thereof, are useful to design and analyze optimization and sampling algorithms. The interplay between optimization, sampling, and gradient flows is an active research area; our…
Maxflow is a fundamental problem in graph theory and combinatorial optimisation, used to determine the maximum flow from a source node to a sink node in a flow network. It finds applications in diverse domains, including computer networks,…
We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and $\mathbb{R}$. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve…
Graph-limit theory focuses on the convergence of sequences of graphs when the number of nodes becomes arbitrarily large. This framework defines a continuous version of graphs allowing for the study of dynamical systems on very large graphs,…
The degree-based entropy of a graph is defined as the Shannon entropy based on the information functional that associates the vertices of the graph with the corresponding degrees. In this paper, we study extremal problems of finding the…
We study geometric properties of the gradient flow for learning deep linear convolutional networks. For linear fully connected networks, it has been shown recently that the corresponding gradient flow on parameter space can be written as a…
We study the distributed optimization problem over a graphon with a continuum of nodes, which is regarded as the limit of the distributed networked optimization as the number of nodes goes to infinity. Each node has a private local cost…
We study a class of graphon particle systems with time-varying random coefficients. In a graphon particle system, the interactions among particles are characterized by the coupled mean field terms through an underlying graphon and the…
This work addresses the problem of simulating Gaussian random fields that are continuously indexed over a class of metric graphs, termed graphs with Euclidean edges, being more general and flexible than linear networks. We introduce three…
Graph Neural Networks (GNNs) show strong expressive power on graph data mining, by aggregating information from neighbors and using the integrated representation in the downstream tasks. The same aggregation methods and parameters for each…
The paper surveys recent progresses in understanding the dynamics and loss landscape of the gradient flow equations associated to deep linear neural networks, i.e., the gradient descent training dynamics (in the limit when the step size…