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The purpose of this article is to approximately compute the eigenvalues of the symmetric Dirichlet Laplacian within an interval $(0,\Lambda)$. A novel domain decomposition Ritz method, partition of unity condensed pole interpolation method,…
Many problems in machine learning can be expressed by means of a graph with nodes representing training samples and edges representing the relationship between samples in terms of similarity, temporal proximity, or label information. Graphs…
The eigendeomposition of nearest-neighbor (NN) graph Laplacian matrices is the main computational bottleneck in spectral clustering. In this work, we introduce a highly-scalable, spectrum-preserving graph sparsification algorithm that…
Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data…
We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair $(\lambda_{*},x_{*})$ of a given analytic regular nonlinear eigenvalue problem (NEP). In terms…
Graph Partitioning is a critical problem in numerous scientific and engineering domains including social network analysis, VLSI design, and many more. Spectral methods are known to produce quality partitions while minimizing edge cuts for a…
We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation…
In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study…
We propose a new method for embedding graphs while preserving directed edge information. Learning such continuous-space vector representations (or embeddings) of nodes in a graph is an important first step for using network information…
In this paper, we cast the scribble-based interactive image segmentation as a semi-supervised learning problem. Our novel approach alleviates the need to solve an expensive generalized eigenvector problem by approximating the eigenvectors…
The aim of this paper is to propose a novel framework to infer the sheaf Laplacian, including the topology of a graph and the restriction maps, from a set of data observed over the nodes of a graph. The proposed method is based on sheaf…
Graph Convolutional Networks (GCNs) have proven to be successful tools for semi-supervised classification on graph-based datasets. We propose a new GCN variant whose three-part filter space is targeted at dense graphs. Examples include…
Many of today's problems require techniques that involve the solution of arbitrarily large systems $A\mathbf{x}=\mathbf{b}$. A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor…
Online social networks are being increasingly used for analyzing various societal phenomena such as epidemiology, information dissemination, marketing and sentiment flow. Popular analysis techniques such as clustering and influential node…
We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly…
Graph autoencoders (GAEs) are powerful tools in representation learning for graph embedding. However, the performance of GAEs is very dependent on the quality of the graph structure, i.e., of the adjacency matrix. In other words, GAEs would…
Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as…
This paper provides a construction method of the nearest graph Laplacian to a matrix identified from measurement data of graph Laplacian dynamics that include biochemical systems, synchronization systems, and multi-agent systems. We…
It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of…
This book is about solving matrix nearness problems that are related to eigenvalues or singular values or pseudospectra. These problems arise in great diversity in various fields, be they related to dynamics, as in questions of robust…