Related papers: Spectral Partitioning with Blends of Eigenvectors

Clustering is concerned with coherently grouping observations without any explicit concept of true groupings. Spectral graph clustering - clustering the vertices of a graph based on their spectral embedding - is commonly approached via…

Our problem of interest is to cluster vertices of a graph by identifying underlying community structure. Among various vertex clustering approaches, spectral clustering is one of the most popular methods because it is easy to implement…

While orthogonalization exists in current dimensionality reduction methods in spectral clustering on undirected graphs, it does not scale in parallel computing environments. We propose four orthogonalization-free methods for spectral…

Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network…

Graph clustering aims at discovering a natural grouping of the nodes such that similar nodes are assigned to a common cluster. Many different algorithms have been proposed in the literature: for simple graphs, for graphs with attributes…

Vertex clustering in a stochastic blockmodel graph has wide applicability and has been the subject of extensive research. In thispaper, we provide a short proof that the adjacency spectral embedding can be used to obtain perfect clustering…

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by…

Modern graph or network datasets often contain rich structure that goes beyond simple pairwise connections between nodes. This calls for complex representations that can capture, for instance, edges of different types as well as so-called…

We study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which…

Nonlinear reformulations of the spectral clustering method have gained a lot of recent attention due to their increased numerical benefits and their solid mathematical background. However, the estimation of the multiple nonlinear…

Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses…

Finding a maximum clique in a given graph is one of the fundamental NP-hard problems. We compare two multi-core thread-parallel adaptations of a state-of-the-art branch and bound algorithm for the maximum clique problem, and provide a novel…

We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the…

An exact, number-conserving solution to the generalized, orbit-dependent pairing problem is derived by introducing an infinite-dimensional algebra. A method for obtaining eigenvalues and eigenvectors of the corresponding Hamiltonian is also…

Spectral clustering has found extensive use in many areas. Most traditional spectral clustering algorithms work in three separate steps: similarity graph construction; continuous labels learning; discretizing the learned labels by k-means…

In the first part of this paper, we survey results that are associated with three types of Laplacian matrices:difference, normalized, and signless. We derive eigenvalue and eigenvector formulaes for paths and cycles using circulant matrices…

Analysis of the convergence rates of modern convex optimization algorithms can be achived through binary means: analysis of emperical convergence, or analysis of theoretical convergence. These two pathways of capturing information diverge…

Spectral clustering has been one of the widely used methods for community detection in networks. However, large-scale networks bring computational challenges to the eigenvalue decomposition therein. In this paper, we study the spectral…

The extraction of clusters from a dataset which includes multiple clusters and a significant background component is a non-trivial task of practical importance. In image analysis this manifests for example in anomaly detection and target…

The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It…