相关论文: Landscape Correspondence of Empirical and Populati…
We revisit the theoretical performances of Spectral Clustering, a classical algorithm for graph partitioning that relies on the eigenvectors of a matrix representation of the graph. Informally, we show that Spectral Clustering works well as…
We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the…
Spectral clustering is one of the most popular methods for community detection in graphs. A key step in spectral clustering algorithms is the eigen decomposition of the $n{\times}n$ graph Laplacian matrix to extract its $k$ leading…
In an era of unprecedented deluge of (mostly unstructured) data, graphs are proving more and more useful, across the sciences, as a flexible abstraction to capture complex relationships between complex objects. One of the main challenges…
We consider three distinct and well studied problems concerning network structure: community detection by modularity maximization, community detection by statistical inference, and normalized-cut graph partitioning. Each of these problems…
Using the diagrammatic method, we derive a set of self-consistent equations that describe eigenvalue distributions of large correlated asymmetric random matrices. The matrix elements can have different variances and be correlated with each…
This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly…
We consider the problem of embedding the nodes of a hypergraph into Euclidean space under the assumption that the interactions arose through closeness to unknown hyperedge centres. In this way, we tackle the inverse problem associated with…
Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper…
In these notes we discuss tools and concepts that emerge when studying high-dimensional random landscapes, i.e., random functions on high-dimensional spaces. As an illustrative example, we consider an inference problem in two forms:…
Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for…
We study a simple embedding technique based on a matrix of personalized PageRank vectors seeded on a random set of nodes. We show that the embedding produced by the element-wise logarithm of this matrix (1) are related to the spectral…
The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used in spectral clustering and community detection. However, in real-life applications the number of clusters or…
The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle…
We analyse the eigenvectors of the adjacency matrix of a random inhomogeneous graph constructed from a specified degree sequence. We assume that the empirical degree sequence has bounded mean and variance. We show that near the edges of the…
Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of…
These notes offer a unified introduction to spectral methods for the study of complex systems. They are intended as an operative manual rather than a theorem-proof textbook: the emphasis is on tools, identities, and perspectives that can be…
The present work is concerned with community detection. Specifically, we consider a random graph drawn according to the stochastic block model~: its vertex set is partitioned into blocks, or communities, and edges are placed randomly and…
Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…
This paper is concerned with the spectral properties of matrices associated with linear filters for the estimation of the underlying trend of a time series. The interest lies in the fact that the eigenvectors can be interpreted as the…