No abstract

As abstract representations of relational data, graphs and networks find wide use in a variety of fields, particularly when working in nonEuclidean spaces. Yet for graphs to be truly useful in in the context of signal processing, one ultimately must have access to flexible and tractable statistical models. One model currently in use is the ChungLu random graph model, in which edge probabilities are expressed in terms of a given expected degree sequence. An advantage of this model is that its parameters can be obtained via a simple, standard estimator. Although this estimator is used frequently, its statistical properties have not been fully studied. In this paper, we develop a central limit theory for a simplified version of the Chung-Lu parameter estimator. We then derive approximations for moments of the general estimator using the delta method, and confirm the effectiveness of these approximations through empirical examples.

Covariance matrix estimates are an essential part of many signal processing algorithms, and are often used to determine a low-dimensional principal subspace via their spectral decomposition. However, exact eigenanalysis is computationally intractable for sufficiently high-dimensional matrices, and in the case of small sample sizes, sample eigenvalues and eigenvectors are known to be poor estimators of their population counterparts. To address these issues, we propose a covariance estimator that is computationally efficient while also performing shrinkage on the sample eigenvalues. Our approach is based on the Nyström method, which uses a data-dependent orthogonal projection to obtain a fast low-rank approximation of a large positive semidefinite matrix. We provide a theoretical analysis of the error properties of our estimator as well as empirical results, including examples of its application to adaptive beamforming and image denoising.

Spectral methods requiring the computation of eigenvalues and eigenvectors of a positive definite matrix are an essential part of signal processing. However, for sufficiently high-dimensional data sets, the eigenvalue problem cannot be solved without approximate methods. We examine a technique for approximate spectral analysis and low-rank matrix reconstruction known as the Nyström method, which recasts the eigendecomposition of large matrices as a subset selection problem. In particular, we focus on the performance of the Nyström method when used to approximate random matrices from the Wishart ensemble. We provide statistical results for the approximation error, as well as an experimental analysis of various subset sampling techniques.

In many applications, it is convenient to represent data as a graph, and often these datasets will be quite large. This paper presents an architecture for analyzing massive graphs, with a focus on signal processing applications such as modeling, filtering, and signal detection. We describe the architecture, which covers the entire processing chain, from data storage to graph construction to graph analysis and subgraph detection. The data are stored in a new format that allows easy extraction of graphs representing any relationship existing in the data. The principal analysis algorithm is the partial eigendecomposition of the modularity matrix, whose running time is discussed. A large document dataset is analyzed, and we present subgraphs that stand out in the principal eigenspace of the timevarying graphs, including behavior we regard as clutter as well as small, tightly-connected clusters that emerge over time.