Applications of low-rank approximation: complex networks and inverse problems

The use of low-rank approximation is crucial when one is interested in solving problems of large dimension. In this case, the matrix with reduced rank can be obtained starting from the singular value decomposition considering only the largest components. This thesis describes how the use of the low-rank approximation can be applied both in the analysis of complex networks and in the solution of inverse problems. In the first case, it will be explained how to identify the most important nodes or how to determine the ease of traveling between them in large-scale networks that arise in many applications. The use of low-rank approximation is presented both for undirected and directed networks, whose adjacency matrices are symmetric and nonsymmetric, respectively. As a second application, we propose how to identify inhomogeneities in the ground or the presence of conductive substances. This survey is addressed with the aid of electromagnetic induction measurements taken with a ground conductivity meter. Starting from electromagnetic data collected by this device, the electrical conductivity profile of the soil is reconstructed with the aid of a regularized damped Gauss{Newton method. The inversion method is based on the low-rank approximation of the Jacobian of the function to be inverted.