Data Filtering for Cluster Analysis by ℓ0\ell_0-Norm Regularization

A data filtering method for cluster analysis is proposed, based on minimizing a least squares function with a weighted $\ell_0$-norm penalty. To overcome the discontinuity of the objective function, smooth non-convex functions are employed to approximate the $\ell_0$-norm. The convergence of the global minimum points of the approximating problems towards global minimum points of the original problem is stated. The proposed method also exploits a suitable technique to choose the penalty parameter. Numerical results on synthetic and real data sets are finally provided, showing how some existing clustering methods can take advantages from the proposed filtering strategy.Comment: Optimization Letters (2017

Perron vector optimization applied to search engines

In the last years, Google's PageRank optimization problems have been extensively studied. In that case, the ranking is given by the invariant measure of a stochastic matrix. In this paper, we consider the more general situation in which the ranking is determined by the Perron eigenvector of a nonnegative, but not necessarily stochastic, matrix, in order to cover Kleinberg's HITS algorithm. We also give some results for Tomlin's HOTS algorithm. The problem consists then in finding an optimal outlink strategy subject to design constraints and for a given search engine. We study the relaxed versions of these problems, which means that we should accept weighted hyperlinks. We provide an efficient algorithm for the computation of the matrix of partial derivatives of the criterion, that uses the low rank property of this matrix. We give a scalable algorithm that couples gradient and power iterations and gives a local minimum of the Perron vector optimization problem. We prove convergence by considering it as an approximate gradient method. We then show that optimal linkage stategies of HITS and HOTS optimization problems verify a threshold property. We report numerical results on fragments of the real web graph for these search engine optimization problems.Comment: 28 pages, 5 figure