Linear convergence of CQ algorithms and applications in gene regulatory network inference

Abstract

In the present paper, we consider the varying stepsize CQ algorithm for solving the split feasibility problem in Hilbert spaces, investigate the linear convergence issue and explore an application in systems biology. In particular, we introduce a notion of bounded linear regularity property for the split feasibility problem, and use it to establish the linear convergence property for the varying stepsize CQ algorithm when using some suitable types of stepsizes, which covers most types of stepsizes used in the literature of CQ algorithms. We also provide some mild sufficient conditions for ensuring this bounded linear regularity property, and then conclude the linear convergence rate of the varying stepsize CQ algorithm for many application cases. To the best of our knowledge, this is the first work to study the linear convergence rate of CQ algorithms. In the aspect of application, we consider the gene regulatory network inference arising in systems biology, which is formulated as a group Dantzig selector and then cast into a split feasibility problem. The numerical study on gene expression data of mouse embryonic stem cell shows that the varying stepsize CQ algorithm is applicable to gene regulatory network inference in the sense that it obtains a reliable solution matching with biological standards.