Scalable algorithms for signal reconstruction by leveraging similarity joins

Signal reconstruction problem (SRP) is an important optimization problem where the objective is to identify a solution to an underdetermined system of linear equations that is closest to a given prior. It has a substantial number of applications in diverse areas including network traffic engineering, medical image reconstruction, acoustics, astronomy and many more. Most common approaches for SRP do not scale to large problem sizes. In this paper, we propose multiple optimization steps, developing scalable algorithms for the problem. We first propose a dual formulation of the problem and develop the Direct algorithm that is significantly more efficient than the state of the art. Second, we show how adapting database techniques developed for scalable similarity joins provides a significant speedup over Direct, scaling our proposal up to large-scale settings. Third, we describe a number of practical techniques that allow our algorithm to scale to settings of size in the order of a million by a billion. We also adapt our proposal to identify the top-k components of the solved system of linear equations. Finally, we consider the dynamic setting where the inputs to the linear system change and propose efficient algorithms inspired by the database techniques of materialization and reuse. Extensive experiments on real-world and synthetic data confirm the efficiency, effectiveness and scalability of our proposal.