A two sided Bourgain Tzafriri Restricted Invertibility Theorem
The Bourgain-Tzafriri Restricted Invertibility Theorem guarantees the existence of large subsets of a family of norm one vectors that form Riesz bases for their linear spans. The challenge is to choose a large subset while making sure that the resulting lower Riesz bound does not deteriorate too much. An upper Riesz bound of the selected Riesz bases is inherited from the frame bound of the original system of vectors. In this talk, we shall present an algorithm based on recent work by Spielman and Srivastava that allows us to choose one vector at a time while controlling both, the lower and the upper Riesz bound of the resulting system.