An improvement of the parameterized frequent directions algorithm

Abstract

Matrix sketching is a technique used to create summaries of large matrices. Frequent directions (FD) and its parameterized variant, $\alpha$-FD are deterministic sketching techniques that have theoretical guarantees and also work well in practice. An algorithm called the iterative singular value decomposition (iSVD) has been shown to have better performance than FD and $\alpha$-FD in several datasets, despite the lack of theoretical guarantees. However, in datasets with major and sudden drift, iSVD performs poorly when compared to the other algorithms. The $\alpha$-FD algorithm has better error guarantees and empirical performance when compared to FD. However, it has two limitations: the restriction on the effective values of its parameter $\alpha$ due to its dependence on sketch size and its constant factor reduction from selected squared singular values, both of which result in reduced empirical performance. In this paper, we present a modified parameterized FD algorithm, $\beta$-FD in order to overcome the limitations of $\alpha$-FD, while maintaining similar error guarantees to that of $\alpha$-FD. Empirical results on datasets with sudden and major drift and those with gradual and minor or no drift indicate that there is a trade-off between the errors in both kinds of data for different parameter values, and for $\beta \approx 28$, our algorithm has overall better error performance than $\alpha$-FD.