Fast Deterministic CUR Matrix Decomposition with Accuracy Assurance

The deterministic CUR matrix decomposition is a low-rank approximation method to analyze a data matrix. It has attracted considerable attention due to its high interpretability, which results from the fact that the decomposed matrices consist of subsets of the original columns and rows of the data matrix. The subset is obtained by optimizing an objective function with sparsity-inducing norms via coordinate descent. However, the existing algorithms for optimization incur high computation costs. This is because coordinate descent iteratively updates all the parameters in the objective until convergence. This paper proposes a fast deterministic CUR matrix decomposition. Our algorithm safely skips unnecessary updates by efficiently evaluating the optimality conditions for the parameters to be zeros. In addition, we preferentially update the parameters that must be nonzeros. Theoretically, our approach guarantees the same result as the original approach. Experiments demonstrate that our algorithm speeds up the deterministic CUR while achieving the same accuracy.