Algorithmic Solution for Non-Square, Dense Systems of Linear Equations, with applications in Feature Selection

We present a novel algorithm attaining excessively fast, the sought solution of linear systems of equations. The algorithm is short in its basic formulation and by definition vectorised, while the memory allocation demands trivial, because for each iteration only one dimension of the given input matrix $\mathbf x$ is utilized. The execution time is very short compared with state-of-the-art methods, exhibiting up to $\mathcal{O}(10^3)$ speed-up and low memory allocation demands, especially for non-square Systems of Linear Equations, with ratio of equations versus features high (tall systems), or low (wide systems) accordingly. The accuracy is high and straightforwardly controlled, and the numerical results highlight the efficiency of the proposed algorithm, in terms of computation time, solution accuracy and memory allocations demands. The parallelisation of the algorithm is also presented in multi-threaded and GPU accelerators' setting. The paper also comprises a theoretical proof for the algorithmic convergence. Finally, we extend the implementation of the proposed algorithmic rationale to feature selection tasks.